Thursday, 25 September 2014

Just teach, dammit!

Twenty years ago a student had some advice for me:
It's the professor's job to know the theory. It's the student's job to know the facts. You should just tell us the facts and show us how to use them.
That's what the student told me, and this is what I heard:
I don't want to know why things work, I just want you to show me how to do it!
"YESSS!" says the student.

"AArgh!" say I.

I really enjoyed teaching math at university. The students were mostly receptive, most of them worked pretty hard, and I got along with them very well. However, there were always a few that fell outside this norm, and for those few there were typically two things that annoyed them.

The first thing that really yanked their chain was having to learn a proof. The process caused them great agony, and adding to their stress was the fact that a proof often had no immediate use beyond the theorem it was attached to. The other thing that irritated them, not quite so mightily,  but still quite a bit, had to do with what they said they wanted, that is, with what they called the how-to-do-it part of math. They would resist learning a new way of solving a problem when they had an old way at hand, and this was true even when the new way was more efficient. Learning proofs and learning alternate approaches, those two things really rankled them. 

Now, I was actually a pretty good teacher, and I've had some success. As I said, I enjoyed teaching very much, and I'm sure the students knew that and responded to it. However, it was difficult holding my exasperation in check when I met students who did not want to know why something was true or who were unwilling to try different approaches. They just didn't get it. Worse, it seemed like they were not even remotely interested it getting it. 

Interpreting the students' behaviour in a charitable light, I would guess that they were asking for help, but I have to say that it bothered me a lot when they reacted in such an anti-intellectual way. I wonder if they picked up that attitude somewhere, or if their reaction was an inborn one. Does such a response originate in the parenting, or in the school system, or is it really an innate human characteristic? Psychologists tell us that youngsters have an insatiable curiosity, and watching my children and grandchildren grow up tells me the same thing, so it is hard for me to accept that reacting so negatively is part of the human condition. 

So where did the negativity come from? The complete answer is probably quite complicated, and it is outside my expertise. The only way I can understand it is by extrapolating from my own personal experiences. 

Unlike many of my colleagues, becoming a mathematician was not a smooth ride for me. In high school and university I was very good at math but even during those times I did not always like it. There are still parts of math that cause me difficulties (namely arithmetic), and when I think about my past I am quite surprised that I did become a mathematician. 

At the very start, in elementary school, I had some difficulty with arithmetic. Lots of difficulty. I really disliked it and I avoided it whenever I could.  There is no question that I had a bad attitude. 

Some of my troubles arose because my memory for numbers is not always trustworthy. Although I was not aware of this until I was an adult, it certainly must have been a contributing factor in my younger years. However, I somehow became at least marginally competent in arithmetic, and after much reflection I don't believe that an unreliable memory was the main cause of my difficulties. I think my difficulties and my resentment were rooted in the math curriculum and the way we were expected to learn it. 

We learned math in the good old fashioned way,  that is  1) by memorizing the addition and multiplication tables, 2) by practicing such things as adding in columns and performing long division, and  3) by memorizing some procedures to solve some specific problems. 

Some of my classmates thrived under this regime, but a good number of us did not. For me, arithmetic never fully took hold. I never became skilled at it. I found it boring, I found it confusing. The good old fashioned way did not work for me, and it also did not work for a lot of my classmates (so it couldn't have been just my wonky memory that caused my troubles). 

With the good old fashioned way there was an official and unalterable route to the answer. We were taught "This is how you do long multiplication. This is how you do long division. This is how you add a column of numbers." That, together with the practice, the drills, and all that memorization delivered a very strong hidden message:  "You don't need to understand why it works, you just need to learn how to do it." 

Some people, many of them my colleagues, call this "learning the fundamentals". Well, if that's the case, I personally never learned the fundamentals. In retrospect, I eventually did acquire the basics, but not because of the good old fashioned way. I learned them because I was lucky enough to find some ways to work around the procedures that I couldn't master. It would have been a lot easier for me if I had been taught those work-arounds without having to devise them myself.

Things are different in elementary school today. Here's a thumbs up to the teachers and the Education profs who are trying to make kids math life so much more interesting and less punitive. 

Some of you know where I am heading with this. I am distressed with that Alberta back-to-basics petition. Read it, and see if you don't think that they want to reinstitute the "good old fashioned way". Our newly appointed Education minister and the petitioners talk about understanding the basics, but I'm not sure about their commitment to the understanding part.  Despite their disclaimers, I think that, deep down, many of the people who signed the petition want to banish understanding from the classroom by getting rid of anything that offers alternatives and exploration. 

I can't remember much about elementary school math except for the memorization and the drills. It wasn't until I attended high school that I was encouraged to tinker and to try different approaches, and it was there that some remarkable teachers began changing my attitude. Here are three episodes that I remember about learning math in high school. I know these won't seem very novel to today's teachers, but read on anyway. (But be prepared for a little math.)

Episode 1. Mr. Troughton's quiz 

In our first year of high school, in our first day of class, in our first ever Algebra course, our teacher, Mr. Troughton,  gave us a short math quiz. He read the questions aloud, and we wrote down our answers and handed them in. Here was the first question. Although very well known, it was new to us. 
A bottle and a cork cost $1.10. The bottle cost a dollar more than the cork. How much did the cork cost?
I confess that I was sometimes a bit of a smart-ass, and, with a satisfied smirk, I wrote down my answer:  "The cork cost ten cents!!"

Next day, when Mr Troughton handed back the answer sheets he asked "If the bottle cost $1.10, and the cork cost ten cents, how much did the bottle cost?" 

"A dollar," we said.

And then he asked  "So how much more did the bottle cost?" (was he looking directly at me?) 

Ha! I was supposed to be one of the smart ones.

That was first time ever in a math class I was faced with a problem that I had not been taught how to solve. This was a brand new experience, and was a bit of a shock. In mathematics, isn't the teacher supposed to show us how to do the problems? Are we not supposed to always get the method for working out the solution? It took a few more years to realize that the answers will forever be "No."  

Episode 2. Mr. Stirling's geometry challenge

The next year there was Mr. Stirling, who taught us grade 9 Geometry. This was our first exposure to the subject, and early in the course he showed us how to bisect an angle with a compass and a straight edge. Then he challenged us to divide an angle into quarters, and after we figured that out he mentioned, rather off-handedly, that nobody had been able to trisect an angle. He promised that we would become famous if we could do it.

I know external rewards are frowned upon, but the truth is that the prospect of fame was a very tempting lure, and we bit. We spent the next few days exploring the trisection problem. It consumed our lunch hours, and Mr. Stirling let us continue working on it during class.  In the process, we became skillful with the geometric instruments, and we learned what geometric constructions were legal and what types were not. We learned quite a bit of geometry by playing around with it on our own.

[ starting the mathy part . . . .

One bright boy claimed he had a solution. Using a protractor, he measured and drew a trisecting line. We objected. "You can't uses a protractor, you can only use a compass and straight edge," we said. 

"Never mind" he said, "Watch: if you bisect the angle, you get two half-angles, and one of them contains the trisecting line." His argument continued like this:

"Bisect that half-angle and you get two quarter-angles. One of the quarter-angles contains the trisecting line. Now bisect that quarter angle, and you've got a couple of  one-eighth angles and one of them contains the trisecting line. If you keep on like this, you get a one-sixteenth angle, then a one-thirty-secondth angle and you keep getting closer and closer to the trisecting line, so eventually we should get the trisection."

Mr. Stirling gently pointed out that this was not a legal solution because you must be able to finish in a finite number of steps. Nevertheless, the student had discovered something very interesting, namely that 1/2 - 1/4 + 1/8 - 1/16 + 1/32, etc, would get us as close to 1/3 as we desired. Quite a feat on his part, I think, and there is a lot of mathematics going on here.

 . . . . ending the mathy part ]

We never did find the answer to the trisection problem, and I learned many years later that the problem was not merely unsolved, but that the construction has actually been proven to be impossible. I don't know if Mr. Stirling knew that (but I sure hope he didn't).

Episode 3. Mister Watson's Intermediate Algebra course

Mr. Watson walked into the room with the textbook in his hand and sat down at his desk.  We all knew who he was, so he didn't bother to introduce himself. I can't remember what he said about the course but I do remember that there was an uncomfortable silence. We looked at him, but he just sat there. Minutes passed. More minutes passed. Tension grew. Finally, someone put up a hand and asked "Sir, when are you going to start teaching?" 

He smiled and said something like: "Don't you have a textbook? Open it at the beginning, read chapter one, and try solving the problems at the end.  If you have trouble, I'm at the front of the room. Bring your work here and I'll help you."

So we had to learn on our own, and we had to read the textbook by ourselves. Occasionally (maybe a bit more frequently that I remember) there would be a passage in the text or a problem that was difficult, and if Mr. Watson noticed that a lot of us were stuck at that point, he would discuss it with the whole class. 

This continued throughout the course. Sometimes he would teach for the entire period, and sometimes never at all. But what I remember most is working on the problems by myself with hardly any help or instruction from him. 

We finished the course material early, many weeks before end of the school year. This lengthy span of non-course days, however, was not entirely goof-off time. Mister Watson filled the time with a mix of different things. Mostly he lectured about different topics, but what I particularly remember is that he brought math puzzles to the room.

One puzzle stood out, and I'd like to leave it for you. There's a good chance you are familiar with it, but, as was the case with the grade 8 quiz question, it was completely new to us. Here's the puzzle (and no, I'm not going to tell you the solution):
In the following sum, each letter stands for a digit, and different letters represent different digits. The leading digit is never zero, so neither S nor M are zero. Find what each letter stands for.

These three episodes did not turn me into a mathematician, but they did help me realize that I was very good at math and that parts of it could be very interesting. And they did help quash that budding anti-intellectual attitude that I had picked up in elementary school.

Here endeth the lesson.

Sunday, 7 September 2014


You shouldn't have to memorize the 12-times table. You can just multiply by 10 and by 2 and add the results. That was my very vocal argument to my grade three teacher many years ago. My friend, whom I'll call Tough-boy, joined the fray, but the  teacher was having none of it and she sent us outside the classroom to wait until the principal came by. 

That was sobering. In our young eyes, the principal was a vile humourless creature who enjoyed terrorizing schoolchildren. He regularly walked the corridors of the school, and as he walked he announced his whereabouts by bashing the hall lockers with a thick leather strap. As Tough-boy and I waited on the second floor, we could track his progress as he proceeded along the corridor on the floor below. I was very frightened. Tough-boy, who had been in this situation several times before, offered some practical advice: "Let your hand go limp just before the strap hits. It won't hurt nearly as much."  

We were strapped because we were being disruptive in class. It was intended to teach us to not argue with the teacher. It worked, but in my mind I believed that I was being punished because I couldn't master the 12-times table. 

Skip ahead a generation or two. A few months ago I asked my grandson (grade 5) if he had to memorize his multiplication tables. He said "Yes, of course." I asked him a few questions such as "What's 9 times 3? What's 4 times 7?" which he answered correctly, and then I told him that I always had trouble remembering 7 times 8. Without hesitation he said "It's 56." And then, without prompting, he said "You know how I remember it?  Because 6 times 8 is 48 and so 7 times 8 is 48 plus 8."  Apparently he was not disciplined for using this strategy. 

Here in Alberta there is dissatisfaction with the way elementary mathematics is being taught. A large number of people have signed a "back-to-basics" petition which they hope will eradicate strategies like the two that I have just described. I'm sure that my grade three teacher would approve, for the petition is really an endorsement of the way I was taught arithmetic: in those days "the basics" meant memorizing "facts" and following "rules", with such knowledge to be acquired almost exclusively by rote. 

For me, the basics did not work very well. As often as I tried to memorize the 12-times table, it never stuck with me for more than an hour or so. My ability to recall numbers was iffy, so I continued to use my "10 plus 2" method, and I hid this strategy from the teacher as best I could. I left grade three with a significant dislike of arithmetic. To borrow some words from the Alberta petition, I was "repulsed by math".

The back-to-basics crowd has persuaded our government to revise the curriculum, and students will now be required to recall multiplication facts up to 9 times 9. Actually, that's not an unreasonable demand. However it is not clear to me that students weren't already expected to do this anyway. But I wonder if my grandson's strategy for "memorizing" 7 times 8 will now be unacceptable. If so, I imagine that he might continue to use his strategy and just shut-up about it like I did.

The curriculum changes have not satisfied everyone, and there is a push for more.  (Check out some of the the articles curated in Egan Chernoff's Matthew Maddux Education  blog.) The argument is that intensive rote learning will improve the students' arithmetic skills which in turn will lead to a greater understanding of mathematics. 

Frankly, I don't buy it. The argument seems to be based more on opinion than evidence. For example, long multiplication is an application of the distributive law, and so one might conclude that mastery of the long multiplication algorithm in arithmetic would lead to ready understanding of the distributive law in algebra.  Sean Carter, who teaches grade 9 math in Australia, reports that this did not happen in his class. See his blog about it here.

Back in my day, we were told that there were good reasons for memorizing the 12 times table. Here are two justifications that I can remember: first, it would make it easier for us to comparison shop because things were often sold in dozens, and second, in order to convert feet to inches you had to multiply by twelve. Hmmph! At least we weren't told that we that needed to memorize the 12-times table on the off chance that we might some day visit Britain and have to convert shillings to pence.