Identifying Seeds of Algebraic Thinking in Children's Thinking

• In Event: 43.064 - Exploring Diversity and Synergy Across Research Programs Within Early Algebra

Abstract

Objectives: When characterizing learners’ reasoning processes, a prevalent practice involves considering a target concept (“variable” or “equivalence”), described in mathematical terms, and then describing partial progress toward understanding that concept (e.g., Ron, Dreyfus & Hershkowitz, 2010). While this approach is valuable, it does not engage directly with the nature and form of children’s knowledge, including potential sources of knowledge. In this presentation, we provide a complementary approach, beginning from schematizing individuals’ knowledge-in-use.

Perspective: We are informed by a complex knowledge systems perspective that takes as fundamental that (1) it is productive to model knowledge as a complex system of local abstractions of experience and (2) learning is a process of tuning this complex system towards expertise (diSessa, Sherin & Levin, 2016; Hammer, Elby, Redish & Scherr, 2005).

Data Sources: We will present data from two studies illustrating our methods of locating “seeds of algebraic thinking”: (1) a study of pre-school and early elementary children engaging in collaborative board game play (Authors, 2018) and (2) an analysis of a pre-algebra student refining a guessing and checking approach into a deterministic, and essentially “algebraic” algorithm for solving problems with an underlying linear structure (Author, 2018).

Results: Analysis of both data sources revealed “seeds” of algebraic thinking- knowledge structures that are invoked in algebraic reasoning, but they themselves have their source in students’ experience across a wide range of situations. Observed seeds included examples such as “balance,” “open value,” and “changing an input (X) in some way results in a change of a corresponding type in the output (Y).” We argue that these seeds can undergird larger chains of algebraic reasoning such as equivalence, variable and covariation.
In particular, analysis of the pre-algebra student’s reasoning processes illustrated the use and coordination of such knowledge structures into the longer chains of reasoning that may otherwise be identified as “strategies.” At the strategy level, a researcher may apply terms such as “proto-algebraic” (Nathan & Koedinger, 2000) to the reasoning process observed because it fit some, but not all, of the criteria that are commonly invoked when discussing algebraic reasoning. When analyzed at the level of specific inferences that were guiding the student’s reasoning, one uncovers knowledge structures that are neither algebra- (nor even mathematics-) specific.

Significance: We argue that looking at the methodological process by which the “seeds of algebraic thinking” were identified can be illuminating for researchers interested in studying students’ early algebraic thinking. In the presentation, we will discuss implications for a more extended line of work on identifying seeds of algebraic thinking. By exploring student thinking at this level, we can begin to understand the naturalistic development of algebraic thinking, for which there is currently little research.

Authors

• Janet Dawn Kim Walkoe, University of Maryland

• Mariana Levin, Western Michigan University