Using Bare-Bones Mathematical Figures to Examine Elementary and Collegiate Instructors' Conceptions of the Standard for Mathematical Practice 3

• In Event: 14.043 - Roundtable Session 1
  In Roundtable Session: 14.043-14 - Analyzing the Common Core Standards for Mathematical Practice

Abstract

Reasoning-and-proving practices are central to mathematics and readily incorporated into the Common Core State Standards, including the third Standard for Mathematical Practice (SMP3) asserting all students should construct viable arguments and critique the reasoning of others. However, what SMP3 looks like at varying ages is not made clear. In earlier grades, children use juxtaposition within explanations and justifications (Piaget, 1972), which has led to recommendations that children compare their strategies with others to aid in long-term development of mathematical argument (Wood, 1999). Ultimately, the fluid proving process involves exploration of patterns, potentially leading to the generation of conjectures to be tested, revised, or proven (Lakatos, 1976). There are obvious differences in the form of mathematical argument one should expect in elementary and collegiate classrooms, but we are interested in how instructors of such grade-levels solicit and facilitate mathematical argument. We, therefore, explored the nature of tasks that elementary and collegiate mathematics instructors described for engaging students in mathematical argumentation. In particular, we looked for differences attributed to student characteristics and development, and similarities attributed to an obligation to the mathematics discipline.

To examine how mathematics instructors at different levels (4 elementary and 4 college) facilitated mathematical argumentation we asked them to consider barebones figures. The term barebones figure refers to a simple mathematical representation without guiding text, specific structuring, or context (e.g., Figure 1). Instructors were provided with four different figures and asked to create mathematical tasks to provide opportunities for students to construct viable arguments and critique the reasoning of others.

Figure 1. Example barebones figure provided to participants.

Mathematical topics addressed depended upon instructors’ teaching levels. All instructors requested explanations in their designated tasks, but there were key differences in the nature of these solicitations. Grade 1 instructors generally included the requirement that students present and compare different solution methods, and using the term argue to refer to students’ (often procedural) explanations of mathematical thinking. Grade 3 and 4 instructors included similar requirements, but had more emphasis on soliciting rationales. By contrast, college instructors required students to ultimately develop more formalized arguments, consistently including requests for conjectures and generalizations.

The barebones figures allowed for examination of these differences without the introduction of irrelevant content to particular grade levels. During the interviews, the only requirement for designed tasks was that they would facilitate students’ engagement in SMP3. Using barebones figures, themselves, was critical to ensure that particular features of SMP3 were not tacitly conveyed or necessarily encouraged. Thus, the barebones figures lacked significant details while allowing for a shared experience across instructors. The nature of such figures and the task design assignment allowed us to examine how elementary and collegiate instructors envisioned argumentation and to identify differences in how SMP3 was conceptualized for particular teaching levels. Our approach to examining SMP3 with barebones figures provided key insights into how instructors conceptualize applying SMP3 in practice. Extending this approach to examine instructors’ task design to attend to other SMPs holds the potential to provide findings of a similar nature.

Authors

• Karl Wesley Kosko, Kent State University

• Kimberly Cervello Rogers, Bowling Green State University