The Algebra Blues
By Dr. Eugene Maier
Whenever I think about a conversation I had with my granddaughter, I get the blues. She was in a seventh grade advanced math class and they were studying algebra. I asked her how it was going. She showed me her homework.
Her solution of a simple equation like 2x + 10 = 16 required 5 lines. First she had written down the equation. Next she had written "2x + 10 - 10 = 16 - 10, subtract 10 from both sides." Then, "2x = 6, collect terms." And so on.
I asked her why she wrote all that down; couldn't she figure out the solution in her head? She said, "Yes, it's 3, but I just can't write that down." I said, "What about writing x = 3 because 2(3) + 10 = 16." She said, "No, I have to do it this way," pointing to her paper.
I asked her if she had any idea why the teacher wanted her to do it that way. She did: "Mr. X says that's what the eighth grade teacher wants and his job is to get us ready for the eighth grade."
I was disheartened. Mr. X was turning algebra into drudgery and destroying whatever number sense his students possessed. Besides that, he was putting the interests of the eighth grade teacher ahead of his students' welfare. I could only wish that Mr. X would undergo some marvelous transformation that would change him into a teacher like Mrs. Y.
In contrast to Mr. X, Mrs. Y puts her students' educational development ahead of their next teacher's expectations. For her, mathematics instruction isn't dictated by what the next teacher wants but by the present knowledge and understanding of her students. She believes addressing the latter will take care of the former and if it doesn't, then the next teacher needs to change their expectations.
For students in Mrs. Y classes, algebra is not a collection of rules and procedures imposed on them by some authority. Rather, it becomes a way of communicating about and dealing with mathematical situations based on their own investigations, investigations she instigates that are designed to naturally lead the students into the topic at hand. Her students talk and write about their thinking. Clarity of expression, both oral and written, is valued, but students aren't forced to turn their work into ersatz axiomatic demonstrations.
Teachers like Mrs. Y are more concerned about their students' current mathematical development than what some future teacher fancies they ought to know. In their classrooms, algebra is not a set of rules and procedures of mysterious origin to be imposed upon their students, but a subject that evolves naturally from a set of experiences. Their rooms are alive with activities that evoke algebraic concepts and procedures. Algebra is not imposed on their students, but it is drawn from their observations and discussions.
Students' insights and intuitions aren't smothered by forcing them to use the teachers' or textbook's way; they are encouraged to use methods that are based on their own understandings and insights. Instead of being taught there is only one way to carry out an algebraic procedure, students are encouraged to find alternate ways of proceeding. They aren't castigated if their methods don't work, or are based on misconceptions. Rather, their efforts are valued, and ferreting out whatever goes awry becomes a learning experience for the whole class. In such a teacher's classroom, algebra is lively and vibrant.
But in my granddaughter's mind, it's tedious and banal. And should there be more Mr. X's than Mrs. Y's, there must be hordes of other students with the same frame of mind. What a melancholy thought.