Playing With Functions

For this assignment in my Creativity in Teaching and Learning class we were asked to develop a playful introduction to our topic area (functions). I came up with two introductions (because I couldn’t decide and both might be effective in different contexts).

The first introductions uses a technology that I’ve relied on several times throughout this course, desmos.com. What I’ve done is created a very basic smiley face in Desmos. The task for students is to create a “non-basic” smiley face using at least three different families of functions, at least ten functions, and at least one animation. They should strive to be as creative as possible. They should look to other pieces of art for inspiration. This is a good introduction to functions because students have a lot of flexibility to “play” with the mathematics. They’ll have to figure out how to shift functions vertically and horizontally as well as stretch and squeeze them. They’ll also have to figure out how to restrict domains so that the graphs are limited to just the face.  Having them do an animation will ensure that they will learn how to use sliders in an organic way (because they have to create an animation, not because I said “please create a slider). My hope is that this will help them see where sliders might be helpful in future problems. I also think, with reflection and future application, that this will provide a good foundation that I’ll be able to connect back to throughout the year.

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Math, Play, and a bit of Desmos Love

I’m sure a lot has been written about how Desmos is an incredibly valuable tool for the math classroom. I’d like to echo those sentiments and give a few observations from my classroom related to Desmos and play in mathematics.

My thoughts for this post stem from this activity. It’s pretty math heavy but the basic idea is that students spend time making conjectures and either verifying or invalidating them. When their conjecture is invalidated, they spend time tweaking their functions in Desmos to match the actual answer. These kinds of activities are becoming commonplace in my classroom and I’ve noticed a few positive shifts in my classroom environment. First, engagement has increased. I had a couple more high fives during this activity (and a bunch more in a previous activity) and 100% engagement in class. To be fair, this is AP calculus so it is the “better” students, but I am hard pressed to find 100% engagement when I give p. 102 #1-15 from the textbook. Second, they learn better! This shouldn’t be a big surprise since we know that allowing students to explore/play with a concept before direct instruction often yields better results (See page 58 of this research and this learning model from Ramsey Musallam if you don’t believe me). My students came up with these definitions of the chain rule without me ever saying the term “chain rule”, giving any direct instruction on the chain rule, and with minimal guidance from me. Don’t get me wrong, we followed it with a formal lesson on the chain rule with several examples and they will have an assignment from the book tomorrow, but I think laying the groundwork with the activity will pay dividends later. There’s also another pay off here…

Students are more motivated to come to class. They enjoy activities in which they are allowed to, if you can believe this, play with math. I don’t think this is possible without Desmos. The power in it is that it breaks down barriers for students. It’s like giving a student a canvas, paint, and brushes and saying “here, work with this for a while, make some conjectures, tinker with it, and tell me what you come up with at the end. ” Maybe I’ve drank a bit too much of the Lockhart cool-aid, but to me this is the essence of mathematics. I don’t agree with all of Lockhart’s points, but I think the link between curiosity, play, and learning is powerful and we can leverage that linkage in ways that create powerful learning experiences.

And if I haven’t convinced you yet, check out this excerpt from one of my student’s blog posts below.

“Writing this statement in question #4 required that I fully understood what was being shown in the lab. It made me think deeper about how I came to that conclusion. I believe that looking at the graphs, analyzing relationships between them, and forming a conjecture in my own words helped me to truly learn and understand the concept rather than just being given the relationships and rules and being asked to memorize them.  By completing this lab, I have improved in my ability to  identify what the graph of a function’s derivative should look like. I feel that I have a much better understanding of the relationship between functions and their derivatives and I know that I can look back at the examples in Lab 6 if I ever need help.”

I’d love to hear your thoughts on this. What are your experiences with learning and play? How does it fit into your content area or your classroom?