Then I ask them what a triomino looks like. We decide that there are only two.
The other triominos are congruent to these two, by rotation, reflection and translation. I then ask my students to find all possible tetrominos. I suggest you try it before reading on.
When I first gave this activity, many years ago, I learned that it took about ten minutes with several interesting and useful discussions about double-counting and that there would be several students who did not discover all five. Eventually, those students did find all five and were ready for the problem I really wanted them to tackle: finding all pentominos. I then asked them if they could construct a square out of the pentominos, and if not, why not? What rectangles could they construct? It is a nice activity. But the nice activity is not the point of this blog.
One year my students immediately solved the tetrominos problem, and I mean immediately. Like in fifteen seconds or as long as it took them to write the answer. This task was not a difficult problem for them at all. This class was not particularly exceptional, and I was bewildered, until I heard one of the students ask another how he did it so fast. His answer: Tetris.
What had happened from one year to the next was the game whose pieces were these pieces. What was once difficult, was now simple.
I have often thought about what makes a problem difficult and another problem simple. I am particularly interested in problems that some students seem to understand with little effort and other students struggle with. I think of this class. I am convinced that what we perceive as talent and insight is often just experience and familiarity with similar ideas.
I ask my honors students how many of them played with Lego when they were little. Many of them giggle and smile. Some admit that they still play with Lego. Often the conversation moves to other toys of the same nature. When I ask the same questions of students who struggle with math, I get blank stares. I think we need to pay careful attention to what sort of experiences our students have had. Even more important, I think we need to build in lessons so that they are previewing topics we know will be difficult later. And they can't just be pencil and paper activities.
Peg Cagle, a middle school teacher from California, gave a fascinating talk about how students are having different experiences growing up than we had. She asked her students, I think she said about 180, how many had climbed a tree. None had. I think that effects their sense of what three dimensional space looks like and feels like.
Have a thoughtful Memorial day.
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