Creative ways to teach math

Representative image. iStock

There is a crisis of creativity in our school education. More so in mathematics. Our children constantly complain about how ‘boring and dull’ the mathematics classes are and for the right reasons. Repeated application of routine algorithms to solve mathematical problems can be either daunting or repulsive depending on the child’s mindset. It is ironic that we are supposed to be teaching them mathematics as it is ‘useful’!

Let us explore creativity in mathematics using a simple but powerful example in mathematical reasoning and problem solving. First, take a look at this simple puzzle.

Three trucks puzzle

Twenty-one gas cylinders are to be loaded onto three trucks. Seven cylinders are full, seven are filled half-way and seven are empty. A full cylinder weighs 50 kg and an empty cylinder weighs 20 kg. How should they be loaded onto the trucks so that each truck is carrying the same weight?

It may look daunting at first but let us approach it in small, manageable steps.

First step: Growth mindset

Children simply assume that there is a single solution to the problem which they need to find out. This indicates a typical fixed mindset. In sharp contrast, someone with a growth mindset would be asking, “Are there multiple solutions? What are they?” This is the beginning of creative thinking.

Second step: Random or systematic?

A common tendency among the children is to attempt it sporadically, mostly trial-and-error. While occasionally it does lead to a solution, this approach does not help them develop a deeper mathematical thinking. It is a much better idea to attempt the solutions more systematically. Let us see how.

Third step: Identify the critical element

Let us first work out the weight of the half-filled cylinder. As the full cylinder weighs 50 kg and the empty one 20 kg, logically, the weight of the gas is 30 kg. The half-filled cylinder will weigh 20 kg plus 15 kg — 35 kg. Each truck has seven cylinders: some full, some half-full and some empty. The weight of the empty cylinder 20 kg. Hence the total weight of the 21 cylinders and gas is (50*7) +(35*7) +(20*7)= 735 kg. So, each truck will need to be loaded with 245 kg.

Here is one possible solution.

First truck: One full cylinder, plus five half-full cylinders, plus one empty

Second truck: Three full, plus one half-full cylinder, plus three empty

Third truck: Three full, plus one half-full cylinder, plus three empty

Here, which is the essential component? How do we separate the significant information from the trivia? On a closer look, the half-full cylinder is the critical component. The weight of 235 kg per truck can only happen when the number of half-full cylinders is 1,3,5 or 7 (7 is ruled out). We have only 1,3 and 5. Seven cylinders can be formed either as 1+1+5 or 3+3+1. There are no more solutions.

Let us get a little more creative and change the question slightly and move over to the boundary values. If there are only three — 1 full +1 half-full +1 empty —cylinders obviously there is no solution.

If there are six, you can solve it in a jiffy.

First truck: One full plus one empty

Second truck: One half plus one half

Third truck: One full plus one empty

If there are 9 cylinders, it is equally easy.

First truck: One full, one half, one empty

Second truck: One full, one half, one empty

Third truck: One full, one half, one empty

Fourth step: Augmentation

This step is a cinch. This can be easily scaled up for any number of cylinders. All higher numbers must be treated as a combination of either 2s or 3s or both. For eg: 2+3=5; 3+3+2+3=10 or 2+2+2+2+2= 10; 19 is five 3s and two 2s combined etc. In other words, the puzzle can be cracked for every number except 1.

Creative thinking with systematic efforts led us to an unknown but exciting territory. So, we need to make decisions regarding the nature of interventions required to foster creativity. The crucial factor is the overall ambience.

Teachers need to strive to create such an ambience in their classrooms to do justice to the objectives of teaching mathematics. They can create a bank containing a range of pictorial, numerical, geometrical, logical problems to engage children and pique their interest in mathematics. They can have a “problem of the week” and allow multiple approaches to solve them. Even partial solutions must be encouraged.

The classroom will have a good, energising discussions and dialogue. At a later stage, children must be encouraged to “pose new problems” and that would be a fantastic indicator of success.

(The writer is a math educator, trainer and author based in Mysuru)