Reference: Critical Thinking in Education
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The most useful aspect of mathematics is that it provides opportunities:
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To think outside the box.
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To help learn something new.
Thinking outside the box
There’s a popular story that Gauss, a famous mathematician, had a lazy teacher in his elementary school. The so-called educator wanted to keep the kids busy so he could take a nap; he asked the class to add the numbers 1 to 100.
But Gauss found the answer in less than 10 minutes, and he interrupted the teachers nap with his answer: 5050. So soon? The teacher suspected a cheat, but when he looked at Gauss’s method, he realized that he had a genius in his class.
Here is what Gauss did. He was required to add the first 100 numbers as follows.
1 + 2 + 3 + 4 + … + 98 + 99 + 100
But he split the numbers in two groups (1 to 50 and 51 to 100), and arranged these numbers as follows
1 + 2 + 3 + … + 48 + 49 + 50
100 + 99 + 98 + … + 53 + 52 + 51
Each row had fifty numbers. He added the corresponding numbers as follows.
1 + 100 = 101
2 + 99 = 101
3 + 98 = 101
…
48 + 53 = 101
49 + 52 = 101
50 + 51 = 101
Gauss found that the final sum would be
101 + 101 + 101 + … (50 times) = 101 x 50 = 5050.
This was thinking outside the box. Mathematics provides many such opportunities.
EXERCISE: Add numbers
(a) 1 to 20
(b) 1 to 50
(c) 1 to 33
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Helping learn something new
Mathematics also provides many opportunities to help learn something new. For example, the feel for numbers is very important and it helps one learn to add very quickly.
Part of the feel for numbers is to know the gap between a number and the next TEN.
This gap can be seen on a number line at the beginning of this essay, where it helps add
39 + 5 = 39 + 1 + 4 = 40 + 4 = 44.
66 + 8 = 66 + 4 + 4 = 70 + 4 = 74.
Here the gap is filled first by the second number and then the rest of the number is added easily to a TEN.
A student and his or her study partner can drill these gaps. One of them calls out a number and the other responds with the gap. Such drill is a lot of fun, when the numbers called out are random.
Number Gap
9 1
8 2
7 3
6 4
5 5
4 6
3 7
2 8
1 9
27 3
49 1
54 6 etc.
The fundamental aspects of mental math can be learned quite quickly with such drilling. But any such drilling must be followed by proper understanding. For example, the student must first understand that multiplication is “repeated addition” before he or she drills the multiplication tables.
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