My Philosophy of Mathematics Education

My goal in teaching mathematics is to recapture the human element in it, that is, to have students do what mathematicians do --- create and solve problems. In order to create problems, mathematicians observe, find patterns, test the patterns they find to see if they can be generalized, and make conjectures about their patterns. They also try to prove that their patterns always work. They read the work of other mathematicians, find the problems posed by them, and try to solve them. Solved problems and patterns often result in formulas.

What students often get are just the formulas. A formula is a desiccated pellet, the dried-up remains of what was once a vital, lively, and exciting process of discovery. The remains of what was once a lively human activity. Someone observes astutely or casually, makes a clever or intuitive conjecture, carefully or sloppily tests the conjecture, and if it is promising, tries to prove it, and if successful, submits the proof to the mathematical community for review. If unsuccessful, the person may submit it to the mathematical community as a conjecture. The process is replete with toil, thought, and deliberation, and buoyed by emotions that accompany anticipation, discovery, disappointment, and insight. All of that is missing when the student is handed the formula.

And not just one formula. Hundreds of them. The fruit of thousands of years of mathematical endeavor. To make mathematical progress, today’s mathematician must know all that’s gone before. But must all students recapitulate the entire history of mathematics? Maybe the essential thing that we should teach students is the endeavor, the mathematical habit of mind.

Not only is the formula devoid of all the toil and emotions that went into it, it may also lack the insight of which it is the fruit. We can bring formulas alive by showing why and how they work, what the formula tells us, in plain simple words, not mathematical jargon. We should use language that informs and excites; we can say “shortcut” rather than “rule.” We should avoid language that puts down (a tender mind), puts out (the flames of excitement), and puts off (understanding).

We should work with what the student brings. We should ask what the student knows. I often begin a teaching session with, “Tell me what you understand,” rather than going off on a didactic lecture. I try to meet the student where she is. And then I try to help her reach a bit, while staying in her Zone of Proximal Development (Vygotsky), asking questions that are within her understanding but that require her to reach, to extend her knowledge and understanding. This requires posing the right level of question to the student.

And then, as we teach new material, we should provide an environment in which children, adolescents, and mature learners can experience and recapture the joys and satisfactions of mathematical discovery. We can set students up to make “discoveries.” What they discover may not be new to mathematics, but it will be new to the student and can do much to engender enthusiasm for the subject. Then the student is off on a mathematical adventure, observing, discovering a pattern, making and testing a conjecture, and exposing the results (verbalizing and explaining them) to his mathematical community for discussion, verification, and determination of novelty. The student will make a few of his own formulas and, in doing so, acquire ownership of the concept of and understanding behind the formula.

I encourage students to work on a problem any way they know, but as they do it, to be on the lookout for a shortcut. I ask them to be very aware of the question they are asking and the results they are getting. I encourage them to detect patterns that will make this sort of problem easier for them to recognize and solve in the future.

We need to provide students with the opportunity to strike a better balance between answering and asking questions. Our current teaching methodologies lean heavily toward answering someone else’s (usually the teacher’s or the book’s) questions. We have to teach students how to develop their own questions. This can begin with single-digit addition (what happens when you add 9 to another number – what one young student called her “9 plusses”) and go to trigonometry (invent a trigonometric identity) and beyond (develop your own theory that creates a new branch of mathematics before you reach age 30 – like Ramsey theory).

Lynn Salvo, December, 2008

© 2006 – 2008 Lynnea C. Salvo

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