This article talks about the different phases involved in Math learning, across the globe, across different stages of school, college. And how each phase is vital to learn math better.
Below is an extract from a post by Fields medalist, Mr. Terence Tao, The author makes these points in the context of higher education in math. But the idea is perfectly valid for school math learning as well.
One can roughly divide mathematical education into three stages:
The “pre-rigorous” stage, in which mathematics is taught in an informal, intuitive manner, based on examples, fuzzy notions, and hand-waving. The emphasis is more on computation than on theory.
The “rigorous” stage, in which one is now taught that in order to do maths “properly”, one needs to work and think in a much more precise and formal manner. The emphasis is now primarily on theory; and one is expected to be able to comfortably manipulate abstract mathematical objects without focusing too much on what such objects actually “mean”.
The “post-rigorous” stage, in which one has grown comfortable with all the rigorous foundations of one’s chosen field, and is now ready to revisit and refine one’s pre-rigorous intuition on the subject, but this time with the intuition solidly buttressed by rigorous theory. The emphasis is now on applications, intuition, and the “big picture”.Fields Medalist, Mr. Terence Tao
Pedagogy should also be designed to focus specifically on intuition in the beginning, to add layers of rigour on to this, and to revisit the intuition at the end of it. Let me outline this with a simple example.
Ask an 9 (or 10)-year old the following question. Dad and son share 10 chocolates. How many do you think Dad should get and how many you should get? Let the kid think about this. One possible answer is 5 each.
Whatever the first answer might have been follow it up with, since Dad is an adult, he should get fewer chocolates than you. Let us try to distribute the chocolates such that you get 4 more chocolates than Dad. Quite a few kids are smart enough to muddle around and land at the correct answer.
Now, say that Dad should get only 2 chocolates fewer than you. What should be the answer? What if Dad were to get 6 chocolates fewer?
End this discussion with the question, what would be the case if Dad has to get 5 chocolates fewer? Let the child think about this for a while.
This is probably done only after class VI. Only after they have a sense of equations and variables.There are totally 10 chocolates. Dad gets x chocolates, you get y. What is x + y equal to? You get 4 chocolates more than how many Dad got. What is the equation? Now, if we solve these two equations, what do we get for x and y. Now, extend the discussion to bring back intuition into the frame.
We saw that if you are distributing 10 chocolates, then Dad cannot possibly get 5 less than how many you got (or can he? Depends on whether the chocolates can be broken down. But that is a separate discussion). Now, if we are distributing 10 chocolates, can Dad get 3 more than you? Can he get 7 less than you?
What if we distributed 11 chocolates? Can Dad get exactly 2 fewer than what you got? Exactly 3 fewer? What are the kinds of numbers that are possible, and which ones are not possible? What has this got to do with even numbers and odd numbers? If a 10-year old can establish the relationship between sum of numbers, difference of numbers, even numbers and odd numbers, it would be a great result (no matter how much prodding it takes)
Math Learning in India
In India, we completely bypass the pre-rigourous phase and therefore have no basis for a post-rigour discussion in Math learning. We drill equations into kids and try to turn them into solvers. Kids love the pre-rigour phase. The curiosity needs to be built and tapped into well during this phase in order for them to appreciate the next two phases. The importance of rigour needs to be reinforced well and early. A lack of appreciation of rigour can be a major lapse.
Dr Terence Tao elaborates on this here
It is of course vitally important that you know how to think rigorously, as this gives you the discipline to avoid many common errors and purge many misconceptions.Fields Medalist, Mr. Terence Tao
The point of rigour is not to destroy all intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition. It is only with a combination of both rigorous formalism and good intuition that one can tackle complex mathematical problems; one needs the former to correctly deal with the fine details, and the latter to correctly deal with the big picture. Without one or the other, you will spend a lot of time blundering around in the dark (which can be instructive, but is highly inefficient).
So once you are fully comfortable with rigorous mathematical thinking, you should revisit your intuitions on the subject and use your new thinking skills to test and refine these intuitions rather than discard them.
Two types of Limited Students
As a teacher, I see two categories of limited students.
1. Kids either completely lack intuition. In which case, they appreciate the ‘solver’ module in maths, and become experts at handling anything routine thrown at them. But they run out of ideas once the set of frameworks is exhausted. Their ability to discover new frameworks is limited. The vast majority of Indian students (I would put the number at higher than 95%), belong to this category. Our evaluation mechanism barely tests anything more than rigour in standardised frameworks.
2. Kids without appreciation of rigour. These are the over-appreciated smartalecks who have a feel for what value to substitute, but think it is beneath them to verify all possibilities. The Sachin Tendulkar who might have refused to practice. Their intuition plateaus out. For most, they do not discover the plateau. Not because the plateau appears so late, but because they never explore that far. These are the kinds that say that the ocean has no floor because they went in hundred yards from the beach and found solid ground throughout.
How to Fix these limitations?
In the first kind, the error in Math Learning is with the pedagogy. It is far tougher to build basic intuition than to build basic rigour. So, our system has reached a state where there is no effort made to build intuition. For the second kind, the error is with the attitude. Either way, one needs both. An appreciation that one needs both is very important.
The ideal state to reach is when every heuristic argument naturally suggests its rigorous counterpart, and vice versa. Then you will be able to tackle maths problems by using both halves of your brain at once – i.e. the same way you already tackle problems in “real life”.Fields Medalist, Mr. Terence Tao
For parents, in the early stages make subjects as intuitive as possible. As David R Garcia points out here, learning happens inductively in the early stages. Our pedagogy should follow that pattern. Plant as many seeds as possible for young kids to ‘discover’ patterns, then wrap rigour around it and get ready for the next round.
Rajesh Balasubramanian, IITM, IIMB, Director PiVerb. Changing the way math is taught, one video at a time.