Math Lab: Revisiting Technology and Imagination

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.

Sherlock Holmes” by Dr. David Pimm

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.


17 thoughts on “Math Lab: Revisiting Technology and Imagination

  1. Pingback: Teacher Challenge Posts – Weeks 1/ 2: Visit These Posts

    • Hi Jenny,

      By all means. I posted it for people to try and educators to share with their students. That particular math lab (Math Lab 1) is actually two, so try either or both.

      There are four other labs as well; some problems to solve; some problems to pose; some research. In addition, I created a list of link to other sites which offer similar tasks or questions.

      Have fun,

  2. How nice to find this post of yours! I already have a teacher’s diploma for swedish and I am right now studying mathematics to be able to teach that as well. I am really interested in rich problems and also how I can use ICT to teach math more effectively. I’ll take a look around your place and I hope to be able to post on mathematics soon as well!

    • Hi Anna,

      I am glad you enjoyed the math lab challenge. As I mentioned in my reply to Jenny, there are many others you can also enjoy, including those on the sites I link to in my math lab main page or even a level up, on my Mathematics Through Technology page, where I have several GeoGebra lessons and constructs. Please visit, enjoy and use as you like to teach your students (when you get some).

      I am a big fan of open-ended math puzzles and “long”-tasks, which actually encourage mathematical curiosity, play and exploration. I will create more of these and post them every now and then during my mathematics weeks. I am still designing my blog, so right now I can only guess this will be every third or fourth week.

      Feel free to subscribe to this blog or visit again to read more math posts.

      Thank you for commenting, Shawn

    • Thank you for your compliment.

      I strongly believe that learning should be a shared experience. Everyone has some knowledge or some skill which others do not; everyone has something to learn.

      This is why I became a teacher.

  3. I found your post from the Edublog Teachers Challenge.

    This is fabulous! I’m placing it in my Diigo bookmarks and will share with the math department.

    Thanks for all the time put into such a well written post.

    Tracy Watanabe

  4. Hi Shawn,
    I know your blog from Teacher Challenge Blog.
    I really enjoy your posting about math. It’s great resource.
    I like your words “Math to me is a series of puzzles and riddles and games.”
    Thank you for sharing.

    • Hi Mieke,

      Thank you for the compliment. My Math Lab post seems to be my most popular so far. Given that it is also my most practical, I am encouraged by its popularity. I am glad you enjoyed it.

      I enjoy learning in a large part because I view the subjects and topics I learn as fun pastimes. Who wouldn’t want to explore the thoughts, imaginations, emotions, talents, perceptions and dreams of other people through art and language? Who wouldn’t want to explore and exploit nature (all things non-man-made) and physical technology (all things man-made) through science and ICT?

      I believe a subject is how it is perceived. If one sees uselessness, she gets discouraged and bored. If he sees usefulness, she gets encouraged and engaged. And there is the perception of fun. Math is a series of puzzles and riddles and games. And they teach it to you! Art and language are series of expression, experience and communication. Who doesn’t do this everyday? And they teach it to you! I can not think of one subject that in non-school contexts is not fun.

      It is up to us teachers to bring that fun back to the subjects we teach in school.

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