Teaching math to pre-service teachers (2)

In my previous post, I described a fundamental incompatibility between the students in Math 160: many students had consciously avoided math for most of their lives, while others were very confident in their abilities and even enjoyed the subject.

I ended the post with this paragraph:

When I began teaching Math 160, I had very specific ideas about how I should present the course. To me, the key to successful instruction was to concentrate on the clarity and explicitness of my lectures. It turns out that I was singing from the wrong songbook

I am reminded of that old but useful proverb: beauty is in the eye of the beholder. Or, in this case, ‘clear and explicit’ lives solely in the mind of the student.  For my Math 160 students, who differed so much from each other, there could be no interpretation of  ‘clear and explicit’ that would be suitable for all of them. Being ‘clear and explicit’ was not the way to engage them.

If you have tried to motivate students about mathematics, you know that the sticking point occurs at the very moment that you begin to talk about it. If you blow that moment, you will lose them for the duration.

So, after a period of time (too long a period), I shifted my attention towards what I could do to introduce new topics in an interesting way. 

For Math 160, there were certain constraints. I needed to make the introduction understandable without it depending upon prior knowledge. In particular, I wanted inexperienced students to be able to comprehend what was happening without a preliminary lecture or review. At the same time, I did not want the introduction to be obvious or trite to those who did have some previous knowledge. And one of my key tenets was that new terminology must be avoided so that students are not intimidated or distracted by strange jargon. And above all, it has to be interesting. 

Here’s how I usually introduced a certain challenging mathematical topic to my Math 160 students. It was my way of saying “Math 160, say hello to modular arithmetic.”

The Keystone Kidnapper

The police of Keystonia have a crack swat team. The team’s current mission is to rescue the prime minister's daughter who is being held hostage by an evil-doing kidnapper. But, through a sequence of mishaps, the swat team has allowed the kidnapper to capture them. He has herded them, along with the young lady, into a room containing seven caskets.

“Your puny intellect amuses me,” said the kidnapper. “You have one hour to figure out how to escape from this room. Sixty minutes after I leave, six of the seven caskets will disintegrate releasing one hundred angry venomous snakes. The other casket contains the key to the door. Find the key before the hour is up and you can escape. I'll give you a clue—the key is in casket number 54321. And I've done you a favour. The caskets are not locked.”

“But,” said the swat team leader, “The caskets are numbered from 1 to 7. None of them is numbered 54321.”

“I’ll give you another clue: start counting,”  and the kidnapper showed them how. 

Casket 1 was 1, casket 2 was 2, and so on until casket 7 which was 7. Then the kidnapper reversed directions: casket 6 was number 8, casket 5 was number 9, and so on until casket 1 which was number 13. Then the count reversed once more, and casket 2 was 14, casket 3 was 15. 

“You get the idea,” said the kidnapper, “Goodbye,” and he left them in the locked room.

“Well,” said the swat team leader, “Let's start counting.”

“Hold it,” said the young lady. “We have less than 60 minutes—that's 3600 seconds, and my calculator says that 54321 divided by 3600 is about 15. We would have to count 15 caskets per second and not make a mistake.” 

Luckily for the swat team, the prime minister's daughter figured it out. 

Which casket contained the key?

With its corny dialog and silly scenario, there is nothing real world about this puzzle, and there is no pretence that there is. But, as long as humans have walked the earth, people have willingly immersed themselves in tales set in unreal worlds. Using story puzzles to introduce new mathematics leans on this. It's a tactic that I used a lot, and for the most part my students let themselves be drawn in.

Spoiler Alert! If you want to solve this on your own, don't read below this line.

Here’s a short summary of how the lesson typically developed and how it fit the constraints that I mentioned earlier.

To set thing up, and to illustrate the counting process, I drew a diagram:

Usually the students asked me to write down a few more rows to clarify the counting process, and if they didn't, I wrote them down anyway, like this:

So far, all students were drawn in, and all had a clear grasp of the problem. And there was no alarming “mathematics” required. And the stronger students showed no sign of leaping to the solution. 

There was often some dead air time as they thought about the situation, and sometimes they had to be prompted with questions about what they noticed. Eventually, they started to see patterns. 

For example, some them noticed that in the leftmost and rightmost columns, successive numbers differed by 12. Or they noticed that in the second column the differences alternated between 10 and 2, and that in the third column the differences alternated between 8 and 4,  etc. 

The observations varied from class to class, and did not always occur in the same order. Sooner or later someone usually picked up on the ubiquity of the number 12, and said something like

“In all of the columns, the numbers increase by 12 for every two rows.”

With more prodding, they described what was happening by saying things like

“In the second column, all of the numbers are divisible by 12 or else their remainder is 2.”

Then, typically, someone will follow this up with:

“In the third column, when you divide by 12 the remainder is always 11 or 3.”

The key word remainder inevitably came up, but I avoided using the term until the students brought it into the conversation themselves. I sometimes had to find a way to push them towards thinking about remainders (by writing a few more counting rows and asking if they had any questions about any of the columns). I definitely never mentioned the words modulus or congruence or residue classes, even though they were grappling with these notions. They really didn't need them at this point, and there would be ample time to define them later.

As soon as someone mentioned remainders, they noticed the same sort of thing happening in other columns, which led eventually to making a chart of them:

Occasionally someone would express worries that this was leading nowhere. It is a difficult comment to handle. But, usually by this point in the course they will have learned that sometimes you need to go further down a road before you can decide where it leads. I would respond by saying something noncommittal like “Hmm.”

They noted patterns in the chart of denominators, for example, that although the remainder 10 appears repeatedly in the fourth column, it never appears in any other column. They usually also pointed out that the chart contained every remainder that you can possibly get when dividing by 12. (This sometimes initiated a conversation about whether zero should be considered to be a remainder.)

And finally a flash of understanding happened and they realized that they could identify the casket that contained the key by determining what column contained the remainder when 54321 was divided by 12. 

As I said, this is a summary, and it would be misleading to say that all classes followed the same script.  There were often lengthy pauses. There were unexpected observations that sometimes took us off-track. (As happened, for example, when someone noted that in many columns the sum of two consecutive remainders is always 14.) I tended to run with such diversions and let them play themselves out. Usually, someone interrupted and suggested something else that got us back on track.

In spite of the occasional detours, the classes always solved the puzzle well before the lecture period ended.  And I always pointed out that they found the casket well within the hour set by the kidnapper, which seemed to please them. 

* * *

The Keystone Kidnapper puzzle satisfied the constraints that I mentioned. The weaker students did not need either a preliminary lecture or new terminology to understand the puzzle. The quirkiness of the setting was captivating enough to conceal the solution from even the stronger students. And even if they knew something about clock arithmetic, they did not see the connection with the puzzle. 

The fact that the students at all levels worked through the puzzle together provided a common frame of reference. It helped overcome the non-uniformity of their backgrounds and provided a common base to build on.  Afterwards, when the new mathematical concepts of modulus, congruence, and residue classes were formally introduced, all the students could relate them to the same concrete setting. 

* * *

I don’t know the original name for The Keystone Kidnapper puzzle. I first saw it many years ago in a book by Martin Gardner I think, where fingers were being counted instead of caskets. (Several solutions for the finger counting version can be found on the internet. Almost all of the good puzzles have solutions somewhere on the web. If you use them, you might wish to alter the setting as well as the puzzle name so that students won’t be able to easily google them.)

As well as using puzzles to help introduce new topics in my courses, I occasionally used math based magic tricks. In the next post I will show how I used a couple of them, along with some pitfalls that have to be avoided.