This happens more often than I’d like: we start by having a normal class discussion and end up trying to resolve something that doesn’t engage the students’ interest, or even their understanding*. It’s the exact opposite of the way I’d like my classroom to feel.*

Last week in 9^{th} grade class we were discussing the following problem, which had been assigned for homework.

Write an equation for a rule a?b, so that the answer is odd only when both a and b are even.

Since this problem admits of multiple answers, I thought we’d hear what several different students came up with and test them. Student #1 offered a rule that was of the form (a+even)(b+even). Everyone agreed it fit the bill. Student #2’s example was less obvious; we’d have to play around with it to see if it worked. So far so good.

Student #3 tried plugging in two even numbers and got an odd answer. Other students were trying similar approaches. “But wait,” I said, in words that I now regret, “remember that this rule needs to produce odd answers *only when* a and b are even. We need to test other kinds of numbers for a and b.”

I waited. The next time a student raised a hand, it was to ask if they could give their own rule. Of course they were disappointed when I replied, “We have to finish checking Student #2’s rule first.” No one else had anything to say, so to help things along I suggested that we try the formula with a and b both odd. One student objected: “the problem is about when a and b are even – we don’t have to do that.” I explained as best as I could why we did, in fact, have to do that.

And then, the hideous cycle repeated itself I don’t know how many times.. I tried to give prompts to students to discover a counterexample that I knew was waiting, and they were genuinely stumped. The more focused students were quiet because they didn’t understand the logic of the problem well enough to understand what a counterexample would look like, and the less focused students continued to raise their hands, enthusiastically wanting to offer their own rules. I cringe to tell you that we spent about 15-20 minutes on this problem before I finally showed them a counterexample, and we moved on. Perhaps worst of all, those students who so desperately wanted to demonstrate their own rules (most of which probably wouldn’t have worked, given their incomplete understanding of the problem) did not even get to do that.

In retrospect, I wish I hadn’t assigned this problem at all. Or I suppose I could have assigned a version of the problem that was equivalent to the students’ interpretation of the problem. Knowing now that the notion of “only when” was not intuitive to these students, even after explanation, I *might* have designed a worksheet with much easier questions designed to illustrate the principle. But I don’t think I would have done that, because ultimately, with another year or two of maturity, these students will come to understand phrases like “only when” mostly on their own. There’s no reason to force the issue now.

And yet I *did* force the issue. Once it became clear to me how difficult they were finding the concept, and how not a big deal that was, I still didn’t pull the plug on the discussion. Why? Because I couldn’t find a graceful way to do it. I could have said, “Yep, you’re right – the rule works,” but I couldn’t stand the thought that we were, officially, as a class, going to make a major error in logic. I could have said, “You know what – this problem is harder than I thought it was. Let’s abandon it.” But that wouldn’t have made any sense to the students, most of whom thought they had solved the problem.

Do you ever find yourself in situations like this? What are your “abort mission immediately” strategies? And what would you have done in my place?

## 8 Comments

Sounds like a tough situation! I might hunt for a “real-world” example of “only when” and see if they understand how to check in that kind of situation. For instance “people may pass through Customs only when they have passports” or something like that. That seems to be my general strategy when I run into a wall like this: ask a different question that has some relationship to where I think we’re stuck, work through that, and then see if there’s any insight into the original question.

Yeah, tricky, I suspect we’ve all been there some time… in terms of “abort”, it would tend to be just me giving the answer, which isn’t ideal, but might be the best of the lousy options. I usually at least try to follow through with an explanation.

One thought in this instance might be to continue to accept suggestions, but categorize them into two sections on the board: one section where it works, and one where it seems to but there are actual counterexamples. (You don’t have to set that up in advance either… just put them into the two groups yourself, and ask the students how you’re doing it. Or if you get one where you’re not sure, ask THEM which group it should go in.) Seems like they just had the one particular rule case to work with here (#2’s). If you can get a few, seeing that they have counterexamples in common might be easier. (Or maybe not, and it’s dubious I’d have thought of this at the time. But maybe for next time?)

The other option instead of “Let’s abandon it” is “Let me think about this again and we’ll revisit it tomorrow”.

I like Joshua’s reply. Instead of leading the students to feel they need to memorize some logic rule that doesn’t make sense to them, they are led to see how the logic rule really makes sense. Hopefully then they can apply the rule to the problem. In general, I like to use real-world examples to teach logic rules. Some of my students enjoyed using imaginary-world examples too, e.g. If a Muncha Monster has red hair, then it has 3 eyes; Muncha Matt has 7 eyes; What do you know about Muncha Matt’s hair? Mimi, I’m curious to see if your students could understand “only when” statements like Joshua’s passport example, and whether they could connect this understanding to the original problem. If you try it, I hope you’ll let us know how it turns out.

To answer your original question, in situations like these, I have stopped the discussion and said something like, “I’m sorry everybody. There’s a bit of background knowledge that you would need to fully understand this problem and I thought that you had already learned about that, but I see I made a mistake. So let’s put this problem aside until we’re ready to tackle it and move on to a different problem now.

I’m taking Keith Devlin’s MOOC about mathematical thinking, and was tripped up by this same kind of reasoning. Here’s the first problem (and it’s a famous task, so maybe you recognize it!):

Four cards are placed on the table in front of you. You are told (truthfully) that each has a letter printed on one side and a digit on the other, but of course you can only see one face of each. What you see is:

B E 4 7

You are now told that the cards you are looking at were chosen to follow the rule “If there is a vowel on one side, then there is an odd number on the other side.” What is the least number of cards you have to turn over to verify this rule, and which cards do you in fact have to turn over?

And then the second problem:

You are in charge of a party where there are young people. Some are drinking alcohol, others soft drinks. Some are old enough to drink alcohol legally, others are under age. You are responsible for ensuring that the drinking laws are not broken, so you have asked each person to put his or her photo ID on the table. At one table are four young people. One person has a beer, another has a Coke, but their IDs happen to be face down so you cannot see their ages. You can, however, see the IDs of the other two people. One is under the drinking age, the other is above it. Unfortunately, you are not sure if they are drinking Seven-up or vodka and tonic. Which IDs and/or drinks do you

need to check to make sure that no one is breaking the law?

I, like most people, flubbed the first one and got the second one right. Luckily, I was able to reason that they were logically identical and go back and change my answer to the first one! But it seems like, especially in unfamiliar contexts, the “only when” concept is really hard for people, even adult people!

Max

PS — I know this doesn’t answer the “how do you bail out gracefully” question, which I wish I had a good answer for! But I thought you (and even your students) might like to see how this situation stumps even the occasional math teacher!

One of the amazing things about your story is that it sounds like the students were not looking for the “teacher stamp of approval.” They were not upset simply because you did not say their answer was right or wrong. Congrats on getting your 9th graders to that point!

Now that they are there, you can leverage it to your advantage. One student offers an incorrect answer, but none of them see the counter-example. Many are eager to move on and discuss their own ideas. You can say something like, “You are all satisfied with this answer? Okay, well, there is something here that I would like to come back to later, but for now we will move on.” As they produce more and more answers to the problem, you could even arrange them on the board so that the ones that meet the “only if” condition are listed together, and the ones that don’t meet it are listed together. At the end, if there is time, see if they can figure out the difference between the two sets of answers. Or, come back to it the next day when you are prepared to discuss this idea. That sort of wishy-washy language can be helpful; you are not agreeing with them, and you are planting the idea in their minds that there is something here worth discussing farther. Maybe some of them will go home wondering what that other thing was, and eventually find it on their own, or maybe not.

Of course, this is my idea while sitting back, reading about it, sipping tea, and having plenty of time to think. Who knows what I would have done in your position. But it’s a great question, thanks for describing it so clearly and sharing it so openly!

Thanks, Debbie. I really like the idea of being nebulous in the moment and returning to the issue when I’ve had some time to think. I’m sure I’ll find opportunities next school year to do just that!

I totally agree with the above – when I was training as a teacher I witnessed something that I continue to build on in my own teaching today: an experienced (and excellent) teacher simply told the students it was his own fault for getting his planning wrong (not their fault for not understanding) and that he would go home and think about it so they could try again the next day. The rest of the lesson they played maths games. The pupils fully appreciated and accepted the teacher’s honesty and it set a good example to them (and me!) about taking responsibility for your mistakes and then fixing them. Job done : )

I like the idea of a problrm parking lot. We just park it till we’re ready to give it another go. The problems are posted wheer they are visible.Someodtimes it’s the kids who are ready first, siometimes it’s me.