Thinking versus Rules

Mathematical Drill or Education to Think?

Recently, I sometimes get invited to talk about „math“ in general, how to teach, and how to learn. Apparently, the German „Jahr der Mathematik“ yields its fruits. Of course, more famous mathematicians than me are currently discussing this topic in numerous books and talks. Nevertheless, I have my own opinion.

Today I came upon a blog about mathematical teaching. I won’t tell you the address, since I was somewhat disappointed by the content. The main argument was that teachers should concentrate on the current subject of a lesson, and not look sideways, nor ask any questions not related to the topic. Otherwise, students may be „stressed“, or feel „anxious“. An hour of repetition and drill applying the same rule over and over again, would be a more relaxed and fun way of teaching.

The problem I have with this approach is that I am constantly encouraging teachers to let their students ask questions, look aside, and avoid drill. This, of course, is in sharp contrast to the above image of a math lesson.


Now, you may argue that the blogger and I are probably teaching different math levels. On the lower level, some say, we have to teach facts, and train procedures. On the university level comes the time to ask questions. But this is not going to work! Once the kids are spoiled by years and years of such math lessons, they are lost for true mathematical thinking. As the blogger said, they feel „anxious“ and „stressed“ when they meet things not met before.

Teachers on the other hand argue that they need to do the procedure business for their bad students. These students would otherwise not be able to get any satisfying grade. I tell you what: These students may learn the procedures, maybe with some additional training, but they gain nothing in the process. Does anybody really believe that those bad students will apply their school math in later life? Forget that. If, however, you show them even only a glimpse of mathematical thinking and asking, they have learned much more than by routine drill of things they do not understand.

Another Sample

Before this blog gets out of control, I will give you another example. Also today, I stumbled over a pretty website collecting rules for divisibility without remainder. E.g., the rule for 7 was this: Double the last digit and subtract from the rest. If this divides by 7, the original number will. E.g., For 154, we compute 15-8=7, and so we know that 154 is a multiple of 7.

This, of course, is a procedure. Am I asking too much, when say, it would have been much nicer, if the author would have shown why this works. Isn’t that much more interesting? Will you ever do a test like this on a number? Moreover, isn’t it as easy to simply divide the number by 7?

Since I can add LaTeX here, let me give the proof:

\(10x+y=0 \mod 7 \Leftrightarrow 50x+5y=0 \mod 7 \Leftrightarrow x-2y=0 \mod 7 \)

Of course, one needs to know how to compute „modulo 7“. Otherwise the proof gets a little bit longer. In any case, you learn much more for your life, your education, and your mind, if you try to work out this proof, than by memorizing the rule itself.

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