Discussing Comparisons Leads to Micro-Changes in Algebraic Knowledge

• In Event: 52.069 - Just a Minute! New Methods for Observing Conceptual Change in STEM

Abstract

My goal in this segment of the symposium is to discuss how small changes in knowledge can be observed over the course of minutes in the domain of algebraic problem-solving. Typical studies of conceptual understanding in algebra examine learning that occurs over the course of weeks or months (e.g., Rakes, Valentine, McGatha, & Ronau, 2010). In contrast, clinical interviews have the power to elucidate, in great detail, not only the product of an intervention but also the process by which students’ thinking changes from moment to moment during an intervention (Ginsburg, 1997).

The specific context that I will examine involves students discussing the similarities and differences among the ways that they each independently solved a complex algebra problem. Despite the growing body of evidence that comparing multiple solutions is beneficial for learning (Woodward et al., 2012), how such comparisons cause conceptual change on a short time scale in adolescent populations has not been extensively researched. One study showed that having students discuss the similarities and differences among researcher-generated solutions led to changes in students’ procedural and conceptual algebra knowledge (e.g., Rittle-Johnson & Star, 2007). Whereas this study provided a tightly controlled set of materials for discussion, investigating a context in which students compare self-generated solutions may be better suited for observing the process of convergent conceptual change (Roschelle, 1992). Comparing self-generated solutions with a peer may provide a setting in which participants can better integrate their individual prior knowledge and the knowledge that is collaboratively created because there is greater opportunity for students to negotiate meaning without 1) a clearly more knowledgeable other or authority figure, and without 2) a clearly preferred solution that devalues alternative solutions. Both of these factors facilitate the documentation of micro-changes in knowledge.

In my talk I will present analysis of data that includes video recordings of several pre-intervention and post-intervention clinical interviews as well as video recordings of students discussing their solutions to a math problem with a peer. Analysis of moment-to-moment changes in students’ gesture and language use over the short timespan of the study will provide evidence for how changes in these external behaviors correspond with changes in students’ mental representations of algebraic rules and concepts.

Furthermore, in line with the broader goals of the symposium, I will explore how this process of “getting on the same page” is equivalent to learning. This will parallel commentary on how researchers negotiate meaning of interviewees’ knowledge during a clinical interview to be able to make claims about learning.

The findings of this study can inform educators of what learning looks like during brief pair discussions, as well as add to the literature of how and under what conditions comparison facilitates learning. Knowing what behaviors to look for and how comparisons can change students’ knowledge will help teachers better prepare their students to meet the new Common Core State Standards in Mathematics that place higher value on students’ conceptual understanding and abilities to reason algebraically.

Author

• Sarah Louise White, Northwestern University