Running head: BEGINNING TEACHERS’ USE OF TASKS
1
Beginning Secondary Teacher’s Use of Tasks
to Support Equitable Spaces
Ayanna D. Perry
North Carolina State University
Author Note
Ayanna D. Perry, STEM Department, North Carolina State University.
Ayanna Perry is now at the Knowles Science Teaching Foundation. This research was a part of a
dissertation project (Perry, 2013) and was partially supported by the National Science
Foundation under Grants No. DUE-0733794 and DUE-1240003. Any opinions, findings, and
conclusions or recommendations expressed herein are those of the researcher and do not reflect
the views of the National Science Foundation.
Correspondence concerning this article should be addressed to Ayanna Perry, Knowles Science
Teaching Foundation, 1000 N. Church St., Moorestown, NJ 08057.
Contact: [email protected]
BEGINNING TEACHERS’ USE OF TASKS
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Abstract
This paper describes how tasks can be used to characterize equitable classroom spaces in early
career high school mathematics teachers’ accelerated and academic courses. Equitable classroom
spaces are observable parts of classroom instruction, resulting from interactions within a
community of learners, focused on particular content with normative ways of being aimed at
reducing the barriers for participation and engagement for all students. To investigate students’
learning opportunities, mathematics teachers’ tasks in three academic and three accelerated
classes were examined. Through a collective case study of these six high school mathematics
courses the researcher describes how teachers’ choice of homework, classwork, and assessment
tasks support students’ opportunity to learn (National Research Council, 2001). Task analysis
was conducted at the course level, which yielded six cases of tasks. Each task was numerically
coded using the Potential of the Task Rubric (Boston, 2012) and statistically analyzed to
investigate students’ opportunities to learn. Then, similarities in task collections showed that
students with persistent opportunities to learn frequently engage with high demand tasks on
homework, classwork, and assessments, while moderate and limited opportunities provide less
frequent opportunities to engage with these tasks. The features of task collections that support
students’ opportunities to learn described in this study may provide a framework for early career
teacher support and instruction of prospective teachers.
Keywords: equitable spaces, beginning teachers, mathematical tasks, task implementation
BEGINNING TEACHERS’ USE OF TASKS
3
Beginning Secondary Teacher’s Use of Tasks to Support Equitable Spaces
Mathematical task choice is an important feature of mathematics teachers’ classroom
practice. The tasks teachers choose to use with their students greatly influence students’ views of
what mathematics is, what it means to do mathematics, and what it means to be a doer of
mathematics (Hiebert et al., 1997). The Common Core State Standards for Mathematics
(CCSSM) assert “all students must have the opportunity to learn and meet the same high
standard if they are to access the knowledge and skills necessary in their post-school lives”
(National Governors Association Center for Best Practices & Council of Chief State School
Officers, 2010, p. 4). Therefore, failing to arm all students with the ability to reason and think
mathematically is an issue of equity, especially in a society where mathematics has been
conceptualized as a gatekeeper (Martin, Gholson, & Leonard, 2010; Moses & Cobb, 2001). This
paper describes how three early career mathematics teachers’ task choices supported students’
engagement with meaningful mathematics in both academic and accelerated courses.
Literature Review
Given that students benefit from engaging with tasks that connect procedural and
conceptual knowledge, provide opportunities to justify and explain thinking, and have multiple
approaches to gain a correct solution, beginning teachers need to choose these types of tasks for
instruction (Hiebert et al., 1997; Stein, Grover, & Henningsen, 1996; Stein & Smith, 1998). The
literature on choosing mathematical tasks is vast (e.g. Boston & Smith, 2009; Doyle & Carter,
1984; Hiebert et al., 1997) but is generally based on Smith and Stein’s (1998) conception of
cognitive demand of mathematical tasks. With respect to beginning teachers’ choice of tasks,
Borko and Livingston (1989) found that beginning teachers with strong content knowledge were
able to choose tasks that supported students’ learning of content but struggled to produce
BEGINNING TEACHERS’ USE OF TASKS
4
examples on the fly to aid students’ understanding. However, they attribute the inability to
produce examples on the spot to a lack of familiarity with the content and the amount of time
required to prepare for new class assignments. Though there is not much research on beginning
teachers’ choice of mathematical tasks, it is regarded as a high leverage practice in teaching
(TeachingWorks, 2012). This area of research is necessary because students who have access to
high cognitive demand tasks have a broader conception of mathematics and possibly more
facility with the subject (National Research Council, 2001).
A brief treatment of equity
The equity principle posed by NCTM speaks generally about equity as “high
expectations and strong support for all students” and references a specific goal that “all students,
regardless of their personal characteristics, backgrounds, or physical challenges, must have
opportunities to study- and support to learn- challenging mathematics (NCTM, 2000, para. 1).
Gutiérrez (2007) further elaborates on this notion of equity when she describes it as: “the
inability to predict mathematics achievement and participation of students based solely upon
student characteristics such as race, class, ethnicity, sex, beliefs, and proficiency in the dominant
language” (p. 2). In addition to the inability to predict students’ achievement based on physical
or social characteristics, Gutiérrez (2012) includes access to physical resources, measurable
outcomes of student learning, inclusion of students’ identities in curricula, and experiences of
social transformation as doers of mathematics as features of equity. Still others define equity as
“obliterating the differential and socially unjust outcomes in mathematics education” (Gutstein &
Secada, 2000, p. 26); “a fair distribution of opportunities to learn or opportunities to participate”
(Esmonde, 2009, p. 1010); and the opportunity to “participate substantially in all phases of
BEGINNING TEACHERS’ USE OF TASKS
5
mathematics lessons (e.g. individual work, small group work, whole class discussion), but not
necessarily in the same ways” (Jackson & Cobb, 2010, p. 3).
Regardless of how equity has been defined, a focus on equity within mathematics
education has resulted from evidence of bias or favoritism in education (DiME, 2007; LadsonBillings, 1995). Researchers have looked at creating equitable opportunities in instruction in
various ways including critical race theory, situated learning theory, and sociocultural theory
(Strutchens et al., 2011). Also, frameworks for teaching have been created to challenge
discriminatory ways of being (e.g. Dimensions of Learning, and culturally relevant
pedagogy)(Gutiérrez, 2007; Ladson-Billings, 1995). Though these approaches have had
successes, there are barriers to widespread implementation. Teaching practice based on these
theories can overwhelm early career mathematics teachers (ECMTs) and teachers lack the time
and resources to do the type of in depth cultural study of their students necessary to create
culturally relevant pedagogy (Sowder, 2007).
One framework that may support equitable classroom spaces is the Opportunity to Learn
framework. These frameworks highlight how common teacher practices (e.g. use of appropriate
tasks) can provide all students opportunities to learn (National Research Council, 2001). In this
paper, opportunity to learn refers to students’ opportunity to use prior knowledge when engaging
with high cognitive demand tasks, provide explanations and justifications during task
implementation, and connect procedural and conceptual knowledge while engaging with tasks.
Methods
The overarching question that guided this study was, “How do early career high school
mathematics teachers’ classroom practices support students’ opportunities to learn?”
The research question discussed in this brief report is:
BEGINNING TEACHERS’ USE OF TASKS
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1. How do the tasks that ECMTs choose support students’ opportunities to engage with high
cognitive demand tasks in class, outside of class, and on assessments?
a. How do the task choices of an ECMT differ when teaching academic versus
accelerated courses?
In this study, an accelerated course is a course described as honors or AP (advanced placement)
where the curriculum is theoretically more rigorous (as defined by detail, pace, etc.) than the
curriculum in an academic course. An academic course is a course described as college
preparatory or regular.
Study Design
An instrumental multi-case study (Creswell, 2007; Stake, 1995) was conducted to
investigate how three ECMTs support students’ opportunities to learn mathematics. Each
participant was observed teaching five lessons in each course. Each also participated in five
interviews about each course and submitted classroom artifacts such as lesson plans and assigned
tasks to the researcher. Data from classroom observations, field notes, individual interviews, and
mathematical tasks and assessments were collected and triangulated for analysis.
This study used a convenience sample that contained three teachers: Mr. Nimrick, Mrs.
Moreland, and Mrs. Kaiser. Each teacher was employed at a senior high school in a large, highneed southeastern school district, but no two teachers were teaching at the same school. Each
teacher was observed teaching in two classes at: one accelerated (e.g. AP Statistics or Honors
Algebra II) and one academic class (e.g. Geometry). Cases were defined at the class level
because “by comparing sites or cases, the researchers can establish the range of generality of a
finding or explanation and at the same time, pin down the conditions under which that finding
will occur” (Borman, Clarke, Cotner, & Lee, 2006, p. 123). This study contains six cases: Mr.
BEGINNING TEACHERS’ USE OF TASKS
7
Nimrick’s Class A, Mr. Nimrick’s Class B, Mrs. Moreland’s Class A, Mrs. Moreland’s Class B,
Mrs. Kaiser’s Class A, and Mrs. Kaiser’s Class B.
Data Analysis
Three types of coursework were analyzed in this study: homework, classwork, and
assessments. The IQA rubric for task potential was used to rank the three types of coursework:
classwork, homework, and assessment tasks (Boston & Smith, 2009). These ranks determined
the cognitive demand of the task. The ranks of the tasks show the opportunity for students to
reason mathematically. A rank of 0 indicates the absence of mathematics in a task. A rank of 1
indicates a recall or memorization task. A rank of 2 indicates mathematical tasks that require use
of procedures disconnected from conceptual reasoning. A rank of 3 indicates mathematical tasks
that may use procedures, but also could support complex thinking or reasoning. A rank of 4
indicates mathematical tasks that require student to devise and apply a unique solution strategy
or tasks that explicitly prompt for justification of procedures used to find solutions. The highest
potential of reasoning present in the task corresponds to the rank assigned (Boston & Smith,
2009). Chi-squared and Mann-U Whitney tests were used to look for significant differences in
the ranks of classroom tasks within and between cases (Jones & Tarr, 2007).
Findings
Case 1: Mr. Nimrick’s Class A
Mr. Nimrick’s Class A was a Geometry course with a total of 16 students. There were a
total of 20 homework tasks, 23 classwork tasks, and 17 assessment tasks assigned in Class A
during this study. Table 1 shows the number of high demand (ranks of 3-4) and low demand
(ranks of 1-2) tasks for each type of coursework. Students only received higher percentages of
high demand tasks when doing homework. Sixty-five percent of the homework tasks were high
BEGINNING TEACHERS’ USE OF TASKS
8
demand compared to 30% of the classwork tasks and 18% of the assessment tasks. Looking
across all tasks, 38% of all tasks were high cognitive demand. The chi-squared test was
performed to test whether the numbers of high and low cognitive demand tasks within the three
coursework types were significantly different (the alternative hypothesis). This test provided
evidence that the number of high demand tasks used across the three types of coursework is
significantly different (29.7, df=2, p.008). This suggests a lack of alignment among the
cognitive demand levels of homework, classwork, and assessment tasks.
Table 1
Cognitive demand of Nimrick's Class A Tasks
Type of
Coursework
Tasks
n (%)
Low Demand
High Demand
Total
Homework
7 (35%)
13 (65%)
20
Classwork
16 (70%)
7 (30%)
23
Assessment
14 (82%)
3 (18%)
17
Total
37 (62%)
23(38%)
60
Note: Percentages are calculated across the row and may not sum to 100 due to rounding.
Case 2: Mr. Nimrick’s Class B
Mr. Nimrick’s Class B is an Advanced Placement Statistics course with a total of 32
students. There were a total of 8 homework tasks, 11 classwork tasks, and 22 assessment tasks
assigned in Class B during this study. Table 2 shows the number of high demand and low
demand tasks for each type of coursework. Students engaged with higher percentages of high
demand tasks across all types of coursework. One hundred percent of the homework tasks were
high demand compared to 75% of the classwork tasks and 54% of the assessment tasks. Looking
collectively, 69% of all tasks were high cognitive demand. For this case, the chi-squared test did
BEGINNING TEACHERS’ USE OF TASKS
9
not show a significant difference in the number of high and low demand tasks used in Nimrick’s
Class B for homework, classwork, and assessments (26.0, df=2, p.05).
Table 2
Cognitive demand of Nimrick's Class B Tasks
Types of
Coursework
Tasks
n (%)
Low Demand
High Demand
Total
Homework
0 (0%)
8 (100%)
8
Classwork
3 (25%)
9 (75%)
12
Assessment
10 (45%)
12 (54%)
22
Total
13 (31%)
29 (69%)
42
Note: Percentages are calculated across the row and may not sum to 100 due to rounding.
Cross-Case Analysis of Mr. Nimrick’s Class A and Class B Tasks
To compare the overall distributions of Class A and Class B tasks, the Mann -Whitney U
test was used (Jones & Tarr, 2007). The null hypothesis is that the task ranks of Nimrick’s Class
A are the same as the task ranks of Mr. Nimrick’s Class B. Therefore, the alternative hypothesis
is that the task ranks of Mr. Nimrick’s Class A and Class B are different and specifically that the
ranks of the tasks used in Mr. Nimrick’s Class B are higher overall that the ranks of the tasks
used in his Class A, which implies higher cognitive demand of tasks used in Mr. Nimrick’s Class
B. There is evidence that Mr. Nimrick’s classes, Class A and Class B, have task distributions that
are significantly different (U=1982 Z-Score =-2.7334, p=0.00634). This implies that the
cognitive demand of the tasks used in Class B, AP Statistics, is higher than those used in Class
A, academic Geometry. This suggests that more of the tasks used in the accelerated course had
the potential to provide students opportunities to explain and justify their thinking and engage
with tasks that connected conceptual and procedural knowledge when compared to those used in
the academic course.
BEGINNING TEACHERS’ USE OF TASKS
10
Case 3: Mrs. Moreland’s Class A
Mrs. Moreland’s Class A is a Geometry course with a total of 26 students. There were a
total of 4 homework tasks, 20 classwork tasks, and 9 assessment tasks assigned in Class A during
this study. Table 3 shows the number of high demand and low demand tasks and items for each
type of coursework. Within the types of coursework, students only received high percentages of
high demand tasks during assessments. Sixty-six percent of the assessment tasks were high
demand compared to 25% of the homework tasks and 15% of the classwork tasks. There appears
to be alignment among the tasks based on the percentages, however the small number of tasks
prohibits using the Chi-squared test for tasks.
Table 3
Cognitive demand of Moreland's Class A Tasks
Type of
Coursework
Tasks
n (%)
Low Demand High Demand Total
Homework
3 (75%)
1 (25%)
4
Classwork
17 (85%)
3 (15%)
20
Assessment 3 (33%)
6 (66%)
9
Total
23 (70%)
10(30%)
33
Note: Percentages are calculated across the row and may not sum to 100 due to rounding.
Case 4: Mrs. Moreland’s Class B
Mrs. Moreland’s Class B is an AP Statistics course with a total of 32 students. There
were a total of 1 homework task, 9 classwork tasks, and 3 assessment tasks assigned in Class A
during this study. Table 4 shows the number of high demand and low demand tasks for each type
of coursework. These data were more difficult to analyze because of the small number of tasks
used in Class B during the study. Nevertheless within each type of coursework, students received
higher percentages of high demand tasks. One hundred percent of the homework and assessment
BEGINNING TEACHERS’ USE OF TASKS
11
tasks were high demand compared to 78% of the classwork tasks. Collectively, 85% of the all
tasks were high demand. Overall, the tasks in Mrs. Moreland’s Class B appeared aligned which
implies that students were provided with more opportunities to engage with high demand tasks
on across all task types.
Table 4
Cognitive demand of Moreland's Class B tasks
Type of
Coursework
Tasks
n (%)
Low Demand
High Demand
Total
Homework
0
1 (100%)
1
Classwork
2 (22%)
7 (78%)
9
Assessment 0
3 (100%)
3
Total
2 (15%)
11 (85%)
13
Note: Percentages are calculated across the row and may not sum to 100 due to rounding.
Cross-Case Analysis Mrs. Moreland’s Class A and Class B Cases
To compare the overall distributions of Class A and Class B tasks, the Mann-Whitney U
test was used. The null hypothesis for this test is that the rank of the tasks used in Mrs.
Moreland’s Class A have the same distribution as the ranks of the tasks used in Mrs. Moreland’s
Class B. The null hypothesis is rejected which provides evidence that Mrs. Moreland’s Class A,
Geometry, and Class B, AP Statistics, have task distributions that are significantly different
(U=3804, Z-Score =3.4702, p=0.00052). This implies that the cognitive demand of Class B tasks
is higher than the cognitive demand of Class A tasks. This also suggests that students in the
accelerated class received more opportunities to explain and justify their thinking based on the
potential of the mathematics tasks chosen for use in the course compared to the students in the
academic course, and that students in the accelerated course received more opportunities to
engage with tasks that connected conceptual and procedural knowledge.
BEGINNING TEACHERS’ USE OF TASKS
12
Case 5: Mrs. Kaiser’s Class A
Mrs. Kaiser’s Class A is a Geometry course with a total of 28 students. There were a total
of 10 homework tasks, 14 classwork tasks, and 3 assessment tasks assigned in Class A during
this study. Table 5 shows the number of high demand and low demand tasks and items for each
type of coursework. Students received higher percentages of high demand tasks in each type of
coursework. Seventy percent of the homework tasks, 71% of classwork tasks, and 66% of
assessment tasks were high demand. Looking across all tasks, 70% of all tasks were high
cognitive demand. In spite of the differences in high demand task percentages, the overall
demand percentages may indicate alignment among the tasks. The small number of tasks
prohibits using the Chi-squared test for tasks. Overall, the tasks in Mrs. Kaiser’s Class A
appeared aligned. Students were provided with similar opportunities to engage with high demand
tasks on homework, classwork, and assessment tasks and overall, 70% of the tasks were high
demand.
Table 5
Cognitive demand of Kaiser's Class A tasks
Type of
Coursework
Tasks
n (%)
Low Demand High Demand Total
Homework
3 (30%)
7 (70%)
10
Classwork
4 (29%)
10 (71%)
14
Assessment 1 (33%)
2 (66%)
3
Total
8 (30%)
19 (70%)
27
Note: Percentages are calculated across the row and may not sum to 100 due to rounding.
Case 6: Mrs. Kaiser’s Class B
Mrs. Kaiser’s Class B is an Honors Algebra II class with a total of 31 students. There
were a total of 9 homework tasks, 15 classwork tasks, and 7 assessment tasks assigned in Class B
BEGINNING TEACHERS’ USE OF TASKS
13
during this study. Table 6 shows the number of high demand and low demand tasks and items for
each task type. Within types of coursework, students received higher percentages of low demand
tasks. Seventy-eight percent of the homework tasks, 60% of classwork tasks, and 71% of
assessment tasks were low demand. Looking across all tasks, 68% of tasks were low cognitive
demand. Because the within coursework type percentages are close in number, the overall
demand percentages may indicate alignment among the tasks. The small number of tasks
prohibits the use of the Chi-squared test. Overall, the tasks in Mrs. Kaiser’s Class B appear to be
aligned. At least 60% of the tasks in each coursework type were low demand. The trend was also
evident in the overall percentages with 68% percent of all tasks being low demand, which means
that students generally received more opportunities to engage with tasks that required the use of
prescribed or single procedures.
Table 6
Cognitive demand of Kaiser's Class B tasks
Type of
Coursework
Tasks
n (%)
Low Demand High Demand
Total
Homework
7 (78%)
2 (22%)
9
Classwork
9 (60%)
6 (40%)
15
Assessment 5 (71%)
2 (29%)
7
Total
21(68%)
10 (32%)
31
Note: Percentages are calculated across the row and may not sum to 100 due to rounding.
Cross-Case Analysis of Mrs. Kaiser’s Class A and Class B Cases
To compare the overall distributions of Class A and Class B tasks, the Mann-Whitney U
test was used. Because a higher percentage of the tasks used in Mrs. Kaiser’s Class A were high
demand, the Mann Whitney U test was used to test whether the distribution of task ranks in Class
A and B were the same or if there was a significant difference in the ranks of the tasks between
BEGINNING TEACHERS’ USE OF TASKS
14
the two courses. There is evidence that Mrs. Kaiser’s Class A and Class B have task distributions
that are significantly different (U=214.5, Z-Score =-3.1722, p=0.00152), and based on the
percentages of high demand tasks computed for each course, the cognitive demand of tasks in
Class A, Geometry, is higher than the cognitive demand of tasks in Class B, Honors Algebra II.
Findings and Conclusion
Analysis of the mathematical tasks resulted in a descriptive opportunity to learn
continuum for tasks. The continuum shows classifies cases of tasks as limited, moderate,
moderately persistent, and persistent. The term limited refers to a lower tendency for students to
have opportunities to justify and explain thinking and engage with high cognitive demand tasks;
the classification of limited does not refer to student achievement or teachers’ ability. The
combinations of features that represent teacher success with supporting students’ opportunities to
learn are labeled moderately persistent and persistent. The term persistent refers to a higher
tendency for students to have opportunities to justify and explain thinking and engage with high
cognitive demand tasks. Further, a characterization of collections of tasks as a struggle to support
students’ learning opportunities may be a consequence of the particular lessons observed. Thus a
characterization only serves to illustrate what students’ opportunities to learn might be if a
particular collection of tasks embodies a teacher’s task choices over the course of an academic
year.
Cases were classified limited or moderate if students engaged with a higher percentage of
low cognitive demand tasks across all task types, while moderately persistent and persistent if a
BEGINNING TEACHERS’ USE OF TASKS
15
higher percentage of high cognitive demand tasks were used across all coursework types.
Learning goals were also used to classify cases and these analyses can be found in the full
dissertation (Perry, 2013).
The tasks used in this study were classified using this continuum. Based on the analysis
of tasks and alignment across coursework, Mr. Nimrick’s Class A, Mrs. Moreland’s Class A, and
Mrs. Kaiser’s Class B seem to represent struggles to support students’ opportunities to learn. In
each of these three collections of tasks, there was evidence that students in those classes had
some opportunities to engage in higher cognitive demand tasks. In Mrs. Kaiser’s Class B, the
majority of the tasks in different coursework types ranked low. Mr. Nimrick’s Class A had two
coursework types that were majority low demand and one that was typically high demand. Mrs.
Moreland’s Class A tasks also shared a similar profile. Two of the three coursework types had
more tasks ranked at a low demand and coursework types were not aligned. While Mrs. Kaiser’s
Class B tasks seem to represent very limited opportunities for students to engage in tasks that
provide higher cognitive demand, there were more moderate opportunities in Mr. Nimrick’s
Class A and Mrs. Moreland’s Class A.
Conversely, Mr. Nimrick’s Class B, Mrs. Moreland’s Class B, and Mrs. Kaiser’s Class A
seem to represent examples of successes to support students’ opportunities to learn. In all three
cases, the presence of higher demand tasks was more persistent. All of the coursework types in
Mr. Nimrick’s Class B had more tasks ranked high demand and were well aligned across
coursework. The tasks used in Mrs. Moreland’s Class B and Mrs. Kaiser’s Class A have similar
profiles.
The opportunity to learn continuum for tasks (Figure 1) can help teachers think
concretely about how the tasks they use support students’ opportunities to use prior knowledge,
BEGINNING TEACHERS’ USE OF TASKS
16
connect procedural and conceptual knowledge when engaging with high cognitive demand tasks,
and provide explanations and justifications while working on tasks. The issue here is frequency
of opportunities opposed to a gross lack of opportunities.
BEGINNING TEACHERS’ USE OF TASKS
17
Opportunity to Learn Con nuum- Tasks
Limited
Moderate
Moderately
Persistent
Persistent
Students engage with
a high percentage of
low demand tasks
within each type of
coursework.
Students engage with
a high percentage of
high demand tasks in
one type of
coursework. The other
types of course work
feature low demand
tasks.
Students engage with
a high percentage of
high demand tasks in
two types of
coursework. The other
features low demand
tasks.
Students engage with
a high percentage high
demand tasks within
each type of
coursework.
Types of coursework
are aligned.
Types of coursework
are not aligned.
Types of coursework
are not aligned.
Types of coursework
are aligned.
Learning goals and
tasks are procedural.
Tasks and goals are
aligned. Or learning
goals and tasks are not
aligned.
Learning goals are
both procedural and
conceptual. Tasks and
goals are aligned.
Learning goals are
both procedural and
conceptual. Tasks and
goals are aligned.
Learning goals are
both procedural and
conceptual. Tasks and
goals are aligned.
Figure 1: Opportunity to Learn Continuum for Tasks
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