classroom structure investigation reflection

Maths’ True Form

I teach mathematics at the high school level, and know all about the various theories surrounding school mathematics. I can still remember the intrigue when the term “Math Wars” was introduced to me through some undergraduate reading. I immediately took to the history of my art, and found a very convoluted and bloody past. The constant pendulum between retention math, new math, back to basics, and now the new-new math is dizzying. Whenever I converse with a colleague about a new way of thinking in math education, I am sure to remind them that we are in a war. It is this idea that has appealed to the more militant teachers (myself included).

I would love to stop this war. It discourages collegiality, and increases excuses. It also acts as a thin veil behind which many educators hide. One side accuses the other of being “useless”; the other retorts with chants of “fuzzy”. One focuses on repeatable and measurable skills, while the other wants to develop thinking skills through mathematical investigations. If we were honest with ourselves:
Both sides need the other.
Students need tools for investigations. I am not saying they need to be transmitted to them, but there needs to be more than an extremist ‘open discovery’. I would love to see an “open” discovery that accurately and completely covers a curriculum. Even the most effective ones are structured in “units” where teachers dictate what problems the students will investigate. The nature of the beast is that teachers have a job to do, and that job is dictated by our standards. We can still approach the standards with a spirit for understanding, but must also ensure we don’t make basic skills the casualty of war.
After my exposure to the issues, it seems as though it was an immeasurable gap between the two sides. There was not much encouragement from either side, and negotiations were fruitless. Teachers sat in Professional Development days shooting down every possible suggestion. Eyes were rolled and the room filled with hurried whispers when a “reform” technique was proposed. The entire room was bursting with “here we go again”. University standards require a rigorous understanding of many branches of mathematics, but the traditional teaching was not raising the scores at that level. The carousel continues to this day.
It was while looking through a number theory book that I found this paragraph. I was trying to find examples of mathematical induction, when I came across this introductory paragraph. It provides an interesting connection between the two armies. It is also very ironic that such a sympathetic view was found in a book written for study at the university level.
The following excerpt is from Elementary Number Theory: Second Edition by Underwood Dudley:
“Mathematics is notoriously a deductive art: starting with a collection of postulates, theorems are deduced by following the laws of logic. That is the way it is presented in print, but that is not the way that new mathematics is discovered. It is difficult to sit down and think, “I will now deduce,” and deduce anything worthwhile. The goal must be in sight: you must suspect that a theorem is true, and then deduce it from what you know.” (page 205)
I was very delighted to hear the ambiguous tone in the author’s voice. Mathematics begins with suspicion. This is the nature with which mathematics must be presented in school. There is a common misconception that teachers who advocate for discovery want students to recreate every formula and theorem–no matter how difficult. We do not desire a re-invention of the wheel. To me, the new-new math is about developing the necessary atmosphere of suspicion in school mathematics. This is a feature that is crucial for investigations and also valued–as evidenced by Dudley–at the university level. Maybe the two camps are closer together than we think.

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