I have to tell you about an incredible activity that I engaged in a few days ago.
This was one of those rare days of late when I just took the day off – no teaching, no marking, no tweeting, no blogging.
I relaxed. And may I say relaxing is highly underrated? You never know you need to until you do it.
On this day, I reworked a math lab called Sherlock Holmes offered by Dr. David Pimm of the University of Alberta. It is a 5 x 4 tangram containing seven pieces with a minimum of side lengths and angle widths given. The task has two parts. First, find the length of each side of each piece. Second, prove that the tangram can never be arranged into a square.
The fun begins when an additional task is given to you: while engaged in solving the tangram, describe and justify either the math processes or the math content you use at each step in your activity. Now you have to pay attention to the task, the Math and what you are doing. School math does not get any richer! Nor does it come closer to the activities professional mathematicians engage in.
Or does it?
Most real-world math problems are open ended; they do not provide specific solution schemes or even all the information necessary to solve the task set before the mathematician. The mathematician must make assumptions in order to address the math task and engage in the Math activity. The Sherlock Holmes task does this too. The student or mathematician must determine what assumptions (there are more than one of them) must be made to solve the task. So, the task is open to interpretation, solution and answer. Now that is math worthy of a Math lab.
Out of the Box
I first worked on this problem about five years back, and I got it wrong! I received high marks because I was meticulous with my activity, but my solution was messy, took several steps and never did solve the task.
I did everything from drawing to geometry to calculations by hand, which was fun. I endorse math-by-hand-first, at least when dealing with unfamiliar math. Too many students do not know Math nor how to solve math unless they have a calculator, and when asked they have no idea why the calculator spits out the correct answer. With each lost comprehension or misunderstanding, these students fall further behind in their Math reasoning and soon learn to hate math and Math.
This is so sad. I love Math and working on rich math tasks. Math to me is a series of puzzles and riddles and games. It is
a way of thinking: of thinking things through imaginatively, logically, thoughtfully and step by step. … It is the process of solving and creating puzzles; that is, of recognizing, understanding, manipulating and describing patterns. Mathematics, then, is about patterns and school mathematics is about riddles, challenges designed to encourage the exploration of [overarching and] underlying patterns. – Me (2005); I credited Geri Lorway, but these are my words.
What other class in elementary and secondary school is premised on playing with puzzles and games?
So, a few days ago I took a day off and I returned to the “Sherlock Holmes” lab. And I solved it! This time, I used GeoGebra, a visual mathematics software, much like Geometer’s Sketchpad except that it is open-source and free. I use GeoGebra a lot and provide several free resources for teachers to use.
I actually used GeoGebra to create the puzzle image above, but I decided, while I was at that, to use the program to also solve the problem.
“Sherlock Holmes” changed
GeoGebra did a superb job. With it I saw more clearly what aspects of the task were fixed and what aspects were free to manipulate. This clarified what assumptions needed to be made and allowed quick experimentation of these assumptions to solve the task and check the answers I came up with.
The best part is I could see the puzzle moving as I manipulated it and I identified the (one silly, arithmetic) error I made five years ago which prevented me from solving the puzzle. I solved the puzzle in three steps using the program. But the activity was not as fun as doing it by hand. There is a certain “magic” and flow in using one’s imagination to problem solve.
And there is something fun and relaxing about playing a challenging game.
So, why don’t you take some time and choose a math lab and play a game? I would love to hear what lab, of mine or others’, you chose, how you fared and what you experienced. Play, enjoy, learn and leave a comment.
This post was inspired by The Rascal Triangle which I read a couple of days after engaging in this activity.Follow @stefras