Now that I have defined what I believe learning through problem solving is, let’s look at some examples.
A good learning through problem solving experience involves a compelling problem from which students can extract meaning and learning. These problems can be simple in their design, or they can be quite complicated. They can take a few minutes for a class to do, or they can take an entire class.
Much of the What Can You Do With This (WCYDWT) on Dan Meyer’s blog fits into my definition of learning through problem solving. Dan Anderson has added some nice WCYDWT stuff on his blog. Their stuff is good, and is often supported with compelling video as a way to further engage students in the problem.
I have shared several ideas on this blog that are nice examples of learning through problem solving. Mozart’s Dice Game, The Speeding Teenager, and The Giant Gummy Bear are three such examples. They are diverse in how they present the problem to students. Mozart’s Dice Game attempts to hook them through one statement from the teacher. The speeding teenager allows them to come up with the question to explore based on a newspaper article. The giant gummy bear can be taken in many directions after students watch the video.
Let’s look at some other specific examples.
One of my favorites is the Coke vs. Sprite question on Dan Meyer’s blog. The video is simple, yet students are compelled to explore after watching it. The math involved is deceptively complicated, and the result is surprising to most students. This is an example of a learning through problem solving that takes some time to come up with, and takes some time for students to work through.
Learning through problem solving can be simpler to create than that, though. Not all of us possess video editing skills. Here are three examples that are simple, easy to create, and take less time in class to do.
Sam Shah posted this one on his blog last year. In it, he asks his calculus students to figure out the area under a curve. What qualifies this as a learning through problem solving experience is that he asks them before they have been taught how. It is a simple idea that leads to powerful thinking and learning.
I spoke with a Math 10C teacher earlier this year who said he had started out spending hours trying to come up with deep, rich explorations and problems for every class. One day, when he was pressed for time, he came up with a simple question on the fly. The class was working on surface area and volume of three-dimensional objects. Instead of showing them nets and having them build the geometric shapes like he would have in the past, he showed them a cone and asked them to draw what the net would look like. It was simple to come up with, but lead to powerful conversation and exploration on the part of the students.
The last simple example I will use is one I have tried myself. More than any other topic, factoring trinomials has frustrated me as a math teacher. In my career, I have tried at least 5 different ways to teach factoring to grade 10 students and experienced very little success with any of them. What I have learned is that, “Factor 2x^2 + 7x – 15” is an incredibly powerful learning through problem solving experience if it is presented in the right way.
Two years ago, as part of my M.Ed., I was taking a course called Quality Teaching and Peer Consultation. The professor was a constructivist, so to him the only thing that could be considered quality teaching was constructivism. Thus, it ended up being a course in constructivist teaching. One day, I asked him how kids could possibly construct their own meaning around factoring trinomials (thereby demonstrating that I was stuck in a traditional model of math education). I was surprised that he knew what I was talking about, and even more surprised when he had an answer for me.
He suggested that I go into class on the day I was going to teach factoring, and put two trinomials on the board. Assuming that they could multiply two binomials (covered the previous day or two), I would ask them to come up with the binomial factors of the two trinomials on the board, and turn them loose. I was skeptical, but I tried it with my Math 10 Pure students.
It worked. After some discussion and argument with each other, they were able to find the two binomial factors for each of the trinomials. I gave them a couple more, and they could extend to negatives, leading coefficients, and larger numbers with no problem and without me “teaching” anything. It was a rich learning through problem solving experience for them because I gave them the problem up front, before they knew how to do it. Later on, I tried to teach them a method, and they rebelled. I’m glad they did. I was intruding on their own, individually constructed methods. I never “taught” factoring again.
Learning through problem solving should be a component of our math lessons. No more should the high school math teacher be standing in front of a room putting example after example on the board. Students can learn these things themselves – even in our highest level classes. It is not hard to come up with a quick problem for students to work on in advance of the day’s learning. I will eventually post some sample unit plans, matching our Alberta curriculum, that show how to integrate this approach daily.