Posts from the ‘Teaching & Learning’ Category

Understanding Mathematics

In schools, we see that many teachers and students try to focus on learning a lot of rules and formulae in mathematics without even trying to understand the ‘reasoning’ behind it. Since they manage to score good marks when they learn like this, they don’t see anything wrong in this way of learning. That is because, neither these students nor most of the school teachers are aware of the perils of this type of learning. Let me just try to explain it with an example.

Suppose there is a question on writing a given set of numbers in ascending/descending order. The kind of numbers they get in school for such a question are: 1/5, 3/7, -2/3, etc…which can be easily done by the way of finding LCM. Probably, the focus in school is to learn the ‘method’ of comparing fractions using LCM. But what if you get some different numbers like 13/18, 2/7, 14/19? If the students try to find the answers using LCM method, then they’ll take a lot of time in multiplying the denominators because they are bigger numbers as compared to the earlier question. And there’s a great probability of making an error in calculation too.

As a school student (or teacher), they might not be much bothered about the second question because they don’t get questions of this difficulty level in school exams. Here lies the problem!
The second type of question might be asked in an aptitude exam later in their life when they apply for a job interview or entrance exam for higher studies. At that time, they just know one method of solving the question….the rigorous LCM method…by which they end up using a lot of time (which they cannot afford in an aptitude test) and making errors. By this time, it’s almost too late and no one is going to teach them fraction-addition at this age.

Realizing this at an early age can be a blessing. While we are learning or teaching fractions in smaller classes, we can teach them alternative ways of comparing three fractions using logic. Let us now consider the second example of fractions again and compare them using logic.

A = 13/18,

B = 2/7,

C = 14/19

Comparing A and B using cross multiplication we get A>B. At this step, all we have to do is 13*7 and 18*2. Without even getting the results, we can look at those numbers and say that 13*7 will be greater than 18*2. So, we can immediately say A>B just by MERE OBSERVATION!

Now we can compare A and C. 18*14 > 13*19. Hence C>A. This is the ONLY CALCULATION one has to do in this question. (if a student is trained properly, they can say C>A by MERE OBSERVATION without actually finding the results 18*14 and 13*19 because average of 18 & 14 and 13 & 19 is 16. In such cases, the numbers nearer to the average, i.e.,18 and 14 in this case, will have a greater product than 13 & 19).
Thus we have got A>B and C>A, hence C>A>B!

In an aptitude test, the time given to solve this question might be 1 to 2 minutes depending on the type of exam. Now, many students (who can’t think…rather, who are not trained to think) might say that the second question will take more than 2 minutes to solve and hence the exam was difficult. But the fact is, they are not able to think logically and solve a problem…they just know one way of solving the problem. The world today does not want such people who cannot Think.

Thus, when we are teaching and learning such topics in school, we should be trying to find logical ways of solving along with conventional methods. This can happen when we shift our main focus of teaching/learning from Scoring Marks to UNDERSTANDING MATHEMATICS.



Learning – The Other Way

Once I happened to meet somebody with whom I was having a conversation on Mathematics. He was telling me his story as to how haunting the subject was for him during his school and college days. He would top in every other subject but Maths…would take his percentage down. Somehow, with the help of his friends (before and during the exams) he used to pass in Maths. When he was in his twelfth standard, he confessed to his father that he won’t be able to clear the exam without having tuition. So his father arranged a Tuition Master for him. This Sir had a stammering problem due to which he didn’t get a good job in schools and made his living by taking private tuitions. Before he started going for the tuition, he had scored 4/200. But after four months coaching, he scored 168/200 in his final exams. Of all the people in the world, he (the student) had received the greatest shock. After he thanked his Sir he asked, “Sir, how come I got so many marks in Maths? I never used to understand this subject no matter how hard my professors & I tried. How did I understand from you?” His Sir replied (stammering), “My boy…the pause taken due to my stammering, gave you time to Think!”

Today, what usually happens in schools & colleges is Teaching; and not Learning. The teachers teach, write something on the blackboard and the students copy them down and study. As a result, they just become photocopying machines, with no Actual Knowledge happening inside them. Maths is one subject, which just cannot be learnt like that. And if someone tries teaching this way, he may succeed in teaching a very few and creating nightmares for the remaining students. Maths cannot be taught. It has to be discovered within us. It’s like music. Every student who learns Classical music, has to discover the correct musical notes within him.

Look at the world around. Don’t we see Maths everywhere? Who taught the birds speed, time & distance? Who taught the honey bee to make every cell in a hive in the shape of a hexagon? Who taught the spider to create a web in a particular model? Innumerable are the examples in the case of animals and birds to prove this fact. In case of humans also, it’s not different. Let us take a common example that we have seen many a times. Imagine a fielder on a cricket pitch. When he runs towards the boundary line to catch a ball flying through the air, all he does is keeping his focus on the ball. But unknowingly, he calculates the speed of the wind, the way in which his palms should come together to catch the ball, how fast should he run, what is the shape of the arc that he has to take to catch the ball, at which spot is the ball going to land, etc. How does this happen?

Our brain has an inbuilt capacity to do Mathematical calculations even without our knowledge. Since it is done so fast, we don’t realize such a happening. Maths is in our blood. We just have to bring it out. And this can be done when we get time to think. Children can do a lot better if they are given time to think. This is what our Maths teacher, Sir P.P.Raman, taught us. He used to give us a particular problem in geometry and go out of the class for a break. By the time he comes back to the class, most of us would have got the solution to the problem, and that too not just one but different solutions. This was how he taught some great things in life – Any problem can be solved in different ways. The clue to the solution lies in the problem itself. Keep meditating and the solution will appear.

I tried experimenting this in my classes of Vedic Mathematics for secondary school students. I stopped teaching the usual way. I started writing down problems and the answer without showing how I derived it. Then all I had to do was, challenge the children to find out the method. Within no time, they started finding out the methods, without being taught. The joy that I saw in their eyes was something that cannot be explained. They started feeling as if they have become Scientists or Mathematicians doing their own discoveries. They never forget these methods of calculating because – it is ‘their discovery’. Not just that, after a few days, some children (who were just average in Maths) came up with new methods of doing some particular calculations which I didn’t know.

This was the beginning of a whole set of students crying out as Maths as their favourite subject. How much confidence they must have gained in themselves after deriving such methods? What a change will it bring in their lives? The effort taken was just a shift in the way of teaching.

Let’s try something different. Let’s give them time & opportunity to Think. And the children will come up with better ideas. Let LEARNING happen, the OTHER WAY!