Curricular Activity Systems: Creating Dynamic Learning Communities
Susan J. Courey, Ph.D.
Pam LePage, Ph.D.
Jody R. Siker, M.S.
San Francisco State University
United States
scourey@sfsu.edu
plepage@sfsu.edu
jrsiker@yahoo.com
Abstract: The purpose of this study was to examine how special education teachers help credential
candidates move away from procedural thinking to actively making meaning in the mathematics
classroom. Teaching and making meaning of difficult mathematics concepts can be influenced by
the interaction of technological, pedagogical and content knowledge TPACK. Special Education
credential candidates learn to make meaning through a communal classroom activity system in
which the university instructor creates zones of proximal development where candidates begin to
make meaning. Dynabook, a web-based tool to help college faculty, including special education
faculty, instruct teacher candidates how to teach proportional reasoning to children in middle
school was the foundation for this communal classroom activity system.
Introduction
The purpose of this study was to examine how special education teachers make the transition from
procedural thinking to making meaning in the mathematics classroom. Teaching and making meaning of difficult
mathematics concepts can be influenced by the interaction of technological, pedagogical and content knowledge
(TPACK; Mishra & Koehler, 2006). Special Education credential candidates learn to make meaning through a
communal classroom activity system in which the university instructor creates zones of proximal development
where candidates begin to make meaning. Dynabook, a web-based tool to help college faculty, including special
education faculty, instruct teacher candidates how to teach proportional reasoning to children in middle school was
the foundation for this communal classroom activity system. Dynabook provides a multimedia toolkit for
instructors, giving them access to videos, interactive problems, definitions, activities and other interesting and useful
tools for instructing, discussing, and modeling how to teach proportional reasoning to children.
Theoretical Framework
Boyd & Bargerhuff (2009) completed an extensive literature review exploring research that intersects middle school
mathematics with special education. They claim that in special education, college faculty are still instructing preservice teachers to teach children to solve problems procedurally; in mathematics education, college faculty are
working with pre-service teachers in student-centered, constructivist ways to teach children to solve problems more
conceptually. These researchers admit that learning procedures in mathematics is important, but that special
education teachers tend to focus on procedures too much, in part because special education methodology is more
likely to emphasize task analysis and specific, measurable objectives, often targeting procedural rather than
conceptual skills. The propensity of special education to use these approaches is not surprising, given some of the
common characteristics of students with disabilities, who often struggle in areas such as short-term memory, visual
and auditory processing, and executive functions. Furthermore, past research in the field of special education has
demonstrated more effective outcomes for students with disabilities when teacher-directed instruction is used
(Kroesbergen & Van Luit, 2003).
Those who write about special education teachers and mathematics instruction also make some of the same
conclusions that are found in research related to mathematics instruction in general education, namely that it is
important for teachers to not only know mathematics content, but they also need to know mathematics content
pedagogy (Griffen, Jitendra, & League, 2009). They need to provide children with opportunities to elaborate their
ideas or to make their reasoning explicit through the use of why and how questions, and by using visuals and
interactive materials to help make abstract concepts more concrete.
Boyd & Bargerhuff (2009) claim that very little has been written about how mathematics teachers are specifically
prepared to work with students with disabilities, and how special education teachers are prepared to teach
mathematics. They suggest teacher preparation programs provide a mathematics methods and intervention course
that includes both a general and special education focus. As pre-service mathematics and special education teacher
candidates develop their understanding of mathematics content, and explore teaching tools and strategies for
teaching this content, they should also consider the accommodations and other interventions students with learning
differences require to support them in mastering challenging content. Boyd & Bargerhuff discuss teaching about
proportional reasoning, a vital mathematics concept during the later middle school years that lays crucial
groundwork for algebra. They suggest that candidates must enrich their own understanding of proportional
reasoning beyond the over-simplified notion of cross-multiplication. While cross-multiplication is important and
valuable, this narrow view provides an insufficient basis for later algebra learning. Instruction and activity to enrich
a math teacher’s understanding of proportional reasoning might be typical in a mathematics methods course but the
consideration of students with disabilities is generally given short shrift. Conversely, the need of special learners is
the focus a special education methods course, but not specific to teaching proportional reasoning or other specific
mathematics content (Boyd & Bargerhuff, 2009).
To learn about proportional reasoning more meaningfully, teacher candidates need to build connections
through a coherent learning progression with adequate support for the affective challenges of maintaining interest
and engagement (Stein, Engle, Smith, & Hughes, 2008). Engagement with meaningful mathematical ideas depends
on the kinds of tasks candidates are given (Schoenfeld, 1985), the tools and representations they are able to use
(Sfard & McClain, 2002), and available supports when they get stuck – and of course on the pedagogical talent of
their instructor. Technology can support new, more engaging tasks and better tools and representations, and can
provide layered supports when students get stuck. Technology cannot substitute for good pedagogy, but it can
encourage and support good pedagogy.
Teachers need to learn to work with new forms of curricular materials, such as digital books. For example,
researchers increasingly focus on “educative curricular materials” – those materials that also educate teachers as
teachers use them with students (Davis & Krajcik, 2005; Remillard, 2005). Emerging technological advances
combined with Shulman’s (1987) work on pedagogical content knowledge (PCK) have lead to the technological
pedagogical and content knowledge (TPACK) framework (Mishra & Koehler, 2006). TPACK is a form of
knowledge that represents how a teacher utilizes the dynamic interplay of technology, pedagogical skills, and
content knowledge to represent concepts in different ways to engage learners. It represents the corpus of knowledge
that an expert teacher utilizes to create an effective learning environment. Knowing what makes concepts difficult to
comprehend and understanding how students struggle to formulate ideas and understanding, an expert teacher
designs a classroom activity system that integrates technology to build on existing knowledge and foster greater
understanding (Mishra & Koehler, 2006).
Method
Participants. A purposive sample of 20 students was recruited from the Mild to Moderate Special
Education Program at a California State University in Fall of 2011. The students who participated were enrolled in
one of two advanced Special Education (SPED) curriculum classes.
Design. Researchers used a mixed-methods design to explore how special education teachers developed
knowledge in the area of mathematics in an advanced curriculum class focused on preparing teacher candidates to
teach junior high children with learning differences in math and reading. It is difficult for researchers to make a
connection between what teachers learn in preparation programs to what children learn in classrooms. A chain of
evidence is needed to establish that certain kinds of content knowledge support useful practices and that those
practices support more positive outcomes for children (Darling-Hammond et al., 2005). For this research project,
we planned to implement research methods that provided links for this chain of evidence. First, we recognize that
teachers must gain knowledge and develop competencies. Second, they must translate their knowledge and
competencies into effective practices. To this end, we collected the following types of data: pre- and post-surveys,
pre- and posttest of pedagogical content knowledge, and video observations of class sessions.
Instruction. Thirteen pre-service and intern special education teacher candidates participated in two threehour classes dedicated to interacting with the ratio section of Proportional Dynabook. Participants had varying levels
of mathematics proficiency and teaching experience. Over two class periods, teacher candidates navigated the
Dynabook starting with an activity that introduced the Ratio section and aspects of UDL. After becoming familiar
with Dynabook, the candidates worked on solving a ratio word problem, discussed the various solutions, watched
instructional videos related to the shifts in proportional reasoning (Khoury, 2002; Labato, Ellis, Charles, & Zbiek,
2010; Lamon, 1999), watched video of a student incorrectly solving the problem, and composed scripts to correct
the student’s misconceptions.
Research Questions. 1) Will the Dynabook help teachers’ learn mathematics in the area of proportional
reasoning? 2) How do teachers learn and develop knowledge about proportionality by using the Dynabook tool?
Results. Early into the first class session, teacher candidates were reluctant to discuss math reasoning and
evasive when asked discussion questions. By the end of the Dynabook sessions, they were sharing their
mathematical thinking by discussing their own solutions to problems. In addition, they showed an increased
understanding of student thinking by writing scripts to address a student’s misconceptions. Teacher candidates were
surveyed about their attitudes toward teaching proportionality and familiarity with terms such as TPACK. They
showed increases in self-efficacy for teaching ratio conceptually to struggling learners. They were also more
consistently confident with addressing the Common Core standards for teaching ratio, such as generalizing from
patterns and making sense of word problems. Teacher candidates reported an increased understanding of, and
familiarity with TPACK.
Conclusion. Initially credential candidates were only interested in learning the one “best” way to teach
ratio and requested a video of a teacher explaining ratio. After working with the Dynabook and creating scripts to
teach a student with misconceptions, they were all willing and enthusiastic to discuss the mathematical thinking
behind the videos. However, participants did not internalize the idea that there are many ways to explain ratio
problems and they would need conceptual understanding to reach students who may need multiple means of
instruction. After utilizing Dynabook in a well-designed curricular activity system, teacher candidates were
reintroduced to concepts such as ratio and were better able to recognize and understand the math content. Many of
them did not realize how much they had forgotten since they originally learned about proportionality in junior high
and high school. The candidates realized that they needed to go back to this curriculum and review and remember
what they learned in those grades. Following their use of the Dynabook, they were able to talk more precisely about
ratio and how to assess students’ understanding of ratio. They were more confident in their ability to teach the
subject. After a relatively quick review, candidates were able to discuss their solutions to ratio problems and
analyze other perspectives that they may not have considered. Moreover, the candidates were enthusiastic during
the discussions, often carrying them over into breaks and after class. They reported increases in understanding
TPACK and felt more confident teaching ratio because TPACK was well explained in the Dynabook. Also, the
Dynabook and pedagogical activities helped them understand TPACK and think about their ability to solve
proportionality problems. In addition, they grew increasingly able to recognize misconceptions in children and
develop ways to address those misconceptions.
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