Misconceptions in School Math 1

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Students internalize several misunderstandings or partial understandings at different stages of learning math at school. I am using the term misconceptions to cover these. Misconceptions created in a particular grade can seriously hamper learning of math in the subsequent grades. In this and the next chapter, we will look at some of these misconceptions.

Why do misconceptions arise?

Mathematical concepts are difficult for young students to understand. Hence many teachers convert concepts into &ldquo;rules&rdquo; which are easier for students to remember and apply. For example, a concept in subtraction is that &ldquo;from a given quantity, only a smaller quantity can be taken away&rdquo;. Students can internalize this concept by working on the &ldquo;take away&rdquo; concept using physical materials. But a teacher instead of doing activities, may just give a rule &ldquo;subtract the smaller number from the bigger number&rdquo;.

But many concepts get modified as the grade levels increase. For example, in subtracting with 2-digit numbers, the above rule has to be applied after understanding the context. While subtracting 25 from 43, many students may use the old &ldquo;rule&rdquo; and write the answer as 22 (5 – 3 =2 &amp; 4 – 2 = 2).

Hence many of the &ldquo;rules&rdquo; either become invalid or have to be modified at higher grades. A student who has memorised a &ldquo;rule&rdquo; at an earlier grade can get confused when it is no longer valid and another &ldquo;rule&rdquo; has to be memorized.

Familiarity & Understanding

On the other hand, most adults have the opposite problem. They are so "familiar" with math ideas & calculations that they start thinking that they understand them.

Let us see some examples of the common misconceptions which are formed in math

On Math as a subject

Whole Number Syndrome
 * 1) Math is a collection of number &amp; computation recipes which need to be memorized.

All of us start learning math with whole numbers, which can be easily identified in our environment. The idea of numbers as &ldquo;whole numbers&rdquo; is deeply embedded in our unconscious memory. So even as we learn different types of numbers as we proceed to higher classes, it does not easily penetrate our thinking. If you ask even a group of educated adults to &ldquo;tell a number&rdquo;, you will invariably find that the answers would mostly be natural single digit numbers, i.e 1 to 9.

It is no wonder that many of the rules that we learn to apply in the case of whole numbers are deeply embedded in us and result in misconceptions at higher classes.

Word Problems
 * 1) When we add or multiply, the value increases 				Not true for multiplying with fractions or adding negative integers
 * 2) When we subtract or divide the value reduces 				Not true for dividing with fractions &amp; subtractions with negative integers
 * 3) To multiply by 10, we just add a &ldquo;0&rdquo; at the end 				Not true for decimal numbers. 3.5 X 10 is not 3.50
 * 4) You cannot take a bigger number from a smaller number 				Not true in integer subtractions
 * 5) Always divide the bigger number by the smaller number 				Not true for fractions</li></ol>
 * 6) The number which comes &ldquo;before&rdquo; another number is the smaller number 				Not true for &ldquo;backward&rdquo; counting</li></ol>
 * 7) The longer the number bigger it is 				Not true for decimals. 3.5 is bigger than 3.49876</li></ol>
 * 8) While adding 2 numbers, we just have to add the numbers in the tens &amp; units places. This fails where the total in a &lsquo;place&rsquo; is more than 9.
 * 9) Use &ldquo;regrouping&rdquo; instead of &ldquo;carry&rdquo; or &ldquo;borrow&rdquo;. Children may wonder as to why we never pay back what we have borrowed.

In chapter 14.3 we have dealt with in detail about how many words which have different meanings in math &amp; daily life can cause a lot of misconceptions for students.
 * 1) Language &amp; Math

Take this counter example – &ldquo;There are 3 chairs for visitors in the principal&rsquo;s office. 4 teachers come in for a meeting with the principal. How many more chairs are required in the office for the meeting?&rdquo;
 * 1) Many teachers try to give a short cut &ldquo;If you find &ldquo;more&rdquo; in a problem, then add&rdquo;.

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