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Looking Back 17 Years

In Summer (1992) issue of Consortium, headlines to Jerrold Grossman's letter read: A Proposal to Ban "Equivalent Equations" From College Algebra. The implications to high school algebra are the same.

First, despite Grossman's empassioned plea, I have seen neither curricular nor textbook changes being made. Is it because he had a weak argument...or was he wrong? I believe it is because we do not care!

As to Grossman's argument, he claimed that our treatment of equivalent equations introduced both pedagogic and mathematical errors. He wrote: "solving algebraic equations, a topic we think we understand but...is much more complicated than we are willing to let on."

His biggest concern deals with the misuse of the "theorem": If A = B, then A+C = B+C when A, B, and C are not numbers but algebraic expressions. That is, by this theorem, A = B and A+C = B+C are quivalent equations. Grossman also repeats the common mantra: Two equations involving a variable x are equivalent if they have the same solution set. To illustrate the possibility of error, Grossman suggest letting A be x, B be 2, and C be 1/(x-2). Then the solution to the equation x = 2 is obviously 2, but 2 is not a solution to x + 1/(x-2) = 2 + 1/(x-2). So is the theorem wrong? Grossman adds: The theorem with this interpretation is too complex for students at this level to appreciate.

To resolve the situation, Grossman suggests: What students really need to learn is simply how to manipulate equations to obtain their solution sets. We must teach them how to transform a given equation E1 into another equation E2; find the solution set for equation E2; and "check" the solutions so obtained by seeing whether they are, indeed, solutions of E1. If the transformation has been done cleverly enough, then no solutions of E1 will have disappeared, and few, if any, extraneous solutions will have been introduced.

And what about the role of technology? Grossman notes that when asked to solve the equation x + 1/(x-2) = 2 + 1/(x-2), the computer algebra systems Derive, Macsyma, Maple, and Mathematica all produced x=2 as the solution! Why not try it on your TI-9*?

How do you feel about Grossman's compaint and suggestion? Are they petty, minor, and symptomatic of the "mortis" side of rigor? How do you get around textbooks' misleading statements/theorems regarding equivalent equations, as they have not changed in this regard?