Why teach mathematics?
[This is part of a presentation my school’s math department made to our faculty in 2016.]
Traditionally, the rationale for learning mathematics has been focused on students acquiring skills – such as learning times tables or solving equations or doing long division. This also has been driven by students asking: “how will we ever use this in real life?” - beleaguered math teachers have given reasons such as balancing a checkbook or understanding your car loan. This is like an art teacher who teaches painting focusing on color names and brush strokes, with a central aim of being able to paint a house. There are useful ways to apply the skills of mathematics, but learning mathematics for that reason alone has been the historical downfall of mathematical education and misses the beauty and deeper value of the discipline.
We want our students to learn what it is to be a mathematician. At the heart of being a mathematician is playing with solving problems and sharing those experiences with others. We want problems that challenge and surprise, problems that expose beautiful mathematical relationships. In this way, our students experience the fun and art of our discipline, and it is also the direction that causes the greatest growth for our students.
In problem-based learning, we give a small collection of 6 to 12 homework problems each night. The students don’t get a lecture preparing them for the problems - they must figure out on their own which tools will help them solve each problem. The problems are often challenging - the student may not be able to solve all the problems – and that is important! We want our students to learn grit. We want them to experience the magical moment of figuring out a puzzle that they couldn’t solve initially. We want them to learn to be creative, resourceful and persistent, to learn how to handle roadblocks and failures comfortably. We recognize that growth occurs as the result of effort!
Another key ingredient of this process is communication of mathematical ideas. When the homework is due, the students take turns presenting homework problems to the class in a seminar setting. This produces a great deal of practice with logical communication – presenting your own ideas, describing and defending your reasoning, listening carefully and critically to others. The students listen much more actively and critically when students present their homework than when the teacher is telling them the homework solutions perfectly. In this way, the students work together discovering mathematics. There are times when the teacher is needed to summarize material or point out important results and methods. The teacher is there as a model mathematician, a guide, and a colleague.