The purpose of this study is to investigate preservice elementary teachers’ choice of examples on comparison of fractions and how they sequenced these examples to promote student understanding. Examples play a crucial role in learning mathematics especially in concepts formation (Vinner, 2011). Thus, choosing appropriate examples is important to scaffold students’ learning of a mathematics concept (Skemp, 1971). However, teachers may not be aware of the significance of examples, whose misuse may even contribute to misconceptions. This attention to how to select and sequence examples is of greater import in preservice programs to facilitate prospective teachers to be cognizant of student thinking. This session addresses the issue of supporting preservice elementary teachers’ development of PCK through a series of activities where they need to choose and sequence examples to be presented to their students around the concept of comparison of fractions. Specifically, we examine what mathematical knowledge is entailed in their choices and what their perceptions are on selecting appropriate examples. Furthermore, this study also assesses whether providing opportunity to examine the choices and sequences of examples support preservice elementary teachers’ development of mathematics knowledge for teaching and focus their attention on student thinking.

Shulman (1986) defines pedagogical content knowledge PCK as comprising “… the most powerful analogies, illustrations, examples, explanations, and demonstrations” (p. 9). Ball, Thames, and Phelps (2008) further refine this notion of PCK when they examine the mathematical tasks of teaching, which includes “… finding an example to make a specific mathematical point” (p. 400). The task of teaching mathematics often involves considering what numbers are strategic to use in an example. For instance, when introducing the coordinate system, using the ordered pair (1, 1) as an initial example is not useful because students would not be able to distinguish the ordinate and the abscissa. When choosing an example, teachers need to predict what will be mathematically useful to develop conceptual understanding.

Ideally, examples provided by a teacher needs to be the outcome of a reflective process of choice, a deliberate and informed selection from the available options, some ‘better’ than others (Rowland, 2008). However, when it comes to novice teachers, they require much guidance and help in appreciating the different roles of examples in mathematics teaching. The extent to which preservice teachers choose examples wisely, or otherwise, has been found to be a significant indicator of their mathematics content knowledge for teaching (Rowland, 2008). Therefore, it is not trivial for preservice teachers to develop such knowledge of choosing examples to inculcate awareness of a procedure or concept in mathematics during their training program.

Data sources for this study are drawn from 60 preservice elementary teachers in the third year of their preparation program enrolled in three sections of an elementary mathematics methods course in a public university in the intermountain region. These prospective teachers have taken a math content course prior to the methods course. A series of mathematics task is presented to the PSTs where they need to choose examples to introduce a topic, analyze sets of examples (exercises), develop an exercise for a purpose, and creative use of textbook exercises. This paper focuses on the topic on comparison of fractions. PSTs’ written works on their choice and rationale are collected and videotapes of the classroom discussions are transcribed. Based on their written work, six PSTs are selected for a follow-up interview to provide in-depth data on their choices and perceptions on the role of examples. These artifacts are analyzed to answer the research questions of the study, i.e., to examine PSTs’ knowledge of choosing and sequencing examples, their perceptions on how the examples chosen and the way they are sequences might affect student learning, and if and how attending to examples help them to focus on student thinking.

Preliminary findings suggest that preservice teachers initially choose examples to present students predominantly by the numerical size or familiarity of the numbers, not necessarily judging the cognitive difficulty for the students. They rely heavily on the authority of the textbooks and how the examples are presented and sequenced. They rarely question the examples and their sequence, and when asked to choose from a pool of examples, their choices often are informed by factors external to the students, such as time limit and emphasis of the textbook. However, after this activity the PSTs begin to pay more attention to the mathematical difficulty and complexity for the students and anticipate how students might feel and experience when working on the examples or problems selected and sequenced.

Studies have shown that preservice elementary teachers’ pedagogical content knowledge is difficult to develop during their preparation program (Suzuka,  Sleep, Ball, Bass, Lewis, and Thames, 2009; Superfine & Wagreich, 2009; Rathouz, & Rubenstein, 2009). However, results from this study suggests that focusing on specific aspects of the PCK can significantly hone in their attention to student thinking and better prepare them to teach more effectively. This study can inform teacher education programs to incorporate important elements of the methods course. This research contributes to the literature by confirming the pedagogic importance of examples upheld in the literature, but also refines and illuminates this category by reference to the classroom practices of novice teachers.

The background, research questions, and theoretical framework that drive this study will be presented, followed by the method and analytical framework employed to answer the research questions. The presenter will then highlight the significant findings from this study and provide evidence in the form of excerpts of preservice teachers’ work and interview transcripts. Finally, implications for teacher educators are offered. The audience will have opportunity to engage in working on an instance of choosing and sequencing examples, and will have opportunity for questions.

This paper addresses the professional learning of preservice elementary teachers in the area of attending to student thinking through a focus on examining choice and sequence of examples around a specific content topic of comparison of fractions to support students’ mathematical learning.

References

Ball, D.L., Thames, M. & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389-407.

Rathouz, M. & Rubenstein, R. (2009). Supporting preservice teachers’ learning: A fraction operations task and its orchestration. In D. Mewborn & H. Lee (Eds.), AMTE Monograph 6, Scholarly Practices and Inquiry in the Preparation of Mathematics Teachers, (pp. 85-104). San Diego, CA: Association of Mathematics Teacher Educators.

Rowland, T. (2008). The purpose, design, and use of examples in the teaching of elementary mathematics. Educational Studies in Mathematics, 69(2), 149-163.

Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14.

Skemp, R. R. (1971). The Psychology of Learning Mathematics. Harmondsworth: Penguin.

Superfine, A.C. & Wagreich, P. (2009). Developing mathematical knowledge for teaching in a content course: A “design experiment” involving mathematics educators and mathematicians. In D. Mewborn & H. Lee (Eds.), AMTE Monograph 6, Scholarly Practices and Inquiry in the Preparation of Mathematics Teachers, (pp. 25-42). San Diego, CA: Association of Mathematics Teacher Educators.

Suzuka, K., Sleep, L., Ball, D.L., Bass, H., Lewis, J.M., and Thames, M.K. (2009). Designing and using tasks to teach mathematical knowledge for teaching, In D. Mewborn & H. Lee (Eds.), AMTE Monograph 6, Scholarly Practices and Inquiry in the Preparation of Mathematics Teachers, 7-23. San Diego, CA: Association of Mathematics Teacher Educators.

Vinner, S. (2011). The role of examples in the learning of mathematics and in everyday thought processes. ZDM Mathematics Education, 43(2), 247-256.