Notch 9: Ed Barbeau’s Classroom Problems, and His Request

Ed Barbeau is a Canadian maths ed legend, and an occasional commenter here. I previously noted Ed’s astonishingly extensive website, which is also listed in the Good Resources post, and I posted a long comment of Ed’s on gifted education.

Ed has recently compiled a list of classroom problems, accompanied by discussions, which he has used “positively” in classrooms. He is now requesting teachers will also give them a try and see how they go, and let Ed know how they go: Ed wants your feedback. 

Ed’s document of problems is here (and is embedded below), and you can email Ed. to provide feedback. (Of course everyone is also free to comment below.) Ed also makes it clear that he is happy for the document to be passed around further afield.

What follows are two (slightly edited) introductions: a short one, which Ed used to introduce the problems on an email group; then the longer introduction from the document, which is worth reading in itself.

I am sharing with you a list of problems that I have found useful with students. However, my sample of situations is not large enough for me to draw any conclusion as to how they would go over in general, so I have been trying to get teachers to try some of them out and let me know what happens. Please feel free to forward them to people who might find them useful. There is nothing new here, but I thought I would gather them in one place.

I think that the problem with many students is that they are detached from the context and meaning of what is going on, and this is a situation that may be made worse by our focus on testing (and grades) and emphasis on basic skills and logical reasoning when these do not arise organically from their own experience.

Rather then being a luxury, I think these sort of problems should be dropped into a class on a regular basis, and that part of the formation of teachers is their learning how to handle them.

INTRODUCTION

When I have had the opportunity to interact with school students, I usually go in with some problem or investigation that will engage them and illuminate some part of the syllabus. I look for situations that almost every student can make some progress in, but which have aspects that go deeper and will keep the quicker students on board.

Both mastery of procedures and logical reasoning are key parts of the curriculum, but these must be pursued in a context in which they make sense and arise naturally as a part of whatever investigation the students are engaged with. An important characteristic of student learning is being sensitive to patterns and willing to change perspective, because success in mathematics is often dependent on developing attentiveness to details and looking at things in a productive way.

School education can often lead to students being able to go through the motions blindlessly. Here are a few examples of how students can get in a rut. Once I was in a Grade 5 (!) class in which a girl had to find the change when making a purchase. Without batting an eyelash, she offered an amount that was greater than the bill tendered (evidently adding rather than subtracting). It was only when I mentioned that she was getting more money back than she was giving, that, after a brief pause, the penny dropped and she got on board.

An algebra student, asked to solve a quadratic equation given in the form (x−a)(x−b) = 0, multiplied the factors and then applied the quadratic formula. Another student, having solved an equation, asked me whether his answer was correct. I asked whether he had plugged the solution into the equation to check, and he seemed to be unaware of the point of solving an equation.¹

¹ At the college level, there was the time when I asked, unwisely as it turned out, on a test for the students to find the Fourier expansion of on the interval , expecting them to write down and recognize this as the answer. (The Fourier expansion is a representation of the function as a sum a series of certain trigonometric functions whose coefficients are given by a particular integral formula.) Some students got bogged down by the integration required for the coefficients, while others wrote down the correct answer and then used it to perform the necessary integrations. An aware student would either have recognized that satisfies conditions that ensure that it had a unique trigonometric expansion which was its Fourier series, and that was a trigonometric expansion, or mentally noted that because of the orthogonality conditions, the Fourier expansion of was itself.

Even when doing an automatic procedure, students need an elf sitting on their shoulders, to keep them aware of what is going on, why it is appropriate, and what might be going wrong.

Here is a situation that older people might be familiar with, which illustrates the role of awareness in a mathematical situation. A customer buys an item that costs 87 cents. She gives the cashier one dollar and twelve cents. Why did this happen and how is it handled?

When people regularly used cash rather than cards in stores, it would be understood that the customer wanted to reduce the amount of loose change she carried. The cashier at the time probably would have thought, “That is 75 cents out of a dollar; here is your quarter change.”

Conceptually, the twelve cents can be thought of as a down payment on the 87-cent cost, leaving the balance of 75 cents to be covered by the dollar. A modern cashier, blind to this view, might either have just entered the cost and amount tendered into the cash register which would then have calculated the change (at which point he might have twigged as to what was going on), or else left the 12 cents on the counter and given the customer 13 cents change from the dollar (leaving the customer with seven coins instead of one). The change in each case is correctly calculated, but what a difference in the environment!

There is an interesting book by Suzanne Simard, Finding the Mother Tree, published in 2022. Dr. Simard is a forest ecologist and has studies trees as a community, in which the members, even of different species, can communicate with and support each other, that makes for a healthy forest. This might serve as model for a school curriculum. Currently, we are likely to pay attention to the individual topics (trees) at the expense of the health of the ecosystems (forest) as a whole, a system in which various topics feed into each other and different areas have structural and analogical connections that can firm up the learning and be exploited.

The problems and demonstrations that follow are ones that to varying extent I have tried with students at university and in schools. It is hard to predict what will happen in the class; this depends on how the students respond and the ability of the teacher to deal with the twists and turns. However, I have generally had a positive experience. Students vary in their background, intuitions, their interest, their ability to think divergently, their technical skill, the perspective they bring to the problem and their ability to articulate their reasoning.

I would like teachers in schools and at educational institutions to try some of the problems with their classes, and would be interested to receive reports from them on what transpired at barbeau@math.utoronto.ca.