Beyond PEMDAS: Teaching Students to Perceive Algebraic Structure

Evidence shows that transforming expressions is a major stumbling block for many algebra students. Using Sfard’s (1991) theory of reification, I highlight the important roles that the process of parsing and the notions of subexpression and structural template play in competent expression transformation. Based on these observations, I argue that one reason students struggle with expression transformation is the inattentiveness of traditional curricula to parsing, subexpressions, and structural templates. However, simply refocusing attention on these ignored aspects of algebra will not alone ensure that students avoid common pitfalls. After examining evidence that students are very prone to overgeneralize…

Contents

Introduction
What is expression transformation?
Evidence that students have difficulty with expression transformation
Simplifying numeric expressions and evaluating algebraic expressions
The non-notational precedence conventions: The hierarchy of operations
Notational precedence conventions: Grouping symbols
Parsing: Implicit activity versus explicit activity
Transforming expressions
Generalization, and the two “faces” of algebra
The rules of algebra
Transforming an expression: Matching given expressions to rule expressions
Subexpressions and structural templates
Transforming an expression: Using subexpressions
Using reference as support for expression transformation
Diagnosing student difficulties with expression transformation
Reification: Mathematical objects as compressed processes
Subexpressions and structural templates as mathematical “objects”
PEMDAS: Structure (or lack thereof) in the traditional curriculum
Diagnoses that support or overlap with mine
The role of overgeneralization
Matz: Overgeneralizing rule-revision strategies
Kirshner & Awtry: Overgeneralizing memorable visual sequences
The vicious circle of reification
The pragmatic value of reification
The vicious circle and algebra: A window for overgeneralization
Competing impulses: The connectionist view of mind
Kirshner on the role of spacing in parsing decisions
Kirshner on connectionism
Landy & Goldstone’s research on formally irrelevant distractions
Other examples of competition in algebra performance
Instructional strategies
Instructional strategies for helping students achieve the process- perception of parsing
Instructional strategies for inducing reification of structure vertically
Instructional strategies for inducing reification of structure horizontally
Some supporting and overlapping curricular recommendations
Conclusion and implications
References

Author: Merlin, Ethan Michael

Source: University of Maryland

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