Never DO Mathematics in a Math Class?

Within our efforts to make major improvements for our students, both in developmental and gateway college math courses, we have been looking at our content and our methodologies. I want to connect the over-used phrase “Do the math!” with our phrase “doing mathematics”, within this process of building a better future.

I’ve commented before that the phrase “do the math!” is frequently used as a propaganda technique, to imply that everybody would reach the same conclusion as the speaker (and often used when the ‘math’ in the statement involves a small set of numbers, far removed from meaningful evidence for any argument). I wonder if our phrase ‘doing mathematics’ serves a similar purpose within the profession.

Part of our problem is that we assume that there is a single meaning for ‘doing mathematics’. Historically, the phrase seems to have grown out of the view of mathematics as seeking patterns, often in a constructivist approach; this ‘doing mathematics’ just means to immerse people in a situation involving quantities with a goal to establish patterns and statements that make sense to the learners. Within professional mathematicians, we have two contrasting meanings — occupational (actuaries, for example) using systems of mathematical knowledge, and researchers using a variety of tools to establish new knowledge or applications.

As a learning tool, the “doing mathematics” has very limited benefits for the student. Most research I have read suggests that learners need a very directive structure for the learning to occur by discovery; this guidance takes the process out of the original meaning of ‘doing mathematics’ into the more appropriate ‘learning mathematics’. Doing mathematics, with the goal of learning mathematics, is a very advanced process — it is what some experts can do; expecting novices to engage in this process is a bit like expecting novices to become good piano players by having them sit at the piano (without any technique, without any theory). Doing mathematics to learn mathematics does happen, often not by design, frequently with great excitement by those involved.

Perhaps the question is:

If students can experience what we experience when we ‘do mathematics’, they will be motivated to learn more mathematics.

Now, motivating students is one of the central roles in our classrooms. Sometimes we focus so much on content and skills that we provide no information on our world of mathematics. If we are the ones doing mathematics, presented in a way that novices can follow, then I can see some real benefits. I have tried to do this in all of my classes; with colleagues, I use the phrase:

Students will see and perhaps do some beautiful and useless mathematics in every math class.

I include ‘useless’ in the description, and actually focus on that. Why? Because it is not reasonable to expect students to understand the eventual usefulness of the mathematics which they can appreciate; to them, it will likely seem “useless” even though it is not to a mathematician. When I do this, I am walking a little beyond the limits of what students currently understand; I am in a beautiful field on the other side of a path, and want to share this perspective with my students. My honest answer for ‘why I do this’ is simple: It is fun! I also have pedagogical reasons; walking a little beyond the current level helps to create an atmosphere of respect and one where learning for learning’s sake is appreciated.

So, my advice is: never have students to mathematics in a math class. Help them learn mathematics, and you should do some mathematics in class so they know why we are mathematicians. We don’t do mathematicians just for the money, or the fame. We do mathematics because we want to.

Developmental Mathematics Revival!

Bringing a new vitality to college mathematics 2011 to 2019