Why math should not be taught abstract in early stages?
by Amrudesh Santhanam
5 + 2 = 7
That's great... But how is this?
5 m + 2 sq.m = 7
Believe me. I have had to face this with some students. Not just high school students. What would you say if an engineering graduate said
F = mx' + k
Where F is force, m is mass, x' is dx/dt and k is stifness?
I wonder where the units are gone? Can any amount of force be equal to momentum or stiffness? Of course not !!! I really feel put down when I see this.I had one other difficulty recently. I was teaching trigonometry to a student and found that he was totally confused when he had to use
pi = 3.14159
pi = 180 (strictly speaking wrong. I will show the correct version down the line)
I also see students struggle with something like
9 + 5 * 3 . They end up solving it as
14 * 3
9 + 15
All this has lead me to think deeply on what problems in teaching methods have they seen in their early stages that lead to these problems. I am quite strongly convinced that all these are the result of teaching math in a very abstract manner in early stages. When I say abstract, let me be clear. We teach them numbers as if numbers had an independent existence and that they can be used without reference to any physical entities. I do know that abstract math is a field in itself and a wonderful field. I too have had a go at it during my college days but I am sure all would agree that for all practical purposes we use numbers to count real entities. I will base my argument on this henceforth.
Also, let me make it clear that this is a phenomenon I have seen in India and I have no clue how it works in other places. I was born here, studied here and work here as a mechanical engineer, (for an international aerospace company though). I teach here too, especially in a rural school run by a non-profit organisation. I have also taught other students in the city and interviewed many many engineering graduates as part of my job and all through, I see these issues. The F = mx' + k is a classical example of an engineering graduate's response during an interview which I was taking of him.
Why we have this problem?
I hope that by this time, you too have a clear understanding of the problems we end up with. Let us see why each of those problems come up.
5 m + 2 sq.m = 7
This is a classical error. I believe this arises primarily because in primary school, we believe that children need to learn numbers and even addition, multiplicaton etc. Good but how? If I am not wrong, one of the first few things a child counts is either it;s own fingers or something it is playing with. Parents teach a kid that it has 4 chocolates with it or 5 fingers in it's one hand. Lovely because at this time we are dealing with physical entities still
When the kid goes to school (or even pre-school) though, we start numbers more formally and we still use physical entities initially. We show them an apple and say "1 apple", two ducks and say "2 ducks" and henceforth. All is fine.
A little later though, we start believing that the kid has got the hang of the concept of numbers and start teaching
1 + 1 = 2
2 + 3 = 5
But wait !!! Where are the units now? We have let go of it for the sake of convenience. OUR CONVENIENCE !!! but what about the kid? Does he/she understand that while
1 apple + 1 apple = 2 apples is true,
2 mangoes + 3 oranges is not equal to 5 anythings
I strongly believe that the kid does not understand this. It is too early to understand the concept of 'abstraction'. We go further and teach
3 * 5 = 15
4 * 5 = 20
9 * 9 = 81
and so on...
All this while, the kid starts believing that numbers have an existence of their own and that we can multiply any number by any number, add any number to any number and so on..
We start teaching physics (as part of science) later and then slowly introduce the concept of units.
Mass is measured in kilograms or pounds or so (depending on which country we live in)
Distance is measured in feet or metres ...
But by this time, the kid is quite confident about mulitplication, division etc (at least with single digit numbers if not more)
We start teaching that
speed = distance traveled / time taken
area of a square = length of each side * length of each side
These again go on in the classroom on the blackboard without kids getting a feel of what speed is or what area means. This is when they start memorising the formulae. Well. and we all know. Memory can fail us quite easily. Then, one fine day, when the teacher asks the kid what is the formula for speed, distance and time, the kid (no longer so much a kid) says
s = d * t
Oh my god !!! Then the teacher scolds the students (not kid anymore) for understanding wrongly. But hey !!! When did the kid 'understand' the formula? He/she only 'memorised' it. "Whats wrong about multiplying 'd' by 't' ? " After all, I have been taught that any number can be multiplied by any other number. So, if
d = 5
t = 3
s = 5 * 3 = 15
Now, we ask ourselves where we have gone wrong and hence got the kid to go wrong.
Problem - 2:
F = mx' + k
(which should have been F = mx'' + cx' + kx, the equation of motion)
This is nothing but an extension of the first problem. Just that is has gone too far. Such an engineer would have to go back to primary school level to understand why he/she is wrong. Its too late now?
Problem - 3:
About pi. As we know correctly,
pi = 3.1415926535..... and so on.. It is also known in short as
pi = 22 / 7 = 3.14
Sin ( pi / 3 ) = Sqrt(3)/2
Here, the student is supposed to know that in this case
pi = 180 (wrong !!! )
but no. Please
pi is not equal to 180
pi radians = 180 degrees (right !!!)
We know which is the right one but try explaining now to the student that while
Sin ( pi / 3 ) = Sin ( 60 ) = Sqrt(3) / 2,
pi is not equal to 180.
Unless they know the difference between 'pi' as just a number and 'pi radians' as an angle, this would be tough. They will keep intermixing these.
I have had to teach my student that
We give 'angle' as an input to the Sine function but a 'ratio' to the Inverse Sin function and why. Its not been easy. He would end up doing
Inverse Sin (90) = 1 (wrong !!!)
Sine of any angle cannot cross 1 or go below -1 and that's not easy to teach now :((
Please do not teach kids numbers without referring to units.
Always, please teach that
5 mangoes + 2 mangoes = 7 mangoes
3 baskets of 5 apples each = 5 apples + 5 apples + 5 apples
= 3 baskets * 5 apples per basket = 15 apples
3 dollars + cost of 4 apples (if one apple = 0.5 dollars) is
3 dollars + 4 apples * 0.5 dollars / apple
= 3 dollars + (4 * 0.5) dollars
= 5 dollars
Else, when they come to write
F (kg m / s^2) = m (kg) * a (m / s^2) (correct !!!)
, they might end up writing
F = m * v (wrong !! Force cannot be equal to momemtum)
Hence, we will also try in Edoola to deliver tests online for primary school kids with units always and
NOT FORSAKE THEIR LEARNING FOR OUR CONVENIENCE !!!
Do feel free to give your comments, especially about your experience if any in similar terms across the globe.