In Teaching vs Training 1 (2 Dec 2012) I identified ten flaws in a particular example of training which is common in mathematics classrooms. The second of these flaws was:

No mathematics is involved in the process.

We perform mathematics, like solving an equation, by using mathematical operations. Through the use of deductive reasoning with mathematics objects, operations, relations, and stipulative definitions we are assured of the validity of the work. If some other form of reasoning (memory, emotion) enters the picture, the results are no longer assured. If non-mathematics operations are introduced, the results are no longer assured. If non-mathematics relations are introduced, the results are no longer assured. If words are interpreted differently than the accepted stipulative definitions, the results are no longer assured.

There is no mathematics operation called move. Consequently the idea of moving b to the other side of the equation has no foundation in mathematics and any results of such action cannot be trusted.

The next step in the process is to divide both sides of the equation by the coefficient a. It would be a weak argument to try to dismiss this idea of division although multiplication by the reciprocal of that coefficient might be more desirable. So we will accept dividing both sides of the equation by the coefficient as legitimate mathematics.

Finally the student believes this result to be the correct “answer” as a result of his memory of a pronouncement by his teacher, not deductive reasoning. Although memory is an amazing mental function, it is not a good mathematics tool and does not provide adequate justification. Notice that in this limited example, the student might not realize what constitutes a solution of an equation. I encounter this absence of understanding nearly every day.

The entire exercise has degenerated into training the student to perform a motor skill instead of obtaining the solution for the equation through intellectual activity. If our only goal is the motor skill, we can train (program) a computer/calculator to perform that activity. In fact we frequently do program calculators and computers to perform routine tasks.

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December 7, 2012 at 10:20 AM |

So how do we train or retrain ourselves to accept the beauty of mathematics and see the world differently. We are trying here visually with art and design thinking that the brain will sort it out in time. I’d like a better more tangible way for my linear sequential brain my children are visual spatial.