Essay about Cloud
One of the most important findings of mathematics education research carried out in
Britain over the last twenty years has been that all pupils constantly ‘invent’ rules to explain the patterns they see around them.
(Askew and Wiliam 1995)
While many of these invented rules are correct, they may only apply in a limited domain. When pupils systematically use incorrect rules, or use correct rules beyond the their proper domain of application, we have a misconception. For example, many pupils learn early on that a short way to multiply by ten is to ‘add a zero’. But what happens to this rule, and to a child’s understanding, when s/he is required multiply fractions and decimals by ten? Askew …show more content…
This approach involves reading 3a + 2b + b + 4a as 3 apples plus 2 bananas plus a banana plus 4 apples, which naturally becomes 7 apples plus 3 bananas or 7a +
3b, which is correct. Unfortunately this would have helped least where I needed help most, and it would also have encouraged the same mistakes!
As Doug French points out, if k is interpreted as kangaroos, when faced with the expression 2k + k + 4, the students’ “instinctive thought is likely to be ‘4 what?’. And the obvious answer is 4 kangaroos, giving 7k altogether” (2002: 11).
This is not the only problem that can arise from thinking of letters as representing objects. If a and b represent apples and bananas, then what does ab mean? Indeed, if letters are used in this manner and we are asked to convert the equivalence a week is seven days into a mathematical formula relating days and weeks, we could easily be led to w = 7d rather than d = 7w (where w is the number of weeks, and d the number of days, in a given period).
Given the popularity of the fruit salad approach among teachers at the school in question, I would not be surprised if such a misconception was the underlying cause of much of the confusion on the topic of expanding brackets. However, my experience with the collection of like terms led me to search for other possible explanations for such errors.
Misconception 2: Operations vs Answers
My conclusion, confirmed by the literature (Tirosh et