Cul
Copyright © 2012 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
ultivating
Algebraic
Representations
NAUTILUS_SHELL_STUDIOS/ISTOCKPHOTO.COM
Teachers learn to appreciate varied student
representations of the Flower Bed problem
after working through it on their own.
r
Leigh A. van den Kieboom and Marta T. Magiera
Reasoning and sense making are at the heart of
mathematics teaching and learning throughout
the K-12 mathematics curriculum (NCTM 1989,
1991, 2000). Focusing on these skills gives students
the opportunity to analyze, interpret, and critically
think about the mathematics they learn. Instead
of memorizing procedures, students make sense
of why procedures work; develop flexibility using
procedures to solve a variety of problems; analyze
the relative effectiveness of a procedure for a given
problem situation; and are able to make sense of, or
interpret, the validity of their solutions. Reasoning
and sense making are of paramount importance for
teaching mathematics with understanding.
Vol. 17, No. 6, February 2012
The terms reasoning and sense making are frequently used to describe the wide range of mathematical processes that relate to one’s ability to think
about and use mathematics meaningfully. Mathematical reasoning frequently denotes one’s ability
to draw conclusions on the basis of assumptions
and definitions. Mathematical sense making refers
to one’s ability to critically analyze concepts, situations, and contexts and draw connections among
them to other knowledge (NCTM 2000, 2009).
The two terms—mathematical reasoning and sense
making—are closely interrelated.
Principles and Standards for School Mathematics
(NCTM 2000) presents a compelling argument
●
MatheMatics teaching in the Middle school
353
why reasoning and sense making are
essential in providing K-12 students
with meaningful opportunities for
learning mathematics. How can
teachers help their students develop
the reasoning and sense-making
skills necessary to see mathematics
as a sense-making discipline? One
opportunity comes from actively
engaging students in problem solving, encouraging them to listen and
question one another (including the
teacher), and justify their mathematical work.
Exploring why aspects of one’s
mathematical point of view not only
helps students develop a deeper
understanding of mathematics but
also helps teachers make instructional decisions. Teachers are able
to cultivate their own reasoning and
sense making by analyzing the thinking processes that their students use
when solving problems. This article
explores a professional development
program in which middle school
teachers solve problems, assign the
same problems in their own classrooms, and analyze their students’
thinking.
Algebraic Thinking
For years, algebra has been viewed
as a “gatekeeper” (Edwards 2000,
p. 26) that prevents students from
accessing higher-level mathematics.
Introducing algebra-based ideas into
the K-8 mathematics curriculum and
focusing on algebraic thinking give
students the opportunity to examine
algebra-based concepts in the context
of arithmetic and affords them a
foundation for future mathematics. Cuoco, Goldenberg, and Mark
(1996) broadly defined algebraic
thinking as a useful way of thinking
about mathematical content. Driscoll
(1999, 2001) specified that the phrase
refers to thinking about quantitative
situations in ways that make relationships between variables obvious. He
354
Teachers often
reach out for ways
to foster and
orchestrate classroom
situations
in support of
middle school
students’ algebraic
thinking.
described algebraic thinking in terms
of three habits of mind:
1. Doing and undoing;
2. Building rules to represent
functions; and
3. Abstracting from computation.
Through these habits, Driscoll
emphasized thinking processes leading, for example, to recognizing and
analyzing patterns, investigating and
representing relationships, generalizing beyond specific examples, analyzing how processes or relationships
change, and developing arguments
for how and why rules and procedures work.
A successful implementation of
algebraic concepts at the K-8 level
strongly relates to teachers’ knowledge of mathematics and pedagogy.
Without a doubt, teachers need
to understand the mathematics
that they teach. However, teachers’
readiness for early algebra instruction
often resonates with their own experiences with a “traditional” approach
to algebra that emphasizes procedural understanding (e.g., knowing
what steps are needed to solve an
equation). These kinds of experiences
limit teachers’ flexibility in thinking
about algebraic concepts in a more
Mathematics Teaching in the Middle School
●
Vol. 17, No. 6, February 2012
developmental way. Thus, teachers
often reach out for ways to foster and
orchestrate classroom situations in
support of middle school students’
algebraic thinking.
Teachers use their pedagogical
knowledge to identify, analyze, and
interpret their students’ mathematical
thinking. Examining students’ mathematical thinking requires teachers
to use reasoning and sense making
to uncover students’ problem-solving
strategies and gain access to students’
understanding of mathematical ideas.
Making sense of others’ thinking
might not always be an easy task.
However, van Es and Sherin (2008)
have found that teachers’ ability to
examine students’ thinking improves
not only when teachers can identify
the ideas that students reveal in their
solutions but also when they use
these ideas as evidence to interpret
students’ mathematical thinking. The
work of Sherin and van Es creates a
strong argument for engaging teachers in reasoning and sense making
in the context of analyzing students’
mathematical thinking.
Cultivating Teachers’
Reasoning and
Sense Making
We use our work with middle school
teachers in a professional development
program to describe the processes
through which teachers foster their
own reasoning and sense-making
abilities. The program’s activities provide an opportunity for participants
to learn how to dig deeper into their
students’ solutions. They also examine
how their students think about and
develop an understanding of algebraic
ideas. The activities encourage teachers to identify, analyze, and interpret
students’ algebraic thinking. They also
allow teachers to think deeply about
the mathematics that their students
are learning. The cycle of professional
development activities includes—
1. solving and discussing solutions to
algebra-based tasks in collaboration with other teachers;
2. implementing the same problems
in the classroom;
3. conducting a mini case study
to analyze a selected student’s
algebraic thinking; and
4. engaging in discussions about
implementing algebra-based problems in the classroom and analyzing students’ thinking.
Before teachers can make sense of
students’ mathematical thinking in
the context of any given problem, they
themselves need to make sense of that
problem. Thus, the first step of the
cycle of activities asks teachers to solve
the problem on their own. They then
share and closely analyze their solutions by discussing different ways in
which they organized or represented
the problem information, engaged in
analyzing and predicting patterns, described an algebraic rule or procedure,
or critiqued the validity of arguments
made to justify a solution. The teachers
keep in mind the connections between
the arithmetic and algebraic concepts
found in the problems. From their
examination of solution paths, they
consider different reasoning approaches and ways of thinking that they can
anticipate from students.
The same problem is then assigned
to their middle school students, and
each teacher conducts a mini case
study. The teachers’ in-depth analysis
of a selected student’s solution focuses
on the reasoning and understanding of
the algebraic concepts embedded in the
task and an examination of the growth
of that student’s algebraic thinking
over time.
A subsequent element of the cycle
involves teachers’ discussions in a
professional development group of
what they learned about their students’ mathematical thinking overall,
and the selected case-study student,
in particular. This activity encourages
teachers to revisit the algebraic thinking addressed in the problem as they
consider the different and sometimes
unexpected solutions that students
provide. Discussing students’ algebraic
thinking fosters teachers’ reasoning
and sense making because, in a group,
they examine and sometimes re-examine their students’ solutions.
We use the Flower Beds problem (see fig. 1) to discuss each of the
events in the cycle in greater detail
in relationship to teachers’ reasoning
and sense making. When the teachers
solved this problem, they determined
that the council needs 402 slabs to
create 100 flower beds. They also
generated several formulas to find
the number of slabs needed for any
number of flower beds, including—
• 4n + 2;
• 6 + 4(n – 1); and
• 6(n) – (n – 1)(2).
different forms of algebraic thinking.
The city council wishes to
create 100 flower beds and
surround them with hexagonal
paving slabs according to the
pattern shown above. (In this
pattern, 18 slabs surround
4 flower beds.)
1. How many slabs will the
council need?
2. Find a formula that the
council can use to decide
the number of slabs needed
for any number of flower
beds.
Source: Shell Center for Mathematical
Education 1984, p. 64
As teachers discussed how to pose this
problem to their students, they used
their own thinking about the problem
to guide their conversations.
As they prepared for classroom
implementation, the teachers shared
ideas about how they could effectively launch the problem. One group
emphasized the understanding of
vocabulary. They discussed the possibility that some of their students
might have a limited understanding of
the real-world context of the problem (e.g., using words such as council,
paving, and slabs). Another group
of teachers expressed a concern that
some of their students might experience difficulty organizing the information in a way that was useful for
discovering the regularity. Teachers’
conversations led to a rich discussion
about how to scaffold students’ work
in a way that helps students identify
and describe the pattern and then use
that understanding to write a formula.
Vol. 17, No. 6, February 2012
Fig. 1 The Flower Beds problem elicited
●
Several teachers suggested guiding
students toward creating a table as a
way to organize the information and
recognize the change in the number of slabs needed to create each
additional flower bed. Other teachers suggested using pattern blocks
to engage the students in creating
the design to give them hands-on
experiences and help them discover
pattern regularity. In the context of
solving the problem and discussing
ways to implement it successfully
in the classroom, the teachers also
considered different strategies that
one might use to reason about the
problem. For example, they noticed
that a student might use figural
reasoning to examine the pattern by
focusing on the first full flower bed
(6 slabs) and then continuing the
pattern by adding four slabs to form
each consecutive bed (see fig. 2a).
MatheMatics teaching in the Middle school
355
Fig. 2 Different forms of thinking led to
Fig. 3 Erik’s discussion of his solution led to the algebraic representation 4(n – 1) + 6.
different algebraic rules for the Flower
Beds problem.
(a) 6 + 4(n – 1)
(b) 6n – 2(n – 1)
(c) 2 + 4n
Some teachers anticipated that their
students might focus on the two slabs
connecting each set of the two flower
beds (see fig. 2b). One teacher group
noticed that the students could reason
about the pattern by focusing on the
two initial blocks and then extending
the pattern by adding four blocks for
each consecutive bed (see fig. 2c).
The teachers discussed how these
different perspectives on the pattern
could influence the algebraic formulas their students generate. Eliciting
different ways of reasoning about the
pattern would be an important part
of helping them write a formula and
engaging students in creating mathematical arguments to validate the
different rules they might use.
Sharing their student case studies
at the subsequent professional development meeting, a group of teachers
discussed the algebraic thinking that
two seventh-grade students, Erik and
Rodney, revealed in their solutions.
The teachers discussed Erik’s
reasoning, which had not been antici356
Fig. 4 Rodney’s incorrect solution, involving proportion, led to a discussion about how
to steer students toward a correct solution.
pated, about numbers and operations
(100 – 1 = 99; 99 × 4 = 396; 396 + 6
= 402) (see fig. 3). They also shared
ideas about how they could help him
go beyond a closed set of numbers
and generalize the pattern to an
explicit formula. Erik represented his
thinking verbally as “total amount of
flower beds - 1; answer × 4; answer
MatheMatics teaching in the Middle school
●
Vol. 17, No. 6, February 2012
plus 6; total paving slabs needed.”
The teachers discussed how his verbal
representation could be linked to an
algebraic representation in the form of
a formula 4(n – 1) + 6.
Rodney’s solution (see fig. 4) provided the opportunity for the teachers
to think about the misconceptions
some students might demonstrate
in the context of this problem. One
teacher noted that, like Rodney, some
students might incorrectly reason proportionally while analyzing the Flower
Beds problem. Rodney’s reasoning was
based on a faulty assumption that
a ratio of 18 slabs to 6 flower beds
is the same as that of x slabs to
100 flower beds. The teachers not only
identified Rodney’s reasoning but also
agreed that asking Rodney to explain
his formula could be a good first step
in helping him to reanalyze his own
thinking and perhaps self-correct it.
Several teachers suggested other
pedagogical approaches focused on
helping Rodney reason about the problem and his solution. They suggested
modifying it so that a proportion
would yield a fraction number answer.
They discussed how asking Rodney to
solve the same problem with a different
number of flower beds could further
prompt his thinking about the pattern.
iMPlications oF
ProFessional develoPMent
We highlight how middle school
teachers, engaged in a professional
development program, cultivate their
own reasoning and sense making by
solving, discussing, implementing, and
analyzing students’ thinking about
algebra-based problems. One teacher
commented on the activities:
This [solving the problem alone and
with a group] helped me to think
through the problems in another way.
I think because I spent time solving
the problems first with the teachers
I learned several different ways to
think about the same problem.
Other teachers explained how this
work helped them focus on the thinking of their students:
I didn’t understand why we found
so many solutions to one problem.
I am more of the “I did it this way”
teacher. It was eye-opening to see
how other people made sense of the
same problem. This will help me
think about my students’ reasoning
and misconceptions.
This [experience] allowed me to
learn how to examine my students’
thinking. . . . In the future, I’ll look
not only to see if a student’s solution
is correct or incorrect but how the
student thought about the problem.
Reasoning and sense making are at
the heart of teaching and learning
mathematics with understanding. Developing these abilities is important for
both students and teachers. Considering a variety of solution approaches to
a given problem; anticipating student
thinking with a goal of planning effective classroom implementation; and
examining students’ thinking to identify and make sense of students’ strategies, conceptions, and misconceptions
when solving algebra-based problems
facilitates teachers’ reasoning and sense
making. Cultivating teachers’ reasoning and sense making through teachers’ professional development has the
potential to enhance the teaching and
the learning processes for students and
teachers alike.
reFerences
Cuoco, Al, Paul Goldenberg, and June
Mark. 1996. “Habits of Mind: An
Organizing Principle for Mathematics
Curriculum.” Journal of Mathematical
Behavior 15:375–402.
Driscoll, Mark. 1999. Fostering Algebraic
Thinking: A Guide for Teachers Grades
6-10. Portsmouth, NH: Heinemann.
———. 2001. The Fostering Algebraic
Thinking Toolkit: A Guide for Staff Development. Portsmouth, NH: Heinemann.
Edwards, Thomas. 2000. “Some Big Ideas
of Algebra in the Middle Grades.”
Mathematics Teaching in the Middle
School 6 (1): 26-31.
National Council of Teachers of Mathematics. 1989. Curriculum and EvaluVol. 17, No. 6, February 2012
●
ation Standards for School Mathematics.
Reston, VA: NCTM.
———. 1991. Professional Standards for
Teaching Mathematics. Reston, VA:
NCTM.
———. 2000. Principles and Standards for
School Mathematics. Reston, VA: NCTM.
———. 2009. Focus in High School Mathematics: Reasoning and Sense Making.
Reston, VA: NCTM.
Shell Centre for Mathematical Education. 1984. Problems with Patterns and
Numbers. Manchester, England: Joint
Matriculation Board.
van Es, Elizabeth A., and Miriam G.
Sherin. 2008. “Mathematics Teachers’
‘Learning to Notice’ in the Context of
a Video Club.” Teaching and Teacher
Education 24: 244-76.
The professional development program
described in this article was funded by
U.S. Department of Education Mathematics and Science Partnership Grant
No. 11-6300-MSP1. The instructional
practices discussed in this article are
not intended as an endorsement by the
U.S. Department of Education.
leigh a. van den
kieboom, leigh.vanden
[email protected],
is an assistant professor in
the College of Education
at Marquette University
in Milwaukee, Wisconsin.
She is interested in helping
teachers learn how to use
a collaborative problem-based approach to
teaching that focuses on developing student
thinking. Her research focuses on the
knowledge that practicing and preservice
teachers need. Marta t. Magiera, marta
[email protected], is an assistant
professor of mathematics education in the
Department of Mathematics, Statistics and
Computer Science at Marquette University.
Her research interests include mathematical thinking, metacognition, mathematical
problem solving, and mathematics teacher
development.
MatheMatics teaching in the Middle school
357