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 first of these flaws was:

A student with virtually no background knowledge can be trained to perform these steps.

I am sure one could train a preschooler to perform these actions when presented with an equation. Does that mean the preschooler has learned anything about solving linear equations? Absolutely not. When my oldest son was in the first grade I trained him to construct the derivative of a polynomial function. Do you think he knew anything about derivatives? Absolutely not. What had taken place was training not teaching. If the goal had been to create a human machine capable of creating derivatives of polynomials, then the training was a success. If on the other hand the goal was to teach an understanding of and an ability to use the derivative of a polynomial, the training was a dismal failure.

The above cited method of training a student to solve a linear equation produces a skill which, in the student’s mind has neither context nor application. The training process contributes to the erroneous, but popular, belief that mathematics consists of hundreds of isolated “formulas” which have no value and will never be used outside the classroom.

There is a more insidious and much more damaging effect of this kind of training. As soon as textbook publishers, authors, administrators, and politicians, become aware of the fact that no background knowledge is required they can begin deleting important mathematics topics from the classroom. So if we don’t need to know about the real numbers and the number line, let’s remove it to make room for other things – maybe training students to create derivatives. If we don’t need to know about sets, let’s remove it and make room for other things – maybe training students to convert irrational numbers to decimals. If we don’t need to know what a solution is, or what kind of mathematics objects have solutions, then let’s delete those topics and make room for other things – maybe training students to use completing the square to solve quadratic equations.

An examination of current textbooks reveals a huge amount of drivel, an inordinate amount of drill and practice, (that’s training) with very little mathematics content and virtually nothing to help the student improve his logical, deductive, organized thought directed at problem solving skills.

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This entry was posted on December 2, 2012 at 2:05 PM and is filed under Teaching vs Training. You can follow any responses to this entry through the RSS 2.0 feed.
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December 2, 2012 at 5:53 PM |

Research in mathematics education has passed through many fashions over the last 7 or more decades but an idea I have always been taken with was exhibited in about 1976 by Richard Skemp as seen at http://math.coe.uga.edu/olive/EMAT3500f08/instrumental-relational.pdf. I seem to observe these effects even today when attempting “IT support” for friends and relatives. I think this is very much what you are talking about in this blog article.

December 6, 2012 at 3:18 PM |

Hi Bob;

I read the article and agree that many of his ideas will be repeated by me. He has some excellent illustrations. Thanks for the link.

Del