In this article, Zack outlines a problem and erroneous solutions she gave her class. The 'chessboard' problem asks students to find all the squares in a 4x4 grid, then extends to a 5x5, 10x10 and 60x60 grid. She then gives a solution that a 10x10 grid has 385 squares, which you can multiply by 6 to get 385 X 6 = 2310 squares. Her students adeptly identify that this is incorrect, and she outlines 5 counterarguments given by nearly every student in her class.
I had a lot of trouble accepting Zack's arguments as, based on my teaching experience, her results seemed nearly impossible! The language Zack gives in her counterarguments is clearly her own, and does not convince me that it is what a Grade 5 student would think about the problem. I would imagine that a 60x60 grid is quite an abstract concept for a 5th grader, and to concieve of numbers of squares in the thousands (apparently, the correct answer is 73 810 squares) seems to be quite a tall order. I felt there must have been quite a bit of coaxing from the teacher/interviewer to encourage the students to come up with these detailed counterarguments - I was not convinced that these would represent Grade 5 students' independant thinking.
Having criticized Zack's methods, I do appreciate her analysis on the value of counterexamples, and teaching these from a young age. She says "Having a diverse assortment of counterarguments is beneficial. The counter-arguments tell us different things about the mathematics in the task." I think encouraging various counterarguments is a great way to help students consolidate different areas of Math across topics.
Encouraging diverse views and strategies for engaging in mathematics is beneficial to everyone, including the teacher. Inclusion of different counter-arguments not only increases students' confidence levels but also recognizes others' viewpoints which may lead to math discoveries. I recall that five or six years ago, there was a topic in the math 11 curriculum which covered conditionals. converses, contrapositives, and counter-arguments. My students were more excited about giving counter-examples to false prepositions than about proving true hypotheses deductively. For example, there was a proposition in one of my lessons which stated, "No triangles have more than one obtuse angle." Two of my students thought it was false and countered it with something like (+160, -120, +140 degrees). They provided a pretty decent counter-example in the sense that angles can be negative in some contexts. They obviously thought of an angle in the clockwise direction in the xy plane without being aware of angles in a real-life context. Their answers led me to talk about angles used in different applications with my students. Hence, counter-arguments provide interesting teaching and learning opportunities.
ReplyDeleteKevin, very interesting example. I know in post secondary math classes some teachers try to introduce `proof by contradiction' last, because it's a technique that students seem to be able grasp easier than others, and they will tend to use it as a crutch when other strategies might be more enlightening. It could very well be, though, that such students become so adept at it through this exact type of education! The mathematician in me is certainly rearing her head this week: I'd be so interested to see what similar early stages of proof techniques could be taught in the earlier years.
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