| I saw this quote in a BBC Program recently. Generally in school we are not encouraged to see mathematics as a visual recreation -- it is taught as a boring lifeless subject.
But in reality it is the most vividly scintilating, uncomprimising, visual feast. To make the students realise this, a visual approach is needed wherever it is possible. This will make the right side of our brain to work with our mathematic left brain to churn out beautiful output.
If we get a simple geometric understanding then the problem is half finished.
He can visualise either by himself or he can plot the graphs using a Graphing calculator.
For example In Linear algebra when we work in simultaneous equations to know whether it has 0,1 or many solutions we use matrix methods. But after this if we draw the curves (or lines in this case) we can clearly see our results in an intuitive way.
Similarly we can cross check the roots of a Quadratic equation with the Graphing calculator by drawing the curve and finding the root. An approach like this should be introduced in our school. In India we can't ask every student to buy a graphing calculator. But just for financial reasons our children should not be deprived to enjoy the beauties of maths.
An viable option will be that, the school can finance one graphing calculator and a projector for every class. This will enable everyone to enjoy the benefits in an economic way. This will make the students know what they are doing.
Why Visualization is important
I will show two very simple formulas which we can derive visually. This idea occured to me 2 years back. A very simple idea that lot of people might have got this.
Addition of positive integers
Take addition of numbers 1 + 2 + 3 + 4 = ?
Of course the formula n * ( n + 1 ) / 2 is there but how did it come to be? If we think geometrically then, say, every number is represented here by *
*
* *
* * *
* * * *
Get a new figure by rotating the first figure 180 degrees in the plane of the screen and join them at the right hand side.
* * * * *
* * * * *
* * * * *
* * * * *
You will get a rectangle like this
* * * * *
* * * * *
* * * * *
* * * * *
So, adding two times our original figure is 4 * 5 = 20. So we have to divide this area by 2
1 + 2 + 3 + 4 = (4 * 5)/ 2 = 20 / 2 = 10
so addition of positive integers is the area of a rectangle formed by two triangles divided by 2.
So it is very easy to see that
1 + 2 + 3 + 4 + ..... n = n * ( n + 1 ) / 2 .
Addition of positive odd integers
Now see the addition of even numbers 1 + 3 + 5 + 7 = ?
Let's see the formula straight from the figure
*
* * *
* * * * *
* * * * * * *
Separate the first three columns and rotate it 90 degree in the plane of the screen and join it in the righthand side.
* * * *
* * * *
* * * *
* * * *
Now join them
* * * *
* * * *
* * * *
* * * *
So the result is 42 = 16.
The addition of first 4 odd numbers
1 + 3 + 5 + 7 = ([7 + 1]/2)2
where 7 is ([2*4] - 1 )
so addition of n positive odd numbers is
1 + 3 + 5 + 7 + ..... 2n-1 = n 2
For most of the addition formulas there is a simple way geometrically like this which we can encourage the students to find themselves.
The Gauss way
The formula which Karl Friedrich Gauss discovered in his 8th Grade class is the best.
Take 1 + 2 + 3 + 4 + ... 100. If we pair it up like
1 + 100
2 + 99
3 + 98
...
50 + 51
We will get 50 pairs of integers with a sum of 101.
50 * 101, which is, (n/2) * (n + 1)-- which gives our general formula
n * ( n + 1 ) / 2.
Every child can become a Gauss if someone lit the fire in the child's brain.
Which brings us back to the quote
A student's brain is a fire to be lit, not a vessel to be filled.
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