Don’t worry, you have come to the right webpage and I promise this is not a treatise on mathematics. Be honest though, did you not sense a chill down your spine, when you glimpsed this algebraic equation? Even if you had been a bright student cracking such equations with aplomb, I bet you might have wondered, “Is this ever going to be useful in my real life?”
Pressure instead of pleasure
Most of us were fed such equations every day at school, when, all we wanted to do was to catch tadpoles. Especially when growing up in India, it was impossible for me to escape the scrutiny of teachers, parents and even neighbours when it came to scoring marks in maths exams. It was seen as the ultimate measure of one’s intelligence. I am not saying that learning maths was underwhelming nor am I questioning its purpose and application. As a matter of fact, there are many sites and blog posts illustrating the usage of such algebraic equations, trigonometry and the like.
My argument is, we scare the kids and inflict pain on them with the way we introduce such concepts, when it could be actually so much fun for the kids to discover and learn intuitively.
I share my recent experience – and the sheer pleasure – when I introduced the concept of algebra to my daughter, who will finish her primary school this year.
Intuition vs techniques & short cuts
Before getting into algebra, I wanted to ensure that her intuitive understanding of arithmetic is strong. For instance, she should be able to appreciate that the arithmetic operation of division (and thus fractions) is the same as the intuitive act of dividing a pizza and sharing the slices among us. I didn’t ask her that simple a question. I asked her this:
Divide 50 by half.
Did you have the number 25 flashing in your mind ? She too fell for it. Well, the answer is 100. The usual mathematical approach to explain the solution has been:
Dividing 50 by (1/2) is the same as (50/1) x (2/1). Thus the answer is 50 x 2 = 100.
However an intuitive way to understand this, is to use a real life example. Imagine 50 chocolate bars and consider you have to share those with a lot of children by cutting each chocolate by half. Now did 100 flash in your mind? I guess you didn’t have to strain your mind. Intuition. Which is our natural gift to learn maths (or anything) with a bit of fun.
Understand infinity by intuition
For instance, the concept of infinity is not easy to grasp by intuition. But let us try the same approach as above: If you slice an orange by half, you get two portions. If you slice it by one-thirds instead, you would get 3 portions. If you cut that fruit into thinner and thinner pieces (one-hundredth or even one millionth) you will be left with a large number of slices. Now attempt a thought leap to cut it into almost invisible slides (thinness=zero), you would get an infinite number of pieces.
Thus 1/0 = ∞
Marcus du Sautoy explains this and much more in his brilliant BBC TV series The Story of Maths. He narrates how the whole world, east and the west, have contributed to the evolution of mathematics across centuries.
Algebra: plums, peaches, weights and the scale
The word Algebra is derived from Arabic “al=jabr” meaning “re-union of broken parts” (source: Wikipedia). Marcus narrates how Chinese traders used an intuitive way to solve real world problems in the ancient times. One such problem is to determine the weight of a plum and a peach while they had the following situation:
One plum and three peaches weigh 15 grams. Whereas, two plums and one peach together would weigh 10 grams. Their approach is as follows:
The trader would place one plum and three peaches on one side of the scale, balancing with 15 grams on the other. Now he would double on both ends of the scale by adding one more plum and three more peaches, which requires 30 grams to balance. Then he would go ahead and remove both the plums as well as one peach from the scale. Left with five peaches on one end, he would find that it takes 20 grams to balance the scale. Thus a peach would weigh 4 grams. It is then straightforward to deduct that a plum would be 3 grams in weight.
This was a fun, intuitive way to solve a real world problem.
Contrast this with the mechanical way of solving equations that I would have to teach her:
Let a be the weight of a plum;
Let b be the weight of a peach:
Equation one: a + 3b = 15
Equation two: 2a + b = 10
multiply equation one by 2:
Equation three: 2a + 6b = 30
Subtracting two from three gives us, 5b = 20, thus b = 4 and then you can use equation one to deduce a = 3
Though this method would not torment her as such, I guess she would have little clue on what was accomplished in the end. At least with plums and peaches approach, even if she fails to solve the problem, she is only going to run away, with a healthy snack in hand!
Maths by story tellers:
I found Dan Meyer‘s blogs and Salman Khan‘s videos to be quite popular. They are some of the the new age gurus who have brought about a paradigm shift in the way maths is taught. When I assumed that this transformation is restricted to the US and developed countries, it was inspiring to watch the story of a math teacher in Morocco (Math in Morocco: Where Math Grows on Trees) which shows the passion with which he teaches an array of mathematical concepts by taking the children through a journey of growing olive trees in the school, measuring its produce, carrying the olives to a nearby traditional olive press powered by a camel.
Must admit, I find myself immersed and lost in these stories a lot more than in teaching her any maths.
We both love stories. Maths can wait!