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Comprehensive Journal of Educational Research Vol. 2(5), pp. 70 - 87, March 2014.
ISSN : 2312-9421
Copyright © 2014 Knowledgebase Publishers.
Research Article
Moving the Standards for Mathematical Practice
Beyond Bullet Points
Scott A. Courtney
Kent State University, School of Teaching, Learning and Curriculum Studies, 401 White Hall, P. O. Box 5190, Kent,
OH 44242-0001.
Accepted 14, February 2014
The movement to adopt the Common Core State Standards for Mathematics impacts not only
school districts and their teachers, but also university teacher preparation programs. In order to
productively implement and sustain the Common Core’s vision of developing mathematically
competent citizens, mathematics teacher preparation programs must support pre-service
teachers’ development of practical conceptions of the Standards for Mathematical Practice. In this
article, the researcher describes pre-service middle school mathematics teachers’ engagements
with activities designed to reveal their emerging conceptions of the mathematical practices.
Using both qualitative and quantitative analysis, those aspects of particular mathematical
practices that appeared to be most influential to participating teachers were identified. Finally, I
argue the need for more and clearer examples of how the mathematical practices might be
exhibited in written work to support teachers’ and students’ development of practical conceptions
of the Standards for Mathematical Practice.
Keywords: mathematics teacher education, Standards for Mathematical Practice, Common Core
State Standards for Mathematics, mathematics processes and proficiencies
INTRODUCTION
The national movement to adopt the Common Core State Standards for Mathematics (Common Core
State Standards Initiative [CCSSI], 2010) has situated school districts and their teachers in positions
primed for change and reform. Along with changes in mathematics content standards and their
progressions, come increased emphasis on mathematical processes and proficiencies—the Standards
for Mathematical Practice (CCSSI, 2010). Table 1 displays the alphanumeric identifier (or code) and title
for each of the eight mathematical practices. For a comprehensive description of each mathematical
practice see Appendix A.
The two consortia awarded federal grants to design the Common Core State Standards for Mathematics
(CCSSM) assessment systems —the Partnership for Assessment of Readiness for College and Careers
(PARCC) and the Smarter Balanced Assessment Consortium—have indicated their respective
assessments will include items and tasks requiring students to apply and connect mathematical content
with the mathematical practices. For example, PARCC (2012) assessments will ―include a mix of items,
including short- and extended-response items, performance-based tasks, and technology-enhanced
items… [designed to] measure student learning within and across various mathematical domains and
practices (p. 5-6). Therefore, providing K-12 students with opportunities to not only engage in problems,
tasks and activities that coherently connect content with the mathematical practices, but experiences at
Scott Courtney. 71
Table 1: Alphanumeric Identifier and Title of the Standards for Mathematical Practice
Mathematical Practices
MP.1
Make sense of problems and preserve in solving them
MP.2
Reason abstractly and quantitatively
MP.3
Construct viable arguments and critique the reasoning of others
MP.4
Model with mathematics
MP.5
Use appropriate tools strategically
MP.6
Attend to precision
MP.7
Look for and make use of structure
MP.8
Look for and express regularity in repeated reasoning
exhibiting evidence of such knowledge and habits of mind in their written work, will become increasingly
important as the PARCC and Smarter Balanced assessments are implemented.
BACKGROUND
Although research related specifically to the mathematical practices is limited, there exists a body of
research pertaining to those processes (NCTM, 2000) and proficiencies (NRC, 2001) that ground them.
Furthermore, there is a growing body of research (e.g., Riordan & Noyce, 2001; Senk & Thompson, 2003)
indicating that students in classrooms that utilize reform curricula (e.g., aligned to NCTM Standards) not
only perform as well on standardized achievement tests as their counterparts in more traditional
mathematics programs, but also outperform these same students on tests measuring conceptual
understanding, applications, and problem solving ability (Schoenfeld, 2007, p. 540). Such results suggest
curricula that focus on the development of powerful processes and proficiencies can positively impact
student achievement. Unfortunately, research also highlights teachers‘ difficulties in operationalizing and
providing students with opportunities to engage in these same processes and proficiencies (e.g., Jacobs,
et al., 2006; Weiss, Pasley, Smith, Banilower, & Heck, 2003).
Transition to the Common Core affects not only K-12 instruction, but also university teacher preparation
programs—programs that will produce the next generation of teachers charged with enacting and
sustaining Common Core‘s vision in their (future) classrooms. Such programs must provide pre-service
teachers with opportunities to experience, develop, and implement instruction and assessments meeting
the demands of CCSSM, and opportunities to reflect on the impact of such instruction on their own and
their (future) students‘ learning.
In this report, the researcher describe pre-service middle school (grades 4-9 licensure) mathematics
teachers‘ (henceforth referred to as PSTs) engagements with activities designed to reveal their emerging
conceptions of the Standards for Mathematical Practice. In addition, I identify those aspects of particular
mathematical practices that appeared to be most influential to participating teachers. Finally, the
researcher argues the need for more and clearer examples of how the mathematical practices might be
exhibited in written work to support teachers‘ (and their students‘) development of practical conceptions of
the Standards for Mathematical Practice.
My reasons for choosing to focus on what it means to exhibit engagement in the mathematical practices
(frequently referred to as MPs) in written work are threefold. Firstly, there is a robust line of research
(e.g., Carpenter & Fennema, 1992; Fennema, Carpenter, Franke, Levi, Jacobs, & Empson, 1996;
Gearhart & Saxe, 2004) indicating the positive impact to student learning of engaging teachers in
activities that focus on analyzing student thinking. Furthermore, The Mathematical Education of Teachers
II (Conference Board of the Mathematical Sciences [CBMS], 2012) asserts that teachers ―need to…be
able to identify instances [of the practices] in their own work on a particular problem and in children‘s
work, and to be able to think explicitly about when, where, and how these types of expertise would occur‖
(p. 50). Therefore, providing teachers with opportunities to focus on what it looks like to exhibit
engagement in the mathematical practices in written work—their own and their students—seems a
productive area of emphasis. Secondly, providing teachers with opportunities to focus on what
engagement in specific mathematical practices (individually and in combination) does and does not look
like in written work serves to complement, reinforce, and enhance their conceptions of what these same
practices look like during verbal classroom interactions. Finally, as indicated above, future PARCC and
Smarter Balanced assessments will include items and tasks requiring students to apply and connect
mathematical content with the mathematical practices, and will measure students‘ learning of the content
Scott Courtney. 72
and practice standards through non-verbal computer-based documentation. As such, (pre-service)
teachers will need to develop conceptions of the mathematical practices that support their capacity to
design and enact instruction and assessments that help students acquire and exhibit increasingly
sophisticated mathematical habits of mind during both verbal classroom interactions and in written work.
This report adds to emerging research into teachers‘ conceptions of the mathematical practices by
exploring the following research question: How do pre-service middle school (grades 4-9 licensure)
mathematics teachers conceptualize engaging in the mathematical practices and exhibiting such
engagement in written work?
METHODS
Participants
Sixteen middle school pre-service teachers enrolled in a middle childhood mathematics methods class
(involving content from grades 6-9) were asked to solve mathematics problems (via ―problem sets‖)
related to the six middle school Common Core content domains (e.g., Expressions and Equations). In
addition, the researcher requested that PSTs solve these problems in a manner aligned with how they
envisioned students might solve the problems and in a manner PSTs believed exhibits engagement in the
mathematical practices in their written work.
The problem set activities were designed and conducted to investigate pre-service teachers‘ emerging
conceptions of the Standards for Mathematical Practice and were conducted as much as teaching
experiments as they were mathematics methods course activities. Such an assertion does not indicate
the activities did not attempt to induce learning; rather, that activities were designed explicitly to create
contexts for which pre-service teachers would articulate and reflect on their conceptions of the
mathematical practices. More specifically, the problem set activities were designed to employ teaching as
a scientific tool—as a method for probing teachers' meanings and ways of thinking (Steffe & Thompson,
2000). As described by Steffe and Thompson (2000), ―A primary goal of the teacher in a teaching
experiment is to establish living models of students‘ mathematics‖ (p. 284). As such, the problem set
activities were designed to bring forth, establish, and explore the conceptual boundaries on teachers‘
ways and means of operating.
Problem Sets and Data
The majority of the problems were chosen from standards-based (i.e., aligned to NCTM or Common Core
Standards) sources, such as the Connected Mathematics Project (CMP2). Data for this report pertains to
two problem sets, one involving the domain of Statistics and Probability, the other Geometry, and
consisted of PSTs‘ written solutions, their choices for which mathematical practice(s) they believed the
problem involved, and what PSTs took as written evidence that any given practice was being engaged in.
At the time these problem sets were assigned, PSTs‘ main experiences with the mathematical practices,
in relation to the course, had involved supporting their conceptions of what the practices look like during
verbal classroom interactions. Such support included watching and discussing video of classroom
instruction correlated to the mathematical practices from the Noyce Foundation‘s Inside Mathematics
website. More specifically, these videos of classroom instruction (grades 6-8) were used to provide PSTs
with examples of ways teachers enact the mathematical practices in their classrooms.
Data Analysis
Analysis was both quantitative and qualitative. Since PSTs‘ identifications involved categorical data (the
eight mathematical practices), quantitative analysis consisted of summary statistics that focused on the
frequencies with which particular mathematical practices were chosen by PSTs on a given problem, a
problem set, or by a particular PST. Data were then examined in greater detail by looking for nuances
that might be missed through descriptive analysis alone. Qualitative analysis involved the identification of
emerging themes, in a manner consistent with grounded theory (Strauss & Corbin, 1998), related to the
descriptions PSTs provided for what they took as written evidence of any particular practice being
engaged in.
Scott Courtney. 73
RESULTS
Results will be divided into three sub-sections: (1) descriptions of the mathematical practices PSTs
identified as being associated with each problem on each problem set, (2) examination of potential
relationships between PSTs‘ conceptions of particular mathematical practices and specific problem
features, and (3) identification and examination of those aspects of particular mathematical practices that
appeared to be most influential to PSTs‘ conceptions.
PSTs’ Identified Mathematical Practices
For the Statistics and Probability Problem Set, only 12 (of 16) PSTs completed the assignment requesting
they solve the problem and identify the mathematical practices they believed students would engage in
and exhibit in their written work. One additional PST completed the assignment (for a total of 13
responses) for the Geometry Problem Set. Figures 1 and 2 (see below) and Appendix B provide samples
from each problem set.
Tables 2 and 3 (below) display those practices PSTs identified for each of the seven problems on the
two problem sets. Specifically, each table indicates PST by pseudonym, problem number (e.g., P1
indicates the first problem), and the mathematical practice(s) chosen. For example, Carrie indicated
Problem #3 of the Statistics and Probability Problem Set involved MP.1 (Make sense of problems and
persevere in solving them) and MP.6 (Attend to precision). A random name generator was used to
designate PSTs.
Table 2: Identified Mathematical Practices by PST and Problem for Statistics and Probability Problem Set
PST
Arthur
Benjamin
Carrie
Eleanor
P1
7
1, 6
4, 8
2, 4, 5, 6
P2
1
7
2, 6
1, 5, 8
P3
8
2
1, 6
1, 5, 6
P4
3
2
7
3, 5
P5
1
Gladys
Jose
Lillian
Melanie
Rolfe
Sara
Terry
1, 8
1, 3, 6
2, 4, 7
2
1, 3, 4, 5
1, 3, 4, 8
1, 2, 5, 6
6, 8
1, 3, 4, 5
2, 3
7
1, 2, 3, 5
1, 4, 6, 8
1, 3, 7
1, 8
1, 4, 5, 6
1, 4, 7
6
1, 2, 3, 5
1, 5, 6
1, 2, 4, 5
Wayne
1, 4, 5, 6,
7, 8
1, 5
1, 4, 5
5, 6
1
1, 2, 3, 5
1, 4, 5
1, 3, 4, 5,
6, 7
1, 4, 5, 6,
7, 8
1
1, 3, 4, 6,
7, 8
1, 5
1, 6
2, 4, 5, 7, 8
1
1, 3, 4, 5
1, 5, 6
1, 2, 5, 7
1, 4, 5, 7
1, 4, 5, 7
1, 4, 5, 7
P6
1, 3
1, 2
2
2, 3, 4, 7
P7
1, 3, 4, 5
4
8
4
1, 3, 4, 6
1, 8
3
1, 3, 4, 5
1, 6
1, 4, 5, 6
1, 8
1, 4, 6
1, 4, 6, 8
4
1, 4, 5, 6
1, 4, 5, 7
1, 5, 7, 8
In Tables 2 and 3, blank cells indicate that no practices were identified for that problem. Task-based
interviews with PSTs indicated such omissions were unintentional and not meant to suggest that no
practices were involved.
As illustrated in Tables 2 and 3, there was a reasonable degree of variability in the mathematical
practices identified among and within problems, and among and within PSTs for each problem set. There
was also a reasonable degree of variability among the combinations and numbers of practices identified.
For example, for Problem #2 of the Statistics and Probability Problem Set (Figure 1), the number of
mathematical practices identified ranged from one (Benjamin identified MP.7) to six (Terry identified MPs
1, 3, 4, 5, 6, and 7).
Scott Courtney. 74
Table 3: Identified Mathematical Practices by PST and Problem for Geometry Problem Set
PST
Arthur
Benjamin
Carrie
Eleanor
Geraldine
Gladys
Jose
Lillian
Melanie
Rolfe
Sara
Terry
Wayne
P1
1, 4
3, 7
1, 4, 5
1, 4, 7
2, 4
1, 2, 3, 4, 6
P2
4
1
4
2, 4, 5, 8
1, 4, 6
1
1, 3, 5, 6
P3
3, 7
2
4
1, 2, 5, 7
2, 4
6, 8
1, 2, 3, 6
P4
3, 7
6
4
2, 3, 5, 7
1, 3
1, 7
2, 3
P5
8
3
2
1, 3, 8
3
1, 7
1, 2, 3
1, 4, 6, 7
1, 2
1, 3, 4, 5, 6
1, 4, 6
1, 2, 3
1, 4, 5, 6
1, 4, 7
4
1, 3, 4, 5
1, 4, 5, 6
1, 2, 5, 6
1, 4, 5, 6, 7
6, 7
3
1, 2, 3
1, 4, 5, 6
1, 3, 7
1, 4, 5, 6
4, 6, 7, 8
1, 3
1, 3, 4, 5, 6
1, 3, 4, 5
1, 2, 3, 6, 7
1, 4, 5, 6, 7
6, 7
3, 8
1, 2, 3, 6
1, 3, 5, 6, 8
1, 2, 7
1, 5, 6, 8
P6
1, 3
4
4
1, 4, 5, 6
2, 4
1, 4, 5
1, 2, 3, 4,
5, 6
1, 2, 7
1, 2
1, 3, 4, 5, 6
1, 3, 6
1, 4
1, 4, 5, 6
P7
3
4
1, 4, 5, 6
2, 4, 7
1, 4, 5
1, 4, 5, 6
1, 3
1, 4
1, 3, 4, 5
1, 5, 6
1, 2, 3
1, 2, 4, 5
Figure 1. Average number of tornados per year (adapted from Lappan, Fey,
Fitzgerald, Friel, & Phillips, 2009b, p. 18). This figure illustrates Problem #1 from
the Statistics and Probability Problem Set.
Figure 2. Determine potential garage dimensions (adapted from
Lappan et al., 2009a, p. 26). This figure illustrates Problem #1 from
the Geometry Problem Set.
Across the fourteen problems, there was not a single instance where any individual mathematical practice
was chosen by all PSTs. The problem that received the closest to a unanimous selection was Problem
#5 of the Statistics and Probability Problem Set (Table 2; Appendix B) in which 10 of 11 PSTs chose
MP.1 (Make sense of problems and persevere in solving them); Benjamin failed to identify any
mathematical practices for this problem. In fact, there was very little identical mathematical practice
Scott Courtney.75
Figure 3. Carrie‘s solution. This figure illustrates Carrie‘s solution to Problem #1 from the
Geometry Problem Set.
Figure 4. Eleanor‘s solution. This figure illustrates Eleanor‘s solution
to Problem #1 from the Geometry Problem Set.
identifications across all fourteen problems; for example, Jose and Sara each identified MPs 1, 4, and 5
for Problem #5 of the Statistics and Probability Problem Set (Table 2; Appendix B).
Problem #1 of the Geometry Problem Set (Figure 2) provides a specific example illustrating the degree of
variability exhibited on a single problem. Although Carrie and Eleanor derived similar solutions to this
problem, Carrie identified the problem as involving MPs 3 and 7, whereas Eleanor identified MPs 1, 4,
and 5. Such disparate identifications suggest the capricious nature of these pre-service teachers‘
conceptions of the mathematical practices.
Comparing these PSTs‘ written work shows that although Carrie (Figure 3) revealed more of her
reasoning for part (a), and provided a more complete description for why particular garage dimensions
will or will not work in part (b) than does Eleanor (Figure 4), Carrie gives no indication for what she took
as evidence that MPs 3 and 7 had been engaged in. In addition, although Eleanor‘s solution (Figure 4) to
part (b) indicates what she considered as appropriate dimensions, her explanation refers to suitable
―angles…for storing a car,‖ rather than focusing on which rectangular dimensions could enclose a car yet
still allow for the car doors to open. Furthermore, Eleanor indicates that she engaged in MP.5 (Use
appropriate tools strategically) in working toward a solution, but does not articulate what tool(s) she
utilized (or anticipates students might utilize) in solving the problem or how she decided which tool
afforded the most usefulness. Unfortunately, time constraints did not allow for me to conduct follow-up
Scott Courtney.76
interviews to probe Eleanor‘s meaning for the phrase ―suitable angles‖ or request that she identify the tool
she utilized to solve the problem.
Although such variation in results might be expected, considering the potential for idiosyncratic
interpretations of the mathematical practices, the interaction and overlap amongst practices (PARCC,
Figure 5. Strength of Mathematical Practice Pairs. This figure illustrates
a network graph displaying the strength of pairs of mathematical
practices for the Statistics and Probability and Geometry Problems Sets
combined (generated with Gephi, www.gephi.org)
2012, p.13), and the limited opportunities these PSTs had to discuss and operationalize the practices, my
intent was to gather data with which to develop a baseline for PSTs‘ interpretations of the mathematical
practices. Such a baseline would then serve to guide future engagements with these and other teachers.
One particular interesting result involved the frequency with which the mathematical practice pairs MP 4
and 5, MP 4 and 6, and MP 5 and 6 were identified. Figure 5 illustrates the frequency (the weight of the
connection; thicker lines indicate greater frequency) with which pairs of MPs were chosen by PSTs for the
two problem sets combined. For example, MPs 4 and 5 occur together in each of the following
combinations: MPs 1, 4, and 5, and MPs 4, 5, 7, and 8.
As Figure 5 indicates, other than certain practice pairs that included MP.1 (e.g., MPs 1 and 5), the pairs
MP 4 and 5, MP 4 and 6, and MP 5 and 6 occurred with the greatest frequency. This result also held true
when each problem set was examined individually. The frequency with which MP 4 and 5 occurred might
be accounted for in light of their relationship (modeling and using tools) in McCallum‘s (2011) higher order
structure to the practice standards. Although this does not provide insight into why the pair MP 4 and 5
(43 occurrences) was identified significantly more frequently than the pair MP2 and 3 (16 occurrences)—
reasoning and explaining (McCallum, 2011).
PSTs’ Mathematical Practice Conceptions and Problem Features
In order to attempt to account for PSTs‘ mathematical practice identifications, the researcher looked for
potential relationships between the problems‘ features, such as a realistic context, and those practices
identified. The main reason for doing so was due to PSTs‘ limited experiences with the mathematical
practices. As such, the researcher anticipated much of PSTs‘ decision making would be based on what
they deemed as being relevant between a problem‘s features and the mathematical practice descriptions
provided in Common Core documents (see Appendix A).
Particular problem features included: whether the problem asked for an explanation, involved a realistic
context, asked students to critique another‘s reasoning or justify their own assertions, included a
mathematical representation (e.g., diagram, table, graph, formula) or object (e.g., triangle), or requested a
Scott Courtney.77
mathematical representation or object be constructed. For example, Problem #2 of the Statistics and
Probability Problem Set described earlier (Figure 1) involves a realistic context (i.e., the average number
of tornadoes per year for several states) and includes a mathematical representation (i.e., bar graph).
Although the data are categorical, determining the standard score of the frequency with which each
mathematical practice was chosen (i.e., the number of standard deviations from the mean that each
practice was identified) provides a way to quantify whether certain practices occurred more or less often
depending on the features of the problem. For example, Problem #7 (denoted P7) from the Statistics and
Table 4: The Frequency with which Mathematical Practices were Chosen by PSTs for the Critique/Justify Problem Feature
MP1
MP2
MP3
MP4
MP5
MP6
MP7
MP8
Mean
SD
Problem Feature – Requests Rep
SPPS: P7
GPS: P1
GPS: P2
7
10
9
0
4
2
1
4
2
8
9
9
4
3
6
4
5
5
1
3
2
4
0
1
Total
26
6
7
26
13
14
6
5
12.875
8.758
Table 5: Standard Score for Mathematical Practices by Problem Feature
MP1
MP2
MP3
MP4
MP5
MP6
MP7
MP8
Explain
SPPS: P1, P3,
P4
GPS: P1, P2,
P3, P4, P6
1.78
-0.44
-0.13
0.89
0
0.13
-0.70
-1.52
Standard Score (SS)
Context
Critique/Justify
SPPS: P1, P2,
SPPS: P6
P4, P6, P7
GPS: P4, P5
GPS: P1, P6, P7
1.80
-0.56
-0.39
1.04
0.27
-0.12
-0.88
-1.16
1.65
-0.34
1.47
-0.52
-0.70
-0.16
-0.34
-1.06
Includes Rep
SPPS: P1, P2,
P3, P4, P5
GPS: P3, P4,
P5, P7
2.12
-0.57
0.03
0.03
0.37
-0.10
-0.57
-1.31
Requests Rep
SPPS: P7
GPS: P1, P2
1.50
-0.79
-0.67
1.50
0.01
0.13
-0.79
-0.90
Probability Problems Set (denoted by SPPS; see Appendix B) and Problems #1 and 2 (P1 and P2) from
the Geometry Problem Set (denoted by GPS; see Figure 2 and Appendix B, respectively) all requested
that teachers construct a representation or object as part of their solution. For these three problems
combined, the average frequency, the average number of times any individual practice was chosen by a
Scott Courtney.78
PST was 12.875 with standard deviation of 8.758. Table 4 illustrates the frequency with which each
mathematical practice was identified for the three problems.
Therefore, for the two problems sets together, the standard score for MP.1 being identified with problems
26−12.875
requesting the construction of a representation or object was SS MP.1 =
= 1.50. Similarly, the
8.758
standard score for MP.5 being identified with problems requesting the construction of a representation or
13−12.875
object was SS MP.5 =
= 0.01.The notation ―SS‖ is used rather than ―z‖ to stress the fact that
8.758
we are not discussing z-scores in the sense of a normal distribution, which is statistically inappropriate for
categorical data.
Table 5 displays the problem features, the problems associated with each feature, and the standard score
(SS) for each mathematical practice.
As Table 5 illustrates, problems involving a realistic context, identified by ―Context,‖ were frequently
associated with MP.4 (Model with mathematics) being identified (SS = 1.04). This was anticipated
considering MP.4‘s description, ―Mathematically proficient students can apply the mathematics they know
to solve problems arising in everyday life, society, and the workplace‖ (CCSSI, 2010, p. 7; Appendix A).
Problems‘ explicitly requesting the learner (i.e., pre-service teacher) to critique another‘s reasoning—a
fictitious character making assertions as part of the problem—or justify their own assertions, identified by
―Critique/Justify,‖ were frequently associated with MP.3 being identified (SS = 1.47). This result was also
expected due to the nature of MP.3 – Construct viable arguments and critiques the reasoning of other
(Table 1; Appendix A).
Problems requiring PSTs to explain their work, their thinking, or their reasoning (identified by ―Explain‖)
were frequently associated with MP.4 being identified (SS = 0.89). This result was surprising, since the
researcher anticipated such problems would motivate PSTs to choose MP.3, ―construct viable arguments‖
and-or MP.6, ―communicate precisely to others‖ (CCSSI, 2010, p. 7; Appendix A). The standard score for
MP.4 was significantly larger for such ―Explain‖ problems in the Geometry Problem Set (SS = 1.08) than
for the Statistics and Probability Problem Set (SS = 0.21). Problems meeting only the ―Explain‖ criteria,
and not also the ―Context‖ criteria—P3 from the Statistics and Probability Problem Set, P2 (Appendix B),
P3, and P4 (Appendix B) from the Geometry Problem Set—were still associated with MP.4 (SS = 0.56),
although less frequently.
Looking at the problem sets together, problems including a mathematical representation or object were
frequently associated only with MP.1 (SS = 2.12). For the Statistics and Probability Problem Set alone,
including a representation (e.g., table, graph) was frequently associated with MP.5, use appropriate tools
strategically (SS = 0.80), suggesting PSTs conceived such representations as tools. Alternatively, for the
Geometry Problem Set, including a representation or object (e.g., a triangle), even for problems not also
asking for an explanation, critique, or justification, was associated with MP.3 (SS = 1.01). The one
geometry problem that met this criteria (P5, Appendix B) asks PSTs to determine whether given pairs of
triangles are congruent—to ―use stated assumptions, definitions, and previously established results in
constructing arguments‖ (CCSSI, 2010, p. 6; Appendix A)—but does not explicitly ask that PSTs
articulate their thinking and reasoning.
Problems requesting PSTs create or construct a mathematical representation (e.g., table, chart) or object
were frequently associated with MP.4 being chosen (SS = 1.50), suggesting PSTs might have focused on
MP.4‘s statement, ―Mathematically proficient students…are able to…map their relationships using such
tools as diagrams, two-way tables, graphs, flowcharts and formulas‖ (CCSSI, 2010, p. 7; Appendix A).
Furthermore, ―Requests Rep‖ was the only problem feature involving a practice other than MP.1 being
identified most frequently (MP.4 and MP.1 each had SS = 1.50), although such ―Requests Rep‖ problems
were nearly a subset of ―Context‖ problems (see Table 5).
Although these findings lack generalizability due to the small number of problems and PSTs involved,
they do suggest that particular problem features influenced PSTs‘ identifications. In addition, these
results fail to make explicit which aspects of any particular feature PSTs focused on. More specifically,
which parts of any particular mathematical practice description were most influential in PSTs‘ choices?
This limitation will be addressed below.
Scott Courtney. 79
Figure 7. Most Influential Bullets for MP.1. This figure indicates those
aspects of MP.1 that PSTs indicated as most relevant to the problems.
Characterizing PSTs’ Conceptions of the Mathematical Practices
As part of the assignment for each problem set, PSTs were asked to articulate (in writing) where and how
they believed a written response might exhibit their identified practice(s) being employed.
For PSTs, engagement in MP.1 was associated with employing or using a given or created mathematical
representation as part of the solution process. Specific instances included: ―When I created [the] tree
Figure 8. Most Influential Bullets for MP.4 This figure indicates those aspects
of MP.4 that PSTs indicated as most relevant to the problems.
diagram to help me come up with the different combinations‖ (Sara) and, ―Occurs by understanding the
box plot info and using it to solve the problem‖ (Gladys). Therefore, if we look at MP.1 in bulleted form
(Figure 7, previous page), the highlighted bullets appear to have been most influential for PSTs.
Furthermore, PSTs‘ responses suggest they were constrained to articulate how any particular
Scott Courtney.80
representation helped them to make sense of the problem and what sense they actually made.
Rolfe, who chose MP.1 for all 14 problems, indicated, ―The first thing…all students have to do…is to
make sense of problems and persevere in solving them. If a student cannot do this they have little to no
chance of solving the problem.‖ Such a description suggests that for any problem as long as a student
arrives at a solution MP.1 has occurred. Such an interpretation is misaligned to that promoted in recent
CCSSI (2012) documents asserting, ―MP.1 does not say, ‗Solve problems‘…or ‗Make sense of problems
and solve them‘‖ (p. 14); rather, it refers to students building ―their perseverance in grade-levelappropriate ways by occasionally solving problems that require them to persevere to a solution beyond
the point when they would like to give up‖ (CCSSI, 2012, p. 14).
Engagement in MP.4 was associated with using, creating, or interpreting some form of mathematical
representation. Specific instances included: ―Student must make a tree diagram to find all possible
combinations‖ (Arthur) and, ―When I drew my hexagons to explain my answers‖ (Sara). If we look at MP.4
in bulleted form (Figure 8), the highlighted bullets appear to have been most impactful for PSTs. Such a
focus, as suggested by PSTs, emphasizes the tool—the representation or object (tree diagram,
hexagon)—rather than the relationship between the situation and the representation, and the conclusions
they derived from analysis of such relationships.
Engagement in MP.5 (Use appropriate tools strategically) was also associated with using, reading, or
interpreting mathematical representations, which helps account for the frequency with which MP.4 and
MP.5 were chosen in concert (Tables 2 and 3, Figure 5). Specific instances included: ―When I used the
graph to conclude my answers (Gladys)‖ and, ―Read and manipulate grid and picture to help you solve
the problem‖ (Rolfe). If we look at MP.5 in bulleted form (Figure 9), the highlighted bullet appears to have
been most impactful for PSTs. As with MP.4, for PSTs, the focus appears to be on the tool
(representation or object), and more specifically, the fact that there was a tool, rather than a focus on how
the tool was strategically employed to support their attempts to make sense of and work toward a solution
to the problem.
Finally, engagement in MP.6 (Attend to precision) was associated with knowing and using mathematical
definitions, terms and symbols, and working with units. Specific instances included: ―When I used clear
definitions‖ (Sara) and ―When dealing with units‖ (Benjamin). If we look at MP.6 in bulleted form (Figure
10), the highlighted bullets appear to have been most influential to PSTs. Although PSTs were requested
to articulate in writing not only where they believed their written response indicated engagement with the
MPs, but also how their response exhibited such engagement, PSTs‘ focus fails to make explicit: (1)
which definitions and meanings they employed, (2) how and why they made their choices regarding the
definitions and meanings they employed, and (3) what about the units needed to be dealt with?
Although the information provided here helps to identify which practice aspects (bullet points) PSTs
focused on as they made their decisions, PSTs‘ limited descriptions suggest they were constrained to
articulate their meanings, thinking, and reasoning. More specifically, PSTs appeared constrained to make
explicit the role the mathematical practices played or could play in their (or their students) attempts to
Figure 9. Most Influential Bullet for MP.5 This figure indicates those aspects of MP.5
that PSTs indicated as most relevant to the problems.
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Figure 10. Most Influential Bullets for MP.6. This figure indicates those aspects of MP.6
that PSTs indicated as most relevant to the problems.
make sense of and work toward solutions to the problems. For example, when Gladys states MP.5 was
exhibited, ―When I used the graph to conclude my answers,‖ she fails to indicate what it was about the
graph and how it was interpreted (or used) that supported her ability to make sense of the problem, or the
potential for someone to not engage in and exhibit MP.5, but still arrive at a solution. Furthermore,
although they were requested to do so, PSTs were constrained to explicate what they would take as
written evidence that specific mathematical practices were or had been engaged in. More specifically,
PSTs focused on identifying where engagement might have occurred, rather than on how such
occurrence might be revealed in their written work and how they might manage such occurrences in
others‘ learning.
Although PSTs‘ focus on specific statements, or practice components that comprise the descriptions of
each mathematical practice (e.g., the bullet points) seems a natural first step toward developing more
robust conceptions, such a focus does little to support their students in developing more sophisticated
processes and proficiencies ―as they grow in mathematical maturity and expertise throughout the
elementary, middle and high school years‖ (CCSSI, p. 8) and has the potential to degenerate the ―bullet
points‖ into checklists. The question remains, then, as to how to support PSTs in developing more robust
conceptions.
CONCLUSIONS AND RECOMMENDATIONS
In this report, PSTs‘ engagements were described with activities designed to reveal their emerging
conceptions of the Standards for Mathematical Practice. In addition, it was described how analyses
suggest specific problem features and particular practice aspects influenced PSTs‘ conceptions. Next, I
would like to move the discussion toward addressing the question of how to move PSTs‘ conceptions of
the mathematical practices beyond the bullet points. That is, how to provide PSTs with opportunities to
develop conceptions that support teachers‘ abilities to manage their students‘ development of the
―varieties of expertise described by the practice standards‖ (PARCC, 2012, p. 6).
The researcher addresses this question by returning to my reasons for engaging PSTs in the task of
identifying the mathematical practices they believed students might engage in and reveal as they worked
to solve mathematics problems. As indicated earlier, one reason involved the implication that under both
PARCC and Smarter Balanced assessment systems students will be assessed on their ability to
demonstrate engagement in the mathematical practices. Such implications are reinforced by
supplementary, consortia-approved documents. For example, Educational Testing Service‘s Coming
Together to Raise Achievement: New Assessments for the Common Core State Standards (ETS, 2012)
asserts, ―Teachers and administrators need to recognize that the mathematical practices are standards,
i.e., students are expected to develop proficiency in them, and practices will be assessed along with the
content standards‖ (p. 10).
Such assertions point to the necessity for students to demonstrate engagement in the mathematical
practices in their written work. As such teachers at all levels will not only need clear images for what such
engagement looks like (and what it looks like to not engage in the practices), but also how similar images
Scott Courtney. 82
can be developed in their students. As indicated in the Mathematical Education of Teachers II (2012), ―To
help their students achieve the CCSS Standards for Mathematical Practice, teachers must not only
understand the practices of the discipline, but how these practices can occur in school mathematics and
be acquired by students‖ (p. 24).
Results from this study suggest that to support PSTs (and potentially in-service teachers) in developing
such robust conceptions, and thereby authentically implement the Common Core Standards for
Mathematical Practice, mathematics teacher education and professional development should provide
teachers with opportunities to engage in activities and discussions centered around teachers‘ conceptions
for how the mathematical practices can be applied and exhibited in the context of solving mathematics
problems and tasks. Furthermore, such activities and discussions must move teachers to articulate how
they conceive of the mathematical practices as being engaged in (by themselves and students) and what
they take as evidence of such engagement, both verbally (during classroom interactions) and in written
work.
To support mathematics teacher education and professional development in providing teachers with
opportunities to engage in such activities and discussions requires resources that explicate the
mathematical practices beyond generalized descriptors or bullet points. Although current resources (both
consortia and non-consortia) provide ever-increasing examples of common core aligned problems, tasks,
and activities of varying levels of cognitive demand (e.g., Illustrative Mathematics, Mathematics
Assessment Project) the significant majority fail to move beyond simply identifying those practices
associated with each task—if they identify the associated practices at all.
In addition, the limited resources that do focus on providing evidence for what engagement in specific
practices or mathematical practice combinations looks like in the context of solving problems and tasks
focus on such engagement during verbal classroom interactions (e.g., Inside Mathematics). Missing from
such interactions is a complimentary focus on how students might demonstrate such engagement in their
written work—a necessary practice that students will need to develop as Common Core assessments are
implemented.
If teachers (and students) conceptualize the mathematical practices as merely implicit to finding solutions
to problems or tasks, then little reason is seen for teachers‘ (or students‘) mathematical practice
conceptions to move beyond bullet points. Alternatively, if the mathematical practices are what constitute
doing mathematics (i.e., content is engaged in via the practices), then the focus must shift to what it
actually looks like, both verbally and in written work, to engage in (or not) individual practices and practice
combinations in the context of working on problems and tasks.
Future research must explore how best to support teachers‘ development of robust conceptions of the
mathematical practices—conceptions supporting teachers‘ capacity to manage the development of
significant mathematical habits of mind in their students. Resources providing clear indications for how
engagement with the practices might look in written work are a positive first step and would be a muchwelcomed tool to not only support such research, but also mathematics teacher education and
professional development.
REFERENCES
Carpenter TP, Fennema E (1992). Cognitively guided instruction: Building on the knowledge of students and
teachers. Int. J. of Educ. Res. 17: 457-470.
Common Core State Standards Initiative (CCSSI) (2010). Common Core State Standards for Mathematics.
Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School
Officers.
Common Core State Standards Initiative (CCSSI) (2012). K–8 publishers‘ criteria for the Common Core State
Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the
Council of Chief State School Officers.
Conference Board of the Mathematical Sciences (CBMS) (2012). The mathematical education of teachers II.
Providence, RI: American Mathematical Society and Mathematical Association of America.
Educational Testing Service (ETS) (2012). Coming together to raise achievement: New assessment for the Common
Core State Standards. Retrieved from http://www.k12center.org/rsc/pdf/Coming_Together_April_2012_Final.PDF
Fennema E, Carpenter TP, Franke ML, Levi L, Jacobs VA, Empson SB (1996). A longitudinal study of learning to use
children‘s thinking in mathematics instruction. J. for Res. in Math Educ. 27(4): 403-434.
Gearhart M, Saxe GB (2004). When teachers know what students know: Integrating Mathematics Assessment.
Theory into practice. 43(4): 304-313.
Scott Courtney. 83
Jacobs J, Hiebert J, Givvin K, Hollingsworth H, Garnier H, Wearne D (2006). Does eighth-grade mathematics
teaching in the United States align with the NCTM Standards? Results from the TIMSS 1995 and 1999 Video
Studies. J. for Res. in Math Educ. 37(1): 5-32.
Lappan G, Fey JT, Fitzgerald WM, Friel SN, Phillips ED (2009a). Covering and surrounding. Upper Saddle River, NJ:
Prentice Hall.
Lappan G, Fey JT, Fitzgerald WM, Friel SN, Phillips ED (2009b). How likely is it? Upper Saddle River, NJ: Prentice
Hall.
Lappan G, Fey JT, Fitzgerald WM, Friel SN, Phillips ED (2009c). Kaleidoscopes, hubcaps, and mirrors. Upper Saddle
River, NJ: Prentice Hall.
Lappan G, Fey JT, Fitzgerald WM, Friel SN, Phillips ED (2009d). What do you expect? Upper Saddle River, NJ:
Prentice Hall.
McCallum
B
(2011).
Structuring
the
mathematical
practices.
Retrieved
from:
http://commoncoretools.me/2011/03/10/structuring-the-mathematical-practices/
National Council of Teachers of Mathematics (NCTM) (2000). Principles and standards for school mathematics.
Reston, VA: NCTM.
National Research Council (NRC) (2001). Adding it up: Helping children learn mathematics. Washington, DC: The
National Academies Press.
Partnership for Assessment of Readiness for College and Careers (PARCC) (2012). PARCC model content
frameworks for mathematics: Grades 3-11. PARCC.
Riordan JE, Noyce PE (2001). The impact of two standards-based mathematics curricula on student achievement in
Massachusetts. J. for Res. in Math Educ. 32(4): 368-398.
Schoenfeld AH (2007). Problem solving in the United States, 1970–2008: Research and theory, practice and politics.
ZDM – The Int. J. on Math Educ. 39(5–6): 537–551.
Senk S, Thompson D (Eds.) (2003). Standards-oriented school mathematics curricula: What does the research say
about student outcomes? Mahwah, NJ: Erlbaum.
Steffe LP, Thompson PW (2000). Teaching experiment methodology: Underlying principles and essential elements.
In R. Lesh & A. E. Kelly (Eds.), Research design in mathematics and science education (pp. 267- 307). Mahwah,
NJ: Lawrence Erlbaum Associates.
Strauss AL, Corbin JM (1998). Basics of qualitative research: Techniques procedures for developing grounded
theory. Thousand Oaks, CA: Sage Publications.
Weiss IR, Pasley JD, Smith PS, Banilower ER, Heck DJ (2003). Looking inside the classroom: a study of K-12
mathematics and science education in the United States. Chapel Hill, NC: Horizon Research, Inc.
APPENDIX A
Standards for Mathematical Practice (CCSSI, 2010, p. 6-8)
MP.1 Make sense of problems and preserver in solving them
Mathematically proficient students start by explaining to themselves the meaning of the problem and
looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They
make conjectures about the form and the meaning of the solution and plan a solution pathway rather than
simply jumping into a solution attempt. They consider analogous problems and try special cases and
simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate
their progress and change course if necessary. Older students might deepen on the context of the
problem, transform algebraic expressions or change the viewing window on their graphing calculator to
get the information they need. Mathematically proficient students can explain correspondence between
equations, verbal descriptions, tables, and graphs or draw diagrams of important features and
relationships, graph data, and search for regularity or trends. Younger students might rely on using
concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient
students check their answers to problems using a different method, and they continually ask themselves,
―Does this make sense?‖ They can understand the approaches of others to solving complex problems
and identify correspondences between different approaches.
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MP.2 Reason abstractly and quantitatively
Mathematically proficient students make sense of quantities and their relationships in problem situations.
They bring two complementary abilities to 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. Quantitative reasoning entails habits of creating a coherent
representing of the problem at hand; considering the units involved; attending to the meaning of
quantities, not just how to compute them; and knowing and flexibly using different properties of operations
and objects.
MP.3 Construct viable arguments and critique the work of others
Mathematically proficient students understand and use stated assumptions, definitions, and previously
established results in constructing arguments. They make conjectures and build a logical progression of
statements to explore the truth of their conjectures. They are able to analyze situations by breaking them
into cases, and can recognize and use counterexamples. They justify their conclusions, communicate
them to others, and respond to the arguments of others. They reason inductively about data, making
plausible arguments that take into account the context from which the data arose. Mathematically
proficient students are also able to compare the effectiveness of two plausible arguments, distinguish
correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what
it is. Elementary students can construct arguments using concrete referents such as objects, drawings,
diagrams, and actions. Such arguments can make sense and be correct, even though they are not
generalized or made formal until later grades. Later, students learn to determine domains to which an
argument applies. Students at all grades can listen or read the arguments of others, decide whether they
make sense, and ask useful questions to clarify or improve the arguments.
MP.4 Model with mathematics
Mathematically proficient students can apply the mathematics they know to solve problems arising in
everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition
equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a
school event or analyze a problem in the community. By high school, a student might use geometry to
solve a design problem or use a function to describe how one quantity of interest depends on another.
Mathematically proficient students who can apply what they know are comfortable making assumptions
and approximations to simplify a complicated situation, realizing that these may need revision later. They
are able to identify important quantities in a practical situation and map their relationships using such
tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships
mathematically to draw conclusions. They routinely interpret their mathematical results in the context of
the situation and reflect on whether the results make sense, possibly improving the model if it has not
served its purpose.
MP.5 Use appropriate tools strategically
Mathematically proficient students consider the available tools when solving a mathematical problem.
These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a
spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient
students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions
about when each of these tools might be helpful, recognizing both the insight to be gained and their
limitations. For example, mathematically proficient high school students analyze graphs of functions and
solutions generated using a graphing calculator. They detect possible errors by strategically using
estimation and other mathematical knowledge. When making mathematical models, they know that
technology can enable them to visualize the results of varying assumptions, explore consequences, and
compare predictions with data. Mathematically proficient students at various grade levels are able to
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identify relevant external mathematical resources, such as digital content located on a website, and use
them to pose or solve problems. They are able to use technological tools to explore and deepen their
understanding of concepts.
MP.6 Attend to precision
Mathematically proficient students try to communicate precisely to others. They try to use clear definitions
in discussion with others and in their own reasoning. They state the meaning of the symbols they choose,
including using the equal sign consistently and appropriately. They are careful about specifying units of
measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate
accurately and efficiently, express numerical answers with a degree of precision appropriate for the
problem context. In the elementary grades, students give carefully formulated explanations to each other.
By the time they reach high school they have learned to examine claims and make explicit use of
definitions.
MP.7 Look for and Make Use of Structure
Mathematically proficient students look closely to discern a pattern or structure. Young students, for
example, might notice that three and seven more is the same amount as seven and three more, or they
may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7
× 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property.
2
In the expression x + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize
the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary
line for solving problems. They also can step back for an overview and shift perspective. They can see
complicated things, such as some algebraic expressions, as single objects or as being composed of
2
several objects. For example, they can see 5 – 3(x – y) as 5 minus a positive number times a square and
use that to realize that its value cannot be more than 5 for any real numbers x and y.
MP.8 Look for and express regularity in repeated reasoning
Mathematically proficient students notice if calculations are repeated, and look both for general methods
and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating
the same calculations over and over again, and conclude they have a repeating decimal. By paying
attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2)
with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity
2
3
2
in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x + x + 1), and (x – 1)(x + x + x + 1)
might lead them to the general formula for the sum of a geometric series. As they work to solve a
problem, mathematically proficient students maintain oversight of the process, while attending to the
details. They continually evaluate the reasonableness of their intermediate results.
APPENDIX B
Sample Problems from the Statistics and Probability Problem Set
(5) Jamal spins the spinner shown below several times. The table shown below displays the results of his
spins. Adapted from Lappan et al (2009d, p. 15)
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a) How many times did Jamal spin the spinner?
b) Determine the percent of Jamal‘s spins that landed in the blue region. What percent of Jamal‘s
spins landed in the red region?
c) Theoretically, what percent of Jamal‘s spins are expected to land in the blue region over the long
run? What percent are expected to land in the red region?
d) Compare the experimental probability of the spinner landing in each of the two regions with the
theoretical probabilities. Explain any differences in these probabilities.
(7) Monica is designing a birthday card for her little sister. She has one sheet each of orange, yellow,
pink, and purple paper. Monica also has a black, a red, and a green marker. Suppose that she chooses
one sheet of paper and one marker at random to make the card. Adapted from Lappan et al (2009b, p.
30)
a) Make a tree diagram to find all of the possible card color combinations.
b) What is the probability that Monica chooses purple paper and a red marker?
c) What is the probability that Monica chooses pink paper? What is the probability that she does not
choose yellow paper?
d) What is the probability that she chooses a green marker?
Sample Problems from the Geometry Problem Set
(2) For each of the descriptions given below, attempt to draw two triangles that are not congruent. If you
cannot draw two such triangles explain why. Adapted from Lappan et al (2009a, p. 47)
a) The triangles each have a base of 8 cm and a height of 5 cm. If two such non-congruent triangles
can be drawn, do they have the same area?
b) The triangles have an area of 18 square centimeters. If two such non-congruent triangles can be
drawn, do they have the same perimeter?
c) The triangles each have sides of length 6cm, 8 cm, and 10 cm. If two such non-congruent
triangles can be drawn, do they have the same area?
(4) Answer the following questions involving triangle ABC shown below. Adapted from Lappan et al
(2009c, p. 60)
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a) Frank states that ΔAOC and ΔAOB are congruent. Is Frank correct? Explain why or why not.
b) Identify all equal lengths and all congruent angles. Justify your statements.
(5) For each pair of triangles shown below, determine whether the two triangles are congruent. Use only
the information provided. Adapted from Lappan et al (2009c, p. 70)