KSA #04 Subject Knowledge · KSA #06 Planning · KSA #09 Instructional Strategies · KSA #15 Professional Learning

Variation Theory and it’s Application to Teaching and Learning Math

So, truth be told last night I was feeling exhausted, overwhelmed and defeated. This week there has been something every week and the start to a new semester, so work has been insane. It’s basketball season and I’m coaching our senior boys, we had our first games this week and practices and now a tournament. Last night I came home feeling overwhelmed with respect to this course and felt as thought I’d been neglecting it, in that I hadn’t completed any of the readings yet.  With some serious self talk, and using Jody as my sounding board, I buckled down and read “What is Made Possible to Learn When Using the Variation Theory of Learning in Teaching Mathematics”. As I read this article I started to feel better, it made sense. Then I got to the example in Lesson 2 and was inspired. I’d already had my lesson for today planned, but this article made it SO MUCH BETTER.

We started percent today. My plan was to review percent of a number (5% of 10, 25% of 75, etc.) and then have them play a game to review. It was going to be golden, right? Perhaps haphazard is a better word now that I’ve read this article.

After reading the article I thought, I need to be intentional about what is varied and what is invariant. I quickly grabbed my notebook and jotted down some ideas. My original plan was to have everything be done mentally, and I stuck with that plan, no calculators were used today at all. As I went through class, I did a couple for them and then had them fill in the rest. Each time students shared their responses I recorded their answers in a different colour. Once we came to consensus as a class that we agreed with the answer provided, we engaged in a number talk and I asked for as many explanations for why this answer worked “How did your classmate get that answer?” Each time a student would give their reasoning, I would notice it would spark ideas or the desire for others to participate. “Well I looked at it this way,” or, “Couldn’t we look at the percent this way too?” etc, etc, etc.

First I varied the percent and the number was invariant (10 x 1%, 5%, 10%, 20% and 50%). Next, I varied the number and the percent was invariant (1% x 10, 20, 25, 35). Then, I kept the number constant and varied the percent again, but this time the number was chosen so that decimals would be part of their solutions (15 x 1%, 5%, 20%, 25% and 75%). Finally, I used a decimal as my number and kept that constant and again varied the percent (5.5 x 1%, 5%, 20%, 25%, 50% and 75%).

As we went through this, some students started to see patterns and some students would talk about how “If I know 10% and want 20% I can just multiply 10% by 2 and I get 20%). When we were looking at 75% of 15, one student said “I found 1% and multiplied that by 5 and then found 10% and multiplied that by 7 and then added those together because I know that 5% + 70% is 75%”. What was even better was that students that NEVER EVER EVER EVER want to contribute in class today were. Even better? Can it get better? It can! After I had them play a game with a 10 sided dice (using these as the percent benchmarks) and then drawing two cards to make a number to multiply the percent by (hmm… so I guess both pieces were varied in the game, I’m just really realizing this now as I reflect. Now I wonder, is this okay?). Again, I had students who typically struggle that were applying this understanding. They had new numbers to use, so they couldn’t just use the benchmarks on the board, but they were able to transfer that process to the game that they were playing.

It will be interesting to see what they remember from Monday and are able to do. Truth be told, I would have been haphazard with the lesson before this reading (yes, I know that we’ve read other articles saying something similar, but it was the timely read of this one that gave me the spark to make the change in my lesson today). I would have tossed up random numbers and random percent, without a thought of the order or what I was doing. I really do think that the intentionality in the examples I chose and the way that we moved through the lesson allowed all students to be successful in the task.

This was the spark that I needed to keep me going. I was so excited about it I was inviting teachers from all around the school in to see my boards, listen to the kids conversations and share the theory behind what I was doing. One teacher even asked me if I was willing to share the article with her so that she could read it. Perhaps this will ignite some common language within our team as we try to move forward in our teaching as a unit, rather than just individual teachers!

What is made possible to learn when using the variation theory of learning in teaching mathematics?

Until next time,
Mr. B



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