Friday, April 10, 2015

Have You Checked Your Work?

"Have you checked your work?"
I reckon this is one of the most common questions that teachers ask their students. And it is usually more of a rhetorical question as we ask it when we know that students have made a glaring error or have come up with an answer that makes no sense.
And usually the students look briefly at the work and say "Yep!" as if to say "Yep, it's still there!"
Checking your work is a vital piece of Polya's four step approach to problem solving. In many ways, I think it should feedback into the first three parts (Understand the Question, Make a Plan, Carry Out the Plan). My sense of things is that this maybe the weakest link for a lot of students.
I have been wondering why this is and now I'm thinking that maybe it is the question itself ("Have you checked your work?") which is causing the problem. Maybe there are better prompts to get the students to reflect on what they have written.
As a case in point, I was in a Grade 9 class recently and they were working in groups on this problem: try it yourself before you read on!!
What was interesting is that many students stopped after they found an answer. They didn't think to consider of there was more than one answer. It was almost as if they stopped out of habit: I have an answer, so now I'll just wait for further instructions.
Biting my tongue, I managed to avoid saying "Have you checked your work?" Instead I asked "Which two sides are equal?" They pointed to the 3x-4 and the 5x-8. I then asked "Are these the only possibilities?" 
Immediately, this had a much better effect than "Have you checked your work?" The students realised that the 3x-4 could be the same as x+6, or the 5x-8 could be the same as the x+6. There were some comments along the lines "They don't look the same!" but they were reminded that they were told that the diagram is not to scale!

But even if they had considered each of the three possible isosceles triangles, and had done the algebra correctly, there was one geometric error that kept on coming up and was not seen by the students: notice the 2cm, 2cm, 8cm triangle:
"How do you know this is an isosceles triangle?"
"Because two sides are the same."
" do you know it is a triangle?"
"Errr...because it has three sides that are joined together, (said in that 'D'uh' tone!).
"OK...if you can draw me that triangle, then I'll give you twenty dollars!"
Sometimes when we ask students to check their work, they might just check the algebra (which in this case was correct) but not realise that what this leads to is an impossible shape. Which is why I really like this question as it will force students to appreciate the triangle inequality: any two sides of a triangle must add up to more than the third side.
But to get students into this habit of thoroughly seeing if their solutions make total sense, we must provide them with rich questions that force them to look for different cases, or to examine the validity of their solutions.
If, however, we only provide them with questions where there is only one answer (and probably just one way to get this), then they won't get into the habit of effectively reflecting on their work. Thus, we cannot expect them to be complete problem solvers.
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The problem above is adapted from the University of Waterloo's CEMC's Problem of the Week. 

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