Mathematical Questions Without Numbers

by Pool Builders on 04-14-2013 in Articles

A Teaching Approach To Help Students Solve Problems In Mathematics

Students, as a whole, believe that a problem in Mathematics must have numbers before it can be solved. Students must come to understand that numbers are not important in deciding how a problem should be solved. They only become important after the student has decided how to solve the problem.

Let me give you an example. This question was put to a group of teachers at a professional development work shop. The teachers, initially, reacted the same way as the students did when I gave it to my classes. Here is the question, the reactions of my students and the teaching that took place afterwards.

The Swimming Pool Problem: How long does it take to fill a swimming pool with a bucket?

The immediate reaction from most students and, initially, the teachers is to say, "It can't be done. There is no data." That is, you need the dimensions of the pool and bucket and how long it takes to fill the bucket. However, all of these are immaterial to actually solving the problem.

These are the steps in the solution that you need to do. They don't change because the dimensions change.

  • Work out the volume of the pool.
  • Work out the volume of the bucket.
  • Work out how long it takes to fill a bucket and pour it into the pool and return to the tap.
  • Work out how many buckets it will take to fill the pool.
  • Work out the total time by multiplying the time to fill a bucket by the number of buckets used. You would probably divide by 60 to find the number of hours to give a realistic answer.
  • Now you could use the dimensions to find the actual answer.

As part of the training of students to solve problems, teachers need to teach students to recognise the action words (verbs) that lead them to the operations they would use in solving the problem.

Here is one way to proceed.

Step 1:

To begin the process, select a series of normal written exercises in a Maths text book. Ask your students to read each question and write down which of the four operations (addition, subtraction, multiplication, division) they would use to solve each question. If the students were to use an operation more than once, they should indicate how many times they would use it. In lower grades, use only one operation at a time. Then, as you move up the school, increase the number of operations and the complexity of the questions used.

Step 2:

Select another set of exercises. This time remove the numbers and replace them with an empty box. Then, have the class repeat the process in step 1.

Step 3:

You then extend the complexity a bit further by insisting they put the operations in the order of their use. The questions are not solved. (This might be a good time to revise the concept of order convention and consolidate it in the students' minds.)

Step 4:

In upper primary and high school classes, use problems that would require the use of rules or formulae. Here they would simply write out the steps that they would need to take to solve the problem. This might mean the listing of formulae or rules they might use. The Swimming Pool Problem, I used as an example earlier in this article, is the sort of question I mean.

What these steps are aiming to teach the students to do is to work out what are the steps they need to solve the problem. When you begin a new topic with new exercises, have your students read the questions first and list the operations or steps they would use. Then, go through the early exercises with them to confirm what operations/steps they would take before you ask them to actually solve the problem and obtain an answer. This way, students will have the confidence to try to solve the new problems.

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