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Problem: (Source)
At a track meet at Hayward Field, a runner does one quarter mile lap in 1.25 minutes. Assuming she starts and stops at the same place, what is her average speed?

This is a pretty straightforward application of the definition of average speed.

a

The concept-based solution starts by responding directly to what is ASKED FOR using the concept appropriate to the problem. This response implements the previous verbal statement in mathematical form. The concept then ASKS FOR the quantities needed. The problem solver supplies these quantities and the problem is solved.

The forward inference solution infers the starting expression and presents first the quantities the concept will need.

The forward inference approach is logical nonsense. Instead of stating the concept that will be used, and this is what we wish to teach, the student is instructed to go through a bunch of busy work first. This destroys any hope of placing emphasis on ideas. The "first list the givens" approach is tragic and is on high on the list of reasons students are trained to be poor problem solvers. The simple change to concepts first instead of givens first would cure the I can't solve word problems syndrome.

In solving this problem the FIRST thought one has is the definition of average speed. Yet, in forward inference, this is not the first thing presented. The student is not led through the solution in the same order as the writer solved the problem.

This inverted presentation, resulting from forward inference, is typical of presentations found in textbooks and problem solving instructional material. The FIRST idea that the writer had, motivating the steps in the solution, is not revealed to the reader. The reader is thus not aware of the reason for the first steps shown in the solution in a forward inference presentation.

A rather common complaint of students is that they don't understand how a problem solution was started. The reason for this is quite simple. The motivating ideas for the initiation of the solution are locked in the writer's mind and not accessible to the reader.

A forward inference solution proceeds as if the reader is already expert in the subject matter. Such a solution emphasizes mechanisms as opposed to concepts.

Concept-based problem solving provides a common sense approach to solving problems.

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The presentation was developed using SureMath.
Copyright 1995-2000 Howard C. McAllister
Adapted from a problem solution presented by David Mason. Return to Problem