This year I have made a lot of great mistakes to learn from and this blog has been a great place to reflect. Here is the latest greatest mistake:
In Geometry we were talking about classifying triangles by side length and angles. I asked students to classify an many different triangles as possible. About 2/3 of them did it using pictures paired with words, while the other 1/3 used only words.
This alone is really interesting to me. So much of math seems to revolve around definitions that are created using words. But at the same time I can define parallel lines by drawing them, or classify an angle as 90 degrees with a simple symbol.
I went along with students and created the following collection of triangles.
As I asked for each triangle, students were pairing perfect definitions to them, which made creating a visual representation easy.
After we came up with 7, I didn’t say much else and started them on a worksheet.
About 5 minutes into the worksheet students ran into this:
We stopped for a little and talked about it. Triangle BDC is equilateral and isosceles. Our visual representation didn’t include the fact that all equilateral triangles are also isosceles. An easy fix was just putting *at least two congruent sides under our visual representation.
Is a picture enough for a definition? Is a question that I am pulled both ways on. On one end my students were accurately describing each triangle with congruent marks and could translate those images to words. During class I felt that students had a strong enough understanding to move on. Also, I bet that most people wouldn’t write out the definition of isosceles triangle as A three sided polygon with at least two congruent sides. At most the definition would be A triangle with at least two congruent sides.
On the other end, most pictures don’t show the special cases, so the fact that my students didn’t see the special case adds a lot of value to written definitions.
I still have a bitter taste in my mouth from written definitions in calculus, so I am reluctant to pick a side here. Maybe there is a place in mathematics for both….
Let me know what you think.