I used to do an act when teaching air resistance. I would write ‘A’ on one piece of A4 paper and ‘B’ on another. I’d then say that I was going to drop them from the same height and ask the class to vote for which one they thought would hit the floor first. Whichever one they chose, I’d say I was backing the other. Then I’d screw that one up into a ball and let them both go. I always won.

I’ve never considered if this was motivating. It was intended as a bit of fun. I would only do it with a class I knew well because, otherwise, it’s a risk. You risk implying that you think the students are stupid and some may take offence.

I thought about this when reading Dan Meyer’s latest post. He’s been touring maths conferences, asking maths teachers to take a bet: he thinks of a number between 1 and 100 and they get ten guesses. Each time they get it wrong he tells them if the real answer is higher or lower. The catch is that it’s not a whole number and so Meyer always wins.

I’d groan if someone played this trick on me. But there are many other ways it could go down if a teacher did it in class. Students might conclude that their maths teacher is an otherworldly geek; a definitional pedant. They may be annoyed by the trick because, again, it potentially implies that they are stupid.

According to Meyer, it is intended to build motivation by giving a reason to students for why we might have different categories of number such as natural numbers and rational numbers. This fits with Meyer’s pet theory about maths motivation which is the idea that maths is an aspirin and students need to experience the headache to appreciate it.

If we think about the specifics, what are we trying to motivate these kids to do here? Do we want them to learn the names of different classes of numbers? If so, that’s pretty low level and at best incidental to the maths I’m keen for students to learn. It’s also not hard to convince kids of the need for labels because categorising things is human. And for interest I tend to go with a schtick about infinities of different sizes; the infinity of natural numbers is smaller than the infinity of real numbers and so on.

Do we perhaps think that the trick will motivate students about maths more generally? If so, they’re going to be disappointed when they get to quadratic functions.

Ironically, Meyer’s game seems like a solution in search of a problem.

This is a reflection of a much bigger issue in maths education. Some folks are holding on to motivation for dear life. If you believe in constructivist maths teaching, for instance, then that can be hard to maintain when the evidence stacks up against its effectiveness. So you can rationalise your position by claiming it’s motivating.

But the theories of motivation that these claims rest upon tend to be homespun. They fail to distinguish between a passing or ‘situational’ interest in a specific activity and a long term or ‘personal’ interest in mathematics. You also have to swallow that solving problems is inherently more fun than listening to people explain things. But problems can be frustrating and explanations can be interesting.

To really sort this out we would need to run robust trials. That’s a problem because the literature is full of research where students are asked if they found the new thing interesting. And that’s not robust.

So anyone can grow their own unproven theory of motivation. For what it’s worth, I’m not convinced that you can ensure students develop a personal interest in *any* subject. That’s because we are humans with independent tastes and desires. However, I believe that the greatest *threat* to developing a personal interest is a lack of success. I reckon that constantly being hit with the fact that you can’t do something would bum you out. In that case, we need to avoid gratuitous struggle and instead teach maths really well so that our students experience success. That, if anywhere, is the place instructional theories and motivational theories intersect.

Agree.

Success leads to motivation and motivation leads to success but probably in that order.

That trick could be good to illustrate the point of reading the question. I don’t really know how it leads to number theory though.

I agree with Mitch. Kids need to learn to question and not take everything at face value.

Can you imagine how annoying and sad that would be in this context. Anytime someone wants to ask a question of the student they are expected to analyze it for some weaselly way to turn it into a trick question.

In the case of the pick a number it becomes a trick question because people have a reasonable expectation that we are talking about the natural numbers.

Further this wouldn’t work as a good way to train someone to question everything as most people don’t ask trick questions. So they will learn that a particular person asks trick questions but unless everyone starts doing it they will have the normal expectation that everyone else is not such a pain and a normal interpretation of what they say will work.

We don’t know the rest of the details, context is king. You can pull a trick on people without offence if you tell them so in advance. The field of stage magic is predicated on this principle.

“In the case of the pick a number it becomes a trick question because people have a reasonable expectation that we are talking about the natural numbers.”

And that’s the problem: there SHOULD be no “reasonable” expectation. Just like “I have a reasonable expectation that the 50-page contract I’m signing with my insurance agent has no hidden charges.” Or “I have a reasonable expectation that all drivers at the intersection have seen me when I step out to cross the road.” Or “I have a reasonable expectation that everything Donald Trump asserts is the truth – after all, he IS the US President!”

“Further this wouldn’t work as a good way to train someone to question everything as most people don’t ask trick questions.”

I’ll have a word about that with my friend who is a detective. I’m sure he’ll say something along the lines of – not every question is straightforward and not every answer is fully truthful.

I find that that the best way a student can slow down learning in a class without getting into trouble is to question everything. After every statement by the teacher they ask “why?” or, if they are more sophisticated, “What if … some unlikely event?” and “What about … some trivial exception?”. Progress slows to an absolute crawl if you don’t nip it in the bud pretty quickly (I generally offer to explain everything to the student after class, in their own time, at which point they lose all interest in the need to have the explanations).

No-one has time to question everything. No-one can go through life refusing to take anything they are told at face value.

I think students reasonably have an expectation that the Algebra I am teaching them is correct, and that they should not question my procedures if they want to learn, simply on the basis that I might be wrong.

Imagine trying to explain parallel line geometry if the students insist that I prove the parallel iines don’t meet — as the proof is impossible, I couldn’t even if I wanted to.

Sure, we need to be open that we can be, and sometimes are, wrong. And it helps if teachers give examples as they teach — I usually open my discussion on negative numbers with a bit of the history of them, and how for centuries it was agreed that they were not relevant. But that’s different from “not taking anything at face value”.

“I generally offer to explain everything to the student after class, in their own time, at which point they lose all interest in the need to have the explanations”

You’ve solved the problem, then. I disagree that it needs to slow the lesson down, if you’re careful to set the ground rules early and consistently with all your students.

“Imagine trying to explain parallel line geometry if the students insist that I prove the parallel iines don’t meet.”

A brilliant challenge! How I would love to work with a group of students like that, rather than a SLANTing group of zombies. Simplest solution would be to say “fantastic – why don’t you prove that I’m wrong! Create a proof that shows the lines DO meet!”

So John you’re saying they should challenge me, but I should not use class time to answer their challenges?

To me that sounds like a con. I’d be one of those teachers that say good things on the surface, but don’t deliver in practice.

Telling 14 year olds to go prove something known to be unprovable seems vaguely unprofessional to me.

That depends very much on the age and (particularly) the context.

To Chester:

“Telling 14 year olds to go prove something known to be unprovable seems vaguely unprofessional to me.”

Is that really how you see it? How unfortunate.

“I reckon that constantly being hit with the fact that you can’t do something would bum you out.” That seems as much unfounded a motivational theory as what the post is arguing against?

Furthermore, if learning (long term memory etc) is the goal, then ‘proven’ strategies like retrieval practice, interleaving, spacing (perhaps even productive failure) all introduce a low level of succes/performance during practice for the benefit of learning in the long term. That seems at odds with ensuring succes (at least in the sort run which tends to be the time horizon of young learners).