It is important to be able to accurately complete computations in an adequate time frame but also important to understand. In Schoenfeld’s article Good Teaching, Bad Results, he gives a couple of examples of how students were able to answer mathematical problems using procedures and operations but without a deep understanding of the ‘underlying substance’. (p.5)
In one example students were given problems such as
274 + 274 + 274 or 812 + 812 + 812 + 812 + 812 (p.5)
3 5
A lot of the students who could use all four of the basic arithmetic operations solved these problems by working out the addition and then dividing rather than looking at the problem and seeing that the number they had to divide by was the same as the number of addends and the addends were the same numbers. If they had used their number sense and understood the problem being asked they would have completed the problem without having to work out all the operations. “By virtue of obtaining the correct answer, the students indicated that they had mastered the procedures of the discipline. However, they had clearly not mastered the underlying substance.” (p. 5) Schoenfeld explains that this shows that being able to perform the appropriate operations does not necessarily indicate understanding.
The second example Schoenfeld gives is Wertheimer’s example of ‘the parallelogram problem’. Students had been taught how to find the area of a parallelogram by cutting off and moving a triangle to change the parallelogram to a rectangle which they could easily calculate. The students could easily do this problem but when they were asked to find the area of a parallelogram that was not in a standard position they couldn’t do it. The students had memorized a formula but did not understand the reasoning.
Understanding why we do the steps we do in math and why they work is a very important aspect of understanding mathematics. From my own experience in school I remember doing ‘long division’ where I knew how to put a number ‘into’ another and how to bring down the digits until I got the correct answer but didn’t really understand what division was. This made it more difficult when given a ‘story problem’ in which I needed to figure out what operation to use to solve the problem.
In our provincial curriculum in Newfoundland and Labrador I am proud to say we do now teach our students to understand division and not rely on step by step formulas to do computations only.
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