In my previous column I challenged the use of tricks for performing calculations. My point was if you don’t understand what you are doing the answer you produce is worth nothing. Not everybody agrees with me.
Many teachers tell me that as long as the child gets the correct answer they are happy. In this column I want to challenge the assumption that the correct answer shows that the child is doing what we think they are doing. I also want to make the point that errors may be as useful to teachers as correct answers are exciting: errors give us a window into a child’s way of thinking. By contrast, correct answers don’t necessarily mean the child is doing what we think he or she is doing.
Some time ago I watched a young grade two boy doing some addition sums. I am going to reproduce the approach he used by showing his work in a number of stages in the illustration to the right.
This boy worked on a worksheet on which there were more than 40 addition sums and he got all the answers right. What he was doing was unusual in terms of the column-addition algorithm in that he was adding from left to right. But that did not cause him to get any wrong answers because in none of the questions did either the units or the 10s complete or bridge the 10. In classroom-speak: none of the sums involved a “carry”.
On the face of it this boy was getting the right answers and appeared to know what he was doing. And yet I had a nagging doubt. I was dissatisfied and decided to check the boy’s understanding by asking him to do two further addition sums for me. See the illustration to see what he did.
Before I continue, let me be clear about something. This boy was as bright as a button. He did all the single-digit arithmetic in these addition sums in his head. He did not need counters, neither did he use his fingers. But he also did not hesitate when adding eight and five in the third sum to write down 13. He treated the sums I had given him as more of the same that his teacher had given him earlier.
But does he understand?
I am sure some who are reading this feel I was unfair to the boy because I asked him to do addition sums involving the bridging of 10s before he had been taught how to do this. I disagree. The point is not whether I was asking something that he had been taught or not — the point is whether or not the boy was adding 23 and 45 in the first sum at all. He also did not think of his answer as 68 but had produced two answers: six and eight.
My claim is that he was actually doing two sums: one in the left-hand column and one in the right-hand column. He calculated 2 + 4 = 6 and wrote down his answer. He then calculated 3 + 5 = 8 and wrote down that answer. I claim that he had not done 40 two-digit plus two-digit additions sums but 80 single-digit addition sums.
On the face of it, it looks as though he added 28 and 45 and got an answer of 68. But this is not the case. When I asked him to add 28 and 45 he did not get an answer of 613 (even though it may look like that); his working shows that he added two and four to get six and then did a second sum adding eight and five to get 13. He wrote down his answers and went on to the next sum.
The sceptical reader may argue that, with time, this boy will learn how to “get the right answer”. I have two responses. The first is: I use a grade two example to make a point because it is more obvious and more easily accessible to a wider audience.
There are many similar examples across the grades and phases that I could use. Recently a colleague told me how he noticed exactly the same in a grade seven class where children were adding decimals in columns.
The second comment is that, although I agree that we may be able to coach this boy to get the “correct answer” for addition sums involving bridging (or carrying, if you prefer), the point remains that those “correct answers” will not necessarily indicate that the boy understands what he is doing any more than the “correct answers” at the beginning of this article did.
I want to draw attention to the fact that watching the boy doing his work caused me to have doubts about the “understanding” that the “correct” answers demonstrated.
The “error” in the second set of sums confirmed my suspicion. Correct answers do not necessarily indicate a child is doing what we think he or she is doing and errors can help us to understand a child’s thinking better than a seemingly correct answer does.
Celebrate errors as insights into a child’s thinking and be suspicious of correct answers. Ask: “Does the child really know what he or she is doing?” and reflect on whether or not you are teaching for understanding or for correct answers.
Aarnout Brombacher is a private maths consultant. For more information go to www.brombacher.co.za