Teaching math to pre-service teachers (3)

Previously, I posted about teaching a class of students who, for all intents and purposes, lived in different mathematical time zones. Every new mathematical topic presented a challenge, most especially when it was first introduced. Mention a new math concept right off the bat and I would lose the weaker students. Make it so simple as to be trivial and I’d lose the stronger ones. 

I needed a way to introduce new mathematical material that appealed to all, one that was non-threatening to the weaker students, yet just perplexing enough to capture the interest of the stronger students. I wanted an opening with a mathematical floor so low that it didn't even appear to involve any math at all, but which at the same time contained the core idea(s) underlying the new material.

One tactic that worked for me was to introduce things via a good puzzle, and I gave an example in the previous post. Another tactic, which I'll write about here, was to start things off with a math-based magic trick. This was not quite as successful as using a puzzle, but it did work. 

As was the case for initiating a topic with a puzzle, before doing the trick I never announced what mathematics was involved.

Here, for example, is a trick that, although simple in the extreme, permits an approach to a topic that usually causes eyes to glaze over.  

The 1001 trick

You place three playing cards face down on the table.

You invite a volunteer to write a three-digit number on the board. She writes 732.

You ask the volunteer to write the six-digit number that is formed by repeating the three-digit number alongside itself. You may have to explain that abc becomes abcabc. 

She writes 732732 on the board.

Then you ask the class to choose one of the three cards. You turn over the indicated card and it is a Jack. We are using the cards as counting numbers so that  Ace = 1, Jack = 11, Queen = 12, and King = 13. 

You ask them to divide the six-digit number by the Jack: 

732732/11 = 66612.

You ask them to indicate another card. You turn it face up and it’s a 7.  You ask them to divide the new number by 7.

66612/7 = 9516.

Now you tell them that when they divide that number by their original number, the result will match the card that is left. So they divide 9516 by 732:

9516/732 = 13.

You turn the remaining card face up and it’s a King!


Explanation 

It doesn’t matter what three-digit number is chosen, but it does matter what three cards are used—they must be a 7, a Jack and a King.  (After doing the trick, I always told the students this fact. I would usually repeat the trick, having them select different cards from the 7, Jack, and King.)

The trick works because 

732732 = 732 x 1001,  

and   

1001 = 7 x 11 x 13. 

So, dividing 732732 first by 11, then by 7, and then by 732 amounts to this:

732732 x (1/11) x (1/7) x (1/732) 

which is really this:

732 x 7 x 11 x 13 x (1/11) x (1/13) x (1/732).

Doing the trick a few times lets the students see that the order in which they selected the cards does not make a difference. In the past, most of my students had heard the words “commutative law” and “associative law”, but many hadn’t remembered these laws or seen them in action. The trick would not work if those laws weren't true. So I used this trick as a jumping off point to discuss the basic laws of arithmetic.

(I should mention that the fact that 1001 = 7 x 11 x 13 is the basis of an interesting test for divisibility by 7, 11, or 13, so the trick had some relevance later in the course.)

 * * *

Here's another simple trick that I have used to introduce the same topic (I can't recall where I first saw it):

Your favourite number

Everyone has a favourite digit. You can’t have 8 because that’s mine. The digits that are left form the magic number 12345679. 

Ask a volunteer to tell you his/her favourite digit. 

Jan tells you 6. You tell her that her magic multiplier is 54, ask her to multiply the magic number by her magic multiplier,  and this happens:

12345679 x 54 = 666666666.

Henry tells you 4. You tell Henry his magic multiplier is 36 and this happens:

12345679 x 36 = 444444444.

Explanation

You get the magic multiplier by multiplying the favourite digit by 9.

It so happens that 111111111/9 = 12345679, so multiplying the favourite digit, n, by the magic multiplier (9 x n) does the following:

12345679 x (9 x n) 

= 111111111 x (1/9) x 9 x n 

= 111111111 x n =  nnnnnnnnn.

And the trick would not work if the associative law of multiplication did not hold.  

* * *

The cup of coffee swindle

You shuffle a deck of cards and place the top card face down on the table. You predict that card’s value will be the same as a randomly chosen number.

With your back to the board so you cannot see, a volunteer writes any three-digit number on the board. It can be any number as long as the digits are different. 

The volunteer is asked to scramble the digits to form a different number. The smaller number is then subtracted from the larger and some arithmetic is performed, all of this being done out of your sight. 

Apparently unknown to you, the computation results in the number zero. (Your back is still turned and you do not see the answer.)  The students are sure that you have made a mistake. However when the volunteer turns the prediction card face up, it is revealed to be totally accurate—its face is completely blank.

Explanation

(You can get a blank card from any magic supply shop or you can manufacture one yourself). 

Of course, the shuffle is fake—it’s done so that you retain the blank card on the top of the deck. And any three-digit (or two-digit or four-digit) number may be used as long as not all the digits are the same.

Here are the instructions that you give for the computation after the smaller number has been subtracted from the larger one: 

“Add  the digits of the result. Don’t tell me your answer. If your answer has two or more digits, add the digits of your answer together. Keep doing this until you get a single digit. Tell me when you get a single digit, but don't tell me what that digit is.

“Now, subtract that single digit from 9. Don’t tell me the answer, because that’s the number that the card is predicting.”

Here’s how the computation would proceed if the randomly chosen number was 128:

Scramble the digits:  812 

Subtract the smaller number from the bigger one:  812 - 128  =  684

Add the digits of the answer:   6 + 8 + 4 = 18

Two digits, so add the digits again:   1 + 8 = 9

Single digit, so subtract the answer from 9:   9 - 9 = 0.


That was how I sometimes introduced a module about digital roots and divisibility tests. My students all knew the tests for divisibility by 2, 5, and 3, but strangely, they hardly ever knew the test for divisibility by 9. 

In case you don’t know what a digital root is, it’s just the sum of the digits of the number repeatedly done until you end up with a single digit. If the result is 9 (or zero), the number is divisible by 9. 

This trick uses the fact that, no matter what number you start with, the subtraction always results in a number whose digital root is 9 (or zero). And after the trick, the students were usually interested in seeing why that happens and why the test for divisibility by 9 works.


I usually began my presentation of this trick by saying 

“I have won many cups of coffee with this test because it works almost all of the time, even though people want to bet me it can’t be right. But it’s not fair to swindle people so I won’t take any bets this time.” 

But that's not the reason I call it The cup of coffee swindle. The first time I used this trick, I didn’t preface it with anything, and one student bet me a cup of coffee that I was wrong. When I turned up for the next lecture, there was a cup of Timmy’s sitting on the table, but the student wasn’t there. She was embarrassed and angry, and she skipped the next three lectures. I feel bad about it to this day.

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. 

Teaching math to pre-service teachers (1)

They’re going to be teachers, not mathematicians. 

In Alberta, the government certifies elementary teachers, but not with regard to what they are qualified to teach. Elementary teachers are generalists, and all of them are expected to be able to teach mathematics.  

In the past, the Faculty of Education at the University of Alberta encouraged Elementary Education students to take one or more half-term courses offered by the math/stats department. Some opted for a rudimentary Calculus or Statistics course (neither of which is still offered by our department). Others took Math 160, a course which I taught many times. (It went by the spiritless and dismaying name Higher Arithmetic, and it was restricted to Elementary Education students.)

Back then, math was a bad thing for aspiring elementary teachers. Previous encounters with it had intimidated them. Math avoidance had become part of their lifestyle. The exasperating fact was that they were actually very smart, and abundantly capable of understanding the math that they needed. 

Of course, not all of my students had spent their lives sidestepping math. Some were very strong mathematically, and they actually liked the subject. 

So, there was a large variance in the students’ readiness to take on more mathematics, and it raised some concerns: Should I cater to the math avoiders and reteach them “the basics”, or should I dip right into the “higher level” topics for the stronger students? Or is there some middle-of-the-road compromise? 

Here are three options which appear to offer some resolution and which I occasionally found quite tempting. 


Option 1. Treat Math 160 as a sieve. 

In a sieve course, the professor usually has in mind standards that should be achieved, and he or she sets up the course accordingly. The standards are generally quite high and inflexible, and only those students who approach a predetermined level of attainment get a passing grade.

Rationale: Somehow, the weaker students gained admission to our university, and that’s not our fault. Teach the course at the appropriate level and let the students sink or swim. Weed out the weak ones before they become teachers.

Counterargument: It’s not always clear what the the words “appropriate level” mean. Besides, this won’t actually weed out the so-called weaker students. Remember, many of them are expert at math avoidance. If pushed, they’ll find a way to circumvent the math requirement. And when they become teachers they’ll still believe that math is a bad thing and will transmit their aversion to another generation. 

Even worse, a sieve math course will weed out some brilliant arts and humanities teachers.

Caveat: I do not advocate a complete absence of standards.


Option 2. Treat Math 160 as a math appreciation course.

By a math appreciation course, I mean one that discusses the accomplishments and personalities of mathematicians and explores how math has played a significant role in civilization.

Rationale:  Such a course shows students the human side of math and de-emphasizes the hard mathematics that frightens so many. It will show students that math is something that is worthwhile learning and will encourage them to learn more. 

Counterargument: Adulation of great mathematicians spreads the myth that you have to be a “math brain” to learn basic mathematics. An appreciation course for students with a weaker math background will be high on memorization, low on developing mathematical thinking & problem solving skills. The course may end up doing exactly what the weaker students are doing, namely avoiding math. 

Caveat: I do think that university math departments should offer courses that deal with the sociological aspects of mathematics, but, I also think such courses should have a fairly heavy dose of mathematics and therefore should be restricted to students with strong math backgrounds. 


Option 3. Treat Math 160 as a remedial course.

Ahh! The teach-them-the-basics approach. By “basics” I mean stuff that usually involves the so-called standard algorithms along with lots of shortcuts and tricks and strange rules like cross-multiplication or FOIL or BEDMAS¹. The course should include lots of drill and practice so that the students become fluent with the material that they should have already mastered.

Rationale: Too many Math 160 students lack mastery of the basic skills—for example, some cannot even carry out the long division algorithm². It’s our job to remediate the situation—the students should know the basics and should be able to perform the standard algorithms automatically. 

Counterargument: For a large number of people, the basic arithmetic facts and routines are very abstract and highly cryptic. Drill, practice, and contrived exercises will not only fail to alleviate their view that math is impenetrable, but will actually reinforce that belief. Math avoiders will become even more reluctant to learn mathematics. And the course would be a great disservice to the stronger students. To them, Math 160 would become a Mickey Mouse course. They would see it as an easy “A”, and would derive no benefit from it.

Caveat: In spite of my displeasure with the pedagogical advice by the back-to-basics people, I do believe that students should understand the fundamentals of arithmetic and be comfortable working with them. 

—

Teaching problems arise in any course where there is great disparity between the levels of the students³. I think the counterarguments above show that none of the three options resolve that situation for Math 160. In reality, all three skirt the issue by tilting the content towards one end or the other of the spectrum of student abilities.  

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.

In a subsequent post I will describe an approach for Math 160 that actually worked (at least for me).

—

[1] Incidentally, when I went to school, BEDMAS was called BOMDAS. You may called it PEDMAS or PEMDAS. The four algorithms don’t always produce the same result. James Tanton has an illuminating essay about the order of precedence for arithmetic operations. 

[2] This shocking state of affairs occurred in the era before there was any commotion about “discovery math”, so don’t go placing the blame there.

[3] Judging by what I see in teachers’ tweets and blog posts, the disparity in K-12 courses is even greater than in university courses.