Jordan Rappaport, Educator, Brown Ridge Public School, YRDSB
Susan Lamon estimates that over 90% of students who enter high school cannot reason well enough to learn mathematics and science with understanding and are unprepared for real applications in statistics, biology, geography or physics (Lamon, 2005, p. 10).
Without opportunities to think and reason, how can we expect students to build understanding?
In the absence of context, and opportunities to explore math concretely and visually before moving to the abstract, how can we expect students to develop flexibility in their thinking and construct understanding?
In this task, a student (let’s call her Amy) was given a piece of metal from her shop teacher measuring 28cm long and 20cm wide. The assignment was to form the box by cutting squares out in each corner and bending the sides up. The shop teacher will give extra credit to any student who forms the box that holds the most. How large of a square should Amy cut from each corner to form the box with the greatest volume?
Students right away went to creating tables and calculating to show the increase in volume as the L and W decreased in order to see the increase in height. This is where the misconceptions ensured as many students sidestepped the critical step of creating a visual and/or model so they could actually experience the mathematics.
Seeing the benefits of creating visual representation #thinkingclassroom
So I let them go, and while working, gently nudged them towards creating a visual representation. At this point, I started hearing comments like ‘this is not making sense’, or ‘my calculations work, but they don’t match up with what we’ve built.
We came together and I drew a visual for students so they could actually see and then reason their way through what happens every time each corner has the same size square cut out. Almost immediately, students recognized how a ‘wrong turn’ led them astray and how a visual to support the practical application was and is critical.
Learning from their mistakes
A few students asked if I knew ‘they were wrong the whole time?’ I said ‘I knew there were some misunderstandings, but you needed to get to a certain point to figure out what those misunderstandings were.’ Another student then chimed in and said ‘of course he knew, he was doing what he always does, walking around, asking us questions and making us think so we can figure it out.
The process that students went through solving the problem was amazing to watch. I didn’t need to do much by way of facilitating dialogue. Instead, I was able to key in on what students were doing (or not doing) and saying. By recording their thinking and strategies on #vnps, I quite literally am able to stand in the middle of the class and monitor the work of each group.
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