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/(x2). Then the solution to the equation x = 2 is obviously 2, but 2 is not a solution to x + 1/(x2) = 2 + 1/(x2). 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 E_{1} into another equation E_{2}; find the solution set for equation E_{2}; and "check" the solutions so obtained by seeing whether they are, indeed, solutions of E_{1}. If the transformation has been done cleverly enough, then no solutions of E_{1} 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/(x2) = 2 + 1/(x2), the computer algebra systems Derive, Macsyma, Maple, and Mathematica all produced x=2 as the solution! Why not try it on your TI9*?
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?
