The Pennsylvania State University
The Graduate School
College of Education
STUDENTS’ CONCEPTIONS OF MATHEMATICS
AS SENSIBLE (SCOMAS) FRAMEWORK
A Dissertation in
Curriculum and Instruction
by
Maureen M. Grady
 2013 Maureen M. Grady
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
August 2013
The dissertation of Maureen M. Grady was reviewed and approved* by the following:
E. Frances Arbaugh
Associate Professor of Education
Dissertation Advisor
Chair of Committee
Glendon W. Blume
Professor of Education
Curriculum and Instruction Graduate Coordinator
Gwendolyn M. Lloyd
Professor of Education
Andrea McCloskey
Assistant Professor of Education
Kai A. Schafft
Associate Professor Of Education/Rural Sociology
*Signatures are on file in the Graduate School
ii
ABSTRACT
This study describes the development of the Students’ Conceptions of Mathematics as
Sensible (SCOMAS) Framework and its application to the study of the conceptions of
mathematics as sensible of students in a secondary mathematics classroom. The SCOMAS
Framework begins with indicators that students conceive of mathematics as sensible and
provides a categorization of these indicators into five types of student activity. Utilizing
indicators and categories of activity the SCOMAS Framework provides a tool for documenting
students’ conceptions of mathematics as sensible.
This study begins with a description of the methods used to develop an initial framework
of indicators from the research literature. It then describes how the initial framework was used
to investigate the conceptions of mathematics in a 9th grade algebra classroom. The study reports
on the use of the initial framework as a tool to describe the dimensions of students’ conceptions
of mathematics as sensible and the changes in those conceptions over the course of an academic
year. Finally, the study reports on how analysis of the classroom data helped shape the initial
framework into the final SCOMAS Framework.
iii
TABLE OF CONTENTS
List of Figures .......................................................................................................................... vi
List of Tables ........................................................................................................................... viii
Acknowledgements .................................................................................................................. ix
Chapter 1 Introduction ............................................................................................................. 1
Research questions ........................................................................................................... 4
Rationale for the study ..................................................................................................... 6
Theoretical considerations ............................................................................................... 8
Chapter 2 Literature Review .................................................................................................... 13
Definitions ........................................................................................................................ 14
The landscape of the literature on beliefs and conceptions .............................................. 17
Studies of a broad spectrum of beliefs and conceptions .................................................. 18
Studies focused on conceptions of mathematics .............................................................. 19
What conceptions do students have about the nature of mathematics? ........................... 26
What can this study add?.................................................................................................. 30
Chapter 3 Development of the Initial Framework of Indicators .............................................. 31
Generating indicators from the literature ......................................................................... 31
Initial testing of the list of indicators ............................................................................... 40
Literature in the framework categories ............................................................................ 44
Explaining ................................................................................................................ 45
Strategizing............................................................................................................... 46
Making Connections ................................................................................................ 47
Assuming Authority: ................................................................................................ 48
Other Observation Frameworks: ...................................................................................... 50
Chapter 4 Research Methods for Phase 2 ................................................................................ 52
Selecting the setting ......................................................................................................... 52
The setting ........................................................................................................................ 53
Participants ....................................................................................................................... 54
Daily pattern in the classroom.......................................................................................... 55
Data collected ................................................................................................................... 56
Preexisting data ........................................................................................................ 57
Data collected during spring observation ................................................................. 58
Analysis of classroom data............................................................................................... 62
Analysis of student interviews ......................................................................................... 65
Trustworthiness ................................................................................................................ 66
iv
Chapter 5 Findings ................................................................................................................... 68
Findings related to expecting explanations ...................................................................... 68
Types of Evidence .................................................................................................... 68
Providing explanations ............................................................................................. 69
Seeking explanations ................................................................................................ 86
Summary .................................................................................................................. 90
Findings related to expecting connections ....................................................................... 91
Types of evidence..................................................................................................... 91
Connections within mathematics.............................................................................. 92
Connections between mathematics and other contexts ............................................ 102
Summary .................................................................................................................. 111
Findings related to strategizing ........................................................................................ 112
Types of evidence..................................................................................................... 112
Strategizing when problem solving .......................................................................... 113
Alternative strategies ................................................................................................ 117
Role of memory in mathematics .............................................................................. 123
Summary .................................................................................................................. 127
Findings related to mathematical authority ...................................................................... 127
Types of evidence..................................................................................................... 127
Assuming authority .................................................................................................. 128
Ceding authority ....................................................................................................... 131
Mathematics as authoritative .................................................................................... 132
Summary .................................................................................................................. 133
Findings related to stating that mathematics makes sense ............................................... 134
Chapter 6 Discussion ............................................................................................................... 135
Assertions based on findings ............................................................................................ 135
Research question 1 .................................................................................................. 135
Research question 2 .................................................................................................. 136
Significance of the study .................................................................................................. 136
Evolution of the framework ............................................................................................. 141
Changes to the indicators ......................................................................................... 142
Changes to the framework........................................................................................ 143
Challenges in studying students’ conceptions of mathematics ........................................ 144
Limitations of survey items ...................................................................................... 145
Limitations of interviews ......................................................................................... 151
Implications of this study ......................................................................................... 154
Suggestions for further study ........................................................................................... 155
References ................................................................................................................................ 157
v
LIST OF FIGURES
Figure 1-1. Beliefs and conceptions and their relation to productive disposition. .................. 6
Figure 1-2. Multilevel Mediational Framework of Instructional Quality and Effectiveness. . 9
Figure 1-3. A portion of Reusser’s (2001) Framework. .......................................................... 10
Figure 1-4. The work of teaching............................................................................................ 11
Figure 1-5. The work of teaching conceptions....................................................................... 12
Figure 2-1. Relationship between beliefs and conceptions .................................................... 16
Figure 2-2. Framework of Student Conceptions of Mathematics. .......................................... 20
Figure 2-3. Intercorrelative relations between the dimensions ............................................... 22
Figure 2-4. Relationship between conceptions. . .................................................................... 22
Figure 2-5 Conceptions of Mathematics ................................................................................ 23
Figure 3-1 – Initial Framework ............................................................................................... 43
Figure 4-1. What the video shows........................................................................................... 57
Figure 4-2. Conceptions cards for student interviews. ............................................................ 61
Figure 4-3 Studiocode coding scheme. ................................................................................... 63
Table 4-1 – Sample incident detail .......................................................................................... 64
Figure 4-4 – Comparison of Fall and Spring Combined Timelines. ....................................... 65
Figure 5-1. Warm-up problem – Find the area........................................................................ 71
Figure 5-2. Warm-up problem – What fraction is this? ......................................................... 79
Figure 5-3. Warm-up problem. What’s my area? ................................................................. 80
Figure 5-4. Board problem: finding f(x)................................................................................. 81
Figure 5-5. Warm-up problem. Angles of a perfect pentagon. ............................................. 82
Figure 5-7. Clock face labeled in degrees. .............................................................................. 93
Figure 5-8. Expression with exponents to simplify................................................................ 97
Figure 5-9. What does this trapezoid look like?..................................................................... 103
Figure 5-10. Demonstrating height equals wingspan. ............................................................ 106
vi
Figure 5-11. Side view of bridge............................................................................................ 109
Figure 5-12. Area of concave hexagon. ................................................................................. 118
Figure 5-13. An alternative dissection of the concave hexagon............................................. 119
Figure 5-14. A second alternative dissection of the concave hexagon. ................................. 119
Figure 6-1. The work of teaching……………………………………………………………137
Figure 6-2. Students’ Conceptions of Mathematics as Sensible (SCOMAS) Framework ..... 144
vii
LIST OF TABLES
Table 3-1 – Sources used to generate indicators. .................................................................... 33
Table 3-2 – Representative sources for indicators .................................................................. 37
Table 3-3. List of indicators .................................................................................................... 39
Table 3-4 – Sources of Indicators............................................................................................ 40
viii
ACKNOWLEDGEMENTS
This dissertation has been such a central part of my life for the last several years that
everyone around me has become an accomplice in this work, willing or unwilling. Some
volunteered for the journey. My adviser, Fran Arbaugh, knew even better than I what she was
taking on and stepped forward anyway. I will always be grateful for her support, her
encouragement, her patience, her insight, and her humor. She was, for me, the perfect adviser
who came along at the perfect time. My dissertation committee members were also more aware
than I of the journey for which they volunteered yet each readily agreed to serve and have been a
valuable source of challenge, assistance, and support. The teacher and students participating in
this study, although all volunteered, probably had little idea what they were getting themselves
into. Despite this, they have been unfailingly cheerful and gracious. They not only allowed me
into their classroom, they took over running recording equipment, eagerly participated in
interviews, and made me feel like a welcome part of their community.
For every person who volunteered to travel this journey with me, there is one who came
along because they were a part of my life and this was the path on which I was traveling. My
family and friends have listened to my whining, been patient with both my physical and mental
absences, and provide the emotional and practical support that I needed to continue along rocky
patches in the road. When I moved into town to begin my doctoral program, I’m sure my new
neighbors had no idea what living near a graduate student was going to do to their lives. I was
blessed with great neighbors. When I was too distracted by my work to tend to details of my life
they mowed the lawn, fed baby goats, relit the furnace, brought me food, moved boxes, and fixed
plumbing. The help of my family, friends, and neighbors made my work possible.
ix
Finally, there are the people at the Mid-Atlantic Center for Mathematics Teaching and
Learning. Kathy Heid enticed me onto the path and she and the other faculty and staff supported
and challenged me along the way. My fellow graduate students have proved to be great
travelling companions. We have shared joys and sorrows, excitement and frustration, success
and failure.
It is only with the help of all of these fellow travelers that I have made this journey and I
thank each one of you for your contribution.
x
Chapter 1 Introduction
Recent calls for reform in mathematics education have espoused ambitious goals for how
students should interact with school mathematics. The National Council of Teachers of
Mathematics (NCTM) has proffered a vision of school mathematics in which “Students
confidently engage in complex mathematical tasks,” “draw on knowledge from a wide variety of
mathematical topics,” and “value mathematics and engage actively in learning it” (NCTM, 2000,
p. 3). The Standards for Mathematical Practice of the Common Core State Standards for
Mathematics (CCSS-M) call for students to “make sense of problems and persevere in solving
them,” “reason abstractly and quantitatively,” “construct viable arguments and critique the
reasoning of others,” “model with mathematics,” “use appropriate tools strategically,” “attend to
precision,” “look for and make use of structure,” and “express regularity in repeated reasoning”
(National Governors Association and Council of Chief State School Officers [NGA & CCSSO],
2010, pp. 6-8). These visions and standards go far beyond a traditional view of school
mathematics in which students are expected to learn procedures and definitions by listening to
the teacher, taking notes, and then practicing the procedures. Cobb, Wood, Yackel, and McNeal
(1992) described these two very different classroom mathematics traditions and, following
Richards’ (1991) terminology, called them school mathematics and inquiry mathematics. These
calls for mathematics education to look more like the inquiry mathematics of mathematicians
and less like the school mathematics that is probably familiar to many Americans have been
echoed by many prominent mathematics educators (e.g., Cobb & Bauersfeld, 1995; Hiebert et.
al., 1997; Romberg, Carpenter, Dremock, 2005; Schoenfeld, 1994) and might be best described
by Harel (2010) who wrote, “[t]he ultimate goal of instruction in mathematics is to help students
1
develop ways of understanding and ways of thinking that are compatible with those practiced by
contemporary mathematicians” (p. 91). This is, indeed, an ambitious goal.
Part of this ambitious goal for mathematics education is for students to develop
mathematical proficiency along five different strands. The National Research Council’s (NRC)
Adding It Up: Helping Children Learn Mathematics (2001) document contains a description of
the five interwoven strands of mathematical proficiency:
•
conceptual understanding—comprehension of mathematical concepts, operations,
and relations;
•
procedural fluency—skill in carrying out procedures flexibly, accurately, efficiently,
and appropriately;
•
strategic competence—ability to formulate, represent, and solve mathematical
problems;
•
adaptive reasoning—capacity for logical thought, reflection, explanation, and
justification; and
•
productive disposition—habitual inclination to see mathematics as sensible, useful,
and worthwhile, coupled with a belief in diligence and one’s own efficacy. (p. 5)
The authors of this document argue that each of these strands is essential for mathematical
proficiency and that each develops in conjunction with the other strands. The metaphor used in
Adding It Up is that of a multistrand rope in which the strands are twisted together and weakness
in any strand weakens the entire rope. In this view of mathematical proficiency no strand stands
alone and all are important for the development of the others.
At the heart of the ambitious view of mathematics learning is the conception of
mathematics as a sensible, connected system that can be reasoned about, understood, and
2
expanded upon. The term conception is used here to denote a truth held by students that is
assumed to be primarily cognitive, as opposed to affective, in nature. (For further elaboration,
see the section on definitions of beliefs and conceptions.) If students do not have a conception of
mathematics as a sensible, connected system it is unlikely that they will, as the CCSS-M
Standards for Mathematical Practice demand, “look for and make use of structure” in
mathematics (NGA & CCSSO, 2010, p. 8) or develop ways of understanding and thinking about
mathematics “compatible with those practiced by contemporary mathematicians” (Harel, 2010,
p. 91). Despite the critical importance of such a conception of mathematics, both research and
anecdotal evidence reveal that many students in the United States and other countries lack such a
view of mathematics (Muis, 2004).
Also central to this ambitious view of mathematics education is the assertion that this
type of mathematical proficiency is critical not just for gifted or college-intending students but
for all students. The NCTM calls for “ambitious expectations for all,” asserting that “there is no
conflict between equity and excellence” (NCTM, 2000, p. 3). Adding It Up asserts that,
“mathematics is a realm no longer restricted to a select few. All young Americans must learn to
think mathematically, and they must think mathematically to learn (NRC, 2001, p.1) and CCSSM states that, “all students must have the opportunity to learn and meet the same high standards
if they are to access the knowledge and skills necessary in their post-school lives” (p. 4). Given
the emphasis on equity in mathematics education, it is important that we study the conceptions of
mathematics as sensible of low achieving mathematics students as well as those of students who
have been more successful in mathematics.
To date, the research in this area has been based predominantly on students’ self-reports.
Absent from the literature is an empirically based tool that allows researchers and teachers to
3
assess students’ conceptions of mathematics as sensible through observation. Thus, the aim of
this study is to establish, through systematic and rigorous inquiry, a framework of observable,
action-based indicators that students conceive of mathematics as sensible. This was
accomplished through a two-phase investigation. In Phase 1, I conducted a thorough and
systematic investigation of the existing literature on conceptions of mathematics in order to
identify indicators. I then arranged those indicators into a preliminary framework. In Phase 2 of
this study, I used the framework as a research tool to examine student conceptions in a secondary
mathematics classroom in which students appear to have developed a conception of mathematics
as a sensible system. I then adapted the original framework based on my analysis of classroom
data, culminating with the Students’ Conceptions of Mathematics as Sensible (SCOMAS)
Framework, a framework that has its roots in both the research literature and classroom data.
Research questions
The purpose of this study was to develop a framework for indicators that students see
mathematics as sensible. In particular, this study used the current literature on conceptions of
mathematics to develop a initial framework of indicators generally accepted as indicating that
students have a conception of mathematics as sensible, used this framework to examine the ways
in which students in one purposefully chosen secondary mathematics classroom conceive of
mathematics as sensible, and modified the initial framework based on findings from this
classroom. This study, then, addressed the overarching question: What do students’ actions in a
mathematics classroom tell us about students’ conceptions of mathematics as sensible?
Specifically, the following research questions guided this study:
4
•
What constructs exist in the current literature on students’ conceptions of mathematics
that can be used to construct indicators of students’ conception of mathematics as
sensible?
The findings from this research question were used to create a framework for studying students’
conceptions as they were displayed in a classroom setting and this investigation was guided by
the following research question
•
In what ways is the framework a viable tool for documenting students’ conceptions? In
what ways does it illuminate students’ conceptions of mathematics as sensible?
In these research questions and throughout this study I use the term “sensible” in relation
to conceptions of mathematics to denote a view that mathematics is a connected, coherent system
in which there are reasons for such things as rules, procedures, and formulas, whether or not the
individual yet knows or understands the reasons. This view of mathematics contrasts with a
view of mathematics as consisting of arbitrary, disconnected rules and procedures; within such a
conception of mathematics, individuals must be told everything they need to know and
remembering this knowledge plays a key role in the learning of mathematics.
The conception of mathematics as sensible is one part of the productive disposition strand
of mathematical proficiency and is related to conceptions about the nature of mathematical
knowledge and the nature of mathematical activity as described by Grouws, Howard, &
Colangelo (1996). Figure 1-1 shows an overview of the structure of conceptions and beliefs
related to mathematics education and where productive disposition and conceptions of
mathematics as sensible fit within that structure.
5
Figure 1-1-1. Beliefs and conceptions and their relation to productive disposition.
Of the four categories of beliefs and conceptions related to mathematics education
common in the literature the categories conceptions of mathematics and beliefs about self in
relation to mathematics, are part of the productive disposition strand of mathematical
proficiency. The category conceptions of mathematics, according to Grouws, Howard, &
Colangelo (1996) breaks down into five components the first three of which, composition of
mathematical knowledge, structure of mathematical knowledge, and status of mathematical
knowledge are components of the construct that I refer to as the conception of mathematics as
sensible.
Rationale for the study
Students’ conceptions of the nature of mathematics are important because they are part of
what educators have defined as mathematical proficiency. The productive disposition strand of
mathematical proficiency begins by stating that part of what it means to be proficient in
mathematics is to have “the tendency to see sense in mathematics” (NRC, 2001, p. 131).
Students’ conceptions of mathematics are also important because a view of mathematics as
6
sensible plays an important role in the other strands of mathematical proficiency. A student’s
conception of mathematics is linked to that student’s ability to and likeliness to strategize about
mathematical problem solving (Wong, Marton, Wong, Lam, 2002), that is, in the development of
the strategic competence strand of mathematical proficiency. The adaptive reasoning strand, in
which students are expected to provide explanation and justification, relies on a sufficiently
sensible system of mathematics to expect that such explanations and justifications exist. Finally,
the interwoven nature of the strands of the mathematical proficiency means that no strand can
fully develop separate from other strands. Thus, the development of productive disposition,
including the need for students for see mathematics as sensible, is critical for the development of
all strands of mathematical proficiency.
Given the importance of the development of students’ conceptions of mathematics as a
sensible system and the consensus that many students lack such a conception, it is important that
we find ways to study students’ conceptions of mathematics as sensible. Researchers have made
considerable progress in understanding the types of conceptions that students have about
mathematics, mathematics learning, and themselves as mathematics learners (e.g., Kloosterman
& Stage, 1992; Op ’t Eynde, De Corte, & Verschaffel, 2002; Schoenfeld, 1989). However, the
conception of mathematics as a sensible system has not been sufficiently operationalized to
identify what might be taken as evidence of this conception in students. Without such a way to
identify whether students conceive of mathematics as sensible and the different ways in which
students demonstrate this conception, research on instructional practices and on the development
of such a conception is limited to describing practices that might reasonably be expected to help
develop conceptions of mathematics as sensible. In this study, I use the extant literature on
students’ conception of mathematics as sensible and data about student actions in the classroom
7
to develop a framework for studying this conception and for conceptualizing several different
categories in which students provide indicators of such a conception. It is expected that this
study will provide a valuable tool for researchers, teacher educators, and teachers to use for
assessing students’ development in this critical area of mathematical proficiency.
Theoretical considerations
This study is conceived as an initial stage for a larger research agenda seeking to connect
instructional practices to the development of productive disposition in mathematics students. In
discussing the theoretical considerations of this study I will begin with the theoretical
underpinning for my larger research agenda. I will then narrow the focus to the theoretical basis
for this particular study.
Central to my research agenda is the assumption that instructional practices have an
impact on student learning. This assumption does not imply that instructional practices are the
only factor or that the impact of instructional practices on student learning is a simple, direct
relationship. In the process–product paradigm of educational research there was an assumption
that we could identify the processes in teaching that lead to desirable products in student learning
and teach all teachers to use these processes. Educational researchers have come to see that
teaching is a far more complex endeavor than could be described by a series of teaching moves;
it involves teacher characteristics, students characteristics, environment, culture, support
systems, and more. Figure 1-2 displays a model of some of the variables related to student
outcomes in school and how they interact.
8
Figure 1-2. Multilevel Mediational Framework of Instructional Quality and Effectiveness (Reusser,
2001).
One danger in a view that is this complex is that we begin to see teaching as so complex
and contextualized that every classroom, every teacher, and every student are seen as dissimilar
to others and, thus, there can be no recommended practices, no guiding principles for practice—
in essence, no craft or science of teaching. This research agenda, then, will focus in on the
classroom instruction as it affects one particular aspect of the “multi-dimensional outcomes of
instruction” (Reusser, 2001). The choice to focus on classroom instruction rests on two
assumptions. First, classroom instruction is one of the features in the complex system over
which educators can hope to exert substantial influence. Second, as shown in the design of the
model in Figure 1-2, classroom instruction is, in many ways, a node through which other factors
9
are influenced and exert influences on learning. Ingvarson, Beavis, Bishop, Peck, and Elsworth
(2004), in their study of teaching and learning in Australian schools, found that “teacher
practices were consistently the most powerful predictors of mathematics achievement growth,
and with growth in affective student outcomes” (p. 71). For these reasons, the research agenda
will focus on the portion of the model shown in Figure 1-3.
Figure 1-1-2. A portion of Reusser’s (2001) Framework.
Having situated the agenda in a specific portion of the model that represents the
relationship between instructional practice and student learning, we are still left with the problem
of trying to understand and study that connection. Hiebert and Grouws (2007) state that one of
the major challenges faced by researchers who seek to study connections between teaching and
learning is that “theories that specify the ways in which the key components of teaching fit
together to form an interactive, dynamic system for achieving particular learning goals have not
been sufficiently developed to guide research efforts that can build over time” (p. 373). They
also identify several methodological challenges facing researchers, including: a) the difficulty of
10
accounting for relevant factors in a setting in which there are so many factors, and b) the
challenge of creating appropriate measures of learning and teaching.
Lampert (2001) proposed a “representation of the work of teaching” (p. 30) in which she
identified the “problem spaces” of teaching and learning as those elements of practice that
connect teacher to student, teacher to content, student to content, and teacher to student practice
(see Figure 1-4).
Figure 1-1-3. The work of teaching (Lampert, 2001, p. 33)
The arc of practice across the three arrows from teacher to student, practice, and content is
intended to indicate that these problem spaces of teaching and learning are not independent
problem spaces but that they interact with one another. The arrow between student and content
represents what Lampert refers to as “studying”—using this term in a broad sense to indicate the
ways in which students engage with the content. In this study, the content under consideration is
not a particular topic or subject in mathematics but rather the conception of mathematics as
sensible (see Figure 1-5).
11
Figure 1-1-4. The work of teaching conceptions.
In brief, for this study I will focus on the problem space between student and conceptions
of mathematics as sensible, seeking to identify and categorize indicators that students are
engaged in the practice of learning to see mathematics as a sensible system. These practices
might include ways in which students talk about mathematics, the types of questions that
students see as reasonable to ask about mathematics, and how students engage in doing
mathematics. It is hoped that a better understanding of this problem space between students and
their conceptions of mathematics will permit teachers to look more closely at an important
outcome of their instructional practices and provide researchers with a tool to more directly
examine students’ conceptions of mathematics.
12
Chapter 2 Literature Review
This study examines students’ conceptions about mathematics as sensible. I will review
the literature on conceptions related to mathematics education with special attention to the
literature on conceptions of the nature of mathematics. I begin by trying to bring some clarity to
the definition and use of the terms conceptions and beliefs and the relationship between them. I
then discuss studies that examined the full spectrum of beliefs and conceptions related to
mathematics education before focusing in on the first category, conceptions about mathematics
(see Figure 1-1 for a layout of the categories). I outline the methods used to investigate
conceptions of mathematics and the research findings about common student conceptions of
mathematics. I end this section by describing the work that has been done linking student
conceptions to student outcomes and the results of studies that have sought to identify influences
on the development or change of student conceptions.
Over the past three decades, many mathematics education researchers have studied the
nature and content of teachers’ and students’ beliefs and conceptions and their relationship to
learning and teaching. Although the focus of this study is on students’ conceptions, the research
methods used to explore the nature and content of beliefs and conceptions of teachers and
students are so similar that, when considering the nature and content of beliefs and conceptions, I
will not distinguish between research about teachers and research about students. The distinction
will not become important until the discussion of the effects of beliefs and conceptions, at which
time I focus on effects on student learning. Before going further in the discussion, it is necessary
to discuss definitions of beliefs and conceptions.
13
Definitions
Despite Pajares’ (1992) call to clean up the “messy construct” of beliefs in educational
research and Philipp’s (2007) attempt to organize and define terms related to beliefs,
conceptions, knowledge, and affect, a lack of consensus still remains about the differences and
the relationship between beliefs and conceptions. A number of working definitions of beliefs
currently exist in the mathematics education literature. A common thread in these definitions is
the notion that beliefs are about what an individual holds to be true about a particular subject.
More problematic is the question of whether beliefs are more cognitive or affective in nature.
Schoenfeld (1992) defines beliefs as “an individual’s understandings and feelings that shape the
ways that the individual conceptualizes and engages in mathematical behavior” (p. 338). Ponte
(1994) states that beliefs have “a strong affective component and evaluative component” (p. 5).
Philipp (2007), in his table of terminology, places beliefs towards the affective end on the
spectrum but makes clear that, “beliefs are more cognitive, are felt less intensely, and are harder
to change than attitudes” (p. 259). At the other end of the spectrum, Sumpter (2009) states that,
“beliefs are primarily cognitive” (p. 5). She responds to Schoenfeld’s (1992) definition by
stating “I like to exclude emotions from the definition of belief since the same belief may be
connected with different emotions for different individuals” (Sumpter, 2009, p. 5).
The relative roles played by affect and cognition are also critical considerations when
researchers attempt to define conceptions and to differentiate between beliefs and conceptions.
Researchers who differentiate between beliefs and conceptions based on the role of cognition
and affect generally identify conceptions as more cognitively based than beliefs. Pehkonen
(2004) states, “In the case of conceptions, the cognitive component of beliefs is stressed, whereas
in basic (primitive) beliefs the affective component is emphasized” (p. 3). Ponte (1994) states
14
that conceptions are “essentially cognitive in nature” (p. 1994) and Philipp (2007) places
conceptions closer to the cognitive end of the spectrum than beliefs.
Pehkonen (1995), having defined conceptions as essentially cognitive in nature, states
that,
in accordance with Saari (1983), we explain here conceptions as conscious beliefs, i.e.
we understand conceptions as a subset of beliefs. Thus, for us, conceptions are higher
order beliefs which are based on reasoning processes for which the premises are
conscious. (p. 12)
Interestingly, although Philipp (2007) seems to agree that conceptions are more cognitive in
nature than are beliefs, he joins many other researchers (e.g., Dahlgren Johansson & Sumpter,
2010; Sumpter, 2009; Thompson, 1992) in asserting that conceptions is the more general term,
that is, conceptions are “viewed as a more general mental structure, encompassing beliefs,
meaning, concepts, propositions, rules, mental images, preference, and the like” (Thompson,
1992, p. 130). Ponte (1994) separates beliefs and conceptions by their cognitive versus affective
nature and states that “both beliefs and conceptions are part of knowledge” (p. 5). Figure 2-1
shows the relative relationships beliefs and conceptions envisioned by various researchers.
15
Figure 2-1. Relationship between beliefs and conceptions
Clearly, there is a lack of consensus in the literature about the definition of conceptions
and beliefs and the relationship between these two constructs. Some researchers choose not to
address the relationship between conceptions and knowledge and use only one of the terms or to
use the terms interchangeably. Even Thompson (1992), who defines beliefs as a subset of
conceptions, indicates that the distinction may not be very important and that her use of the terms
is often a matter of convenience. She states, “Though the distinction may not be a terribly
important one, it will be more natural at times to refer to a teacher’s conception of mathematics
as a discipline than to simply speak of the teachers’ beliefs about mathematics” (Thompson,
1992, p. 130).
Throughout this review, I will use beliefs and conceptions somewhat interchangeably,
tending to use the language used by the particular researchers. However, in the larger study, I
choose to follow Ponte’s view of beliefs and conceptions and treat them as two somewhat
separate constructs with beliefs being more related to affect and conceptions being more
16
cognitive in nature. Since this study focuses on students’ views of the nature of mathematics,
and I take these views to be essentially cognitive in nature, I refer to these views as conceptions.
I agree, however, with Thompson (1992) that the distinction may not be critical since it may be
impossible, in any specific case, to determine how much a view is affective in nature and how
much the view is cognitive in nature.
The landscape of the literature on beliefs and conceptions
Although studies about beliefs and conceptions related to mathematics education utilize a
variety of schema, the research on beliefs and conceptions can be grouped into four categories:
conceptions of mathematics, beliefs about self in relation to mathematics, beliefs about
mathematics teaching, and beliefs about mathematics learning. Due to the complex relationship
between these categories and the multiple ways of interpreting questions and indicators,
however, the distinction between these categories is not always clear. For example, one
problematic category used in some studies (e.g., Kloosterman & Stage, 1992; Schommer-Aikins,
Duell, & Hutter, 2005) is beliefs about problem solving. In the Indiana Mathematics Beliefs
Scales (Kloosterman & Stage, 1992), one of the beliefs is “Word problems are important in
mathematics” (p. 115). This statement describes the nature and content of mathematics, so it
would seem to fit into conceptions of mathematics. However, one of the indicators within the
belief is “Math classes should not emphasize word problems” (p. 115) which could easily be
interpreted as a belief about mathematics teaching rather than about mathematics. Despite the
challenges of defining categories across such a range of beliefs and conceptions, many
researchers have chosen to look across these broad categories.
17
Studies of a broad spectrum of beliefs and conceptions
Most of the researchers who have studied such a broad range of beliefs and conceptions
have used questionnaires or surveys, usually ones that relied heavily upon Likert-scale type
items. Many of these questionnaires are based on two popular instruments, the Indiana
Mathematical Beliefs Scales (Kloosterman and Stage, 1992) and Schoenfeld’s (1989)
questionnaire. The Indiana Mathematical Beliefs Scales, designed to measure high school and
college students’ beliefs about problem solving, is designed around five beliefs “related to
motivation and thus achievement on mathematics problem solving” (Kloosterman & Stage,
1992, p. 109) and a sixth belief about the usefulness of mathematics, measured by the preexisting
“Fennema-Sherman Usefulness Scale.” The six beliefs from the scale are: “I can solve timeconsuming mathematics problems,” “There are word problems that cannot be solved with
simple, step-by-step procedures,” “Understanding concepts is important in mathematics,” “Word
problems are important in mathematics,” “Effort can increase mathematical ability,” and
“Mathematics is useful in daily life” (Kloosterman & Stage, 1992, p. 115). This instrument has
been adapted and used in studies seeking links between students’ beliefs about mathematics and
mathematics achievement (e.g., Mason, 2003; Schommer-Aikins, Duell, & Hutter, 2005; Stage
& Kloosterman, 1995; Steiner, 2007) and in studies examining the effect of various interventions
on students’ beliefs (e.g., Lee, 2006; Mason & Scrivani, 2004; Taylor, 2009). Schoenfeld’s
(1989) questionnaire
contains multiple-choice questions related to attributions of success or failure (sections 1
and 2); students’ perceptions of mathematics and school practice (sections 3, 4, 5, 7, and
8); their views of school mathematics, English, and social studies (section 6); the nature
of geometric proofs, reasoning, and constructions (sections 9 and 10); motivation (section
18
11); and personal and scholastic performance and motivation (section 12). (Schoenfeld,
1989, p. 342)
Variations on Schoenfeld’s questionnaire have been used to develop an overall picture of
students’ beliefs about mathematics (Erickson, 1993; Schoenfeld, 1989) and to examine the
effects on beliefs of an intervention engaging students in problem solving (Higgins, 1997).
Most researchers concerned with examining beliefs and conceptions across the spectrum have
used similar questionnaires (e.g., Carter & Norwood, 1997; Colby, 2007; Franks, 1990; Kaya,
2007; Malmivuori & Pehkonen, 1996; Sumpter, 2009; Verschaffel et al., 1999; Wilkins & Brand,
2004) although a few have used interviews, observation, or a mixture of techniques (e.g., DiazObando, Plasencia-Cruz, & Solano-Alvarado, 2003; Frank, 1988; Garofalo, 1989; Raymond,
1997; Telese, 1999; Wong, 2002). These broad looks at students’ and teachers’ beliefs and
conceptions have helped to provide an overall picture of the nature and content of beliefs and
conceptions and have provided the necessary information for researchers to focus more closely
on individual aspects of beliefs and conceptions.
Studies focused on conceptions of mathematics
Within each of the larger categories related to conceptions about mathematics learning
there are subcategories and specific beliefs. For example Op’ t Eynde, De Corte, and
Verschaffel (2002) subdivided beliefs about the self category into four subcategories: selfefficacy, control, task value, and goal orientation. Grouws, Howard, & Colangelo (1996)
proposed a framework for the first category, conceptions of mathematics, which comprises four
themes, some with several dimensions (see Figure 2-2).
19
Figure 2-2. Framework of Student Conceptions of Mathematics (Grouws, Howard, & Colangelo, 1996, p.
36).
The remainder of this review will focus on the first and second themes of Grouws, Howard, &
Colangelo’s framework: the nature of mathematical knowledge and the nature of mathematical
activity.
Many researchers have focused on conceptions of the nature of mathematical knowledge
and have developed different schema for organizing their findings. Most of the work developing
organizational schema has focused on the conceptions of college students or mathematics
teachers. Ernest (1988) identified three philosophies of the nature of mathematics:
First of all, there is the instrumentalist view that mathematics is an accumulation of facts,
rules and skills to be used in the pursuance of some external end. Thus mathematics is a
20
set of unrelated but utilitarian rules and facts. Secondly, there is the Platonist view of
mathematics as a static but unified body of certain knowledge. Mathematics is
discovered, not created. Thirdly, there is the problem solving view of mathematics as a
dynamic, continually expanding field of human creation and invention, a cultural product.
Mathematics is a process of enquiry and coming to know, not a finished product, for its
results remain open to revision. (p. 2)
Ernest viewed these philosophies hierarchically, with instrumentalist representing the lowest
level of the hierarchy. Dionne (1988 as cited in Mura, 1993) defined three similar perceptions:
traditionalist, formalist, and constructivist. Dionne, however, conceived of teachers’ perceptions
as comprised of some percentage of each of the three perceptions. Torner and Grigutsch (1994
as cited in Pehkonen, 2004) combined these two schemas and proposed the categories:
“toolbox”-aspect, systems-aspect, and process-aspect. Grigutsch and Torner (1998) used the
results of surveys of university mathematics teachers from German-speaking countries to
identify five dimensions of the teachers’ views of mathematics, which are similar to those
identified in earlier studies and then took the further step of examining how those beliefs fit
together and the ways in which teachers’ views in one dimension may be correlated to views in
other dimensions. Figure 2-3 shows how the views group into dynamic and static views of
mathematics. Positive and negative correlations between teachers’ views in the dimensions are
indicated by (+) and (-) signs.
21
Figure 2-3. Intercorrelative relations between the dimensions (Grigutsch and Torner, 1998, p. 26)
Grigutsch and Torner’s five dimensions of views of mathematics (process, application,
formalism, schema, Platonism) are related to Grouws, Howard, & Colangelo’s (1996) first two
categories (composition and structure of mathematical knowledge) while their grouping into
dynamic and static views helps to clarify the relationship of the dimensions to Grouws, Howard,
& Colangelo’s third category, status of mathematical knowledge.
Petocz et al. (2007), in examining the responses of undergraduates majoring in
mathematics or related field to the question “what is mathematics,” grouped responses into five
categories, which they organized into the nested model shown in Figure 2-4.
Figure 2-4. Relationship between conceptions. Petocz et al. (2007).
22
The innermost category, number, is a conception of mathematics as being about numbers and
calculations. The components category correlates with Torner and Grigutsch’s (1994 as cited in
Pehkonen, 2004) conception of mathematics as a “toolbox” containing “a collection of isolated
techniques” (Petocz et al., 2007, p. 446) to be used as necessary to solve problems. The
modeling and abstract conceptions are placed with the same “ring” in the figure to indicate that
an abstract view of mathematics and a view of mathematics as modeling reality are separate
categories but not ones that could nested within each other. The outer level, life, is a conception
of mathematics as “a way to understand how life works” (Petocz et al., 2007, p. 447).
Essentially the inner two levels of conceptions in this model appear analogous to Ernest’s (1988)
instrumentalist conceptions. The descriptions of the outer three conceptions seem to indicate
more of a focus on the role of mathematics than the focus on the nature of mathematical
knowledge evident in some of the earlier writers.
Crawford, Gordon, Nicholas, and Prosser (1998a, 1998b) used a very similar fivecategory structure in their study of the mathematical conceptions of undergraduate students.
However, their categories, while analogous to those used by Petocz, et al. (2007), are clearly
hierarchal and are sorted into two main groupings: fragmented conceptions and cohesive
conceptions. Figure 2-5 shows the categories and their groupings.
Figure 2-5 Conceptions of Mathematics (Crawford, Gordon, Nicholas, & Prosser 1998a, p. 458)
23
This simple division into fragmented and cohesive conceptions is particularly useful for the
present study as Crawford, Gordon, Nicholas, and Prosser’s cohesive connections is an important
part of what is referred to in this study as a conception of mathematics as sensible.
The most common approach used for determining students’ and teachers’ conceptions of
the nature of mathematics is to ask them directly about these conceptions. Researchers have
used variations on the question “What is mathematics?” in a number of different settings. Reid,
Petocz, Smith, Wood, and Dortins (2003) asked 22 “late stage” undergraduate mathematics
majors “what do you think mathematics is about?” (p. 165) followed by several other openended questions. From the results of these interviews they developed a framework (see Figure 24) that they used to analyze data from much a larger, international study in which students were
asked in a survey about their conceptions of mathematics (Houston et al., 2010; Petocz et al.,
2007). Franke and Carey (1997), doing research on 1st graders’ conceptions of mathematics,
adapted the question and asked “Some of the kindergartners are wondering what it will be like to
do math in first grade. What would you tell them about the kinds of things that you do in math?”
(p. 11). Mura (1993, 1995) asked mathematics professors and college-level mathematics
educators for their definition of mathematics as well as for a list of the 10 most influential books
in the development of the discipline of mathematics. Responses in this study sound a note of
caution about the usefulness of direct questions about what mathematics is. Mura found that
33% of mathematicians skipped the question and at least one, in responding the question “How
would you define mathematics?” responded “I wouldn’t” (Mura, 1995, p. 394). Mura also found
that both mathematicians and mathematics educators held a wide variety of conceptions of the
nature of mathematics. Mura notes, “mental images are often diffuse, incoherent and partly
unconscious, hence difficult to articulate. No doubt, what each participant in the present research
24
has produced offers but a small portion of his or her ideas about mathematics” (Mura, 1995, p.
396). Frid (1995), in her examination of the conceptions of mathematics of high school students,
sounded a similar note of caution. She found that students’ responses to her opening question
“what is mathematics?” responded with the expected answers about mathematics being about
“numbers, rules, and formulas” (p. 273). However, when she pushed students further, asking
questions about where mathematics came from and how it was developed, students presented a
far richer view of the nature of mathematics. Telese (1999), working with Mexican-American
high school students, had perhaps the most difficulty in asking students directly about their
definition of mathematics. After conducting interviews with students as a follow-up to a Likertscale type inventory, he concluded, “Generally, the students could not define mathematics” (p.
164).
To overcome the limitations of asking students and teachers directly about their
conceptions of the nature of mathematics, some researchers have developed and used multi-item
inventories designed to provide an overall picture of conceptions of mathematics. Grouws,
Howard, & Colangelo (1996), in order to examine the conceptions of mathematics of high school
students, developed the Conceptions of Mathematics Inventory based on their framework of
students’ conceptions of mathematics (shown in Figure 2-2). The inventory is a compilation of
items from pre-existing sources and new items written by the researchers. Crawford, Gordon,
Nicholas, and Prosser (1994) used a survey of five open-ended questions about mathematics,
including “Think about the maths you’ve done so far. What do you think mathematics is?” (p.
1994). The results from this survey of first-year college mathematics majors were used to
develop and test the Conceptions of Mathematics Questionnaire (Crawford, Gordon, Nicholas,
and Prosser, 1998b). This questionnaire was then used in a large-scale study of the conceptions
25
of mathematics of college students and the relationship of those conceptions to the students’
experience of learning mathematics.
A few researchers have taken approaches other than asking about the nature of
mathematics or using inventories to study conceptions of mathematics. Dionne (1988 as
reported in Mura, 1993) asked elementary school teachers to apportion points across three
different conceptions of mathematics: traditionalist, formalist, and constructivist. Wong,
Marton, Wong, and Lam (2002), as part of their multipart study of the conceptions of
mathematics of students in Grades 3–9, observed students as they engaged in solving a variety of
nonroutine mathematics problems. Presmeg (2003) used in-depth interviews about the nature of
mathematics and students’ experience of mathematics beyond school to examine the conceptions
of mathematics of high school and college students. The alternative research methods used by
these researchers add a detail and richness to the picture of the conceptions of mathematics
provided by the rest of the research.
What conceptions do students have about the nature of mathematics?
The results from studies that examine students’ conceptions of mathematics cannot help
but be disappointing to anyone concerned about students viewing mathematics as sensible.
Large-scale studies consistently find that the majority of students view mathematics as utilitarian
but comprised of essentially disconnected “numbers, rules, and fragments” (Crawford, Gordon,
Nicholas, & Prosser, 1994). Petocz et al. (2007) found that more than 50% of college students
pursuing majors in mathematics or mathematics related fields conceived of mathematics as either
merely “manipulation with numbers” (p. 445) or as a “collection of isolated techniques” (p. 446).
Crawford, Gordon, Nicholas, & Prosser (1994) surveyed 300 college freshmen and found that
26
“over 75% of students conceive of mathematics as a fragmented body of knowledge” (p. 457).
These bleak percentages are reinforced by common student conceptions of mathematics found by
other researchers.
Researchers have found that students tend to view mathematics as being primarily about
computation and following rules (Frank, 1988; Garofalo, 1989; Mason, 2003; Schoenfeld, 1989;
Wong, 2002), that only the mathematics that is tested is important (Garofalo, 1989), and that
various components of mathematical knowledge are unrelated (Crawford, Gordon, Nicholas, &
Prosser, 1994; Muis, 2004). Students also tend to believe that all mathematical problems are
solvable (Reusser, as cited in Mason, 2003), all mathematics problems should be quickly
solvable (Frank, 1988, Schoenfeld, 1992), that mathematics is essentially an individual pursuit
(Schoenfeld, 1992; Sumpter, 2010), that the goal of mathematics is get the correct answers
(Atallah, Bryant, & Dada, 2010; Frank, 1988; Wong, Marton, Wong, & Lam, 2002), and that
getting the correct answer is the same as understanding (Wong, 2002). Several researchers have
found that students do tend to see mathematics as useful (Crawford, Gordon, Nicholas, &
Presser, 1994; Petocz et al., 2007; Wong, 2002) although Steiner (2007) found that over 40% of
college students in her study “did not believe that mathematics beyond basic mathematics was
useful to everyday life” (p. vi). Wong (2002) found that students tended to believe that
mathematics did involve thinking, receiving comments like, “Mathematics is for training logical
thinking” and “Mathematics is a thinking exercise” (pp. 222-223).
One reason why the conceptions that students have about the nature of mathematics are
important is that there is evidence that these conceptions have an effect on students’ achievement
in mathematics and the way that students study mathematics. Several studies have linked
students’ conceptions and beliefs related to mathematics, as measured by variation on the Indiana
27
Mathematical Beliefs Scales (Kloosterman and Stage, 1992), to student achievement. Stage and
Kloosterman (1995) and Steiner (2007) found that college students with certain conceptions of
mathematics and mathematics learning tended to have lower grades in mathematics courses or on
the final examination for their mathematics course. Schommer-Aikins, Duell, and Hutter (2005)
likewise found that more productive conceptions and beliefs about mathematics and mathematics
learning were positively correlated with problem-solving performance, reading scores, and
overall grade point averages for middle school students. Malmivuori and Pehkonen (1996), also
working with middle school students, identified particular factors from the conceptions inventory
that contributed to variance in achievement test scores. They found that self-confidence about
mathematics, an orientation toward effort and self-regulation, a clear self-image in mathematics,
and similar affective beliefs related to mathematics were positively correlated with achievement
scores. All of these findings linking results from the Indiana Mathematical Beliefs Scales
(Kloosterman and Stage, 1992) to student achievement tended to focus on beliefs and
conceptions related to self and learning rather than conceptions related to the nature of
mathematics. Only Crawford, Gordon, Nicholas, and Prosser (1994, 1998a, 1998b) focused on
conceptions of mathematics. They found that college students who had fragmented conceptions
of mathematics were also more likely to engage in surface approaches to studying mathematics.
Given the evidence that students’ beliefs and conceptions related to mathematics
education influence important student outcomes, it is worthwhile to consider influences on the
development and change of those beliefs and conceptions. Several researchers have examined
beliefs and conceptions of different subgroups and found that there are differences in
conceptions among different segments of the population. Grouws, Howard, & Colangelo (1996)
studied the conceptions of mathematically talented students and those of average students
28
enrolled in an algebra class. They found that “Mathematically talented students consistently
viewed mathematics as a coherent system with meaningful connections between and among
concepts, principles, and skills, but the algebra students’ responses were more varied and
reflected a less connected view of mathematics” (p. 19). Wilkins and Ma (2003) studied beliefs
and conceptions of students across middle school and high school and found that conceptions of
the nature of mathematics tended to remain stable but that older students tended to have a less
positive attitude toward mathematics and be less likely to believe in its usefulness than did
younger students.
Several studies have examined the impact of classroom interventions on students’
conceptions and beliefs related mathematics education. Taylor (2009) studied the effect of a 5week summer course for middle school students, which “was designed to necessitate
collaborative problem-solving and emphasized both the debriefing of final answers and the
sharing and discussing of mistakes and roadblocks” (p. 107), on students’ beliefs and
conceptions. She found small but significant changes in five indicators from the Indiana
Mathematical Beliefs Scales (Kloosterman and Stage, 1992) related to effort, self-efficacy, the
importance of understanding mathematical concepts, and the usefulness and relevance of
mathematics. Mason and Scrivani (2004) examined the effect on beliefs of fifth-grade students
involved for 3 months in a “novel classroom environment where students face non-traditional
and open challenging problems, reflect on their nature and discuss the different strategies that
could be adopted to solve them” (p. 172). They found that these students had “more advanced”
beliefs about themselves as mathematics learners and about mathematics and mathematical
problem solving. Verschaffel et al. (1999) and Lee (2006) also examined effects of interventions
on student beliefs and conceptions but their findings focused on categories of beliefs not related
29
to students’ conceptions of mathematics. Only one study reported results directly related to
students’ conceptions of mathematics. This study found that high school students in classes
using a traditional mathematics textbook were more likely than students in classes using a reform
mathematics textbook to “view the structure of mathematics as a coherent system of concepts,
principles, and skills rather than as a collection of isolated pieces” and “more likely … to view
the process of doing mathematics as valuing, exploring, comprehending, and exploring concepts
and principles rather than simply implementing procedures and finding results” (Colby, 2007,
pp. 108–109). This difference was magnified when classes using the reform textbook were
taught using more reform-oriented instructional practices.
What can this study add?
Researchers have made good progress indentifying and classifying students’ and
teachers’ conceptions and beliefs related to mathematics education. However, almost all of the
research of students’ conceptions of mathematics has relied on some form of self-reporting.
Even Likert-scale surveys are a form of self-reporting (Presmeg, 2003) and require that
respondents be aware of their own conceptions and interpret the items in a way similar to that
intended by the researcher. There are, to my knowledge, no studies that seek indicators of
students’ conceptions in students’ actions. This study will offer important information and tools
by examining action-oriented indicators of students’ conceptions about mathematics.
30
Chapter 3 Development of the Initial Framework of Indicators
The first phase of this study was to use the existing literature on students’ conceptions of
mathematics as sensible to create an initial framework for analyzing classroom data. In this
chapter I will describe the process used to produce the initial framework and introduce the list of
indicators and initial framework.
Generating indicators from the literature
The process of examining beliefs and conceptions can be challenging. Although simply
asking about beliefs seems an obvious strategy, Munby (1982) notes, “we may not be the best
people to clearly enunciate our beliefs and perspectives since some of these may lurk beyond
ready articulation” (p. 217). Munby also notes an additional difficulty with simply asking about
beliefs; beliefs are imbedded in a perspective and without context of the perspective from which
a question is asked, it is difficult to know how to answer it. On the other hand, imbedding the
question in a perspective tends to presuppose the belief desired by the person asking the question
rather than giving free rein to the beliefs of the person answering the question. Likert-scale
questionnaires, multiple interview questions, and open-response surveys are many of the ways
that researchers have sought to address the issue inherent in self-reporting of beliefs and
conceptions. However, all of these instruments are still, in a sense, self-reports and are also
subject to the same concerns about interpretation of meaning and context raised by Munby. Both
to address the concerns with self-reporting of conceptions and because conceptions have been
shown to affect student problem solving, I chose to examine students’ conceptions of
31
mathematics by looking for indicators in what students say and do while engaged in doing
mathematics.
This phase of the study began with a thorough examination of the literature on students’
and teachers’ conceptions of mathematics. To identify the relevant literature I started with
chapters from both the Handbook of Research on Mathematics Teaching and Learning (Grouws,
1992) and the Second Handbook of Research on Mathematics Teaching and Learning (Lester,
2007). I examined the sources cited in the chapters and the sources cited in the original works. I
also used Google Scholar to identify studies that had cited any of the sources from the handbook
chapters and examined any that were relevant to conceptions of mathematics. I then searched the
Internet for studies relating to conceptions of mathematics, beliefs about mathematics, and
mathematics as sensible. For each new source, I identified and examined the sources cited and
the works that had cited that source.
I chose, for this portion of my study, to include both studies of students’ conceptions of
mathematics and studies of teachers’ conceptions of mathematics. Part of this decision was
based on the relative scarcity of studies related to conceptions of mathematics. Another factor
was the similarity between the two types of studies. The studies often used very similar research
methods and questionnaire items. By including all of the studies, I hoped to generate the most
comprehensive list of indicators possible.
In this set of studies I did not find indicators that might be used to examine students’
conceptions of mathematics as sensible in any of the literature on conceptions of mathematics or
teaching for sense-making. However, many of the studies contain questionnaire items, interview
questions, and anecdotes about students who do and do not seem to see mathematics as sensible.
There are also, in the literature, some characteristics that seem to set gifted students apart and
32
appear to be related to a conception of mathematics as sensible. In building a set of actionoriented, observable indicators that students conceive of mathematics as sensible I built off of the
questionnaire items, interview questions, and student characteristics used in other studies.
Because, as I argued in my introduction, many of the Standards for Mathematical Practice from
the CCSS-M are related a conception of mathematics as sensible, I also used some of these
standards in the development of indicators. Table 3-1 shows the list of sources used to generate
indicators.
Source
Brown et al., 1988
Carter & Norwood, 1997
Garofalo, 1989
Grouws, Howard, &
Colangelo, 1996
Higgins, 1997
Kaya, 2007
Kloosterman & Stage, 1992
Leder and Forgasz, 2002
NGA & CCSSO, 2010
Raymond, 1997
Schoenfeld, 1989
Schommer-Aikins, Duell, &
Hutter, 2005
Stodolsky, 1985
Sumpter, 2009
Telese, 1999
Wong, Marton, Wong, Lam,
2002
Type of Data
Likert-type scale items adapted from Fourth National
Assessment of Educational Progress (Dossey, Mullis,
Lindquist, & Chambers, 1988)
Likert-type scale items
Characteristics of students
Likert-type scale items
Likert-type scale items from Schoenfeld, 1989
Likert-type scale items
Likert-type scale items
Likert-type scale items
Standards for mathematical practice
Descriptors of teachers’ beliefs
Likert-type scale items and open-ended survey items
Likert-scale type items from Kloosterman & Stage,
1992 and Fennema & Sherman, 1976
Characteristics of classroom interactions
Likert-scale type items from Leder and Forgasz, 2002
Likert-type scale items adapted from Fourth National
Assessment of Educational Progress (Dossey, Mullis,
Lindquist, & Chambers, 1988)
Students’ statements in interviews
Table 3-1 – Sources used to generate indicators.
Many of the studies identified as potential sources for indicators conceptualized students’
conceptions more broadly than as only conceptions of mathematics. The researchers often
33
examined students’ conceptions of mathematics, their conceptions of learning, beliefs about how
mathematics should be taught, and their conceptions of their own self-efficacy. Therefore, the
first stage in building the list of indicators was to identify the items in the source materials that
were related to students’ conceptions of mathematics rather than to their conceptions of other
aspects of mathematics learning such as mathematics teaching or self-efficacy with respect to
mathematics. Items such as “If a student is confused about math, the teacher should go over the
material again more slowly” (Carter & Norwood, 1997) or “I am good at mathematics” (Brown,
et al., 1988) were not included because they did not directly address students’ conceptions of
mathematics as sensible.
In order to create an initial set of indicators about students’ conceptions of mathematics
as sensible, I examined all of the questionnaire items, questions, anecdotes, and characteristics
that were related to conceptions of mathematics as sensible. I sought similarity between
statements and grouped them into similar conceptions. I then considered what student action
would demonstrate such a conception. For example, if a student were to agree with statements
like, “It doesn’t really matter if you understand a math problem if you can get the right answer”
(Schommer-Aikins, Duell, & Hutter, 2005, p. 297) he or she might seek explanation for why an
answer is correct. A student who disagrees with statements like, “Math problems can be done
correctly in only one way” (Higgins, 1997, p. 15) might demonstrate that by seeking alternative
ways to solve a problem. Altogether I generated a list of 22 indicators based on the conceptions
of mathematics as sensible found in the literature. Table 3-2 shows the indicators and a sample
of the original sources from which the indicators were adapted.
1
Indicator
Source
Students discussing how
to solve a problem rather
than seeking the “right
“There are word problems that cannot be solved with
simple, step-by-step procedures” (Kloosterman &
Stage, 1992,, p. 115).
34
steps.”
2
Students seeking
explanations for why an
answer is correct or
incorrect.
3
Students seeking or using
alternative solution
strategies.
4
Students expressing a
role for common sense in
doing mathematics.
5
6
Students being willing to
try to solve a problem for
which they have not been
taught a procedure.
Students offering
suggestions for how to
solve a problem for
which they have not been
taught a procedure.
“There is always a rule to follow in solving math
problems” Telese, 1999, p. 159).
“Mathematical task (sic) should be solved with a
specific method” (Sumpter, 2009, p.6).
“There is always a rule to follow in solving
mathematics problems” (Brown et al., 1988, p.347).
“A person who doesn’t understand why an answer to a
math problem I correct hasn’t really solved the
problem” (Kloosterman & Stage, 1992, p. 115).
In math, knowing why an answer is correct is important
(Telese, 1999, p. 159).
“In addition to getting a right answer in math, it is
important to understand why the answer is correct”
(Kaya, 2007, p. 217).
“It’s not important to understand why a mathematical
procedure works as long as it gives a correct answer”
(Schommer-Aikins, Duell, & Hutter, 2005, p. 297).
“Knowing how to solve a problem is as important as
getting the solution” (Brown et al., 1988, p. 347).
“When I cannot remember the exact way my teacher
taught me to solve a math problem, I know some
other methods that I can try” (Kaya, 2007, p. 217).
“Math problems can be done correctly in only one way”
(Higgins, 1997, p. 15).
Math problems can be done correctly in only one way”
(Schoenfeld, 1989, p. 352).
“A mathematical problem can always be solved in
different ways” (Brown et al., 1988, p. 347).
“Real math problems can be solved by common sense
instead of the math rules that you learn in school”
(Higgins, 1997, p. 15; Schoenfeld, 1989, p. 353).
“My own reasoning is not a safe strategy” (Sumpter,
2009, p.6).
“There are word problems that cannot be solved with
simple, step-by-step procedures” (Kloosterman &
Stage, 1992, p. 115).
“To solve math problems, you have to be taught the
right procedure or you can’t do anything” (Higgins,
1997, p. 15; Schoenfeld, 1989, p. 353).
“To try to create your own solution to a mathematical
task is impossible” (Sumpter, 2009, p. 6).
“To try to create your own solution to a mathematical
task is impossible” (Sumpter, 2009, p.6).
“To solve math problems, you have to be taught the
right procedure or you can’t do anything” (Higgins,
1997, p. 15; Schoenfeld, 1989, p. 353).
35
7(-)
8(-)
9(-)
10
11
12
13
14
- Students invoking
memory of procedures
for solving problems.
- Students using poor
memory as a reason why
they cannot do a
problem.
- Students seeking to be
told whether or not an
answer is correct.
Students checking
answers by examining
reasonableness or using
an alternative strategy or
representation.
Students inventing their
own mathematics
problems to solve.
Students seeking
explanations for things in
mathematics rather than
simply accepting them as
truths to be remembered.
Students justifying
mathematical statements.
Students seeking
connections between
mathematical topics.
Students adapting
“Memorizing steps is not that useful for learning to
solve word problems” Kloosterman & Stage, 1992,
p. 115).
“Learning math is mostly memorizing” (Telese, 1999,
p. 159).
“It is important to remember every step of a method”
(Sumpter, 2009, p.6).
“Learning math is mostly memorizing” (Telese, 1999,
p. 159).
“When I cannot remember the exact way my teacher
taught me to solve a math problem, I know some
other methods that I can try” (Kaya, 2007, p.217).
“If I can’t remember how to solve it (which method to
use), I can’t proceed” (Sumpter, 2009, p.6).
“When you get the wrong answer to a math problem
…you only find out when it’s different from the
book’s answer or when the teacher tells you”
(Higgins, 1997, p. 15; Schoenfeld, 1989, p. 353).
“You can from the answer decide whether you have
solved the task correctly or not” (Sumpter, 2009,
p.6).
“Mathematically proficient students check their
answers to problems using a different method, and
they continually ask themselves, ‘Does this make
sense?’” (CCSI, p.6)
“Make up my own math problems to solve” (Telese,
1999, p. 159).
“A lot of things in math must simply be accepted as try
and remembered: there really isn’t any explanation
for them” (Carter & Norwood, 1997, p. 63).
“Justifying the mathematical statements a person
makes is an extremely important part of
mathematics” (Brown et al., 1988, p.347).
“Mathematically proficient students … justify their
conclusions” (CCSI, p. 6).
“Mathematics is made up of unrelated topics” (Brown
et al., 1988, p. 347).
“Mathematics? is an unrelated collection of facts, rules,
and skills” (Raymond, 1997, p. 557).
“Mathematical knowledge consists mainly of ideas and
concepts and the connections between them”
(Grouws, Howard, & Colangelo, 1996, p. 38).
“Finding solutions to one type of mathematics problem
36
15
16
17
18
19
20
21
22
solution methods for one
type of problem to help
them solve another.
Students recognize
similarity between
mathematics problems.
cannot help you solve other types of problems
(Grouws, Howard, & Colangelo, 1996, p. 38).
“Mathematical tasks often look similar” (Sumpter,
2009, p.7).
“Mathematically proficient students… consider
analogous problems” (CCSI, p. 6).
Garofalo, 1989
Students recognizing that
size and quantity of
numbers is unrelated to
problem difficulty.
Students using multiple
“Diagrams can help thinking” (Wong, Marton, Wong,
representations in
Lam, 2002, p. 32).
problem solving.
Students explaining
“Students explain how to solve math problems”
answers.
(Telese, 1999, p. 159).
Students engaging in 2Stodolsky, 1985
way conversations about
mathematics (versus
information flowing from
expert to novice).
Students strategizing
“Mathematically proficient students start by explaining
about solution methods
to themselves the meaning of a problem and looking
for entry points to its solution. They analyze givens,
constraints, relationships, and goals. They make
conjectures about the form and meaning of the
solution and plan a solution pathway rather than
simply jumping into a solution attempt” (CCSI, p. 6).
Students
“Mathematically proficient students make sense of
decontextualizing and
quantities and their relationships in problem
recontextulizing as they
situations. They bring two complementary abilities to
solve problems
bear on problems involving quantitative
relationships: the ability to decontextualize—to
abstract a given situation and represent it
symbolically and manipulate the representing
symbols as if they have a life of their own, without
necessarily attending to their referents—and the
ability to contextualize, to pause as needed during
the manipulation process in order to probe into the
referents for the symbols involved” (CCSI, p. 6).
Table 3-2 – Representative sources for indicators
37
Whereas many of the questionnaire items and anecdotes tend to focus on indicators that
students do not see mathematics as sensible, I tried to rewrite them as positive indicators. There
are three items, marked in Table 3-3 by a (-), that are negative indicators. These were indicators
that I had difficulty rewording as a positive indicator but were so prominent in the literature that
I decided that they needed to be included in the framework.
Indicator
1
Students discussing how to solve a problem rather than seeking the “right steps.”
2
Students seeking explanations for why an answer is correct or incorrect.
3
Students seeking or using alternative solution strategies.
4
6
Students expressing a role for common sense in doing mathematics.
Students being willing to try to solve a problem for which they have not been taught a
procedure.
Students offering suggestions for how to solve a problem for which they have not been
taught a procedure.
7(-)
- Students invoking memory of procedures for solving problems.
8(-)
- Students using poor memory as a reason why they cannot do a problem
9(-)
- Students seeking to be told whether or not an answer is correct.
Students checking answers by examining reasonableness or using an alternative
strategy or representation.
5
10
11
12
Students inventing their own mathematics problems to solve.
Students seeking explanations for things in mathematics rather than simply accepting
them as truths to be remembered.
13
Students justifying mathematical statements.
14
15
Students seeking connections between mathematical topics.
Students adapting solution methods for one type of problem to help them solve
another.
16
Students recognize similarity between mathematics problems.
38
17
Students recognizing that size and quantity of numbers is unrelated to problem
difficulty.
18
Students using multiple representations in problem solving.
19
20
Students explaining answers.
Students engaging in 2-way conversations about mathematics (versus information
flowing from expert to novice).
21
Students strategizing about solution methods
22
Students decontextualizing and recontextulizing as they solve problems
Table 3-3. List of indicators
In developing the list of indicators I generally did not include ideas that appeared in only
one source. Table 3-4 shows the sources for each indicator. There were two indicators that were
included despite being mentioned in only one source. The indicator, “Students recognizing that
size and quantity of numbers is unrelated to problem difficulty” was included because, although
only one study actually discussed this as a characteristic of students’ engagement with problem
solving, several other researchers cited this single study and treated this characteristic as an
important characteristic. The indicator, “Students inventing their own mathematics problems to
solve” was initially dropped from the list because it came from a single source. It was added
back in because it was an action that was observed during the pilot study.
Indicator #
1
2
3
Brown et al.,
1988
Carter &
Norwood,
1997
Crawford,
Gordon,
Nicholas, &
Prosser,
1998b
x
x
x
x
4
5
6
7
8
9
1
0
x
1
1
1
2
x
1
3
x
1
4
x
1
5
1
6
1
7
1
8
1
9
x
2
0
2
1
2
2
x
x
x
Garofalo,
1985
x
39
x
Grouws,
Howard, &
Colangelo,
et al., 1996
Higgins,
1997
Kaya, 2007
Kloosterman
& Stage,
1992
Leder and
Forgasz,
2002
NGA &
CCSSO,
2010
Raymond,
1997
Schoenfeld,
1989
SchommerAikins,
Duell, &
Hutter, 2005
Stodolsky,
1985
Sumpter,
2009
Telese, 1999
Wong,
Marton,
Wong, Lam,
2002
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
Table 3-4 – Sources of Indicators
Initial testing of the list of indicators
After compiling the list of indicators from the literature, I used the list of indicators that
students conceived of mathematics as sensible to code five classroom videos (each about 45
minutes long). Three of the videos were from an Integrated Mathematics II class taught by Mr.
Wingate (a pseudonym), the teacher described in this study. One video was a TIMMS video
from a U.S. classroom. The fifth video was of a classroom at a local high school taught by Mr.
A, an experienced mathematics teacher. The purpose of this pilot study was to test whether the
40
indicators generated from the literature could be identified in classroom recordings and whether
there were indicators of conceptions of mathematics as sensible that seemed to be missing from
the list.
The list of indicators proved helpful for identifying incidents and most of the incidents
were relatively easy to classify using the indicators. There are, however, three important
limitations to the list of indicators. The first limitation is that the list of indicators is designed to
provide evidence that students conceive of mathematics as sensible not to identify when students
do not have such a conception. In the pilot data, there were some classes in which there were
many indicators that students conceived of mathematics as sensible. There were two classes, the
one from the TIMMS videos and the class taught by Mr. A, in which there were very few
indicators to code. Because of the design of the indicators we can draw some conclusions about
the conceptions of students in the classes in which there were indicators, but it would be a
mistake to assume that, because there were no indicators to the contrary, that students in the
other classes did not conceive of mathematics as sensible. With the exception of the three
negative indicators in the list of indicators all of the indicators identify when such a conception is
likely to be present, not when it might be absent. Drawing conclusions about the lack of
students’ conception that mathematics makes sense would require a different list of indicators.
The second limitation of the list of indicators is a lack of flexibility in coding incidents.
In the pilot data there were several incidents that seemed to provide evidence that students
conceived of mathematics as sensible but that did not match any of the indicators. For example,
in one incident, the teacher is handing out a worksheet and commenting on some of the problems
on the sheet. A student interrupts with a mathematical joke.
T:
One, two, three, four – easy. Five … five, you’ll enjoy.
41
S:
One, two, seven, eight, nine.
T:
Is that your phone number?
S:
No, it’s how I count.
In this incident it appears that the student chooses to take some of the teachers’ words out
of context and interpret the teacher’s “One, two, three, four, … five … five as a nonstandard
counting sequence. He teases the teacher by presenting his own nonstandard sequence. This
joke, relying as it does on the basic sequential structure of the number system would seem to
indicate that the student conceived of mathematics as sensible enough that he noticed when the
teacher’s statement could reinterpreted as violating that structure and used humor to mirror the
violation. The lack of any way to code this incident using the existing list of indicators pointed
to the need for either more indicators or another way to code the data.
A third limitation of the list of indicators became evident when trying to code transcripts
from Mr. Wingate’s classroom. In that classroom there were often so many indicators that
working with a randomly organized list of 22 items was very cumbersome. Because of this
difficulty, I sought a way to organize the list of indicators. I extracted five themes which could
be phrased as and used as broadly defined indicators. These themes became the five guiding
statements for the framework of indicators.
•
Students who conceive of mathematics as sensible expect that things in mathematics can
be explained so they may seek or provide explanations.
•
Students who conceive of mathematics as sensible expect connections so they may seek
or express connections.
•
Students who conceive of mathematics as sensible believe that you can reason through
problems so they may strategize about how to do mathematics.
42
•
Students who conceive of mathematics as sensible believe that mathematics is
authoritative and may assume mathematical authority.
•
Students who conceive of mathematics as sensible may state that something in
mathematics makes sense or may talk about common sense playing a role in doing
mathematics.
Organizing the list of indicators into categories solved both the problem of dealing with
22 separate items and the need for a more flexible way to identify indicators. The indicators
were arranged into an initial framework around five major categories: strategizing, expecting
connections, expecting explanations, assuming authority, and stating (see Figure 3-1).
Figure 3-1 – Initial Framework
43
This initial framework was then used to code the classroom data, as described in Chapter
4. The general categories made it easy to find specific indicators for coding. The general
categories also provided the flexibility to code incidents using just the category rather than
naming a specific indicator. For example, the student joke cited earlier could be coded as seeing
mathematical connections even though it did not match any specific indicator in the category.
This initial framework made it much easier to identify incidents in which students’ actions
indicated that they saw mathematics as sensible and it also provided the flexibility to code some
critical incidents in the classroom that might otherwise have been missed because they did not
match a specific indicator.
Having identified categories for use in the framework, I now examine some of the
literature related to the four major categories: explaining, strategizing, making connections, and
assuming authority. After looking at literature related to these four categories, I discuss several
other observation frameworks and their connection with my initial framework.
Literature in the framework categories
Doing a focused review of the literature in mathematics education related to the
categories of explaining, strategizing, and making connections is made almost impossible by the
fact that the categories are so broad and so central to what it means to do mathematics that they
are, in some way, imbedded in virtually every policy and standards document. The NCTM
Process Standards (NCTM, 2000) explicitly address strategizing, explaining, and making
connections within mathematics. The CCSS-M Standards for Mathematical Practice (NGA &
CCSSO, 2010), while less explicit, also embed these categories throughout. For example
“construct viable arguments” (p.6) is an important indicator from the explaining category while
44
“explain correspondences between equations, verbal descriptions, tables, and graphs” (p. 6) is an
important part of making connections. Likewise, Adding It Up (NRC, 2001) has references to
indicators in the categories of strategizing, explaining, and seeking connections embedded in
multiple different strands of mathematical proficiency. For example, explaining why is
described as a central part of adaptive reasoning, ““One manifestation of adaptive reasoning is
the ability to justify one’s work” (p. 130) and making connections is central to the strand of
conceptual understanding, “facts and methods learned with understanding are connected” (p.
118). The central role the actions signified by these categories within the policy and standards
literature means that these categories are also, in some way, related to most studies about the
learning of mathematics. The challenge is that most studies accept the importance of these
categories and move directly to discussion of how to engender such behaviors in students. There
is seldom much discussion about why the behavior is important or what the action might indicate
about students’ conceptions of mathematics.
Explaining
Much of the literature related to students explaining in mathematics focuses on
explaining why. There is broad agreement that justification is a central part of mathematics and
shoudl be a central part of school mathematics (Zaslavsky, Nickerson, Stylianides, Kidron, &
Winicki, 2012). There has been substantial research on the ways in which students do (or do
not) engage in justifying and how teachers might engage students in justifying. While some the
literature focuses quite narrowly on mathematical proof, some researchers have worked with a
more expanded vision of explaining why. For example, Harel and Sowder (1998) described
several different “proof schemes” as a way to discuss types of justifications. Similarly,
45
Beckmann (2002) makes a distinction between the goals, methods, and rigor of “explaining
why” and “proving.” These distinctions help to make it clear that explaining why may include
a far broader range of student actions than just engagement in formal mathematical proving.
In addition to a discussion of explaining why they believe that something is true,
there is also literature about the importance of students explaining how they solved mathematical
problems. Beckmann (2002) states, “Certainly, making sense of mathematics and engaging in
mathematical reasoning are intimately connected to explaining mathematics.” The literature on
classroom discourse and student activity in reform mathematics classrooms is rich in descriptions
of the ways in which students engage in this type of behavior and ways in which teachers can
help facilitate this activity (e.g. Boaler, 2008; Baxter & Williams, 2010; Lampert, 2001; Sherin,
2002; Stein, Smith, Henningsen, & Silver, E., 2000). There are several researchers who make a
clear distinction between explaining how and explaining why and use both categories in
describing their work (e.g. Beckmann, 2002; Inagaki, Hatano, & Morita, 1998; Kinach, 2002)
Literature on both explaining how and explaining why in mathematics classrooms provides
descriptions of what such discussion might look like, testifies to its importance in mathematics
learning, and provides valuable insights into how teachers might encourage it in their
classrooms.
Strategizing
Much of the current literature about strategizing in mathematics is contained in
studies of constructs such as adaptive expertise (Hatano & Oura, 2003) and strategy adaptivity
(Verschaffel, Luwel, Torbeyns, & Van Dooren, 2009) which focus on flexibility and adaptability
in the use of strategies. Despite the emphasis on the importance of strategizing and adaptive
46
reasoning in mathematics education, there is little research focused on this student behavior. As
with literature on explaining there are examples and suggested instructional practices for
strategizing embedded in much of the literature on reform teaching practices. However, in 2009
Verschaffel, Luwel, Torbeyns, & Van Dooren commented that, “...there is a basic belief in the
feasibility and value of striving for strategy flexibility/adaptivity. However, this basic belief, as
well as some accompanying presuppositions about when and for whom and how to strive for it,
have not yet been subjected to much systematic and scrutinized theoretical reflection or
empirical research” (p. 345) and there appears to have been little progress made in the
intervening years.
Making Connections
There is strong agreement that connections within mathematics are a critical part of what
it means to know and do mathematics (Burton, 1999). The development of conceptual
understanding in mathematics has been widely studied and written about and most of this work
focuses on making connections within mathematics and between different mathematical
representations. Studies in this area suggest many instructional practices to help students
develop connections within mathematics including the use of rich mathematical tasks (e.g.
Ayres, Sawyer, & Dinham, 2004; Boaler, 1998; Stein, Grover, & Henningsen, 1996; Stigler &
Hiebert, 2009), engagement of students in classroom discussion about mathematics (e.g. Boaler,
1998; Cobb et al., 1991; Cobb, Wood & Yackel, 1993; Hiebert & Wearne, 1993; Pape, Bell,
Yetkin, 2003; Stein & Lane, 1996), and explicitly focusing students’ attention on concepts and
connections (e.g. Anthony & Walshaw, 2009; Corcoran & Silander, 2009; Hiebert & Grouws,
2007; Hodara, 2011; McNaught & Grouws, 2007).
47
In addition to an almost universal recognition of the importance of students making
connections within mathematics, there is wide agreement that students also need to see and make
connections between mathematics and other contexts. The CCSS-M Standards for
Mathematical Practice (NGA & CCSSO, 2010) devote an entire practice standard to using
mathematics for modelling and most literature on mathematical sense-making includes seeing
mathematics as relevant and useful in real life and in other academic contexts as an important
component of sense-making (e.g. Hirsch, Coxford, Fey, & Schoen, 1995; Leinwand, 2000;
Schoenfeld, 1992). Despite the agreement of the centrality of make connections in doing
mathematics there is little attention in the literature to the relationship between students learning
to make connections and students’ conceptions of the nature of mathematics.
Assuming Authority:
Mathematical authority is a construct generally seen as an important dimension of the
interactions in a mathematics classroom. It has been conceptualized somewhat differently by
different authors. Stein, Engel, Hughes, Smith (2008) conceptualize student authority as a
classroom norm stating that, “The idea behind student authority is that learning environments
should be designed so that students are “authorized” to solve mathematical problems for
themselves, are publicly credited as the “authors” of their ideas, and develop into local
“authorities” in the discipline” (p. 332). Wilson & Lloyd (2000) conceptualize it as a more
individual concept, basing their work on an understanding of authority which stresses “the
central role of individuals developing an inner voice, one in which authority is seen as an internal
rather than an external factor” (p. 151). Although slightly different in focus, the various
48
conceptualizations of mathematical authority all tend towards a more democratic view of the
locus of authority than the traditional model of teacher as sole authority figure.
In the literature on mathematical authority there are two assumptions about where
mathematical authority properly resides. Some authors discuss authority as resting upon
community-established standards (e.g. Brodie, 2012; Davis, 1997). Others see authority as
residing more in the discipline of mathematics and mathematical reasoning. This view is
exemplified in Adding It Up (NRC, 2000) which describes one advantage of adaptive reasoning
as, “Students who disagree about a mathematical answer need not rely on checking with the
teacher, collecting opinions from their classmates, or gathering data from outside the classroom.
In principle, they need only check that their reasoning is valid” (p. 129). Schoenfeld (1994)
strives to reconcile these two conceptions of mathematical authority. He states, “One might say
that the ultimate authority is the mathematics itself…Nonetheless, mathematical authority is, in
practice, exercised by human hands and minds” (p. 61). Whether mathematical authority is seen
as more focused on community or individual or whether that authority resides in the mathematics
or is embodied in community norms, there is broad agreement in the mathematics education
community that the concept of mathematical authority is critical in mathematics education.
The four major categories of the initial framework, strategizing, seeking connections,
explaining, and seeking authority are all behaviors that are widely accepted as central to students
engaging in mathematics is a sense-making way. There has been some progress identifying
instructional practices which may be useful for encouraging these actions in students. What is
still needed is a better understanding of how these particular actions might fit together and how
they are related to students’ conceptions of mathematics as sensible.
49
Other Observation Frameworks:
There a number of frameworks in the literature designed for coding classroom
observations. In this section I will discuss two observation protocols and their relation to my
initial framework. The Reform Teaching Observation Protocol (RTOP, Piburn & Sawada, 2000)
was chosen because of its extensive use in mathematics education research. The Classroom Visit
Protocol was chosen because it is the most recently published such framework available.
The RTOP (Piburn & Sawada, 2000), as the name suggests, focuses on reform practices
in mathematics classrooms and has been widely used by educational researchers. Although my
study was not designed to focus on reform practices and the curriculum, and structure of the
classroom in this study are quite traditional in nature, the main categories within the initial
framework (strategizing, explaining, seeking connections, and assuming authority) are ones that
are often associated with reform mathematics instruction. The RTOP contains several items
closely resembling indicators from my initial framework. For example, one of the indicators
from my initial framework is “Students using multiple representations in problem solving” and
an item from the RTOP is “Students used a variety of means (models, drawings, graphs, concrete
materials, manipulatives,etc.) to represent phenomena” (p. 16). Despite the similarity of several
items in the two instruments, they were designed to serve very different purposes. The RTOP
was designed to observe teaching in classrooms, focusing particularly of reform teaching. By
contrast, my framework was designed to observe student behaviors in a mathematics classroom
and was not intended to focus on a particular type of instruction. Because of these differences
my initial framework has far more items focused on student behaviors. All 22 indicators in my
50
initial framework focus on students’ actions as opposed to 5 or 6 out of the 25 items in the
RTOP. The RTOP also contains many items not included in my initial framework because the
items focus on other aspects of the classroom such as classroom norms and teacher actions.
The Classroom Visit Protocol (CVP, Grouws et al., 2013) was developed to “document
the use of textbook materials and classroom activities (p. 430). Of the three parts in the CVP, the
one most closely related to my initial framework is the Classroom Learning Environment
instrument. This instrument contained items relating to strategizing, making connections, and
sharing of mathematical authority. However, the items in the CVP are focused on opportunities
provided for students rather than on the actions of the students. The CVP is a good tool for
examining the instruction and curriculum use in the classroom, including instruction relating to
some of the categories in my initial framework, but does not provide a way to examine the ways
in which students engage in doing mathematics.
51
Chapter 4 Research Methods for Phase 2
The major focus of the next portion of the study is to use the framework in a classroom
setting to identify and document action-oriented indicators of student conceptions of
mathematics as sensible. The goal is to test the usefulness of the framework for creating the type
of “thick description” envisioned by Ponterotto (2006), both “describing and interpreting”
students’ behavior in this classroom context. This use of the framework provides an opportunity
to test and improve the framework through use of classroom data. This qualitative study
generates a theory about students’ conceptions of mathematics as sensible that is grounded in
both the literature and classroom data.
In this chapter I will first describe the classroom setting and study participants. I will
describe the data that was collected and the process used in analyzing the data. I will finish with
a discussion about the trustworthiness of the data and the analysis process.
Selecting the setting
This study is an instance of a study being designed around a case of intrinsic interest
(Creswell, 2007). For five years I had the opportunity to teach down the hall from one particular
teacher’s classroom in which students seemed to share a characteristic that is all too uncommon
in many high school students—these students seemed to expect mathematics to make sense.
That is, they expected that there are reasons for rules and procedures (even if they did not
understand those reasons yet), they approached unfamiliar problems as if there was a good
chance that reasoning from what they already knew would be an effective solution strategy, and
they expected mathematical topics to be connected to one another. Although I had encountered
52
such a view of mathematics in students before, it was striking that this view seemed to
characterize almost all of my students who had previously been a part of this classroom, even
students with a history of low achievement in mathematics. Because of this unique
characteristic, I purposely selected this classroom as a setting to study because I wanted to know
more about the students’ conceptions of mathematics as sensible and I had reason to believe that,
in this class, there would be evidence of such a conception.
The setting
The classroom that I studied is a mathematics classroom in a combined middle school
and high school in central New Hampshire with slightly less than 500 students in grades 7–12,
all from the same town. The community is more than 98% white with a median household
income more than 25% above the state average. Almost 50% of the adult town residents have
earned a bachelor’s degree or higher. The school is overcrowded but well-funded and prides
itself on employing top quality teachers. Student scores, district-wide, on state-wide testing are
consistently above the state average for all subjects tested. The school runs on an alternating
block schedule in which classes meet every other day for 90 minutes at a time.
Until the school year during which the observations for this study took place, the high
school mathematics sequence consisted of Integrated Mathematics 1, Integrated Mathematics 2,
Integrated Mathematics 3 or Advanced Algebra, Pre-calculus, AP Calculus 1 (AB), and AP
Calculus 2 (BC). For Integrated Mathematics 1 and Integrated Mathematics 2 there were
“honors” sections, “regular” sections, and “supported” sections with additional special education
support. The school was, during the academic year in which observations for this study took
place, in the process of transitioning to a more traditional Algebra 1, Geometry, Algebra 2
53
sequence and was offering Algebra 1 for the first time, also with regular and honors sections, and
with a supported section called Algebra 1A in which students studied the first half of Algebra 1
one year and the second half the following year.
Supported sections of mathematics courses are generally small and are cotaught by a
mathematics teacher and a special education teacher. Many of the students in these sections have
been identified as having substantial learning disabilities and have Individual Educational Plans
(IEPs) involving mathematics. In the Integrated Mathematics 2 course, the textbook and course
competencies are the same as for both the regular and honors sections of the course. For the
Algebra 1A there is more variation.
Several years ago, the State of New Hampshire began requiring that credits for high
school courses be given based on demonstrated student competencies in course material rather
than on Carnegie units. Each school district was charged with determining competencies for
each course and determining how these would be measured. In the mathematics department at
this high school, the department determined a number of competencies for each course. The
teacher involved in this study uses short pencil – and – paper tests to determine each student’s
achievement of each competency. Students may retake tests as needed until they have
demonstrated mastery of all of the competencies for the course.
Participants
Students involved in this study were enrolled in a supported section of Algebra 1A during
the entire 2011-2012 academic year. All 18 students enrolled in the class agreed to participate in
the video portion of the study. Thirteen agreed to be interviewed. Placement in the class is by
permission only and is limited to students who have experienced difficulties with mathematics in
54
the past or who have a documented learning disability related to mathematics. Most of the
students are freshmen and were enrolled in a supported Pre-Algebra class during the previous
school year. Many of them experience difficulties in other subject areas as well and are more
likely than the school population as a whole to face disciplinary action for breaking school rules.
The teacher involved in this study, Mr. Wingate, has a bachelor’s degree in mathematics
and approximately 35 years of experience teaching high school. He has been teaching supported
sections of mathematics classes for more than 10 years. He also teaches the school’s AP
Calculus classes and a class in desktop publishing.
During the spring observation period there was also a student intern in the classroom.
She was an undergraduate students in an education program at a nearby university engaged in an
early field experience. Her primary roles were to observe and to assist individual students or
small groups engaged in independent work. Because the spring observations took place near the
end of her internship she also led class discussion of warm-up problems on several occasions.
Throughout this document study participants will be referred to using pseudonyms.
Daily pattern in the classroom
Although there were some days in which special mathematical activities changed the
pattern, most days in this classroom are very similar. When students arrive in the classroom
there are four or five warm-up problems on the board. These warm-up problems are not the
quick, straightforward review exercises usually associated with warm-ups. They generally cover
a broad range of topics that the students have already seen. One day the warm-ups required
students to find the area of an unusually shaped region, the area and perimeter of a rectangle with
side lengths containing variables, and slopes of several lines. Another day the students were
55
asked to find the area of a concave hexagon, find the value of f(3) and f(10) for the function
f(x) = x2, solve a linear equation, and find the next values in the sequence 1,1,2,3,_,_. For these
students, most of these problems were familiar but far from trivial. Students usually worked on
the warm-up problems for 10 to 15 minutes, sometimes with an interruption for a whole-class
discussion of strategy if students seemed to have difficulty knowing how to begin a problem.
Once students had finished the warm-ups and handed them in to the teacher, the teacher would a
lead a discussion about the warm-up problems and their solutions. The discussion focused
around students explaining their solutions to the class but the teacher also often extended the
discussion of the problems. Going over warm-up problems often took 30 minutes or more. The
45 minutes or more often spent on warm-up problems carried much of the content of the course.
After the warm-up problems there was generally 15 to 20 minutes in which the teacher discussed
new material or reviewed material that students still needed to work on. This was usually
followed by work time in which students worked on their own or with other students on practice
problems from a worksheet.
Data collected
The corpus of data for this study was collected in a single class over the course of the
2011-2012 academic year. The primary data is a set of video and audio recordings of the
classroom (Yin, 2011). These record are supplemented by field notes (Hays & Singh, 2011), focus
group style interviews (Morgan & Spanish, 1984), and classroom artifacts (Delamont, 2002).
56
Preexisting data
The teacher in this study, Mr. Wingate, regularly video records his lessons for his own
purposes. He also regularly collects such classroom artifacts as worksheets, quizzes, and tests.
He has video recordings and classroom artifacts for the supported Algebra 1A class for
September and October of 2011 that I am able to access. The video camera is placed at the back
of the classroom somewhat above the students’ heads and focused on the center of the
whiteboard as shown in Figure 4-1.
Figure 4-1. What the video shows.
This position captures the teacher presenting the lesson, most of the whole-class
discussion, and some of the student to student discussion. The primary consideration for the
positioning of the camera was to create the least possible disturbance to the class. Because of the
location above students’ heads the audio of the classroom discussion was excellent. However,
the camera placement did not provide a complete record of what was written on the board or of
students’ and teacher’s actions when they were not standing in the middle of the classroom.
Sometimes students sitting near the camera turned the camera to capture these events but this
57
was not the norm. Although this stationary placement limited some of the available data it also
provided “blind spots” in the room in which students who did not mind having their voices
recorded but preferred not to be in the video could still present solutions and information to the
class. Pre-existing data to be used in this study consists of video recordings and related
classroom artifacts for five consecutive 90-minute class periods beginning with the third class
period of the year. The participating teacher and I chose to begin with the third class period of
the year rather than the first day of school because the first several classes of the school year
were not full-length class periods and were devoted primarily to dealing with administrative
details rather than mathematics instruction.
Data collected during spring observation
During the late spring I observed five consecutive 90-minute classes and collected video
recordings, audio recordings, classroom artifacts, and field notes for each of these observations.
The participating teacher and I choose the timing of these lessons to be close to the end of the
academic year while avoiding the school’s spring break, the state-wide testing, school-wide
testing, and review for final exams.
Video recordings
The video recordings collected during the spring observation are similar to the preexisting videos described previously.
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Audio recordings
The audio recordings were collected using five individual recorders placed on student
tables throughout the classroom. They recorded the conversations between students as they
worked on mathematics problems and the discussion between the teacher and individual students
or small groups of students. An additional audio recorder recorded conversations between the
teacher and students from a wireless lapel microphone worn by the teacher.
Classroom artifacts
I collected copies of worksheets, quizzes, tests, and student handbooks. I took
photographs of the classroom set-up and obtained other artifacts related to the class such as
charts of course competencies, class notes taken by the special education teacher, and lesson
plans.
Field notes
Field notes consisted of descriptions of each class as it was conducted. I gave special
attention in the notes to comments, gestures, and actions that may not be captured by the video or
audio recordings and work on the white board that was unlikely to be captured by the video. I
used the field notes to record observer comments and initial impressions to be revisited later
during data analysis.
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Interview data
Group interviews/focus groups with students took place during the same time period as
the spring classroom observations or on the last day of classes before final exams. I interviewed
13 of the 18 students in the Algebra 1A class. There were five group interviews with two to
three students in each group. One of the students, who tended to have difficulty communicating
and interacting with other students, was grouped with the instructional assistant with whom he
usually worked rather than with other students. The interviews were structured around a card
placement activity (Coxon, 1999). Students were given index cards with statements about the
nature of mathematics. They were asked to discuss the statements and make a group decision
about where to place each card along a continuum from “strongly agree” to “strongly disagree.”
Figure 4-2 shows the cards. . (The cards provided to students did not contain the citations
shown on the cards below.)
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Figure 4-2. Conceptions cards for student interviews.
Group interviews were 15 to 25 minutes long. A video recording was made of the interviews
with the camera focused on the card placement. In addition, two audio recordings of each
interview were made to provide a back-up data source.
I chose to use an interview task as a way to triangulate the data from the classroom
observations. Guion, Diehl, & McDonald (2011) identify such “methodological triangulation” as
one of 5 types of triangulation commonly used by qualitative researchers “to check and establish
validity in their studies” (p. 1). I chose to work with groups of students rather than conduct one on one
interviews because I expected the groups to be “useful in accessing the ‘hard to reach’ and the potentially
recalcitrant” (Barbour, 2007, p. 29). While my groups lacked the minimum of 4 members often
considered necessary to qualify as a focus group (Krueger & Casey, 2009) the overall purpose of using a
group was similar. My goal was to provide focus for the discussion and then listen in to the conversations
of the students. I chose the card sorting activity as a tool to stimulate discussion, expecting that the need
to reach a group decision about card placement would stimulate discussion about the meaning and
importance of the statements for the participants. A possible limitation to this type of interview is results
and discussion can easily be influenced by dominant individuals (Krueger & Casey, 2009). I tried to limit
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this issue by ensuring that all students in the group provided input on decisions and having a sufficient
number of groups so that there could be a range of opinions across groups as well as within groups.
Analysis of classroom data
The primary data from the classroom were the video recordings of two weeks of classes
in September and 2 weeks of classes during April and May of the same academic year. A total of
ten 90-minute classes were analyzed. Field notes and classroom artifacts were used, as
necessary, to clarify what was seen and heard in the video recordings. The video recordings were
analyzed using Studiocode software to identify and code indicators that students saw
mathematics as sensible (see Figure 4-3). This software permits a user to identify and label
segments of video as it is watched so, as I viewed the video I first tagged any indicators with the
labels from the major categories and then went back to add labels for specific indicators. Using
Studiocode I could then adjust the beginning and ending points of video clips, code additional
indicators in future passes through the video, and sort video clips by indicator or category. Use
of this type of software eliminated some of the cumbersome and time-consuming tasks of video
editing (Rich & Hannafin, 2009) making it more practical to analyze a larger set of data and
allowing for easy access to any coded video clip for further analysis (Jacobs, Towanaka, &
Stigler, 1999). In addition, coding directly from the video rather than from a transcript allowed
me to consider a greater range of communication with such nuances as gestures, facial
expressions, and tonal inflections all contributing to a better understanding of the classroom
communication. A serious limitation to this type of coding, however, is that it permits only
limited opportunity for flexibility in coding since a coding framework must be established in
advance. This type of analysis would not be well suited to open coding of data. In this study the
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limitations were addressed by both the initial, literature-based set of indicators and the use of the
more general categories of the initial framework when specific indicators did not closely match
the data.
Figure 4-3 Studiocode coding scheme.
Each video was viewed at least twice, stopping to insert a code whenever an indicator
appeared. Incidents were coded with both the specific indicator (shown in the small outer
rectangles in Figure 4-3) and with the larger categories (shown in the larger rectangles in figure
4-3). The 14 incidents that reflected larger categories but did not fit individual indicators were
coded with only the categories. Incidents that exemplified more than one indicator were
assigned multiple codes. Altogether, there were 86 coded incidents in the Fall data and 132 in
the Spring data.
Once the incidents were identified and coded, they were compiled into tables that
described the details of the incident. The tables included the time stamps from the video, the
coding of the category and the specific indicator, the context in which the incident occurred, a
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description of the role of prompting in the incident, and a description of the incident (see Table
4-1 for an example).
110901a #
110901a 1
13:47:53
14:14:37
category indicator
strategize Discuss
how
context
Warmups with
neighbors
prompt
Teacher is at
board
correcting
issue of one
student, this
student raises
hand and
volunteers the
strategy
Event
Student suggests, “so wouldn’t you
have to find the, like take the
triangle and find the perimeter of
just the square and then like … er, I
mean the area” as a strategy for
finding the area of the figure
Table 4-1 – Sample incident detail
After coding the video for each class, the audio recordings from the spring observations
were reviewed. There were no incidents captured on audio recordings that had not already been
coded from the video recordings. However, there were several cases in which the audio
recording provided more detail about an incident and those details were added to the descriptions
of incidents in the tables.
After all video was coded, several types of summaries and compilations were created.
Compilations of video clips were created: one video for each class period containing all coded
incidents and a video for each category containing all the incidents with like coding. These
video compilations provided easy access to individual incidents. They, used in conjunction with
the incident detail tables, were used to look for patterns across incidents coded with the same
indicator or the same category.
Coding timelines visually displaying both the frequency and duration of incidents with
different codes were created for each class period and for the cumulative data for spring and fall.
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These timelines were used to look for patterns in when incidents occurred during class periods,
the relative length of episodes for different codes, and the coding categories that tended to occur
together. The combined seasonal timelines (see Figure 4-4) were used to look for changes in the
patterns of frequency, duration, and type of coding between the beginning and the end of the
academic year.
Figure 4-4 – Comparison of Fall and Spring Combined Timelines. The top (yellow) stripe represents
explaining. The second (blue stripe) is strategizing. The third (red) stripe is authority. The bottom
(green) stripe is connections.
Analysis of student interviews
The card sorting activity from the student interviews provided both data about how
students actually placed the conceptions cards along the strongly agree/strongly disagree
continuum and verbal comments from students about the statements. Each group’s level of
agreement with a statement was evaluated by assigning the placements on the continuum a value
from 0 (strongly disagree) to 10 (strongly agree). In cases in which groups had disagreed and
split their placement of the card, the placement values were averaged. This system allowed for
comparison of placements between groups and a way to examine the average and variation of the
class’s ranking of particular statements. In order to analyze the comments made by students
during the interviews, verbatim transcripts of the interviews were created. Representative
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comments, along with numerical data on group agreement with statements, were then compiled
into a table to allow for easy comparison of comments across groups and across statements. The
conceptions statements were mapped to the main categories in the framework and students’
comments related to particular categories of conceptions of mathematics as sensible were linked
with the classroom indicators for those same categories. Together these data points were used to
look for themes in students’ conceptions of mathematics. In a number of cases students’
interpretations of the conceptions statements or comments made while discussing a particular
statement were related to a category in the framework other than that intended by the
conceptions statement. In those cases, the students’ statements were included in analysis of the
appropriate category.
Trustworthiness
One important check on the trustworthiness of any qualitative study is triangulation of
data sources. To provide triangulation for the first phase of the study, I had at least two sources
in the literature for 20 out the 22 indicators developed (see Table 3-4). In addition, seven
volunteers with expertise in mathematics education examined and provided feedback on the
development of three representative indicators from the source literature.
For the second phase of the study seven volunteers with expertise in mathematics
education provided feedback on the initial coding scheme for the classroom data and its
application to a short video clip from the data. In addition, three volunteers with expertise in
mathematics education were trained to use the framework for coding and asked to code several
short portions of the classroom video. These volunteers and I then discussed the coding of each
incident in the video portions. Working together, there were no substantial differences between
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the codes that I had assigned and what the group agreed was appropriate. Triangulation of
classroom data about students’ conceptions of mathematics was provided by the use of both
classroom observations using the framework and student interviews related to conceptions of
mathematics. Finally, the participating teacher was asked to read and comment on the
manuscript. This member check (Yin, 2011) helped to ensure the accuracy of my data and that
my interpretation of that data was deemed reasonable by the teacher involved in the study.
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Chapter 5 Findings
Analysis of classroom interactions using the framework of indicators provided substantial
insight into the ways in which students in this class conceive of mathematics as sensible. The
framework provided both indicators that identified important incidents and an organizational tool
for examining student conceptions within several categories. The insight gained from the
indicators combined with information gathered in the student interviews provide a rich picture of
the conceptions of students in this class and the ways that those conceptions may have grown and
changed over the course of an academic year.
Findings related to expecting explanations
Students in this class expect that mathematics is sensible enough that it can be explained.
They demonstrated this by providing explanations for their own mathematics and by seeking
explanations for mathematics presented to them. They also expressed strong agreement with
statements about the existence of and importance of explanations in mathematics.
Types of Evidence
There were four indicators and three conceptions statements related to students’
conceptions that mathematics was sensible enough to have explanations. Two of the indicators
were related to students’ engagement in providing explanations: justifying, that is, explaining
why something works or is true and explaining answers, that is, explaining how a problem was
done. The other two indicators were related to students’ seeking explanations: seeking
explanations for ideas and procedures and seeking explanations for why an answer is correct or
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incorrect. The conceptions statements asked students to discuss whether they agreed or
disagreed that: “Knowing how to solve a problem is as important as getting the solution”
(Brown, et al., 1988), “It is not important to understand why a procedure works as long as you
get the right answer” (Kloosterman & Stage, 1992), and “A lot of things in math must simply be
accepted as true and remembered; there really isn’t any explanation for them” (Carter &
Norwood, 1997).
Providing explanations
Two of the indicators relating to explanations in mathematics involved student
engagement in the act of providing explanations. The explain answers indicator was used when
students explained how they arrived at a conclusion, whereas the justifying indicator was used
when students explained why a conclusion or action was valid. Students displayed these
behaviors frequently during both the fall and the spring observations.
Explaining how
Explaining how they solved a problem or arrived at a mathematical conclusion seemed to
be a central feature of how the students in this class engaged in mathematics. There were 20
coded incidents of explaining how during the fall observation and 16 during the spring. During
the first two classes observed in the fall, most of the explanations were prompted by either the
teacher or by another student asking such questions as, “How did you do it?” After these classes,
students were seldom prompted to explain their answers. The teacher usually sought student
volunteers by asking questions like, “Who wants number 2?” or sometimes simply, “Number 4?”
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Students took this as a cue to begin describing, step by step, how they solved the problem. The
students’ emphasis on explaining how is highlighted by the fact that they usually provided an
explanation before giving the answer to the problem. Students in this class explained how they
solved a problem an average of three to four times per class, with that average remaining fairly
consistent between the fall and spring observations.
Some of the explanations provided by students were fairly complete but rather
mechanical. For example, during one of the spring observations, the second warm-up problem
was: Solve for x: 3(2x – 9) = 5x + 4. When a student was asked to, “do number 2” she provided
the following explanation for how she did the problem.
To begin with I distributed before I did the solving. I did 3 times 2x is 6x. 3 times
negative 9 is negative 27. And from there equals 5x plus 4. … minus 5x on each side …
the double 5x’s cancel out … and 6x minus 5x equals 1x minus 27 equals 4. … plus 27
on each side. 27s cancel out. 1x equals 31. Divide each side by 1. The 1s cancel out.
31 divided by 1 equals, x equals 31.
As the student spoke, the student intern in the class worked out the problem on the board. This
type of complete, detailed but very mechanical explanation occurred occasionally but was not the
norm. It occurred most often with routine problems such as solving equations and was more
likely to occur when the discussion was being led by the student intern rather than by the teacher.
Far more common was an explanation embedded in an interchange between the teacher
and one or more students. The following exchange, which occurred during a fall observation,
was typical of this type of exchange. One of the warm-up problems asked students to find the
area of the trapezoid shown in Figure 5-1.
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Figure 5-1. Warm-up problem – Find the area.
After one student argued that if you completed the triangular portion of the figure you
would have a rectangle of the same size as the rectangular portion of the figure, the following
exchange took place.
Teacher:
You’re on the right track Jim but you don’t have the right numbers as yet.
Beth next. Yup. You know I’m going to call on everybody every day.
Beth:
Since the bottom, like, it’s the 16 but they’re two different shapes so you
would, the top of the rectangle is six centimeters so the bottom of it is six
centimeters
Teacher:
Ah. Do you see what Beth is telling us? (The teacher underlines the
bottom of the rectangular portion in red.) That red length has to be six.
How does she know that?
Jim:
Because it’s parallel to the top.
Teacher:
Yeah. Because it’s, here let’s pull the rectangle out (teacher draws a
rectangle off to the side). That’s six so that has to be six, right? (Teacher
writes a six on the top and bottom of the rectangle as he speaks.) Isn’t that
the way rectangles work? Okay, keep going Beth. I like your thinking.
Beth:
And then since you have 10 left on the triangle you can tell
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Teacher:
Ah. So you’re saying that this length - here let me draw the whole shape.
(Teacher draws the remainder of the trapezoid using the rectangle he has
already drawn.) You’re saying that this length (he underlines the base of
the triangle) has to be what?
Beth:
10
Teacher:
10. Because together
Beth:
they make 16.
Teacher:
They make this 16.
In this exchange Beth provides an explanation for how she determined that the length of the base
of the triangle was 10 as the teacher restates, clarifies, and illustrates the explanation.
Most of the incidents coded as students explaining answers took place during whole class
discussion while the class was going over warm-up problems or other problems that the teacher
had written on the board for students to work on. In addition to these episodes in whole class
discussion, there a many fragments of conversation between two or more students as they
worked independently to indicate that students also explained answers to one another. One of
these came during a spring observation in which students were working on finding the equation
of a line when given two ordered pairs.
Jenny:
Did you get ½, 1 over 2?
Beth:
Uh, yeah.
Jenny:
How did you do it?
Rachael:
So it would be 2 …
Jenny:
Why would it be 2?
Rachel:
’Cause you have to put in y into y equals mx plus b. So you pick a y
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Jenny:
You picked, you picked 2, what’s, what’s , oh!
Rachel and Jenny: It’s point 5.
Rachel:
And then times …
Jenny:
Oh, so you just changed the ½ to point 5
Rachael:
Yeah
It seems evident in this clip that Rachael is explaining her work to Jenny. However, like so
many of the student-to-student exchanges, it is fragmented and it is difficult to pick this
conversation out of the ones that surround it. It is also often difficult to know what problem the
students are working on. The frequency of these types of clips makes it evident that students are
explaining their work to other students. Unfortunately, the difficulty identifying these episodes
on the recordings means that they are probably seriously underrepresented in the coding of the
data.
It is very rare in this class to hear a single student provide a complete, cohesive
explanation for how they solved a problem. Both explanations provided during whole class
discussions and those among a few students, display a classroom pattern of discourse in which
explanations are formed in a joint effort of teacher and students. Explanations are not treated as
a performance by an individual but as the work of the whole group. This is seen in the episode
with Beth, previously described. In this episode it is not Beth explaining and everyone else
listening. It is Beth explaining a piece, the teacher representing the work on the board as he
reiterates her points, and another student jumping in to justify one of Beth’s assertions. After the
excerpt above, this conversation continues with several other students entering the discussion to
explain how they solved other pieces of the problem and then how they put everything together
to answer the original question. This class’s common practice of crafting group explanations
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seems to indicate that, for these students, explaining how is not only a central part of what it
means to do mathematics but that it is a viewed as a shared endeavor.
Students’ comments during interviews also provide evidence of the central role that
explaining plays in their mathematics. The best evidence for this is seen in students’ reactions to
the statement, “Knowing how to solve a problem is as important as getting the solution” (Brown
et al., 1988). Overall, students agreed with the statement, although many of them stopped short
of strongly agreeing. The group that placed the card the lowest was split on where to place the
card. Two of the members advocated for placing the card in the middle of the continuum.
I:
(Reading the card) “Knowing how to solve a problem is as important as
getting the solution.”
Joanne:
I mean, I guess that is in the middle for me because, obviously finding the
solution is important but knowing how to be able to figure out how to find
the solution is important too because you’re going to use the procedure if
you know how to use it more then you will find the solution. [Joanne
placed the card about midway along their agree/disagree continuum.]
Zach:
I’m still, I’m still thinking of the 50/50 part because it’s just, altogether
it’s just, well, you know, if you know how to do the, the problem, like
solve the problem it’s, and you know the answer to it, it’s just like saying,
uh, how do I put it? Um, if you that, I’m just going to go to a few
examples. Like, if you like, um, like 2 plus 2 and you know the answer to
it, like, very quickly, like just off the top of your head, you were already
taught like over and over and over, it’s good to memor-, it’s good to
memorize the math because, so you can do so much stuff with math. It’s
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uncountable. So, like, it’s good to know and also like, if you know how to
do the procedure it’s, you don’t, if you know how to do the procedure but
you don’t know what the answer is it’s always 50/50 if you get it right or
wrong. So I’m, yeah, I’m in the middle.
Zach seems to be arguing for the need for a balance between understanding explanations
and the value of automaticity and recall of basic facts in mathematics. Joanne’s concern for a
balance between the value of knowing how and the value of having the correct answer was
echoed by a student from another group who defended her group’s placement of the card near
Strongly Agree by stating, “I think it’s important to understand it before you can get just the
answer but you also need to be able to get the answer or else you’re probably going to fail math.”
The reaction of the third member of Joanne and Zach’s group was more typical of most other
students’ responses to this statement. The above discussion continued:
I:
Okay. So you two are definitely in the middle. Are you in the middle or –
Jerry:
I’m kind of thinking that, definitely figuring out the problem is way more
important than, um
Zach:
Finding the answer.
Jerry:
Yeah so, definitely
I:
So where would you put it? “Knowing how to solve a problem is as
important as getting the solution.”
Jerry:
I’m going to go, like, probably here [Zach placed the card close to the
strongly disagree end of the continuum.] because I’m a little bit
disagreeing with how it’s saying it’s just as important as figuring out the
answer.
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I:
You’re saying it’s more important.
Jerry:
Yeah, it’s more important. It’s definitely more important to know how to
do it than figuring out the answer. The answer would probably come last.
It’s not, solution, or, er, problem-solving thing – it’s, ‘cause I want to
know how to do it before I can find the answer.
Zach:
It’s just like saying breakfast before lunch or kind of last getting dessert.
Although Jerry places the statement closer to disagree, it is not because he disagrees with the
intent of the statement but because he feels that the statement is not worded strongly enough.
For him understanding how to do a problem is almost a prerequisite for being able to find the
correct answer. Zach seems to agree with him, using the analogy of breakfast before lunch and
dessert last.
The conception that understanding how to do a problem is not only as important as
getting the answer but that it necessarily precedes being able to get the correct answer was
echoed by members of several groups. For two of the groups the discussion was very short and
emphatic.
Group 6:
I:
Let’s start with, (Reading the card) “Knowing how to solve a problem is
as important as getting the solution.” “Knowing how to solve a problem is
as important as getting the solution.”
Beth:
Well, if you didn’t know how to solve the problem
Rob:
Yeah, you
Beth:
you can’t get solution.
I:
Okay.
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Beth:
Like, I would put that on strongly agree.
Rob:
Yeah.
Group 3:
I:
(Reading the card) “Knowing how to solve a problem is as important as
getting the solution.”
Rachel:
Well, you have to know how to get the solution,
Sarah:
Yeah, you,
Rachel:
In order to solve it you have to know what you are doing and then, to get
the solution, you have to know how to solve it.
I:
So you’re basically saying you really can’t solve it without knowing how.
Sarah & Rachel: Yeah.
As was true for Jerry in an earlier group, for these students knowing how a problem is solved is a
necessary precursor to getting a correct answer. The insistence of these students on the
importance of knowing how a problem is solved and their tendency to work together to produce
explanations for how a problem is solved are indicators that these students conceive of
mathematics as sensible enough that there are explanations for how problems are solved.
The speed with which explaining became a classroom norm makes it likely that the
tendency these students had to engage in explaining how they did mathematics existed before the
current school year. During classes at the beginning of the school year, students, when asked to
“do” or “go over” a problem sometimes presented only their answer. The teacher would then ask
for an explanation. The following exchange occurred as the class is going over answers to the
warm-up problems and is typical of the way in which the teacher prompted explanations at the
start of the school year.
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Teacher:
Number two, what’s my area? Eddie, what did you say?
Eddie:
I got 50.
Teacher:
you got 50, because?
At this point the student explained how he found the answer. After the first couple of
classes of the school year, this type of exchange seldom occurred because when asked about the
answer to a problem or to “do” a problem, students generally began by explaining how they did
the problem, presenting their answer only after the explanation was complete. Students’
reactions to conception statements about the importance of knowing how seem to take the
importance of explanations for how to do a problem as a given. None of the students provided a
reason for why understanding how was important, it simply was. One student attributed to
teachers, in general, an emphasis on how to do a problem. He stated, “Teachers want you to
know how to do it and they don’t really care about if you get the right answer. Like they want
you to get the right answer but it’s more about learning.” A belief in the importance of
explaining how they got an answer to a mathematical problem seems to be very strong in this
group of students. The unquestioning acceptance of the conception statement, the tendency to
engage in the behavior without prompting almost from the very beginning of the school year, and
the ease with which explaining answers became a class norm, indicate that this tendency likely
predates their involvement in this study.
Explaining why
Students in this class engaged in explaining why they arrived at mathematical
conclusions as often as they explained how they solved problems. There were 22 coded
incidents of justifying during the fall observations and 17 during the spring. However, there was
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a marked change between the fall and the spring observations in the role that teacher prompting
played in students’ tendency to engage in this behavior. During the fall observation, almost all
episodes of justifying were initiated by questions from the teacher. The following incidents are
typical of the type of prompting that was very prevalent during the fall observations. In this first
episode students are going over the warm-up problem shown in Figure 5-2.
Figure 5-2. Warm-up problem – What fraction is this?
Teacher:
Okay. Number one, quickly. What fraction is this? What did you say,
Jenny?
Jenny:
Five twelfths.
Teacher:
Five twelfths is correct. Why five?
Jenny:
Because five out of the 12 is shaded in or taken away or.
Teacher:
Are they the ones we want or the ones we don’t want?
Jenny:
They are the ones we want.
Teacher:
Okay.
Jenny:
I was just saying they weren’t taken away.
Teacher:
Okay. So five. And why 12?
Jenny:
Because there was 12 at the beginning.
Another example of this type of exchange occurs during the third fall observation, again
as the class is going over a warm-up problem. (See figure 5-3.)
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Figure 5-3. Warm-up problem. What’s my area?
Teacher:
What’s my area? What shape is this? Rachel?
Rachel:
It’s a triangle.
Teacher:
And the formula for the area of a triangle?
Rachel:
Length times width.
Teacher:
And then what?
Rachel:
And then divide it by, or, yeah, times it by two or… I don’t know. I’m
confused now. I’m confusing myself.
Jim:
Divide it by two.
Beth:
Divide it by two.
Rachel:
Yeah, divide it by two, that’s what I meant.
Teacher:
[The teacher writes
Beth:
Because it’s half of
Rachel:
Because you do length times width and then there’s like another triangle
on the board.] Why do we divide by two?
like
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Jerry:
On top of
Rachel:
Like there but not there
Teacher:
Yeah. Let me go back to my square mile. [He points to a square that he
had drawn on the board earlier to represent a square mile.]
Rachel:
It’s half of that.
Teacher:
A triangle is half of a rectangle. [Teacher draws the diagonal of the
square.] Half is going to be thrown away, [Teacher shades in half of the
square.] half is going to be kept. So, we divide by two to say we’re
throwing half away and we’re keeping half.
These incidents illustrate the role of teacher prompting in justifying in the beginning of
the school year. We can contrast this with students’ usual behavior at the end of the school year
as illustrated in the following incident. The teacher solicits students suggestions for a function
and for values at which they will evaluate the function (see Figure 5-4 for the photo of what the
teacher writes). As the teacher fills in the values, the following discussion occurs.
Figure 5-4. Board problem: finding f(x).
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Teacher:
[Pointing at the first parenthesis, currently blank.] Numbers?
Sarah:
Six.
Mike:
Five. Four.
Sarah:
Three. Two, one.
Beth:
Negative two.
Mike:
Negative one. Zero.
Jenny:
Four.
Mike:
Negative zero.
Rachel:
One.
Sarah:
Is there a negative zero?
Jenny:
There’s only one zero ‘cause it’s in between the negative and the positive
numbers.
Here, Jenny offers Sarah not only an answer to her question but a justification for why it
must be true. By the end of the school year it has become commonplace for students to provide
an unprompted justification when they make a mathematical statement.
In the following excerpt a student, while working to solve the problem shown in
Figure 5-5, tries to justify to the teacher why the hint provided is not helpful.
Figure 5-5. Warm-up problem. Angles of a perfect pentagon.
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Rob:
If your hint is, “what’s a 360”, 360 is a circle. That’s not a circle so you
couldn’t use 360 because there’d be parts of it that
Teacher:
If you started here, …
In this instance the student does not finish his justification because the teacher interrupts with an
explanation. The important point here is that the student does not stop with stating that the hint
does not work; he tries to justify why it will not work.
In another episode, again from the spring semester, a student has explained how she
solved the equation 7x – 9 = 5(3x + 1). A second student volunteers an alternate approach.
Sarah:
I got the same answer but a different way.
[The teacher asks her to hold off for a minute while the class finishes solving the equation
and arrives at a numerical value for x.]
Sarah:
Just kidding around. It’s similar but not completely
Teacher:
Okay. But I’m still curious to see what you did.
Sarah:
So I had the whole, the whole 7x minus 9 equals 5 parentheses … [As the
student speaks the teacher erases a section of the board and rewrites the
problem. (See figure 5-6.]
Figure 5-6. Warm-up problem: linear equation
Teacher:
Okay, watch this girl like a hawk. See if you can see if she makes a
mistake.
Sarah:
Minus 3x from both sides
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[The teacher writes -3x underneath each side of the equation.]
Sarah:
4x minus 9 is
Teacher:
You’re in trouble. You’re in trouble because it’s not 3x.
Beth:
Yeah.
Rob:
You have to follow PEMDAS.
Jenny:
You have parentheses.
At this point the teacher follows up on the justification provided by Rob and Jenny for why it is
not correct to begin solving this problem by subtracting 3x from both sides of the equation. In
this incident, as in the ones already described, the students offer justifications without direct
prompting from either the teacher or from other students.
Students in this class provide justifications for mathematical statements at a rate of about
four per class period during both the fall and spring semesters. However, during the fall
observation all but two of the student justifications were prompted, most often by the teacher,
whereas almost three-fourths of the justifications provided by students during the spring semester
were, like those illustrated above, unprompted. The tendency of students in this class to engage
in justifying and the change over the school year in the role of prompting in students’
engagement in justifying seems to indicate that, over the course of the school year, explaining
why things in mathematics must be true became an important part of mathematics for these
students.
The importance of justifying is also evident in students’ reactions to the statement, “It is
not important to know why a procedure works as long as you get the right answer” (Kloosterman
& Stage, 1992). Students uniformly disagreed with the statement, sometimes vehemently and
indignantly. One student stated, “I feel like there’s no point in doing it if you don’t understand
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why you are doing it.” Interestingly, in expressing their disagreement with this statement, almost
every student provided reasons why the statement was not true. Students expressed concerns
that without know why a procedure works they would not be able to do it themselves or to apply
it in other settings. One student stated, “You have to know how it works in order to do it again.”
Another stated, “If you get everything right, yeah, you’ll get a good grade but you if you try to
apply it to … work or something it’s not going to help.” In addition to these practical reasons
students provided for why it is important to know why things work one student provided another
reason. He stated, “I wouldn’t enjoy math if I didn’t know why, like what I need to do to get the
answer, like why I was doing it.” The interviews with students were conducted at the end of the
school year and provide evidence that students think they should provide explanations for why
things are true in mathematics.
The students’ perception of the importance of justifying in doing mathematics seems to
have changed over the course of the school year. This change is evidenced by the marked
increase in students’ unprompted justifications between the fall and the spring observations. The
explanations given by students in reacting to the statement about the importance of knowing why
are in contrast to the unreasoned but equally strong reaction to the statement about the
importance of knowing how and may indicate that the importance of knowing why is a relatively
new belief for them. Further evidence of this is provided by one student who, when asked what
in the sorting of conceptions statements might have been different if this activity had been done a
year earlier, stated that last year she would have agreed much more that understanding why was
not important. The combined evidence of students’ engagement in the behavior of justifying, the
changing role of prompting in students’ justifying, and students’ statements about the importance
of understanding why demonstrate that students in this class see mathematics as something
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sensible enough that they can justify their ideas. The evidence further suggests that this
conception is one that has developed over the course of this school year.
Seeking explanations
By the end of the school year, students had come to expect that there were, at least
sometimes, explanations for ideas and procedures and explanations for why things in
mathematics were correct or incorrect. During the fall observations there were no indicators that
students sought explanations for ideas, procedures, or answers in mathematics. However, during
the spring observation students asked for explanations for procedures nine times and
explanations for why something was correct four times. In some cases, students asking for an
explanation not only asked, but persisted in asking until it was explained to their satisfaction.
For example, when the teacher, in the course of reducing the fraction x/x3, “reduces” (divides
out) an x from the numerator and denominator and replaces the reduced x with a one, a student
asks why it reduces to one instead of to zero. Other students treat the answer as obvious but the
student keeps asking the question until she receives an answer that she understands.
T:
What’s left on top?
Mike:
Zero.
Sarah:
One.
Rob:
One
T:
One. Thank you, Rob.
Beth:
Why?
T:
Because when things cancel, it’s something over itself. What’s 8 over 8?
Jess:
One.
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Beth:
Zero.
Rob:
One.
Jenny:
ONE!
T:
If you have a pizza divided into up into 8 slices and you eat 8 slices, how
much pizza -
Mike:
You ate one pizza.
Beth:
I know but you don’t have any left.
Jenny:
That’s not what he’s asking!
Beth:
Sorry for not being smart.
T:
No, no, no. This isn’t telling us how much is left. This is telling us how
much went down your gullet. Okay?
Beth:
Everyone is just so stressy.
T:
So, something over itself, something over itself makes one.
Even though student interactions in this episode became somewhat heated, Beth persisted until
she received an explanation that satisfied her.
Students in this class seek explanations not only from the teacher from one another.
About half of the incidents in which students seek explanations are ones in which students ask
other students for an explanation. For example, as students are practicing simplifying algebraic
fractions one student, probably working on a problem in which the entire numerator reduced out,
asks other students, “Why do you have to put in a fraction bar?” For this student it appears that
putting in a vinculum was a procedure and the student wanted to know why the procedure
needed to be followed.
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In interviews, students were asked to respond to the statement, “A lot of things in math
must simply be accepted as true and remembered: there really isn’t any explanation for them”
(Carter & Norwood, 1997). Although many students had difficulty untangling the meaning of
the compound sentence, the overall consensus was that most things in mathematics did have
explanations. Students’ difficulty interpreting the sentence was demonstrated by several students
who focused only the role of memory in learning mathematics. One student stated, “There are
some things in math that you don’t obviously have to know. Like, you’re not going to remember
everything from school.” One group was upfront about their difficulty interpreting the
statement. When the statement was first presented one student in the group stated, “I don’t really
understand.” When the group was asked at the end of the interview if there were any statements
that they wanted to revisited, one group member pointed to this statement and said, “I’m still
kind of confused about that one.”
Despite the difficulties that some students had interpreting the statement, most groups did
use it as a springboard to discuss whether things in mathematics have explanations. Several
students noted that there were some things in mathematics that did not have explanations. One
group presented as an example the formula for the area of a circle.
Jim:
I don’t really understand [the statement].
I:
Okay. There are some things in math that have to accept to be true. Mr.
Wingate tells you that, um, you know, anything raised to the zero is one
and you just have to accept that that’s true, there’s no explanation for it.
Jim:
Oh, all right, yeah.
I:
So there are times when there just isn’t an explanation.
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Mike:
Well like the formulas for like finding the area of a circle, we don’t really
know how that works. We just know it.
I:
Okay. Do you think there is an explanation and you just don’t know it yet
or is there just not an explanation?
Mike:
I just don’t think there is an explanation.
I:
Okay. It’s just one of those things – the mysteries of the universe.
Mike:
Yeah.
I:
Okay.
In spite of their identification of exemptions, students, overall, indicated that they
believed that things in mathematics did have explanations. The following discussion reflects this
overall sense:
I:
A lot of things in math must simply be accepted as true and remembered
because there really isn’t any explanation for them.
Beth:
That’s wrong.
Rob:
Well, I mean right now, anyway, it’s wrong but I’ve walked into some
math classes where I feel like they’re like
Beth:
Well, that’s because you don’t know what they’re doing.
Rob:
I know, yeah. So
I:
But you believe that there is an explanation?
Beth:
Yeah.
Rob:
Yes.
I:
Okay.
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It is evident, both by students’ actions and statements, that the student in this class ended
the school year believing that at least some things in mathematics had explanations. Students,
during the spring semester, stated that things should have explanations and sought explanations
from both the teacher and from each other. Because of the lack of indicators during the fall
observation, there is no evidence that students had such a conception of mathematics at the
beginning of this course. One student evaluated the truth of the statement about some things in
mathematics not having an explanation in light of the current class. He stated, “Right now,
anyway, it’s wrong but I’ve been in some math classes where I felt like …” It is evident that this
student perceives that, in this class, things in mathematics have explanations but that this has not
always been the case in other mathematics classes. This evidence, coupled with the dramatic
change in the students’ tendency to ask for explanations between the fall and spring
observations, lead me to conclude that students in this class believe that mathematics is sensible
enough that they can seek explanations and that this conception likely grew over the course of
the school year.
Summary
Students in this class believe that things in mathematics should have explanations. They
seem to have been socialized to an expectation that they explain how they do mathematical
problems even before this class began. By the end of the school year, students engaged in all four
types of behavior related to explanations and expressed strong agreement with statements
relating to the importance of knowing how and why things work in mathematics. Some students
are firmly convinced that everything in mathematics can be explained while others, citing
examples such as 2*2 equals 4 and the formula for the area of a circle, believe some things in
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mathematics do have not have explanations and must just be accepted and remembered. The
students showed growth over the course of the school year in the number of indicators displayed,
the type of indicators displayed, and the incidence of indicators without prompting. Students in
this class provided explanations when doing mathematics and expected that explanations would
be provided them. These actions indicate that these students conceive of mathematics as
sufficiently sensible that it can be explained.
Findings related to expecting connections
Students in this class conceived of mathematics as sensible enough that they expected
and verbalized connections between mathematical topics and between mathematics and realworld contexts. They also expressed strong agreement with conceptions statements about
connections within mathematics.
Types of evidence
In the framework of indicators there were initially six indicators related to students’
conceptions of mathematics as sensible enough to be connected. Five of these indicators,
recognizing similarity, adapting solution methods, seeking connections, ignoring size of
numbers, and using multiple representations, were related to students seeing connections within
mathematics, that is connections between different ideas, representations, or topics in
mathematics. The fifth indicator, contextualizing/decontextualizing was related to students
seeing connections between mathematics and real-world contexts. During the data analysis an
additional indicator, noticing, was added and used to code instances in which students
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spontaneously indicated that they saw a connection between what was happening in the problem
immediately before them and something else in mathematics.
There were two conceptions statements that were designed to elicit information about
students’ conceptions about connections and mathematics. Students’ comments on their level of
agreement with the statement “Math is made up of unrelated topics” (Brown et al., 1988) elicited
information both on students’ ideas about connections within mathematics and their ideas about
connections between mathematics and real-world or academic contexts. The statement “Math is
made up of ideas, terms, and connections” (Grigutsch & Torner, 1998), which was also designed
to stimulate discussion about connections in mathematics, proved problematic since student
groups had a difficult time interpreting the statement. One group focused on the ideas of “terms”
in mathematics, one group focused on whether mathematics is comprised of “ideas,”, and several
other groups focused on whether the list provided in the statement was complete. None of the
groups used this statement to comment on the connected nature of mathematics so all of the
interview information about students’ conception of connections is from analysis of students’
comments on the statement “Math is made up of unrelated topics.”
Connections within mathematics
Students in this class conceive of mathematics as sensible enough that they believe that
there should be internal connections within mathematics. Both the numbers and types of
connections made by students changed between the fall and spring observations.
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Mathematical connections during fall observations
During the fall observations there were only five student indicators that were coded as
expecting or seeing mathematical connections. Three of these involved students simply
observing that mathematical objects were the same. In the following example, the teacher put
two problems on the board: a) 5/8 + 3/2 and b) 5/8 – 3/2. The class had already gone over the
solution to part a, converting the problem to 5/8 + 12/8.
T:
Letter b, number 2, letter b. Mike, want to do it?
Mike:
Oh, yeah, sure. You get the same thing, like, yeah. You put 5 minus 12
equals negative 7 over 8.
The student noticed that the numbers were the same and, thus, they did not need to redo the
initial steps of the problem.
In another incident, the teacher had drawn a clock face on the board and labeled it with
degree measures rather than the numerals 1 through 12 (see figure 5-7).
Figure 5-7. Clock face labeled in degrees.
After the teacher draws in the hands of the clock and asks students if it makes sense, they offer
several observations about the clock face.
T:
Would that make sense?
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Mike:
Yeah.
T:
What would they be, those numbers?
Beth:
12, 1, 2, 3, 4, 5, 6
Mike:
How many minutes there are?
Jenny:
There’s not 12 numbers on the clock.
Eddie:
There’s 360 degrees all the way around.
T:
What did you say, Eddie?
Eddie:
There’s 360 degrees all the way around and 180 is half of 360, so that
mean’s it’s 12.
T:
So, this clock is in degrees, right? [T writes “DEGREES” under the clock
face.] Because we know that there are 360 degrees
Zach:
[speaking at the same time] That’s a right angle.
T:
all the way around. Sorry?
Zach:
It’s 90 degrees and it’s a right angle.
T:
90 degrees makes a right angle. [T draws in a right angle between 360 and
90 on the clock face] Yes, because the 360 is also zero, right?
Throughout this incident students were noticing similarities between the clock face that the
teacher had drawn and what they knew about degree measure. In each case, the students appear
to be noticing a similarity between two mathematical objects.
The other two incidents during the fall observations that were coded as making
mathematical connections involved students making connections between different mathematical
problems or approaches to a problem. In one instance, the teacher has written the problem
7/3*4/5 on the board. The following exchange ensues:
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T:
What about that? Do you know how to multiply them?
Eddie:
Just go across.
Joanne:
Like you have to go 7 and 5 and 4 and 3
Zach:
12 over..
T:
[T draws a horizontal arrow above the problem.] Straight across. Straight
across.
Joanne:
Since when?
Beth:
Since forever.
T:
[Teacher draws a horizontal arrow below the problem.] As long as I’ve
been teaching.
Mike:
48 over 15.
Rachel:
48 over 15.
Beth:
You’re getting it confused with division.
In this instance Beth connects Joanne’s incorrect strategy for multiplying fractions with an
algorithm for dividing fractions.
In the other instance the teacher is having a student talk him through the problem
5/8 + 3/2.
T:
What do you like for a common denominator?
Joanne:
8
T:
Do you write two 8s or one?
Joanne:
One.
T:
Write it like that?
Joanne:
Yeah.
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T:
Okay.
Eddie:
It doesn’t matter.
In this episode Eddie seems to recognize that the two different representations are the same and
both are viable approaches to the problem. The students in each of these cases are making
mathematical connections but the connections are limited in nature. The limited number of
indicators involving connecting mathematics internally and the limited nature of the connections
made may indicate that students did not come in to this class expecting mathematics to be a
connected system.
Mathematical connections during spring observations
During the spring observations there were 13 coded indicators related to students seeing
connections within mathematics. In addition to the almost threefold increase in the number of
indicators, all but one of the indicators from the spring semester went beyond simply noticing
surface-level similarities between mathematical objects.
Four of the indicators were related to connecting multiple representations of the same
object. For example, as the class is going over the answer to the problem (5x2y3)2, several
students note that they, incorrectly, got a coefficient of 10 rather than 25 in the expanded form.
One student provides the following explanation for her error:
T:
Just out of curiosity, how many people got 10?
Jim:
Me.
Beth:
Ten what?
T:
Instead of 25?
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Jenny:
I just got that completely wrong. I did 5xy to the second times 2. So I got
5xy to the fourth.
In this episode, Jenny is working to connect how she expanded the problem to the correct
expansion and trying to determine how she must have interpreted the original expression to get
her answer. Although she is not accurate in her assessment, this episode shows her trying to
make a mathematical connection.
In a similar episode, the teacher is discussing a problem from a recent quiz.
T:
(Pointing to the right hand side of the equation shown in figure 5-8.)
Now, I had people who got all the way to the right answer, then did this.
(Teacher writes the reciprocal of the expression shown in the right hand
side of the equation in figure 5-8.)
Figure 5-8. Expression with exponents to simplify
Rob:
You only do that when it’s negative!
Beth:
Only if it’s negative.
In this episode Rob and Beth not only recognize the error that students were making but
connected the incorrect answer to a problem situation in which it would be the correct answer.
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In each of these cases the students were making connections between different problems and
representations.
Three of the indicators related to seeing connections were about students noticing
something in the mathematical object under consideration that was related to another
mathematical idea. In one instance the teacher had plotted a set of data points and a student
observed, “you have an outlier.” The class had not been talking about outliers and this appeared
to be a case of a student observing a characteristic of the mathematical representation and
connecting it to a concept learned in another context. In another instance, looking at the same
plot, students were asked what they observed about the plot. One student observed that it
seemed to have a positive slope. Again, the concept of slope had not been discussed in this
context; indeed, it had only been discussed in the context of graphs of lines and, at this point, no
best-fit line had yet been drawn. In the initial coding structure there was no indicator for coding
these incidents; they were coded simply as seeing connections, I added an indicator, noticing
mathematical connections, to provide a way to code incidents where students appeared to be
noticing particular aspects within the current mathematical context and making a connection to
mathematical terms and ideas from another context.
The other important type of indicator that students were seeing connections within
mathematics was the recognition of the underlying similarity of mathematical objects. There
were five indicators of this coded during the five days of the spring observation. In one instance
a student had asked the teacher about the practical purpose of something like
Mike:
When are we ever going to use this?
T:
This?
Mike:
Yeah.
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.
Sarah:
Every day.
T:
Are you kidding? In the grocery store, if you buy, if you buy 20 bottles of
Tide and it says
Rachel:
You wouldn’t though.
Rob:
We do.
T:
(He writes on the board as he continues to speak.) 4x7 washes, right, this is
what it says right on the package. x to the 7th washes and if you buy 20
bottles, that’s now good for 20 times x to the 7th washes. That’s very
important in the grocery store.
Rob:
Doesn’t it say it does like 40 full loads or something compared to the, like,
bargain brand which only does like
Beth:
It likes cause like, then that other brand
Sarah:
Or just buy the biggest box.
[As several students continue to discuss the best strategy for purchasing
laundry detergent, the teacher begins to draw a picture of speed boat on
the board and explaining details of his sketch to the class. “they have
smoke coming out here, a driver here, a big propeller here …”]
T:
The speed of the boat - now every boat designer knows this, I’m not just
making this up – the speed of the boat raised to the seventh power is equal
to something, I don’t know what, times the horse power. (As he speaks he
writes the equation (speed)7=□hp on the board.) So if you want a boat to
go fast, you can’t just put the horse power up by 10 or 20. It makes no
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difference. If you want a boat to go fast, the horsepower has to go up by a
bunch.
Mike:
I meant that problem, right there.
T:
What problem.
Rob:
Yeah, that’s what that is.
Here Rob appears to be connecting the equation for the speed boat with the original expression
by recognizing the similarity in the underlying exponential structure of the expressions.
In another, similar, incident the class had just finished finding the value of g(x) = x2 – 1 for
various values of x. When then presented with the problem of finding values for h(x)=1/x, one
student is unsure how to proceed. Another student points out that “it’s the same thing.” The
student seems to be pointing toward each expression’s role as a function rule and, thus, the
similar approach that one would use to find values of the function. Although there were no
indicators during the fall observation in which students appeared to make connections based on
the underlying mathematical structure of representations, during the spring observations, there
was approximately one indicator of this type during each class period.
During interviews conducted in the spring semester, students also stated that they
believed that topics and ideas in mathematics were connected. Groups of students were asked
about their level of agreement with the statement, “Math is made up of unrelated topics.” Of the
students who interpreted this statement as relating to mathematical topics being connected to
other mathematical topics, all agreed with the statement. Typical of the reactions was this
exchange:
I:
Math is made up of unrelated topics.
Jenny:
That’s not true.
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Larry:
Not true at all. [Jenny places the card at the strongly disagree end of the
continuum.]
I:
Tell me more about that.
Larry:
That’ because all of them are intertwined somehow, whether it be
multiplication, division, addition …
Jenny:
Well, like sometimes you need to know how to do certain things before
you can do other things.
Interestingly, there were two groups that began by stating that they only slightly
disagreed but very quickly talked themselves into strongly disagreeing. The following
discussion is one such episode.
Beth:
No, I think some interconnect with each other.
Rob:
Yeah.
Beth:
Sometimes, like, in order to, like for exponents, in order to figure out the
problems you have to subtract which is linked to subtraction or you have
to multiply if it’s in parentheses.
Rob:
Well like, yeah, PEMDAS
Beth:
That connects two different topics so … I do disagree with that. [Beth
places the card about ¾ of the way along the continuum for strongly
disagree.]
I:
Okay. But not all the way to strongly disagree?
Beth:
Right.
I:
[Rob firmly slaps the card at the strongly disagree end of the continuum.]
Or is it?
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Beth goes on to agree that the card belongs all the way at strongly disagree. The fact that
all of the groups strongly disagreed with the statement is further evidence that students in this
class saw mathematics as sensible enough that they expected there to be connections within
mathematics. The two groups that began the discussion more ambivalent and talked themselves
into strong disagreement may indicate that this was an idea they had not considered before and
may be a new conception for them that has developed over the course of this academic year.
During observations at the beginning of the school year, there were few indicators that
students conceived of mathematics as sensible enough to look for connections between
mathematical topics. In addition, the indicators that were in evidence tended to rely on linking
surface-level characteristics of mathematical objects. During observations at the end of the
school year, there were two to three indicators during each class period that students were
making connections between mathematical topics and objects. In addition, by the end of the
school year, students expressed strong disagreement with the statement, “Math is made up of
unrelated topics.” It is evident that the students in this class see mathematics as sensible enough
to be internally connected. Further, it seems likely that this conception of mathematics has
grown over the course the school year.
Connections between mathematics and other contexts
The indicator contextualizing/decontextualizing was designed for coding connections that
students made between mathematics and real world contexts. I use the terms in a manner similar
to that of the Standards for Mathematical Practice of the CCSS-M (NGA & CCSSO, 2010) but
expand the ideas to code any attempt to connect the mathematics to real-world referents as
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contextualizing and any attempt to represent or reason mathematically about a real-world
situation as decontextualizing.
Attempting to connect mathematics to real-life contexts was one of the most common
ways that students in this class demonstrated that they conceived of mathematics as sensible.
During the spring semester there were 21 coded indicators that students were connecting
mathematics to other contexts: two during the fall observations and 19 during the spring
observations. Some of these connections were as surface-level as simply mapping visual images
from their mathematics to other objects. For example, when a problem called for students to find
the area and perimeter of the trapezoid shown in figure 5-9, the following exchange took place:
Figure 5-9. What does this trapezoid look like?
T:
There’s no, no real such shape is there?
Mike:
No.
Jenny:
There could be.
T:
Not in real life, though. Not in real life. (The teacher walks away from
the board.)
Larry:
Actually, there is. A certain kind of plane – if you put the nose downward
– that’s the shape.
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T:
Uh huh.
Marie:
It kind of looks like a building.
T:
Kind of looks like a building, kind of. Except kind of a funny roof, huh?
(The teacher comes back to the sketch on the board.
Mike:
It’s as though they’re still making it.
T:
Yeah. It’d be good for snow. It would be bad, it would be bad for the guy
who has to go up and fix the antenna.
Eddie:
I think there is a building like that.
T:
What do you mean there’s a building like that?
Sarah:
It’s going to pop up in a second. (Sarah and several other students are
using their phones to check the internet. One student is using the
computer connected to the interactive white board.)
T:
(Pointing toward something coming up on the interactive white board.)
There’s a little man with fire.
Sarah:
Isn’t there a building
T:
A building shaped like that? A funny shape for a building isn’t it?
Sarah:
It is a shape of a building.
Beth:
Yeah, in New York!
T:
In New York City, yes. Yeah, what’s the name of it, Beth?
Beth:
I don’t know.
Mike:
It’s still the roof.
Sarah:
The, uh, type in a name, the Citigroup Center.
T:
Yeah, it is. The Citigroup Center
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Jenny:
It says it right on top of the page.
T:
Well, you knew it was in New York, I thought maybe you knew.
Beth:
No, I didn’t. I saw it in New York when I was there.
T:
Yeah, yeah. It’s a very unusual building. It’s got a very unusual roof and
looks exactly like this although obviously it’s bigger than 90 centimeters
tall.
In this episode, as in others, students seek real world examples which match the shape of
mathematical objects, seeming to take their quest very seriously.
In another incident, the teacher draws an oval around a set of clustered data and identifies
it as a pickle. Students object to the characterization, stating that it looks more like a lemon or a
football. Contextualization of this nature suggests that students see some connection between
mathematics and real-world objects but does not seem to provide evidence that students are
actually making any mathematical connections beyond identifying potentially similar geometric
shapes. This type of contextualizing accounts for approximately one-third of the coded incidents
in which students sought connections between mathematics and real-world contexts.
Most of the other episodes of contextualizing were incidents in which students were
talking about or questioning how a particular mathematical idea or conclusion applied in a real
world context. In two episodes students suggested ways to model a mathematical conclusion.
For example, when the best fit line of a data plot of class members’ arm span versus height
showed that the quantities were approximately equal, one student volunteers to demonstrate the
relationship.
T:
The longer my wingspan, the taller I am. Is that true?
Rob:
But it’s supposed to be the same.
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Jenny:
That’s the same as your height. (Other students voice agreement.)
T:
You’re right. And you know that. This (he holds his arms straight out to
the side) is how tall I am.
Joanne:
I’m off by a little bit.
T:
Yeah, we’re all off by a little bit but this (he wiggles his hands) is how tall
I am. That’s how tall Jenny is. That’s how tall Sarah is.
Jenny:
Ready, we can do it.
T:
(Teacher points at individual students.) That’s how tall you are. That’s
how tall you are.
Jenny:
Ready, watch this. Joanne, come on. Can we show them? (Jenny starts
towards the front of the room.)
Jenny:
See, go like this (She extends her hands to her sides.) and go like this. (She
bends sideways with arms still extended and touches one hand to the
floor.) Can you put your hand … (The teacher places his hand at the tip of
her extended fingers. Jenny stands back up and moves so that she is
standing directly under the teacher’s hand.) (See figure 5-10.)
T:
It really does work. (Several other students come to the front of the room
to try it.)
Figure 5-10. Demonstrating height equals wingspan.
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Three other incidents involved students stating interpretations of mathematical
representations in terms of real-world contexts. For example, after the class has created a data
plot of class members’ hand span versus their height, the teacher asks if there is a relationship. A
student replies, “The taller you are, the bigger hands you have.” The remaining 11 incidents in
which students consider how mathematical ideas and conclusions apply in real-world contexts
seem to require students to go beyond just visual connections of how to model or directly
interpreting representations. In these incidents students often extend the mathematical ideas and
try to reason about how they might apply. For example, when the class creates a bell-shaped
curve to represent the distribution of heights across a human population and discusses how that is
related to the fact that this particularly tall teacher cannot buy pants at a local store, one student
remarks, “I just figured out why they order more of one thing than another – because they use the
average. The business orders more of what customers are ordering.” In another incident, the
teacher has just reiterated the interpretation of a slope of 2 in a graph of height versus hand span
to mean that, “if you gain 2 inches of height, you gain 1 inch of [hand span].” A student puzzles
over whether this means that someone who is 80 inches tall would have a hand span of 40
inches.
This incident serves not only to illustrate an incident of a student striving to make
connections from mathematics to real-world contexts but the fact that this process was not
always effortless or without error.
Although there were very few indicators during the fall observations that students looked
for or saw connections between mathematics and real-world contexts, by spring it was an
element in all class discussions. In fact, if real-world connections were not introduced, or were
not persuasive enough for the students, students asked about them. In two instances during the
spring semester students asked the teacher what the practical application was of the mathematics
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that they were studying. One incident, described earlier, was of a student asking about the
practical application of exponents. In another incident a student, after a long class discussion
about the relationship between hand span and height asked, “what is the point of all this?
…We’re in math, like I get that, but just to randomly do something?” Although there may be
times when questions of this nature are used by students more as a way of trying to sidetrack the
teacher than as a genuine request for information, this student’s willingness to engage and the
embedded assurance that he was not objecting to learning the mathematics involved seem to
indicate that the student genuinely expected there to be an connection and was concerned about
knowing that connection.
Interestingly, although students often sought to connect mathematics to real-world
contexts, they very seldom applied or suggested applying mathematics to real-world contexts or
saw the mathematics inherent in real-life situations. There were only three incidents coded as
decontextualizing: two during the fall observations and one during the spring. The two
indicators during the fall observations were both related to the same incident in which the teacher
was using board markers and erasers to illustrate ideas about like terms. When the teacher holds
up the objects and points out that one marker plus two erasers is still just one marker and two
erasers, one student observes, “that’s just like our x and y’s.” Another student remarks, “it’s like
negative and positive.” These two indicators demonstrate that students are attempting to connect
a physical situation to mathematical ideas.
The incident coded as decontextualizing during the spring observations was the only
incident in which a student suggested applying mathematics to a real-life context. Soon after the
class had finished discussing an area problem which required students to find the area of an
object by find the area of a larger figure and subtracting out unused portions of the figure (see
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figure 5-11), a student suggested that they should use a similar process to find the amount of
wasted space in a shipping box.
Figure 5-11. Side view of bridge.
Rob:
Wait, Mr. Wingate. We should calculate the area of wasted space that
Apple uses in the boxes when they send you ear buds. ‘Cause they sent
me complimentary ear buds and, no, but I got this box that was like the
size of this desk and I was like, “did they send me a free car with these ear
buds?” And I like opened it and it was bubble wrap. And then below the
bubble wrap was another box and I opened that box and in that box was
bubble wrap. And then, finally, in that box was the white Apple box. But
then inside that box was Styrofoam and then when you broke the
Styrofoam apart, it was finally the ear buds.
T:
Wow. How big was the outermost box?
In this incident it appeared that the student was attending the extra “space” subtracted out the
bridge area problem and considering how the same type of mathematical strategy might be used
for a real-world volume problem. The fact that there were so few incidents of decontextualizing
and so many of contextualizing seems to indicate that students in this class expect to see
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mathematics play out and relate to their world but may not have developed a sense of the
usefulness of mathematics as a tool for understanding and analyzing their world.
Students’ comments during the interviews provide some insight into the connections that
students see and expect between mathematical topics and between mathematics and other
contexts. Although there were no conceptions statements intended to elicit information about
these connections, three out the five groups who discussed the statement, “Math is made up of
unrelated topics” interpreted the statement to mean that mathematics was unrelated to real-world
topics or to other school subjects. Of these groups only one agreed with the statement but they
did so because they believed that many different, unrelated topics were connected to
mathematics.
I:
Math is made up of unrelated topics.
Jenny:
Yes, definitely. Because we use science, we use
Jim:
True, we use science
Jenny:
We use math for science
Mike:
History
Jenny:
Or
Jim:
Yeah, like for geography
Jenny:
Yeah. So, definitely.
I:
Okay.
Jenny:
So we’ll put it like close to the top. A little bit at the top. (The top for
them was the strongly agree end of the continuum.)
I:
Okay.
Students in this group clearly saw mathematics as related to other academic topics.
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The conception that mathematics is connected to real-world settings also came up for
some groups when discussing other conception statements. For example, one group, in their
discussion about the importance of understanding why a procedure works, had the following
discussion:
I:
Anything to add over there?
Larry:
Only that if you get everything right, yeah you’ll have a good grade, but if
you try to apply it to, I don’t know, work of something like
Jess:
to real life situations
Larry:
it’s not going to help.
The students in this group clearly expected that mathematics should be connected to real world
contexts.
During the fall observations of this class there were very few indicators that these
students expected or saw connections between mathematics and real world contexts. However,
by the end of the school year both the number of indicators and the students’ comments during
interviews demonstrate that students had come to conceive of mathematics as sensible enough
that is should be connected to real world contexts.
Summary
Students in this class saw and expected connections between mathematical topics and
between mathematics and real world contexts. Although there were some indicators of this
conception during the fall observation, there were more than triple the number of such indicators
during the spring observations. During the spring observations such indicators were present
during every class period and were a part of every class discussion. It seems likely that the
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conception of mathematics as sensible enough that mathematics should be connected both
internally and to other contexts grew substantially over the course of this academic year.
Findings related to strategizing
Students in this class conceived of mathematics as sensible enough that they strategized
about mathematics; seeking and offering alternate solutions to mathematics problems and talking
about strategies for approaching problems for which they had no ready procedure. Overall, their
actions with regard to engaging in strategizing in mathematics were supported by their
discussion of conceptions statements although there is still some question about the role that
these students believe memory plays in doing mathematics.
Types of evidence
There were, initially, five indicators for coding students’ engagement in strategizing.
Two of these indicators, “Strategize about solution methods” and “Discuss how to solve
problems rather than seeking the ‘right steps’” were found to be almost impossible to separate
both in coding and in analysis of incidents. They were, thus, combined into a single strategizing
indicator. Two other indicators in this category, “Use poor memory as a reason why they cannot
do a problem” and “rely on memory of procedures for problem solving” were two of only three
negative indicators in the framework. They were included both because of their prominence in
the literature and because of their presence in the pilot data for this study. The fifth indicator
was “Seek and use alternative solution strategies.”
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There were four conceptions statements related to strategizing on which students were
asked to comment. Two were related to the role of memory in doing mathematics: “Math is
memorizing and applying definitions, formulas, facts, and procedures” (Grigutsch & Torner,
1998) and “You don’t have to have a good memory to be good at math.” One statement, “A
math problem can always be solved in different ways” (Brown et al., 1988) was designed to get
students to discuss the role of alternative solution strategies in mathematics. The fourth
statement, “There is always a rule to follow in solving math problems” (Telese, 1999; Brown et
al., 1988) was intended to provide students with an opportunity to discuss the role of strategizing
when solving problems rather than simply following procedural rules.
Strategizing when problem solving
Students in this class engaged in strategizing during both the fall and the spring
observation periods. However, it was sometimes difficult to differentiate between coding for this
indicator and coding for the indicator, explain how from the expect explanations category of the
framework. The decision between the two codes was made based on an assessment of the timing
of the students’ action, the specificity of students’ discussion of the problem, and the students’
likely intent. If the indicator came before actual solving of the problem, it seems clear that the
students was strategizing about how to solve the problem rather than explaining how he or she
had already solved it. This accounts for most of the incidents coded as strategizing. However,
even if the student was in the midst of solving or had already solved the problem, statements
which seemed to generalize the solution method were coded as strategizing since they were
designed to provide a strategy for approaching problems of this type. For example, in one
instance students were working independently to find the equation of a line given two ordered
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pairs. After a student found the equation she summarized the process for another student saying,
“Okay, so this number, that is always going to be your x and then, once you take that, or times
that times that, that’s your b, so we stop there.” Although the student is talking about she solved
a particular problem, she generalizes the result into a strategy for working problems of the same
type. Similarly, even specific statements about how a student had already solved a problem were
coded as strategizing if the context was one in which a general strategize was expected. The
following incident illustrates this.
T:
How do we simplify something like 39/65? Do we have a strategy for
doing it? … What’s your strategy Joanne?
Joanne:
Well, I added 9 and 3 and then …
Joanne goes on to explain how she did this particular problem. However, since the call was for a
general strategy and both the teacher and the class treat Joanne’s explanation as general, it was
coded as strategizing.
Using these coding guidelines, there were 16 coded instances of strategizing during the
fall observation and 25 during the spring. In addition to the more than 50% increase in the
number of incidents from fall to spring, there was a distinct difference in the role that prompting
seemed to play in students’ engagement in strategizing. During the fall observations six of the
16 incidents of strategizing were directly prompted by the teacher. One such incident took place
when the class was discussing how to identify which two sides of a right triangle should be used
when finding the area. As a group they had already determined that they should begin by finding
the area of the related rectangle by multiplying length times width. The teacher then asks, “How
do I know that I am looking at the length and width and not some other line?” A student
strategizes, “If I make an L and line it up and make the length times the width.” Here the teacher
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has asked for and received a general strategy. Two other incidents during the fall observation
were prompted not by the teacher by other students. Both took place as students were working
on problems involving ordering a series of fractions from smallest to largest. Several students
asked their peers how to approach the problem. One student suggested converting the factions
into decimals. Another volunteered that she would start by putting the fractions into a reasonable
order and then checking the order by checking each pair. Altogether, half of all incidents of
strategizing during the fall observation were prompted by either the teacher or by other students.
During the spring observation, of the 25 episodes of strategizing, only four had any evident
prompting, all by the teacher. Both the increase in number of coded episodes of strategizing and
the changing role of prompting seem to indicate that the students in this class have grown in their
conception of mathematics as something sensible enough that strategizing is a useful tool.
Students’ reactions to the conceptions statement designed to address the role of
strategizing in problem solving, “there is always a rule to follow when solving math problems”
(Telese, 1999; Brown et al., 1988), present a complex picture of their conceptions about
mathematics. For two of the groups of students, this statement provided little evidence about
their conceptions of strategizing because they used the statement to talk more about the role that
rules play in the structure of mathematics than the role of rules in problem solving. Both of these
groups agreed with the statement and based their agreement on the importance of general rules
such as the order of operations. The following conversation serves as an example:
Beth:
You can’t just randomly look at the problem and do what you want. It’s
like order of operations,
Rob:
Yeah
Beth:
you have to do it the right way
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Rob:
SADMEP. (He laughs. Note that SADMEP is the acronym for order of
operations, PEMDAS, spelled backwards.)
Beth:
or else your answer will be messed up.
The other four groups of students all agreed with the statement and several described a rulebound way of engaging in mathematics that seems at odds with the strategizing observed in the
classroom. One group was particularly articulate about this conception:
I:
There is always a rule to follow in solving math problems.
Rachel:
Yeah.
Sarah:
Yes, you always have to follow…
Rachel:
You have to follow the rules or else you are going to get it wrong.
Sarah:
Yeah, there’s always a different way, ’cause different problems have
different rules to solve them. Unless you do the same rules for the same
problems you’re not going to get the same answer. It’s going to be
completely different. [The girls work together to place the card towards
the agree end of the continuum.] And it’s completely different math –
completely different math problems that, um, need different rules. I just
said the same thing three times.
I:
That’s okay. Sometimes you have to say it a couple of times to make sure
you said it well.
The idea that there are rules for each problem seems to contradict the tendency of the
students in this class to engage in strategizing about problem solving. Such a contradiction may
provide further evidence that the conception of mathematics as sensible enough to strategize
about may be a new, developing conception for students; one that they have begun to enact but
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have not yet internalized as true. It may also indicate that it is a conception of mathematics that
they see modeled in their mathematics class but may not have heard articulated.
Alternative strategies
Students in this class look for and use alternative strategies in problem solving even
though they are never prompted to do so. This practice is far more in evidence during the spring
observations than during the fall observations. During the fall observation, there were four
incidents in which students offered alternative solution strategies. All four of these were offered
in the context of simplifying or performing basic operations on fractions. For example, when the
class is working together to reduce the fraction 84/96, they divide both the numerator and
denominator by 2 several times and then divide each by 3. A student suggests that they could
have simply divided both by 12 instead.
During the spring observations there were 11 incidents in which students suggested or
talked about their use of alternative strategies. Unlike the incidents from the fall observations,
these incidents covered a number of different mathematical topics including finding the
perimeter of a rectangle, calculating the slope of a line, solving linear equations, solving
problems involving percents, and simplifying expressions containing exponents. About half of
the alternative solution strategies proposed by students during the spring observation were
similar in character to the fall incidents in that they involved simply proposing an alternative
procedure for solving a problem. These included incidents like recognizing that one can talk
about the slope of a given line as either going down 4 and right 3 or going up 4 and left 3 and
questioning whether it is easier, in the case of simplifying the expression
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, to first
simplify the fraction within the parentheses or to deal with the exponent outside the parentheses
first. The rest of the incidents from the spring observations in which students proposed alternate
solution methods went beyond just looking at alternative procedures. They involved students
actually looking at problem contexts from a different perspective. For example, students were
discussing how to solve the problem: Bill’s car has four tires. One of them is flat. What percent
is that? Most students seemed to use a process similar to the one articulated by the student intern
leading the lesson. She explained, “So because there’s four tires, if one of them is only flat, then
you put one over four and then when you find that percent that comes to .25 and you do .25 times
100 and you have 25%. However, one student interjected with, “I figured out the percent of
what’s left… I did 75% subtracted from 100%.” This student’s alternative strategy involved a
new way of considering the original problem context. Another example occurred when the class
was working to find the area of the shape shown in figure 5-12.
Figure 5-12. Area of concave hexagon.
One student, in conjunction with the student intern, had already presented a solution
based on finding the area of the entire rectangle and subtracting the area of the small triangle.
Two other students proposed alternative solutions. One suggested dissecting the figure as shown
by the dotted line in figure 5-13.
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Figure 5-13. An alternative dissection of the concave hexagon.
The other student proposed the dissection shown in figure 5-14.
Figure 5-14. A second alternative dissection of the concave hexagon.
These alternative solutions went beyond looking at alternative procedures to seeking new
ways to approach the problem. Although students readily offered alternative solutions to
problems during both the spring and fall observations, both the number of incidents and the
richness of the type of alternatives offered was substantially higher during the spring
observation.
Students’ readiness to offer alternative solutions matched well with what they said in
interviews. When asked about their level of agreement with the statement, “There is always
more than one way to solve a math problem” (Brown et al., 1988) all groups expressed some
level of agreement. One group placed the statement towards the agree side of the middle of the
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agree/disagree continuum because they were not convinced that there was always more than one
way.
I:
A math problem can always be solved in different ways.
Larry:
I guess I can mostly agree with that. [He places the card about two- thirds
towards agree.]
I:
Okay, why?
Larry:
Because, in some cases, it can, but in others it can’t. It depends on the
situation, I guess.
…
I:
Okay. Can you think of something that there’s only one way to do?
Larry:
Dividing fractions.
Jenny:
Dividing fractions, not yet
Larry:
Not yet, why?
Jenny:
No, not, I’m sure there’s more than one way for that
…
Jenny:
I don’t know. Like adding there’s only one way to add.
I:
Okay.
Two groups placed the statement close to strongly agree but did not place it all the way to
strongly agree because, although they agreed that there was more than one way to solve all of the
mathematical problems that they had encountered so far, they did not want to speculate about its
truth for all mathematics. Jenny’s and Jim’s comments were representative of the remarks made
by these groups.
I:
A math problem can always be solved in different ways.
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Jenny:
Not necessarily.
Jim:
Well, kind of, because there’s other ways to do, like, most of the math
stuff that we do there’s
Mike:
Two ways
Jim:
A lot, yeah there’s a lot, like Florida to Washington, and then like the
other way (Jim is referring to two different procedures for adding and
subtracting fractions.)
Jenny:
Yeah.
Mike:
I, that’s like as far that way as you can go.
Jim:
I don’t, I think it’s like around here (pointing toward a place closer to the
¾ mark of the continuum.) ‘cause we could
Mike:
Like about here?
Jim :
Yeah, ‘cause we can
Jenny:
‘Cause we don’t know, we’re not in other math classes right now so we
don’t, necessarily, know.
I:
So you’re really talking about the “always”? It might not always, so
Mike:
Yeah.
Jim:
Like, sometimes.
I:
Okay. Would you go with sometimes or usually?
Jenny:
Probably usually.
Jim:
Usually.
I:
Okay.
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The other groups expressed strong agreement with the statement. One group was initially
hesitant but then, by giving an example of solving a problem in different ways, seemed to talk
themselves into agreeing with the statement.
I:
A math problem can always be solved in different ways.
Rachel:
Well, that depends on what you are doing.
Sarah:
Yeah. Well, it depends on, basically, being solved in different ways just
means like getting the answer in different ways. Like with the combining,
not the combining, with the, uh, what is it, two step equations, you can
solve it two different ways. You can either subtract the x’s first or you can
subtract the regular numbers first.
I:
Okay.
Sarah:
So that’s two different ways.
Rachel:
Or you can graph, first
Sarah:
Or do like the math
Rachel:
or do the algebra first
I:
Okay.
Sarah:
That’s like a, yeah [Sarah places the card towards the strongly agree end
of the continuum.]
This same group also identified this conception statement as one that they would not have
agreed with as much before this school year. They discussed the fact that last year they were
expected to learn and use only one way to solve any particular problem.
Sarah:
I think that one would be toward, more towards the middle for me.
I:
Oh, math problems can always be solved in different ways.
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Rachel:
Like, last year that would be more towards the end for me more.
I:
Towards which end?
Rachel:
This end. [She points to the strongly disagree end.]
I:
Okay.
Rachel:
Because she taught
I:
So both of you would agree with that less?
Sarah:
She only taught us one and she said don’t worry about any other ways
because you’re just going to get confused…. You had to do it her way.
That’s so frustrating.
The change in the number and type of alternative solutions offered by students between
the fall and the spring observation coupled with Sarah’s expression of frustration at not being
encouraged to use other solution methods seem to indicate that students did begin this school
year with a conception that mathematics is sensible enough that one can and should be allowed
to use alternative solution strategies. However, the change in both the number and type of
alternative strategies between the fall and spring observations suggest that this, too, is an area in
which students have grown in their conception of mathematics as sensible.
Role of memory in mathematics
Students in this class provided mixed and sometimes contradictory evidence about their
conception of mathematics as sensible enough that problem solving does not rely on memory.
Because of its prominence in the literature on conceptions of mathematics, there were two
conceptions statements and two indicators related to the role that memory, as opposed to
strategizing, plays in mathematics. In sorting the conceptions, students within groups often
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disagreed on their placement of both “You don’t have to have a good memory to be good at
math” and “Math is about memorizing and applying formulas, definitions, and procedures”
(Grigutsch & Torner, 1998). Even after discussion, students in three out of the six groups could
not resolve their conflicting opinions about their levels of agreement with the statement “You
don’t need a good memory to be good at math.” For example, in one group of three students I
tore the card into three pieces and allowed each student to place a piece. The pieces were placed
at 1.5, 7.5, and 8.8 on a 10 point scale. When students were asked at the end of the interview if
there were any statements that they wanted to revisit in order to reconsider the placement, the
statements about memory were revisited far more than any other statements with three out of the
six groups going back to the card about needing a good memory.
Of the 12 students involved in the interviews, seven strongly disagreed with the statement
“You don’t need a good memory to be good at math.” Some of the students who disagreed did
so because they took the statement to extremes. They seemed to interpret the card as meaning
that you can be successful doing mathematics with no memory at all. As one student stated,
“…if you don’t remember what you did yesterday, how are you going to keep up with it?” Other
students who disagreed seem to interpret the statement as being about the helpfulness of a good
memory as opposed to the need for one. One students stated, “I think I strongly disagree with
that because you have to be able to, you don’t have to be able to, but it’s, I guess I think it’s
easier to have, like, to remember things you’ve learned to use in the future in math.” Although
more than half of the students talked about the importance of memory for doing mathematics,
none of these students insisted that a good memory was essential or that memory was a central
part of what it means to engage in doing mathematics.
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There were only two students in the interviews who expressed strong agreement with the
statement, “You don’t need a good memory to be good at math.” The other three students rated
the statement somewhere in the middle of the continuum. The kinds of rationale provided by
students who agreed and those who were in the middle were very similar. Most talked about the
helpfulness of memory and the importance of remembering some things but also discussed the
ways that reasoning and practice balanced out the need to remember everything. One student
reported:
Jenny:
Well, I don’t know, I feel like you don’t need the best memory to do math
because you keep on practicing and practicing so I think you just know it.
You don’t necessarily need to remember every single step because – I
have a good memory but I know sometimes I forget what I’m doing in
math but I didn’t already do it –[inaudible] understand it.
Although the audio recording is unclear, it appears that Jenny is discussing not only the role of
practice but the role that understanding mathematics may play in alleviating the need to
remember everything in mathematics. Jerry makes this point more clearly in his comment,
“…even if you sometimes don’t have the greatest memory, you can’t remember things but you
can still figure them out by writing them out and you can just have a different thought
processes…” As with the students who disagreed with the statement that you don’t need a good
memory to do mathematics, of students who agreed there were none that gave the statement a
whole-hearted, unqualified endorsement. The closest any student came was the student who
responded to the statement “Math is memorizing and applying formulas, definitions and
procedures” (Grigutsch & Torner, 1998) by stating, “Not really, because it’s mostly about
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problem solving and figuring stuff out. So it’s not, so it’s not, I mean memorize as you go but
it’s not directly as memorizing.”
Overall, students’ statements in interviews indicate that they tend to see an important role
for memory in the doing of mathematics although they also recognize the importance of
“problem solving and figuring stuff out.” Interestingly, there was no evidence of this role of
memory in their classroom work or discussion. There were no incidents during either the fall or
the spring observations in which students used poor memory as an excuse for not engaging with
a problem or talked about needing to remember a procedure. In fact, in the 4 weeks of classroom
observation there were no episodes in which students discussed memory or referred to the need
to remember anything in mathematics. The all-too-familiar refrains in many mathematics classes
of “I don’t remember how to …,” “I’ll never be able to remember that,” or “Do you remember
how to …” were conspicuously absent both during whole class discussion and when students
were working in small groups. The students’ classroom behavior would indicate that they see
mathematics as sensible enough that it can and should be reasoned about rather than needing to
be remembered. The fact that this behavior was consistent from the very start of the school year
also indicates that this may be a conception that was in place before this academic year.
The seeming contradiction between what students said about the role of memory in
mathematics and their classroom actions is difficult to explain. Some of the contradiction might
be explained by some of their extreme interpretations of the conceptions statements. There may
also be other factors at work such as norms of behavior from previous mathematics classes.
Unfortunately, there is insufficient data to make any reasonable speculation about this seeming
contradiction. What is evident is that students, in practice, conceive of mathematics as sensible
enough that they engage in strategizing rather than relying on memory. It is also likely, given
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the consistent nature of students’ classroom actions, that this conception was in place from the
beginning of the academic year.
Summary
Students in this class see mathematics as sensible enough that they can and do strategize
about problem solving and seek alternative problem solving strategies. The students’ statements
during interviews reinforce this assertion. The increase in the number of episodes in which
students engage in these behaviors and the way they talk about strategizing may indicate that
strategizing is a category in which their conceptions of mathematics have grown over the
academic year.
Findings related to mathematical authority
Students in this class conceive of mathematics as sensible enough that they sometimes
assume mathematical authority and that they accept mathematical facts as authoritative.
However, this practice is not common and there are times when the students cede mathematical
authority to the teacher.
Types of evidence
There were initially five indicators related to mathematical authority. Four of these
indicators were related to students assuming mathematical authority: willingness to attempt
unfamiliar problems, checking their own answers to problems, inventing mathematics problems,
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and engaging in 2-way conversations about mathematics. The fifth indicator, seeking to be told
whether an answer is correct, related to ceding mathematical and was one of only three negative
indicators used in the framework. It was incorporated because of its prevalence in the literature.
Two additional indicators were added to this category during data analysis. The first, addressing
possible mathematical misconceptions of other students, is related to assuming mathematical
authority. The second, seeing mathematics as authoritative, is not directly related to either
assuming or ceding mathematical authority.
There was one conceptions statement related to mathematical authority: “You can’t tell
whether or not an answer is correct until someone tells you.” Although this statement could have
led to discussion about assuming authority, ceding authority to an outside source like the teacher,
or mathematics as authoritative, all of the discussion revolved around assuming mathematical
authority.
Assuming authority
Most of the coded indicators and all of the students’ comments during interviews were
related to students assuming mathematical authority. There were four episodes related to
assumption of authority during the fall observations and eight during the spring.
One episode during the fall and one during the spring were of students checking their
own answers rather than seeking to have the teacher or other outside authority check them. In
one case a student has an answer different from that of another student and decides to go with her
original answer because, “it makes more sense.” In the other episode a student rejects her initial
calculation for the slope of line seemingly because it is far larger than expected. She states that
she got “4500 over 7. I think that’s wrong.”
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The lack of indicators that students are routinely checking their own answers or are
checking them by rigorous means is at odds with what students said in interviews about checking
answers. When asked about their level of agreement with the statement, “You can’t tell whether
or not an answer is correct until someone tells you” all but two of the students disagreed. The
comments from those who disagreed ranged from:
I kind of strongly disagree on this one because there’s… a lot of ways to figure out a
problem and if you get the same answer more than twice for a problem you don’t know
then you can be right. The possibility, your chances of being right are greater. …So I
disagree that… because there are many ways to figure it out and if you come up with the
answer, you know, more than once or twice it’s, it should be right – most of the time.
to a very adamant, “Wrong! Liar!” The one student who placed the statement near the
middle identified several possible ways to check your own answer but then stated that, “like a
teacher or like an assistant teacher, they can check the math to make sure you’re right. It’s
always good to double check.” He went on to state that, “I’m not really confident in math, that’s
why I’m processing it more.” I presume that by processing it more he is referring to thinking
more about his rating of this statement than the other members of his group. The only student
who completely disagreed with the statement was a student who was not interviewed as part of a
group and tends to speak very little. He placed the card on the agree end of the continuum and
did not comment.
There were two episodes during the fall observations in which students identified and
pointed out potential mathematical misconceptions of their peers. In the first a student appeared
to be confused when the teacher talked about thirty-five hundred miles. Another student
interjected, “That’s another way of saying three thousand five hundred. Don’t get confused by
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that.” In another incident some students suggest cross-multiplying to multiply fractions. A
student tells them that they are probably confusing multiplication with division. These two
incidents seem to illustrate students taking responsibility for the mathematics in their classroom.
Because of these two incidents an indicator was added to the assume authority category for
addressing mathematical misconceptions of peers.
Although several of the indicators for assuming authority were fairly consistent across the
fall and spring observations, there were two indicators in which there seemed to be some growth
over the academic year. During the fall semester there was only one incident in which a student
posed his own mathematics problem. During the spring semester there were four episodes. In
the fall episode, the teacher has been discussing the cost of new textbooks and tells the students
the cost of the new calculus textbooks and the number of students in the class. When the teacher
asks students about the total cost, one student has already formulated and solved the problem.
He calls out, “I just did it” and gives the correct solution. In a similar episode during the spring
observations one student, trying to make sense of the statement that “if you gain 2 inches of
height, you gain 1 inch of [hand span],” wonders whether this means that someone who is 80
inches tall would have a hand span of 40 inches. Several students respond that it does not work
that way and a pair of students in the back of the room begins to work independently to find the
correct anticipated hand span of a person 80 inches tall. Although the teacher soon begins
working the same problem at the board, the audio recording makes it clear that these students
have formulated and begun solving the problem on their own.
In addition to an increased willingness to pose their own mathematical problems, during
the spring observation there are several incidents in which students display a willingness to
engage in a problem for which they have not yet learned a procedure. This tendency is not
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evident during the fall observations but occurs on three occasions during the spring observations.
One example of this is when the teacher writes the expression x-2 on the board and asks students
to speculate on its possible meaning. Suggestions include: “you have negative two x’s,” “you
have negative two dollars,” and “negative x times negative x, so two negatives is a positive.” In
another incident the teacher writes the sequence 1,1,2,3,5,_,_ on the board and asks students
what numbers come next. The students have had no recent experience with problems of this type
and have not been given strategies to engage in problems like this; however, they try the problem
anyway.
Both the increase in problem-posing behavior and the increased tendency to engage in
unfamiliar problems may indicate that students are assuming more mathematical authority but
the small number of incidents for any of the indicators relating to assuming authority does not
provide enough evidence for a strong claim. The strongest evidence that students do assume
some authority is their statements during interviews about their lack of a need for someone to tell
them whether and answer is correct.
Ceding authority
The initial framework of indicators included one negative indicator in the category of
assuming authority: seeking to be told whether or not an answer is correct. There were two
episodes in the data coded with this indicator, one during the spring observations and one during
the fall observations. Both episodes included several coded indicators in which students,
working independently on a practice worksheet, asked the teacher or one of the other adults in
the classroom if an answer was correct. While this may be seen as evidence that students are
ceding mathematical authority, in both cases students were working on practice exercises related
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to a topic that was new to them. During the fall observation, students were simplifying
expressions by using the distributive property and combining like terms. During the spring
observation they were simplifying expressions involving negative exponents. In addition to
being new, these types of exercises are very difficult to check except by redoing the problem.
During these same episodes students were also checking answers with other students and seeking
to reconcile different answers. With this as the only evidence of students ceding authority, I am
reluctant to draw any conclusions about students not seeing mathematics as sensible enough that
they need to rely on an outside authority for confirmation of their answers.
Mathematics as authoritative
There was incident during the fall semester in which it appeared that a student granted
authority neither to the teacher nor to himself but, rather, to the mathematics.
T:
In 1896, the state legislature in Indiana introduced a bill that said pi, from
now on, in the State in Indiana, will be 3. Not 3.1415926598. 3. For the
benefit of all school children who will now have an easier calculation to
do.
Mike:
Cool.
Beth:
Okay.
Rob:
But –
T:
But?
Rob:
But that’s not accurate to the rest of the world so when they leave their
little bubble of 3
Beth:
I get it
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Rob:
they wouldn’t know what they are doing.
T:
You’re exactly right, Rob, and, in the end, the bill didn’t pass because
mathematicians went to the state capital and said, “Look, this is not how
the world works.” And so, in the end, the bill didn’t pass.
In this episode, Rob’s objection demonstrates that he recognizes that the mathematics of the
situation is authoritative. This is an incident clearly related to mathematical authority but
unrelated to any of the original indicators. It also seems closely related to a view of mathematics
as sensible enough that it should be internally consistent. It was coded as seeing mathematics as
authoritative. As a single incident, it tells us very little about the students in the class and their
conception of mathematics. It may, however, have value as another indicator in the framework
that students see mathematics as sensible.
Summary
There is some evidence that students in this class see mathematics as sensible enough that
they can assume mathematical authority and see mathematics as authoritative. Students, in
interviews, agree that they have sufficient mathematical authority they do not need to be told
when an answer is correct. However, there are too few classroom indicators to make reasonable
conjectures about students’ conceptions about authority in mathematics or about the possible
growth of that conception over the academic year.
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Findings related to stating that mathematics makes sense
In the framework there is one indicator relating to students stating that mathematics
makes sense: expressing a role for common sense in mathematics. This indicator was developed
from the literature on students’ conceptions of mathematics. It might look something like a
student, after being introduced to the formula for finding the slope of line given two points,
stating, “Oh, it makes sense that you would subtract the y’s because that tells you the rise.”
There were no incidents of this nature in the data from either the spring or the fall observations.
There were also no conceptions statements related to this indicator. Although students in this
classroom gave many indicators that they conceived of mathematics as sensible they do not state
this.
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Chapter 6 Discussion
In this chapter I discuss important results from this study. I first discuss answers to the
research questions posed in Chapter 1. I then discuss the significance of the study and the ways
in which the framework evolved over the course of the study. I conclude the chapter by
discussing the implications of the study and making suggestions for further research.
Assertions based on findings
Research question 1
The first research question for this study was related to the use of constructs in the
existing literature that might be used to develop observable, action-oriented indicators that
students in a mathematics classroom see mathematics as sensible. Survey and interview
questions from the current studies on students’ conceptions of mathematics provided questions
and statements that I was able to reframe into observable, action-oriented indicators. While the
list of indicators gleaned from the literature proved not to be an exhaustive list of possible
indicators, the indicators grouped into five main categories. These categories were found to be
adequate for coding all observed behavior in which students seemed to indicate that they saw
mathematics as sensible. Thus, specific behaviors not covered by the list of indicators could be
coded with a more general designation of strategizing, expecting connections, expecting
explanations, assuming authority, or stating. Constructs from the current literature were
sufficient to permit construction of a useable framework for coding indicators that students in a
secondary mathematics classroom conceived of mathematics as sensible.
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Research question 2
The second research question for this study was: In what ways is the framework a viable
tool for documenting students’ conceptions of mathematics as sensible? The framework of
indicators developed in the first part of the study evolved during data collection and analysis into
a robust, useful, and flexible tool for identifying indicators that students conceived of
mathematics as sensible. The initial framework, by permitting coding by category rather than by
specific indicator when needed, provided the flexibility to code indicators not specified in the
original list of indicators. Further categorization of indicators into more general types provided
the necessary step to make the framework one that is useful not only for coding student actions
but for gaining an overall picture of the dimensions and details of the ways in which students see
mathematics as sensible.
Significance of the study
The importance of understanding students’ conceptions of mathematics and the
limitations of both surveys and interviews for assessing those conceptions, points to a need for a
tool that permits inference of those conceptions from students’ actions. The framework
developed in this study, by providing action-oriented indicators that students see mathematics as
sensible, is such a tool. Furthermore, with its categorization of different aspects of this
conception of mathematics, this framework can be used to study students’ conceptions of
mathematics as sensible along several different dimensions.
This study provides one of the few frameworks designed to examine student actions in
the classroom. There are a number of observational frameworks or protocols designed to look at
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teacher actions or classroom traits (e. g. Grouws et al., 2013; Piburn & Sawada, 2000). Going
back to Lampert’s problem spaces of teaching and learning, most work has focused attention on
the problem space between teacher and student. This framework provides a tool for examining
the problem space between the student and content, in this case students’ conception of
mathematics as sensible.
6-1 The work of teaching (Lampert, 2001, p. 33)
This framework is, to my knowledge, the first to be established that provides a way to
assess productive disposition as indicated by students’ actions. Most previous research on
students’ conceptions of mathematics has relied on students’ self-reports of their conceptions.
This framework, by providing a tool for using students’ actions to examine conceptions, provides
a tool that is not subject to the limitations of self-reporting (Munby, 1982).
This framework is also the first to bring together several critical student practices.
Strategizing, seeking connections, assuming mathematical authority, and explaining are all
prominent in the literature on mathematics education. They are central to the Standards for
Mathematical Practice of the CCSS-M (NGA & CCSSO, 2010), the NCTM Practice Standards
(NCTM, 2000), and the strands of mathematical disposition (NRC, 2001). This study, by
beginning with the literature on students’ conceptions of mathematics and identifying each of
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these practices in that literature, brings these practices together and links each to students’
conceptions of mathematics as sensible.
Beyond contributing to the literature as described above, this framework links research
and practice, most notably by providing a tool that teachers and students can use in multiple
ways. This study provides a new tool to help teachers, teacher educators, and researchers better
understand students’ conceptions of mathematics as sensible. By using action-oriented
indicators based in the existing literature, it is helpful for identifying episodes in which students
give evidence of a conception of mathematics as sensible. By organizing these indicators into
categories and subcategories, the SCOMAS framework provides a multifaceted vision of
students’ conceptions and provides a framework with the flexibility to move beyond the specific
indicators from the literature and include other actions that fit within the framework’s categories.
For example, this framework may be helpful for teachers as a tool to examine the
conceptions of their students. They could, as was done in this study, use it to code video
recordings of their classroom. From this they can see the types of activities indicative of a
conception of mathematics as sensible in which their students do and do not engage. A teacher
might find, for example, that her students regularly engage in making connections, strategizing,
and explaining how they solved problems but that they do not tend to engage in explaining why
mathematical statements are true or why they employed certain strategies. The teacher might
then seek to help students develop more in the weaker areas.
The SCOMAS framework could also be useful to teachers as they examine their own
instructional practices. There is not yet strong evidence that encouraging students to engage in
the actions used as indicators in the framework helps to change students’ conceptions of
mathematics. However, it is reasonable to expect that more frequent engagement in these actions
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will help students see that these actions are a part of doing mathematics and, thus, that
mathematics is sensible enough that such actions are productive.
Teachers may also be able to use this framework as a springboard toward instructional
practices that allow for the kind of more “open mathematics” envisioned by Jo Boaler (1998).
The lack of coded indicators in some of the classes used in the pilot study may be less indicative
of students’ conceptions of mathematics in those classes than of instructional practices that do
not provide opportunities for students to engage in the kind of actions named in the framework.
Teachers, in using the framework to examine their students’ conceptions, may better be able to
visualize the kinds of actions in which students might engage and seek opportunities to engage
students in more open mathematics.
Teachers may also be able to use the SCOMAS framework with secondary school
students. The framework could help students visualize the types of activity that are a part of
engaging in mathematics as a sense-making enterprise. Students could use the framework to
examine the specific actions in which they or the class have engaged and could use the
framework as a way to measure their own development in the different categories. The
framework could provide a way for teachers and students to engage in discussion about what
mathematical activity could look like and make teachers and students co-architects of a new way
of engaging in mathematics in the classroom.
In addition to linking research and practice for teachers and students, the SCOMAS
framework could be useful to teacher educators as they work with preservice mathematics
teachers. A common challenge for pre-service teachers is how to reflect on student learning,
particularly for a strand of mathematical proficiency as difficult to measure by pencil and paper
test as productive disposition. This framework provides a tool that preservice teachers could use
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in observing mathematics classrooms and examining video of their own teaching. In addition to
providing pre-service teachers with a tool for reflecting on student learning this framework, with
its action-oriented indicators, may provide pre-service teachers with a vision of the types of
behaviors in which students might be expected to engage in a mathematics classroom focused on
sense-making.
Teacher educators and others who mentor teachers may find this framework a helpful
tool to use for observing teachers. The specific nature of the indicators in the framework makes
it easy to identify actions through which students give evidence that they see mathematics as
sensible. Meanwhile, the categories within the framework provide a way that teacher educators
and mentors can help teachers reflect on the ways in which classroom instruction may provide or
limit opportunities for students to engage in these actions.
This study and the resulting framework make an important contribution to the field of
mathematics education research. In 2001, the five strands of mathematical proficiency
envisioned by the National Research Council provided a new vision for a multifaceted
knowledge of mathematics. These strands also formed the foundation for the Standards for
Mathematical Practice of the CCSS-M. However, there is still much work to be done on
operationalizing the concepts introduced in these strands. This study looks closely at a central
part of the strand of productive disposition, “the habitual inclination to see mathematics as
sensible…” (National Research Council, 2001, p. 5) and provides action-based indicators and a
categorized framework for examining this concept. The SCOMAS framework provides
researchers with a tool for examining students’ conceptions of mathematics as sensible within
several different categories. It could be used to examine changes in students’ conceptions over
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time and as a way of examining effectiveness of instruction in helping to develop students’
conceptions of mathematics as sensible.
Finally, although not intended as an outcome of the study, this study serves as an
existence proof that even students who have struggled in mathematics can develop productive
dispositions towards mathematics. The vision put forth in Adding It Up (NRC, 2001) is that all
children become proficient in all strands. Despite the importance of productive disposition for
all students there has been little research on its development in students who have struggled with
learning mathematics. The high school students involved in this study all had a history of low
achievement in mathematics and many had learning disabilities in mathematics. The indicators
of a conception of mathematics as sensible displayed by this group of students provides an
existence proof that development of a productive disposition is achievable by a broad spectrum
of students.
Evolution of the framework
Given the careful design of the research and development process, as explained in this
document, I am confident that my final framework both represents action-oriented indicators of
productive disposition and that the framework can be used as a useful tool for documenting
evidence of productive disposition in mathematics classrooms. In the following paragraphs I
describe how the framework evolved through its use in analyzing classroom data. I first describe
changes in the indicators, then changes in the structure of the framework.
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Changes to the indicators
The first step in this process of creating a more flexible, robust framework was to identify
necessary changes to the initial list of indicators. This began with the elimination of the three
negative indicators in the framework: relying on memory for problem solving, using poor
memory as a reason for not being able to solve problems, and seeking to be told whether an
answer is correct. Initially the prevalence of these items in the existing literature led me to
include them in the list of indicators, despite the fact that they were negative indicators rather
than positive ones. However, once indicators were organized into a framework, it became clear
that other items within the same category were already coding the opposite of these negative
indicators. For example, the opposite of relying on memory for problem solving is to engage in
some form of strategizing when confronted with an unknown problem. The opposite of seeking
to be told whether an answer is correct is to try to check one’s own work. Given that the positive
indicators for the same situations were already in the framework, these negative indicators had
no more place in the framework than the negative indicators that could be created for any of the
other statements. Since this was a framework designed to identify ways in which students
conceived of mathematics as sensible rather than ways in which they indicated that they did not,
negative indicators of any kind were not necessary.
Along with the elimination of negative indicators, several specific indicators were added
to the initial framework. These indicators came about when there were incidents in the data that
fit into the general categories of indicators but did not fit well with a specific indicator. For
example, there were several incidents in which students noticed and remarked upon a connection
between the current problem situation and another topic or concept in mathematics. This was
clearly related to expecting and seeing connections but not to any specific indicator in the
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category. For these situations, the indicator noticing mathematical connections was created.
Also added to the expecting and seeing connections category was mathematical humor since
most of this humor relies on understanding the underlying structure of mathematics and the
connections inherent in this structure. Several incidents were not coded with a specific indicator
but simply left coded with a category designation. This proved to be nonproblematic as the
framework continued to evolve.
Changes to the framework
The next stage in the evolution of the framework came about during data analysis as a
way to make sense out of what the different indicators indicated about students’ conceptions.
The indicators within each category were grouped into subcategories of behavior. In the
category of expecting explanations the indicators split into seeking explanations, explaining how,
and explaining why. The indicators in the connections category split into connections between
mathematics and other contexts and connections within mathematics. The indicators in the
assuming authority category split into students assuming mathematical authority and students
recognizing mathematics as authoritative. The indicators in the category of strategizing were
organized into strategizing about problem solving and seeking alternative solution strategies.
Without sufficient data from the current study to examine the possible dimensions of the stating
category, it was left to stand without subcategories. The resulting SCOMAS framework [see
Figure 6-1] moves somewhat away from an emphasis on specific, narrow indicators gleaned
from the literature to a framework which provides a way to look at the various dimensions of
students’ conceptions of mathematics as sensible. Embedded within the subcategories of the
framework are still specific indicators but now, rather than constituting an exhaustive list of
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indicators, the indicators serve as illustrations for the kinds of activity that signal students’
conceptions related to the subcategories. This framework permits us to talk about students
demonstrating that they are seeing connections within mathematics by engaging in actions such
as recognizing similarities between problem situations, adapting problem solving strategies, and
coordinating multiple representations. This study has extended the current literature on
students’ conceptions of mathematics both by providing a framework permitting direct
observation of indicators of students’ conceptions of mathematics as sensible and providing a
conceptual framework of the dimensions of that conception.
Figure 6-1. Students’ Conceptions of Mathematics as Sensible (SCOMAS) Framework
Challenges in studying students’ conceptions of mathematics
Although not intended as part of the study, this study produced some important findings
about the study of students’ conceptions about mathematics. To date, research on students’
conceptions of mathematics has been largely based on the use of surveys consisting of openended or Likert-scale type items. These studies have been useful for identifying different
clusters of conceptions and constructing models of types of conceptions. Researchers have
variously identified such classifications as dynamic versus static (Grigutsch and Torner, 1998),
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modeling versus components versus abstract (Petocz et al., 2007), and fragments versus cohesive
(Crawford, Gordon, Nicholas, and Prosser, 1998a, 1998b). Using these classifications,
researchers have been able to draw conclusions about the prevalence of different types of
conceptions of mathematics within a population.
Limitations of survey items
One limitation of the research to date has been the challenge of identifying the likely
conceptions of mathematics of a small population or an individual. Many of these studies have
relied on survey using Likert-scale type items. Findings from this study raise serious questions
about the usefulness of such items for examining the conceptions of secondary school students.
In my interviews with small groups of students, they were asked to place statements about
conceptions on a continuum from strongly agree to strongly disagree and to discuss their
placement. These interviews yielded some valuable insights about the limitations of such
statements and raised serious questions about their validity as survey questions. Of the 10
conceptions statements used in this study, all presented difficulties for some students or were
modified or qualified by at least one student during the interviews.
Complex statements presented a challenge for most of the students interviewed in this
study. Three of the conceptions statements in this study were consistently misinterpreted or not
well-understood by the students involved. The statements “Math is memorizing and applying
definitions, formulas, facts, and procedures” (Grigutsch & Torner, 1998) and “Math is made up
of ideas, terms, and connections” (Grigutsch & Torner, 1998) both attempted to define
mathematics using a list of terms. Students in this study tended to focus in on particular terms in
the list for comment or to consider each term in turn to determine whether it was related to
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mathematics. There was only one group that actually considered the totality of one of the
statements (“Math is memorizing and applying definitions, formulas, facts, and procedures”) and
discussed whether the list of terms constituted a complete description of mathematics. However,
even this group did not analyze the other statement in the same manner. It seemed that students
in this study had difficulty focusing on and evaluating more than a single idea within a
conceptions statement.
A similar issue arose in relation to the statement “A lot of things in math must simply be
accepted as true and remembered; there really isn’t any explanation for them” (Carter &
Norwood, 1997). Students in this study tended to use this statement to talk about whether a good
memory is necessary in mathematics but not to account for the remainder of the statement.
When prompted, two groups did comment on whether things in mathematics had explanations
but, as illustrated by the following excerpt, they required substantial prompting.
I:
The real question is, are there things in math that you just have to say,
okay, I believe it, I don’t understand why and there’s no good reason why
but someone says it’s true so it is.
Jenny:
Well, there’s that but you also have to understand it. Like, I feel like if
you just say it’s right you really don’t get it. So I kind of like, yeah I kind
of like disagree with that ‘cause you should kind of like … I don’t know.
I:
How about you guys?
Mike:
[inaudible]
I:
Okay. Do things always have, do things always have an explanation?
Jim:
No.
146
I:
Okay. So there are some things that you just have to believe are true
without an explanation?
Mike:
Yeah.
Jenny:
Yeah, I guess.
I:
How about you?
Jim:
I don’t really understand it.
I:
Okay. There are some things in math that have to accept to be true. Mr.
Wingate tells you that, um, you know, anything raised to the zero is one
and you just have to accept that that’s true, there’s no explanation for it.
Jim:
Oh, all right, yeah.
I:
So there are times when there just isn’t an explanation.
Mike:
Well like the formulas for like finding the area of a circle, we don’t really
know how that works. We just know it.
I:
Okay. Do you think there is an explanation and you just don’t know it yet
or is there just not an explanation?
Mike:
I just don’t think there is an explanation.
I:
Okay. It’s just one of those things – the mysteries of the universe.
Mike:
Yeah.
In this short excerpt, the interviewer rephrases the question seven times in order to get responses
from all group members and to verify what the students are saying. Even with this type of
prompting when the group was asked at the end of the interview if there were any statements that
they wished to revisit, one group member identified this statement.
I:
Are there any others that we need to revisit?
147
Jenny:
I’m still kind of confused about that one but I think like
I:
You’re kind of confused about the wording of it?
Jenny:
Yeah, I just like, I don’t [Jenny lapses into silence.]
Despite the multiple rewordings provided by the interviewer, this student still is not confident
that she understands the statement. Like the statements with lists of terms, students had
difficulty interpreting this compound statement.
Students in this study also tended to have difficulty interpreting statements that were
phrased negatively: “It is not important to understand why a procedure works as long as you get
the right answer” (Kloosterman & Stage, 1992), “You don’t have to have a good memory to be
good at math,” and “You can’t tell whether or not an answer is correct until someone tells you.”
In several cases, the interviewer rephrased the statement in the positive to make it easier for
students to interpret. For example, in one group a student struggled to interpret the statement
“You don’t have to have a good memory to be good at math” but, when the interviewer changed
the statement to the positive both group members easily expressed their opinion.
Jenny:
You don’t have to have a good memory – wait, which one is, like if I
I:
Would you like me to restate it to the positive? You need a good memory.
Jenny:
Yeah.
I:
Let’s do it that way. [Interviewer takes the card and writes “You need
good memory” and returns the card to Jenny.] Agree or disagree?
Jenny:
Disagree.
I:
How about you?
Larry:
I definitely agree with that.
148
Similar issues arose with the other negative statements with students having particular difficulty
with what it means to disagree with a statement that was already negative.
Other issues interpreting statements were not systematic but idiosyncratic to a particular
group or individual. One example of this occurred when a group was presented with the
statement, “Math is made up of unrelated topics” (Brown et al., 1988).
I:
Math is made up of unrelated topics.
Jenny:
Yes, definitely. Because we use science, we use
Jim:
True, we use science
Jenny:
We use math for science
Mike:
History
Jenny:
Or
Jim:
Yeah, like for geography
Jenny:
Yeah. So, definitely.
I:
Okay.
Jenny:
So we’ll put it like close to the top. A little bit at the top.
I:
Okay.
In this case, the students did not interpret the statement as a statement about the connected nature
of mathematics but rather about the connections between mathematics and other academic
subjects. In addition, they rated the statement in the opposite direction on the scale from that
which would ordinarily be expected. If students in this group had been rating this statement in a
setting in which they were not providing explanations, their placement of the statement would be
interpreted as these students believing that mathematics is not connected in nature.
149
An individual in one group similarly rated a statement in a manner opposite to that which
might be expected.
I:
So, where would you put it? Knowing how to solve is as important as
getting the solution.
Jerry:
I’m going to go like, probably, here because I’m a little bit disagreeing
with how it’s saying it’s just as important as figuring out the answer.
I:
You’re saying it’s more important?
Jerry:
Yeah, it’s more, definitely more important to know how to do it than to
figure out the answer. The answer would probably come last is my
solution, or problem solving thing ‘cause I want to know how to do it
before I can find the answer.
Jerry’s disagreement with the statement would, in a survey situation, be interpreted as a
conception that solutions were more important than understanding when he is actually saying
that it is more important.
Given the difficulties students in this group had interpreting conceptions statements, it is
clear that, if they were to answer these questions in a survey situation with no opportunity to
explain their answers, discuss their interpretation of the statement, or have the statements
reworded for them, that the survey would yield little valid data about their conceptions of
mathematics. This raises concerns about the validity of survey instruments, in general, for
studying student conceptions. More importantly, it points to the limitation of such instruments
for determining the conceptions of individual secondary school students or small groups of such
students.
150
Limitations of interviews
Results from the interviews in this study also point to the limitations of interviews for
determining the conceptions of mathematics of secondary school students. Although in many
cases students’ statements during interviews affirmed what was found in the classroom
observations, there were several substantial limitations.
One difficulty, as already illustrated in some of the incidents in the previous section, is
the challenge of finding clear ways to talk about conceptions of mathematics. For example, even
simplifying the statement, “A lot of things in math must simply be accepted as true and
remembered; there really isn’t any explanation for them” (Carter & Norwood, 1997) to “do
things always have an explanation?” left some students puzzling about what was being asked.
Similarly, although students readily agreed with the statement, “Knowing how to solve a
problem is as important as getting the solution” (Brown et al., 1988), some students, as
illustrated by the following conversation, saw the statement as nonsensical.
I:
Let’s see, knowing how to solve a problem is as important as getting the
solution right.
Rachel:
Well, you have to know how to get the solution, if you have to solve it.
Sarah:
Yeah, you,
Rachel:
In order to solve it you have to know what you are doing and then, to get
the solution, you have to know how to solve it.
I:
So you’re basically saying you really can’t solve it without knowing how.
Sarah & Rachel: Yeah.
I:
Okay.
Sarah:
Put that up there. We’re kind of going along.
151
These two students could not visualize the dichotomy intended in the statement.
Although some of the difficulty that students had interpreting the conceptions statements may be
attributed to wording of the statements, much of the difficulty may lie in the very nature of
conceptions of mathematics. I suspect many students have never before thought about or been
asked to discuss their conceptions of the nature of mathematics. It is also likely that many
students, even if they had thought about their conceptions, could not clearly articulate those
conceptions.
This same difficulty in understanding and articulating their own conceptions may explain
some of the discrepancies that arose between what students said during interviews and what was
observed in class using the framework. The most illustrative of these discrepancies is the one
between what some students said about the role of memory in mathematics and what was
observed in the classroom. During the interviews individual students often expressed strong
agreement that memory is an important factor in mathematics. They did so in the context of
agreeing with the statement “A lot of things in math must simply be accepted as true and
remembered; there really isn’t any explanation for them” (Carter & Norwood, 1997) or by
disagreeing that “You don’t have to have a good memory to be good at math.” Despite the
emphasis on memory in many of the interviews, there were no instances, during the 4 weeks of
observation data, in which students discussed relying on memory to engage with problems or
used memory as an excuse not to engage with mathematics. The common classroom refrains of
“oh, I remember you just …” or “Oh no, I can’t remember how to do this” were conspicuously
absent from this classroom. Instead, students tended to display strategizing and to seek
connections to other problems whenever they were asked to solve problems. When it came to
152
the role of memory, the indicators observed in the classroom were somewhat inconsistent with
what students said in interviews.
A similar difference arose with respect to students’ discussions about the statement
“There is always a rule to follow in solving math problems” (Telese, 1999; Brown et al., 1988).
Students’ general agreement with this statement and comments about the statement would seem
to indicate that they believed that problem solving in mathematics was about following the
correct rule for solving each type of problem. One student stated, “there’s always a different
way, ’cause different problems have different rules to solve them.” Despite these comments,
there was no evidence in the classroom of students seeking rules for specific types of
mathematics problems. Instead, indicators relating to strategizing and making connections in
mathematics point to students solving problems based on extending and using knowledge of
solution strategies from other problems and attending to the mathematical properties of the
problem to strategize solution methods.
These two discrepancies between what students said interviews and their classroom
actions highlight an inherent difficulty with using interviews to determine the conceptions about
mathematics of secondary school students. It seems that students have difficulty interpreting
conceptions statements in useful ways and have difficulty expressing their conceptions of
mathematics. Although it is possible that there is a true mismatch between students’ conceptions
of mathematics and their mathematical activity, it seems more logical to assume that the
mismatch is between students’ statements about conceptions and their mathematical activity.
153
Implications of this study
Findings from this study have some important implications both for classroom practice
and for research.
Implications for classroom practice
This study has important implications for our expectation that all students can develop a
productive disposition towards mathematics. It is sometimes assumed that such a disposition is
characteristic of gifted students (Krutetski, 1976) but this study provides an existence proof that
such a conception can characterize struggling students as well. Students involved in this study
had struggled with learning mathematics and had a history of being less than successful
mathematics students yet, as individuals and as a class, they provided multiple indicators that
they conceived of mathematics as sensible.
The implication for teachers and students is that
there is reason to expect that every student can develop a productive disposition towards
mathematics.
Implications for research
This study has important implications for how we study students’ conceptions of
mathematics. The findings raise serious questions about the usefulness of both survey research
and interview research for studying conceptions. The SCOMAS framework, by providing
action-based, observable indicators of students’ conceptions provides both an alternative to
research relying on students’ self reports and a way to triangulate data from these other sources.
154
Suggestions for further study
The purpose of this study was to develop an initial framework for studying students’
conceptions of mathematics as sensible. Although the initial indicators were piloted in several
classes, the SCOMAS framework was developed in the context of one carefully selected
secondary mathematics classroom. The usefulness, completeness, and validity of the framework
need to be tested in other secondary mathematics classrooms. It might also be helpful to use the
framework within different grade bands and with different populations of students both to test
the framework and to begin to look at differences that may exist in students’ conceptions of
mathematics as sensible. .
The SCOMAS framework provides opportunities for additional research about the
continued development of conceptions of students over time and about the durability of
conceptions developed during the course of an academic year. One possible direction for future
research is to use the framework to follow a group of students over the course of several school
years, examining the ways in which the students’ conceptions change over time and the effect
that different course material, different classroom settings, and student maturity have on
indicators that students’ see mathematics as sensible.
A critical future direction of this research is looking at the instructional practices related
to the development of students’ conceptions of mathematics as sensible. The SCOMAS
framework provides a necessary tool for establishing the dimensions along which students’
conceive of mathematics as sensible. Given that this is a conception that is an important part of
mathematical proficiency, we next need to try to determine instructional practices that may play
a role in the development of such a conception. A future research study should examine the
155
instructional practices in classrooms in which students’ actions indicate that they conceive of
mathematics as sensible.
Future research in this area needs to include a focus on testing and improving the
framework. It should also include research in which the framework is used as tool to further
investigate students’ conceptions of mathematics as sensible. Finally, the framework should be
used as a tool for investigating instructional practices related to the development of students’
conceptions of mathematics as sensible.
156
References
Atallah, F., Bryant, S. L., & Dada, R. (2010). A research framework for studying conceptions
and dispositions of mathematics: A dialogue to help students learn. Research in Higher
Education Journal, 7. http://www.aabri.com/rhej.html
Barbour, R. (2008). Doing focus groups (Vol. 4). Thousand Oaks, CA: SAGE Publications, Inc.
Beckmann, S. (2002). Mathematics for elementary teachers: Making sense by “explaining why.
Paper presented at the Second International Conference on the Teaching of Mathematics at
the Undergraduate Level, Crete, Greece. Retrieved from
http://www.math.uoc.gr/~ictm2/Proceedings/pap174.pdf
Boaler, J. (1998). Open and closed mathematics: student experiences and understandings.
Journal for Research in Mathematics Education, 29, 41–62. doi:10.2307/749717
Brodie, K. (2010). "Learning mathematical reasoning in a collaborative whole-class discussion"
in K. Brodie (Ed.) Teaching Mathematical Reasoning in Secondary School Classrooms, (p.
57-72). Boston, MA: Springer.
Brown, C. A., Carpenter, T. P., Kouba, V. L., Lindquist, M. M., Silver, E. A., & Swafford, J. O.
(1988). Secondary school results for the fourth NAEP mathematics assessment: Algebra,
geometry, mathematical methods, and attitudes. The Mathematics Teacher, 81, 337–397.
http://www.nctm.org/publications/mt.aspx
Carter, G., & Norwood, K. S. (1997). The relationship between teacher and student beliefs about
mathematics. School Science and Mathematics, 97, 62–67. doi:10.1111/j.19498594.1997.tb17344.x
Cobb, P., & Bauersfeld, H. (1995). The emergence of mathematical meaning: Interaction in
classroom cultures. Hillsdale, N.J.: L. Erlbaum Associates.
157
Cobb, P., Wood, T., Yackel, E., & McNeal, B. (1992). Characteristics of classroom mathematics
traditions: An interactional analysis. American Educational Research Journal, 29, 573–604.
doi:10.3102/00028312029003573
Colby, G. (2007). Students’ epistemological beliefs of mathematics when taught using traditional
versus reform curricula in rural Maine high schools. (Unpublished Doctoral dissertation),
University of Maine, Orono, ME.
Coxon, A. P. M. (1999). Sorting data: Collection and analysis (Vol. 127). Thousand Oaks, CA:
SAGE Publications, Inc.
Crawford K., Gordon S., Nicholas J. & Prosser M. (1994) Conceptions of mathematics and how
it is learned: the perspectives of students entering university. Learning and Instruction, 4,
331–345. doi:10.1016/0959-4752(94)90005-1
Crawford, K., Gordon, S., Nicholas, J., & Prosser, M. (1998a). Qualitatively different
experiences of learning mathematics at university. Learning and Instruction, 8, 455–468.
doi:10.1016/S0959-4752(98)00005-X
Crawford, K., Gordon, S., Nicholas, J., & Prosser, M. (1998b). University mathematics students'
conceptions of mathematics. Studies in Higher Education, 23, 87–94.
doi:10.1080/03075079812331380512
Creswell, J. W. (2009). Qualitative inquiry and research design: Choosing among five
approaches. Thousand Oaks, CA: SAGE Publications, Inc.
Delamont, S. (2002). Fieldwork in educational settings: Methods, pitfalls and perspectives. 2nd
ed. London: Routledge.
Diaz-Obando, E., Plasencia-Cruz, I., & Solano-Alvarado, A. (2003). The impact of beliefs in
student's learning: An investigation with students of two different contexts. International
158
Journal of Mathematical Education in Science & Technology, 34, 161-173.
doi:10.1080/0020739031000071476
Erickson, D. K. (1993, April). Middle school mathematics teachers' views of mathematics and
mathematics education, their planning and classroom instruction, and student beliefs and
achievement. Paper presented at the annual meeting of the American Educational Research
Association, Atlanta, GA.
Ernest, P. (1988). The impact of beliefs on the teaching of mathematics. Paper presented at the
Fourth International Congress on Mathematical Education, Budapest, Hungary.
Frank, M. L. (1988). Problem solving and mathematical beliefs. Arithmetic Teacher, 35(5), 32–
34.
Franke, M. L., & Carey, D. A. (1997). Young children's perceptions of mathematics in problemsolving environments. Journal for Research in Mathematics Education, 28, 8–25.
doi:10.2307/749661
Franks, M. L. (1990). What myths about mathematics are held and conveyed by teachers? The
Arithmetic Teacher, 37(5), 10-12. Retrieved from
http://search.proquest.com/docview/208776472?accountid=13158
Frid, S. (1995). Cultural perspectives of secondary school mathematics. In B. Atweh & S. Flavel
(Eds.), Proceedings of the 18th annual conference of the Mathematics Education Research
Group of Australasia. Darwin, Australia: MERGA.
Garofalo, J. (1989). Beliefs and their influence on mathematical performance. Mathematics
Teacher, 82, 502–505. http://www.nctm.org/publications/mt.aspx
159
Gerson, H., & Bateman, E. (2010). Authority in an agency-centered, inquiry-based university
calculus classroom. The Journal of Mathematical Behavior, 29, 195-206.
doi:10.1016/j.jmathb.2010.10.003
Grigutsch, S. & Torner, G. (1998). World views of mathematics held by university teachers of
mathematics science. Schriftenreihe des Fachbereichs Matematik, Preprint 420, Gerhard
Mercator University, Duisburg. Retrieved from http://www.ub.uni-duisburg.de/ ETDdb/theses/available/duett-05272002-102811/unrestricted/mathe121998.pdf
Grouws, D. A. (1992). Handbook of research on mathematics teaching and learning. New York,
NY: Macmillan.
Grouws, D. A., Howald, C. L., & Colangelo, N. (1996, April). Student conceptions of
mathematics: A comparison of mathematically talented students and typical high school
algebra students. Paper presented at the Annual Meeting of the American Educational
Research Association, New York, NY. Retrieved from
http://www.eric.ed.gov/PDFS/ED395783.pdf
Grouws, D. A., Tarr, J. E., Chávez, Ó., Sears, R., Soria, V. M., & Taylan, R. D. (2013).
Curriculum and implementation effects on high school students' mathematics learning from
curricula representing subject-specific and integrated content organizations. Journal for
Research in Mathematics Education, 44, 416-463. doi: 10.5951/jresematheduc.44.2.0416
Guion, L. A., Diehl, D. C., & McDonald, D. (2011). Triangulation: Establishing the validity of
qualitative studies. (IFAS Extension FCS6014). Gainesville: University of Florida.
Harel, G. (2010). Commentary on On the theoretical, conceptual, and philosophical foundations
for mathematics education research: Timeless necessities. In B. Sriraman & L. D. English
160
(Eds.), Theories of mathematics education (pp. 87-94). Berlin, Germany: Springer-Verlag.
doi:10.1007/978-3-642-00742-2;
Harel, G., & Sowder, L. (1998). Students' proof schemes. In E. Dubinsky, A. Schoenfeld, & J.
Kaput (Eds.), Research on Collegiate Mathematics Education, Vol. III (pp. 234-283).
Providence, RI: American Mathematical Society
Hays, D. G., & Singh, A. A. (2011). Qualitative inquiry in clinical and educational settings.
New York, NY: The Guilford Press.
Heirdsfield, A. (2011). Teaching mental computation strategies in early mathematics. Young
Children, 66(2), 96-102. Retrieved from
http://search.proquest.com/docview/874262989?accountid=13158
Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K. C., Wearne, D., Murray, H, .… Human, P.
(1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth,
NH: Heinemann.
Hiebert, J., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students’
learning. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching
and learning (pp. 371–404) Charlotte, NC: Information Age.
Higgins, K. M. (1997). The effect of year-long instruction in mathematical problem solving on
middle-school students' attitudes, beliefs, and abilities. The Journal of Experimental
Education, 66, 5–28. doi:10.1080/00220979709601392
Hirsch, C. R., Coxford, A. F., Fey, J. T., & Schoen, H. L. (1995). Teaching sensible mathematics
in sense-making ways with the CPMP. The Mathematics Teacher, 88, 694-700. Retrieved
from http://search.proquest.com/docview/204613829?accountid=13158
161
Houston, K., Mather, G., Wood, L., Petocz, P., Reid, A., Harding, A., … Smith G. 2010). Is
there life after modelling? Student conceptions of mathematics. Mathematics Education
Research Journal, 22(2), 69–80. doi:10.1007/BF03217566
Inagaki, K., Hatano, G., & Morita, E. (1998). Construction of mathematical knowledge through
whole-class discussion. Learning and Instruction, 8, 503–526. doi:10.1016/S09594752(98)00032-2
Jacobs, J. K., Kawanaka, T., & Stigler, J. W. (1999). Integrating qualitative and quantitative
approaches to the analysis of video data on classroom teaching. International Journal of
Educational Research, 31(8), 717-724. doi: 10.1016/S0883-0355(99)00036-1
Dahlgren Johansson, A., & Sumpter, L. (2010). Children’s conceptions about mathematics and
mathematics education. In Kirsti Kislenko (Ed.), Current state of research on mathematical
beliefs XVI: Proceedings of the MAVI-16 Conference. Tallinn, Estonia.
Kaya, S., 2007. The influences of student views related to mathematics and self-regulated
learning on achievement of algebra I students. (Unpublished Doctoral dissertation). The
Ohio State University, Columbus, OH.
Kenyon, R. (1989). Writing is problem solving. In P. Connolloy & T. Vilardi (Eds.), Writing to
learn mathematics and science (pp. 73-87). New York, NY: Teachers College Press.
Kinach, B. M. (2002). Understanding and learning-to-explain by representing mathematics:
Epistemological dilemmas facing teacher educators in the secondary mathematics
“methods” course. Journal of Mathematics Teacher Education, 5, 153-186. doi:
10.1023/A:1015822104536
Kloosterman, P., & Stage, F. K. (1992). Measuring beliefs about mathematical problem solving.
School Science and Mathematics, 92, 109–115. doi:10.1111/j.1949-8594.1992.tb12154.x
162
Krueger, R. A. (2009). Focus groups: A practical guide for applied research. Thousand Oaks,
CA: SAGE Publications, Inc.
Krutetskii, V. (1976). The psychology of mathematical abilities in school children. Chicago, IL:
University of Chicago Press.
Lampert, M. (2001). Teaching problems and the problems of teaching. New Haven, CT: Yale
University Press.
Leder, G. C., & Forgasz, H. J. (2002). Two new instruments to probe attitudes about gender and
mathematics. Retrieved from ERIC database. (ED463312/TM033744)
Lee, Y. P. (2006, November). Measuring the impact of SAIL on students’ beliefs and teacher
classroom practices. Paper presented at APERA Conference, Hong Kong.
Leinwand, S. (2000). Sensible mathematics: A guide for school leaders. Portsmouth, NH:
Heinemann.
Lester, F. K., Jr. (2007). Second handbook of research on mathematics teaching and learning.
Charlotte, NC: Information Age.
Malmivuori, M., & Pehkonen, E. (1996), Mathematical Beliefs Behind School Performance. In
L. Puig and A. Guitiérrez (Eds), Proceeding of the 20th conference of the International
Group for the Psychology of Mathematics Education (volume 3, pp. 305-311). Valencia,
Spain.
Mason, L. (2003). High school students' beliefs about maths, mathematical problem solving, and
their achievement in maths: A cross-sectional study. Educational Psychology, 23, 73–85.
doi:10.1080/01443410303216
Mason, L., & Scrivani, L. (2004). Enhancing students' mathematical beliefs: An intervention
study. Learning and Instruction, 14, 153–176. doi:10.1016/j.learninstruc.2004.01.002
163
Merriam, S. B. (1998). Qualitative research and case study applications in education. San
Francisco, CA: Jossey-Bass.
Morgan, D. L., & Spanish, M. T. (1984). Focus groups: A new tool for qualitative research.
Qualitative Sociology, 7(3), 253–270. doi:10.1007/BF00987314.
Muis, K. R. (2004). Personal epistemology and mathematics: A critical review and synthesis of
research. Review of Educational Research, 74, 317–377. doi:10.3102/00346543074003317
Munby, H. (1982). The place of teachers' beliefs in research on teacher thinking and decision
making, and an alternative methodology. Instructional Science 11, 201–225.
doi:10.1007/BF00414280
Mura, R. (1993). Images of mathematics held by university teachers of mathematical sciences.
Educational Studies in Mathematics, 25, 375–385. doi:10.1007/BF01273907
Mura, R. (1995). Images of mathematics held by university teachers of mathematics education.
Educational Studies in Mathematics, 28, 385–399. doi:10.1007/BF01274080
National Council of Teachers of Mathematics. (2000). Principles and standards for school
mathematics. Reston, VA: Author.
National Governors Association Center for Best Practices, & Council of Chief State School
Officers. (2010). Common core state standards for mathematics. Washington, DC: Authors.
National Research Council. (2001). Adding it up: Helping children learn mathematics (J.
Kilpatrick, J. Swafford, & B. Findell, Eds.). Washington, D.C.: National Academy Press.
Op ’t Eynde, P., De Corte, E., & Verschaffel, L. (2002). Framing students' mathematics-related
beliefs. In A. J. Bishop (Managing Ed.), G. C. Leder, E. Pehkonen & G. Törner (Vol. Eds.),
Mathematics Education Library: Vol. 31. Beliefs: A hidden variable in mathematics
education? (pp. 13–37). Dordrecht, the Netherlands: Springer.
164
Pajares, M. F. (1992). Teachers' beliefs and educational research: Cleaning up a messy construct.
Review of Educational Research, 62, 307–332. doi:10.3102/00346543062003307
Pehkonen, E. (1995). Pupils’ view of mathematics: Initial report for an international comparison
project (Research Report 152). University of Helsinki. Department of Teacher Education..
Pehkonen, E. (2004). State-of-the-art in mathematical beliefs research. In M. Niss (Ed.),
Proceedings in the 10th International Congress on Mathematical Education. Roskilde,
Denmark: Roskilde University.
Petocz, P., Reid, A., Wood, L., Smith, G., Mather, G., Harding, A., … Perret, G. (2007).
Undergraduate students’ conceptions of mathematics: An international study. International
Journal of Science and Mathematics Education, 5, 439–459. doi:10.1007/s10763-0069059-2
Philipp, R. A. (2007). Mathematics teachers' beliefs and affect. In F. K. Lester Jr. (Ed.), Second
handbook of mathematics teaching and learning (pp. 257–318). Charlotte, NC: Information
Age Publishing.
Piburn, M. & Sawada, D. (2000). Reformed teaching observation protocol (RTOP): Reference
manual (ACEPT technical report no. INOO-3). Tempe: Arizona: Collaborative for
Excellence in the Preparation of Teachers (Eric Document Reproduction Service, ED 447
205).
Ponte, J. P. (1994). Mathematics teachers’ professional knowledge (plenary conference). In J. P.
Ponte & J. F. Matos (Eds.), Proceedings of the Eighteenth International Conference for the
Psychology of Mathematics Education (Vol. I, pp. 195–210), Lisbon, Portugal: University
of Lisbon.
165
Ponterotto, J. G. (2006). Brief note on the origins, evolution, and meaning of the qualitative
research concept “thick description”. The Qualitative Report, 11(3), 538-549. doi:
10.1177/0011000010388423
Presmeg, N. (2003). Beliefs about the nature of mathematics in the bridging of everyday and
school mathematical practices. In A. J. Bishop (Managing Ed.), G. C. Leder, E. Pehkonen
& G. Törner (Vol. Eds.), Mathematics Education Library: Vol. 31. Beliefs: A hidden
variable in mathematics education? (pp. 293–312) Dordrecht, the Netherlands: Springer.
doi:10.1007/0-306-47958-3_17
Raymond, A. (1997). Inconsistency between a beginning elementary school teacher’s
mathematics beliefs and teaching practices. Journal for Research in Mathematics
Education, 28, 550–576. doi:10.2307/749691
Reid, A., Petocz, P., Wood, L., & Dortins, E. (2003). Mathematics students' conceptions of
mathematics. New Zealand Journal of Mathematics, 32, 163–172.
http://nzjm.math.auckland.ac.nz/index.php/New_Zealand_Journal_of_Mathematics
Reusser, K. (2001, September). Bridging instruction to learning: Where we come from and
where we need to go. Paper presented at the Ninth Conference of the European Association
for Research on Learning and Instruction, Fribourg, Switzerland. Retrieved from
http://www.icme-organisers.dk/tsg28/Reusser/EARLI-KR-Keynote.pdf
Rich, P.J. & Hannafin, M. (2009). Video annotation tools: Technologies to scaffold, structure,
and transform teacher reflection. Journal of Teacher Education, 60(1), 52-67.
doi:10.1177/0022487108328486
Richards, J. (1991). Mathematical discussions. In E. von Glasersfeld (Ed.), Radical
constructivism in mathematics education (pp. 13–52). Dordrecht, the Netherlands: Kluwer.
166
Romberg, T. A., Carpenter, T. P., & Dremock, F. (2005). Understanding mathematics and
science matters. Mahwah, N.J.: Lawrence Erlbaum Associates.
Schoenfeld, A. H. (1989). Explorations of students' mathematical beliefs and behavior. Journal
for Research in Mathematics Education, 20, 338–355. doi:10.2307/749440
Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition,
and sense making in mathematics. In D. A. Grouws (Ed.), Handbook of research on
mathematics teaching and learning (pp. 334–370). Reston, VA: National Council of
Teachers of Mathematics.
Schoenfeld, A. H. (1994). Reflections on doing and teaching mathematics. In A. H. Schoenfeld
(Ed.) Mathematical thinking and problem solving (pp. 53-70). Hillsdale, N.J.: L. Erlbaum
Associates.
Schommer‐Aikins, M., Duell, O., & Hutter, R. (2005). Epistemological beliefs, mathematical
Problem‐Solving beliefs, and academic performance of middle school students. The
Elementary School Journal, 105, 289–304. doi:10.1086/428745
Stage, F. K., & Kloosterman, P. (1995). Gender, beliefs, and achievement in remedial collegelevel mathematics. Journal of Higher Education, 66, 294–311. doi:10.2307/2943893
Stein, M. K., Engle, R. A., Smith, M. S., & Hughes, E. K. (2008). Orchestrating productive
mathematical discussions: Five practices for helping teachers move beyond show and tell.
Mathematical Thinking and Learning, 10(4), 313-340. doi: 10.1080/10986060802229675
Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standardsbased mathematics instruction: A casebook for professional development. New York, NY:
Teachers College Press.
167
Steiner, L. A. (2007). The effect of personal and epistemological beliefs on performance in a
college developmental mathematics class (Unpublished Doctoral dissertation). Kansas State
University, Manhattan, KS.
Stodolsky, S. S. (1985). Telling math: Origins of math aversion and anxiety. Educational
Psychologist, 20, 125-133. doi:10.1207/s15326985ep2003_2
Sumpter, L. (2009). Upper secondary school students' gendered conceptions about mathematics.
Umeå, Sweden: Department of Mathematics and Mathematical Statistics, Umeå University.
Taylor, M. (2009). Changing students’ minds about mathematics: Examining short-term changes
in the beliefs of middle-school students. In S. L. Swards, D. W. Stinson, & S. LemonsSmith (Eds.), Proceedings of the 31st annual meeting of the North American Chapter of the
International Group for the Psychology of Mathematics Education. Atlanta, GA: Georgia
State University.
Telese, J. A. (1999). Mexican American high school students’ perceptions of mathematics and
mathematics teaching. Hispanic Journal of Behavioral Sciences, 21, 154–169.
doi:10.1177/0739986399212003
Thompson, A. G. (1992). Teachers' beliefs and conceptions: A synthesis of the research. In D.
A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 147–
164). Reston, VA: National Council of Teachers of Mathematics.
Verschaffel, L., De Corte, E., Lasure, S., Van Vaerenbergh, G., Bogaerts, H., & Ratinckx, E.
(1999). Learning to solve mathematical application problems: A design experiment with
fifth graders. Mathematical Thinking & Learning, 1, 195–229.
doi:10.1207/s15327833mtl0103_2
168
Wilkins, J. L. M., & Brand, B. R. (2004). Change in preservice teachers' beliefs: An evaluation
of a mathematics methods course. School Science and Mathematics, 104, 226–226.
doi:10.1111/j.1949-8594.2004.tb18245.x
Wilkins, J. L. M., & Ma, X. (2003). Modeling change in student attitude toward and beliefs
about mathematics. Journal of Educational Research, 97, 52–63.
doi:10.1080/00220670309596628
Wilson, M. S., & Lloyd, G. M. (2000). Sharing mathematical authority with students: The
challenge for high school teachers. Journal of Curriculum and Supervision, 15(2), 146-69.
Wong, N. (2002). Conceptions of doing and learning mathematics among Chinese. Journal of
Intercultural Studies, 23, 211–229. doi:10.1080/07256860220151096
Wong, N., Marton, F., Wong, K., & Lam, C. (2002). The lived space of mathematics learning.
The Journal of Mathematical Behavior, 21, 25–47. doi:10.1016/S0732-3123(02)00101-3
Yackel, E. (2001). Explanation, justification and argumentation in mathematics classrooms. In
M. Van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th conference of the
international group for the psychology of mathematics education PME-25, vol. 1, (pp. 1-9).
Utrecht (Olanda).
Yin, R. K. (2011). Qualitative research from start to finish. New York, NY: Guilford Press.
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Maureen M. Grady
PO Box 671, Milroy, PA 17063
717-667-9173, [email protected]
EDUCATION
The Pennsylvania State University, University Park, PA
PhD - August 2013, Emphasis in Mathematics Education, Advisor: Dr. Fran Arbaugh
Methodist Theological School in Ohio, Delaware, OH, Master’s of Divinity - 1990
Rhode Island College, Providence, RI, BA in Mathematics – 1985, Emphasis in Secondary Education
Cornell University, Ithaca, NY, Major: Computer Engineering
PROFESSIONAL EXPERIENCE
Teaching Assistant , 2010-2011, The Pennsylvania State University
Research Assistant, 2008-present, The Pennsylvania State University
Secondary Mathematics Teacher, 1997-2008
Adjunct Mathematics Instructor, 1996-2001, Springfield College School of Human Services
Secondary Mathematics Teacher, 1985-1987
PROFESSIONAL ACTIVITIES
• Reviewer for Journal of Research in Rural Education.
• Session recorder at the Conference on Mathematical Proficiency for Teaching - 2010 and 2009
PUBLICATIONS IN PEER REVIEWED CONFERENCE PROCEDINGS
• Heid, M.K., Karunakara, S., Kinol, D., Grady, M. (2009). The roles of processes in the personal and
classroom mathematics of a beginning secondary mathematics teacher. Proceedings of the conference of
the American Educational Research Association, San Diego, CA.
• Heid, M. K., Grady, M., Karunakaran, S., Jairam, A., Freeburn, B., & Lee, Y. (2012). A processes approach to
mathematical knowledge for teaching: The case of a beginning teacher. Proceedings of the 34th annual
meeting of the North American Chapter of the International Group for the Psychology of Mathematics
Education. Kalamazoo, MI: Western Michigan University.
PRESENTATIONS
• Grady, M. (2013, April). Students' conceptions of mathematics as sensible and related instructional
practices: Ramifications for teachers and teacher educators. Paper presented at the National Council of
Teachers of Mathematics Annual Meeting, Denver, CO.
• Grady, M. (2013, January). Students' conceptions of mathematics as sensible and related instructional
practices: Potential uses of indicators with pre-service mathematics teachers. Paper presented at the
Association of Mathematics Teacher Educators Annual Conference, Orlando, FL.
• Grady, M. (2012, October). Students' conceptions of mathematics as sensible and related instructional
practices: a report of initial research findings. Poster presented at the 34th annual meeting of the North
American Chapter of the International Group for the Psychology of Mathematics Education. Kalamazoo,
MI: Western Michigan University.
• Grady, M. (2012, May). Students' conceptions of mathematics as sensible and related instructional
practices: A status report on doctoral research. Presentation to the Pennsylvania Association of
Mathematics Teacher Educators. Shippensburg, PA: Shippensburg University.
• Heid, M.K., Grady, M., Karunakaran, S., Jairam, A., Freeburn, B., & Lee, Y. (2012, April). Influences on
mathematical process use by a novice teacher. Paper presented at the National Council of Teachers of
Mathematics Research Presession, Philadelphia, PA.
• Heid, M.K., Karunakara, S., Kinol, D., Grady, M. (2010). Factors Influencing the Role of Mathematical
Processes in a Beginning Secondary Teacher's Classroom Teaching. Paper presented at the conference of
the American Educational Research Association, New Orleans, LA
HONORS/AWARDS
• Graduate Fellowship, Mid-Atlantic Center for Mathematics Teaching and Learning, 2008.
• Awarded the Gindelsberger Memorial Award for Excellence in Biblical Scholarship, 1990.
• Awarded the John and Mary Alford Memorial Scholarship, 1987.