“Create Your Own” part 1

Today was the first day of calculus for the fall semester of 2018. As a first-day activity, I wanted to do something that didn’t require any calculus knowledge and could break students out of the mindset that doing math is always about solving particular problems that have been fed to you.

So I initiated a sequence of exercises I’m planning, which I’ve come to think of collectively as “Create Your Own…”. In this case, I gave the following prompt:

The number 1 can be written many different ways, for example 4 – 3 or 10/10.
Come up with ten different expressions that equal 1. Be creative!
Try to have at least four of your solutions involve some kind of algebraic expression, like a variable x.

After they had a few minutes to work individually, I had them share their answers in small groups, and each group picked out what expression by its members they thought was most creative. At the end of class, I collected all of their solutions to look at later in the day.

In having students do this exercise, I learned a lot, and I would definitely do it again, with a few tweaks. Here’s some of what I learned:

• Students judge creativity differently than I do. In looking over the collective work this evening, I saw some excellent examples of splitting 1 into a sum of fractions or decimals and some elaborate expressions involving absolute values or square roots. But the groups often picked examples with the fanciest functions as most creative. Each section had some students come up with cos²(x)+sin²(x) as an answer, and some used logarithms, as in ln(e) or log₁₀(10). And I’m glad those functions were there! It gave us a chance to talk a little about them and for me to give assurance that we would review them at an appropriate time. But 1/10+2/5+1/2 is much more personal, somehow, and I’d like it to have its due.
• This kind of exercise was surprising and unfamiliar. I’m not quite sure how much time I gave for the creative process; I started out in my head with the idea of 2–3 minutes, but that clearly wasn’t enough, so it was probably 4–5. In that time, not everyone came up with ten solutions. (Which is fine! We’d spent an earlier part of the class watching Jo Boaler’s “Four Key Messages” video, which emphasizes that speed isn’t essential in learning math; a couple of students added that comment to their work.) I saw a few get stuck for a while, however, and next time I’ll have some ideas for how to gently prod.
• The notion of “variable” is very strongly connected with “solving an equation”. The vast majority of students interpreted the direction “involve some kind of algebraic expression” to mean “write an equation whose solution is 1.” This led to answers like 2x=2 and x+3=4, and many others (one group gave log₅(x)=0 as an answer!). There was a remarkable amount of creativity in the creation of these equations; I’d like to figure out how to leverage that. But now I also know that the distinction between an “expression” and an “equation” has not yet been made clear, and when we start simplifying algebraic expressions (e.g., to compute limits), we’ll still need to inject some flexibility into our thinking.

The main adjustments I would make next time are:

• Rephrasing the instructions to say that the expressions simplify to 1, rather than equalling 1. Hopefully this will give clearer direction regarding algebraic expressions. Also, I would probably add an algebraic example like (x+1)–x.
• Preparing, nonetheless, for a discussion about what the term “expression” means.
• Giving a more definite and slightly longer period of time, and providing more useful interventions as the students do individual work.

That’s all for now. More updates as warranted.