Integrating Algebra and Proof in High School: Students' Work with Multiple Variables and a Single Parameter in a Proof Context

Abstract

In the United States, researchers argue that proof is largely concentrated in the domain of high school geometry, thus providing students a distorted image of what proof entails, which is at odds with the central role that proof plays in mathematics. Despite the centrality of proof, there is a lack of studies addressing how to integrate proof into other mathematical domains. In this article, we discuss a teaching experiment designed to integrate algebra and proof in the high school curriculum. Algebraic proof was envisioned as the vehicle that would provide high school students the opportunity to learn not only about proof in a context other than geometry but also about aspects of algebra. Results from the experiment indicate that students meaningfully learned about aspects of both algebra and proof in that they produced algebraic proofs involving multiple variables and a single parameter, based on conjectures they themselves generated.

ACKNOWLEDGEMENTS

The authors would like to thank Bárbara M. Brizuela for her thoughtful comments on early versions of the article.

Notes

¹In this context, we use the term multiple variables to describe mathematical problem that involve more than two variables.

²Bloedy-Vinner (1994) presents the following problem: “In the following equation, x is an unknown and m is a parameter: m(x − 5) = m + 2x. For what value of the parameter m will the equation have no solution?” where the parameter m plays the role of an unknown, given the context of the problem and the question that is posed. The dependency between the role of a letter and the context has been also highlighted by Usiskin (1988).

³In a similar vein, De Villiers (1990) gave the following list of roles that proofs play in mathematics: verification (concerned with the truth of the statement); explanation (providing insight into why it is true); systematization (the organization of various results into a deductive system of axioms, major concepts, and theorems); discovery (the discovery or invention of new results); and communication (the transmission of mathematical knowledge).

⁴The meaning (semantics) that students imposed to the expression ultimately helped them to obtain the correct answer even though the syntax was incorrect.

⁵Given space restrictions, all problems that led up to problem 18 could not be included in the table.

⁶Even though Janusz did not write the pair of parenthesis around x + d, the next expression in his work is correct. He may have forgotten to write it down and knew by memory that the sum of x and d was multiplying x + 1.

⁷The term parameter was introduced in the following lesson as a result of their work on problem 18.