Working Backwards

Looking through many resources in preparation for teaching math in the school system, there seems to be a very common pattern that is rarely broken: introduce algorithm, apply it to increasingly difficult pre-determined problems, review, then tack on a "real life" or "word" problem to add application as almost an afterthought. Some resources even skip this last step, and few indeed involve proofs, aside from a couple that show how to derive the quadratic equation.

In mathematics, we have a tendency to ask students to accept algorithms without question or debate. We essentially eliminate critical thinking from our teaching.

This is not the way teachers are trained in my province, yet many resources that are used in our classrooms still follow this sequence, and many teachers drift toward this in their practice. The emphasis remains on the lower levels of Bloom's Taxonomy at the expense of activities that promote higher-order thinking. In my review of the literature, it appears that teachers are most likely to work in this direction for two reasons: this is the way they were taught, and their comfort level with the curriculum is low.

When I say their comfort level is low, I do not mean that they do not necessarily hold a deep conceptual understanding of the topic, but that for various reasons (most often relating to allotted classroom time), they feel the need to get the basics covered as quickly as possible, and for many, teaching algorithms is how they view "the basics" when it comes to math.

But what if we were to reverse this direction, and start with the applied problem?

Critics say that this leaves students high and dry, with the need to reinvent conceptual knowledge that took mathematical superstars many years to develop. They say it leads to confusion when the approach they might try is not the most efficient method.

However, no one is saying that we withhold the algorithms from the students, only that we let them think about the problems that lead to them in order to foster a sense of pattern and deeper conceptual understanding of the processes involved in applying mathematical thinking.

All the memorized algorithms in the world are useless if students never learn when or how to use them outside of math class or standardized testing.

Sure, it takes a little more time for students to think through the "why" of a problem, but feeding them algorithms to memorize and apply does students a disservice. Computers can work through algorithms, and they do it faster and more accurately than people. What we need are people who can reason mathematically, and this requires that we provide a space for applied problem solving and reflection.

It is my gut feeling based on what I've seen with the students I've worked with that students who develop applied problem solving skills aka mathematical reasoning skills, begin to make deeper connections quicker with later topics. In this way, the time invested at the outset may offset the time needed to cover later related topics.

For teachers to abandon the chalk-and-talk and promote these skills will take a leap of faith. It is much more comfortable to stay with the known, particularly when there is a perceived crunch in terms of curricular content demands and the allotted classroom time to cover it. Students may resist since they are used to being given the entire topic at the outset. People tend to resist change.

How many times have we heard the question from our students, "When will we ever use this?". Students, particularly those for whom math does not come easily, need to understand this in order to invest their time and effort accordingly. Resources like this one: 101 uses for a quadratic equation and this one: "Why study math?" can be good places to start. Starting with a relevant real-life problem is also an effective way to connect theory with application.

I issue the following challenge to all teachers who read this blog: choose one topic this year to present this way and see how it goes. Start with a real-life problem, challenge students either individually or in groups to devise a way to tackle it, and share results using Bansho or another similar method in which the different approaches can be grouped in a meaningful way. Discuss which ones work and which don't and why. Finish with a review of those ways that work best. Follow up with some practice problems.

Did you or your students resist? What challenges did you face? Did different students participate than usual for your class? How might you use this to best encourage mathematical reasoning in your students?