Overgeneralising Causes Multiplication Errors
Correctly, there is an emphasis that we can only add (or subtract) things that are the same. For example, 5 ones and 3 ones are 8 ones, 4 tens and 2 tens are 6 tens. So 68 is the correct answer to 45 + 23. We can only add tens to tens and ones to ones. It is very natural, then, to take this idea and use it inappropriately with multiplication.
Such inappropriate generalisation (ie. overgeneralisation) is a common way that students develop misconceptions. When we meet something new, our first reaction is to see if one of our current ideas or skills will be useful. Students need a good reason to abandon an idea or skill that they have used before in favour of a new one. We need to help them appreciate that they do, in fact, need to learn something new.
A common error in multiplication arises from overgeneralising the principle of only adding likes to likes. The 4 calculations below are based on only ‘combining’ (with addition or multiplication) like things. Note that only the addition is correct. The next three multiplications are typical student errors.
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Addition of whole numbers
Correct |
45 + 23 68
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5 + 3 = 8 4 + 2 = 6 (or 40 + 20 = 60)
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Multiplication of whole numbers
Incorrect |
45 × 23 15 800 815
|
5 × 3 = 15 40 × 20 = 800 |
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Multiplication of decimals
Incorrect |
4.5 × 2.3 8.15
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5 × 3 = 15 4 × 2 = 8 |
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Multiplication of fractions
Incorrect |
4 1/2 × 2 1/3 8 1/6
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1/2 × 1/3 = 1/6
4 × 2 = 8
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