The Linear Human Mind

It can't turn-around

It never amazes me that students cannot appreciate when you write something like

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9
= 1 + 2 + 3 + 4 + 5
+ 9 + 8 + 7 + 6
= (1 + 9) + (2 + 8) + (3 + 7) + (4 + 6) + 5
= 4 * 10 + 5
= 45

The second step of the turn-around almost always eludes them.

Actually, another way of going about to do the same thing is:

S = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9
If we invert the sum, we get
S = 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1.

Adding both sides, we get
2S = (1 + 9) + (2 + 8) + (3 + 7) + (4 + 6) + (5 + 5) + (6 + 4) + (7 + 3) + (2 + 8) + (9 + 1)
2S = 2(1 + 9) + 2(2 + 8) + 2(3 + 7) + 2(4 + 6) + 2(5)
Therefore, dividing both sides by 2, we get
S = (1 + 9) + (2 + 8) + (3 + 7) + (4 + 6) + 5
= 4 * 10 + 5
= 45.

I have no idea why it took our kids such a long time to figure out that both ways are the same.

Our learners expect a lot of crutches and hand holding. They want to be told how to think. They don't want to think. And that is worrying me.

Many conundrums in life require time and, if not mere elbow grease, mental grease. Our brains are there to work things out. We need to spend time to work things out.

It can't pick up new tricks

The human mind is also so trained to finding the final solution that it often forgets that the final solution is not exactly the solution we are looking for.

For instance, what is the value of the second last digit of the sum
66669 + 6669 + 669 + 69 + 9?

Many students are very content to solve the entire sum proclaim that the answer is 74 082.

They plain forgot that the answer is looking out for the second last digit.

Teach the child to use a butcher's knife and s/he'll use it even to kill a little chick.

After you teach them some technique to solve questions that involve larger numbers, they immediately resort to the technique without asking themselves if it was necessary.

For instance, how does one solve
8 + 18 + 28 + 38 + 48 + 58 + 68 + 78 + 88 - 9 * 8?

It is amazing that few realise that the answer is as simple as:
8 + 18 + 28 + 38 + 48 + 58 + 68 + 78 + 88 - 9 * 8
= 10 + 20 + 30 + 40 + 50 + 60 + 70 + 80
= 360.

I guess all I can say is creativity and innovation requires a keen eye for detail and an open mind to consider all possibilities.

We may have become too reliant on fixed gauges of success that we have forgotten the true measure of a person is on his/her extended ability to apply what s/he has learnt in school for life and not merely to solve test-style problems.