Use it or Lose it !

This is not a column about sexual health.


I’m talking about the basic math facts — times tables and all that stuff that should be stored in our long term memory:

Will we really forget those facts if we don’t use them? 

Our brains are flexible. Neuroscientists use the word "plastic". They tell us that the brain's circuits can be repurposed, new ones can grow, and that frequent use strengthens neural connections. My guess is that YES, our memory of these basic facts will fade if we don't use them.


But I have a follow-up question.

Do you know that  1 / 1.04 = .96154? 

I do know that! And it's a fact that I don't use.


Here's the story.

Fifty years ago I worked for a large Canadian insurance company. I was a clerk in the mathematics section of the Policy Change department. My job was to compute how much to charge or refund when clients changed their insurance policies.

Among the many options available was the possibility of paying premiums in advance. The company liked it when clients did this, and to encourage them the company offered a discount.

We used an interest rate of 4%.  So, if the premium was, say, $150.00, and the payment was being made a year in advance, the client would pay $144.23, which is what you get from the following computation:

$150.00 / 1.04.

To the nearest five decimal places, 1/1.04 = 0.96154, so we could also calculate the discounted premium this way:

$150.00  ×  .96154.

Actually, 1/1.04 is slightly smaller than 0.96154, but for all practical purposes the two methods are identical. Yet, despite their equivalence, our boss made us use the second one. Instead of dividing by 1.04 we had to multiply by 0.96154. It was not because our boss had some weird mathematical fetish: an official company policy memo explicitly stated that when computing prepayments of premiums, we had to use the factors from the supplied interest tables. Policy trumps mathematics.

After a short time on the job, the "fact" that 1/1.04 equalled .96154 had established permanent residence in my mind and it still remains there even though I haven't used it for over half a century.

* * * * *

This sort of thing is not unusual.

• My parents-in-law, veterans of WWII, could recall their military ID numbers for their entire lives, even though they seldom used them.
  

• My wife and her mother used to play a game when she was a child. They recited the alphabet backwards as rapidly as they could. It's a skill that my wife has not used for over 60 years. Yet yesterday she showed me that she can still reel off the alphabet backwards without hesitation. 

So my intuition that facts fade from memory when they are not used may be wrong. Some of us retain basic facts and practiced skills even if we don't use them for an extremely long time.

* * * * *

On the other hand - 

Sometimes when people say "Use it or lose it" they really mean

Use it and you won't lose it!

Well, we all know that statement is false.

How else to account for my spotty recall of simple multiplication facts? I've used those basic multiplication facts quite a lot since leaving that insurance company, and I'm still not "fluent" with them. I can fire off the squares of the natural numbers up to 12 × 12, but ask me what 12 × 7  or 12 × 11 is and I have to perform a mental computation. And 13 × 13 is 169, but what is 13 × 9?

Some people can recall basic math facts quickly and without conscious effort — they automatically know them without thinking. Like the way I know that 1/1.04 is .96154.

Others will never completely achieve this type of fluency. When someone says "Twelve times seven" it may not provoke the involuntary "Eight-four". Instead, the reaction may be "seventy plus fourteen" having broken 12 × 7 into 10×7 plus 2×7.

To be truthful, I am fluent with most of the basic multiplication facts. But, I really do still have trouble with a few parts of the 6, 7, 8 and 12 times tables. And, as I prepared this post, I actually did have to retrieve 12×7 by that quick mental computation. However, once done, I could instantaneously recall the answer for the rest of the post, and I know that the automaticity will remain for several days, maybe even a couple of weeks. But it’s as impermanent as those facts acquired by cramming for an exam: very soon it will fade from my memory.

My advice 

If my children were young, this is what I would tell them today:

Practice your times tables. Memorize as much as you can. It is worth the effort. Maybe you will never be able to keep all of them in your head. But if you can keep a few there, and if you learn how to be flexible with numbers, you will find ways to derive the more difficult ones very quickly — almost as fast as if you had actually memorized them — and it won’t stop you from learning more mathematics.

As for getting every single one of those basic facts into genuine long term memory in a way that they can be instantly and automatically recalled: forget it — for me, and maybe for quite a few others:

It's • Not • Gonna • Happen

Addendum

Just as I was finishing this post, I came across an article (linked below) by Maria Konnikova in The New Yorker magazine. It's about the tenuous relationship between practice and achievement across a variety of fields (including mathematics). It's really worthwhile reading. 

Link: PRACTICE DOESN’T MAKE PERFECT

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. 

Don't trust the math prof

Have you read Dale Carnegie's How to Win Friends and Influence People? If you are a math prof, I'll wager that your answer is "No".  Maybe that might explain the unpleasantness that occurred during a math conference in Alberta. (The conference was over a year ago. I was not there, so my comments here are second hand, but my information comes from a reliable source.)

In attendance were both mathematicians and school teachers. During the proceedings, a few mathematicians took to the stage and panned both the math curriculum and the teaching methods being used. In effect they dressed down the teachers and told them how they should be doing their jobs.

I guess this happened more out of arrogance than malice. Yes, I do know that  arrogance does not invalidate the professors opinions, but it makes me think twice about any advice they might offer. 

A lot of math profs agreed with the criticisms raised at the conference. I too have opinions about curriculum and teaching methods, but I would expect a school teacher to be very skeptical about my advice. And there would be a good reason to be skeptical: unlike school teachers, I have not been taught how to teach, and neither have most of my colleagues. This does not mean that our opinions are automatically wrong, but it sure casts a shadow over them. 

Here's the problem. Some math profs have done a lot of teaching and have even become quite good at it. And like most people, they like to dispense the wisdom that they have garnered from their experience. Fair enough, but that experience is limited to university courses whose class members are not typical school students but are, in fact, the cream of the crop.  Although the profs may have developed some very good teaching practices, those practices are geared towards university students and likely won't transfer well to a K-12 classroom. 

(Is there not a little irony here? We mathematicians haven't even been trained how to teach at a university, and yet we are willing to issue directions about how to go about it in a vastly different setting.)

But there must be some value in what math profs say about math education: after all, they are experts at mathematics. This is a slippery argument, and you may have come across something like it before. It appears in different forms:

"I'm the CEO of BigOilCorp. Climate change is bunk."   

"I'm a certified marriage counsellor, and I know what I'm talking about. Children should be spanked for bad behaviour." 

"He says that the Edmonton Oilers stink. He has a degree in sports journalism, so he must be right." 

OK, so you may on board with that last one, but the reasoning is faulty. It's called "argument from authority" and in its undisguised form it goes like this:

• So-and-so is an authority about topic X.
• So-and-so makes a statement about topic Y.
• Therefore the statement must be correct.

Being experts at mathematics in no way confirms that our opinions about how to teach it are valid. Appealing to our mathematical expertise is simply an argument from authority. 

But why are math profs so ready to be critical? I have some thoughts about that.

Some say that their children have not learned the basics in elementary school. It's difficult to comment about this because it is so personal, but it is a concern held by a much larger group of people.

What I am about to say may be educational heresy. I think there will always be a substantial number of children who will have difficulties with math. It was true when I was a student, and it was true when my children were students, and it is true now that my children's children are students. 

I don't think the problem is wholly dependent on either the curriculum or the way it is being taught. From talking to my grandchildren, and from what I have learned from school teachers, (and also from a brief examination of the K-6 curriculum), my own conclusion is that students today are being taught the basics, just in a different way than we were. 

On a less personal level, some professors are concerned that students entering university have not mastered the fundamentals. They perceive that students arriving at university from high school today are not as adept at mathematics as they themselves were in the past. As long as I can remember, math profs, including me, have held that view.  (And in fact you can go back 100 years and read the same complaint.)  

When I first started teaching, our department's concern led to an "advisory exam" that we gave to first year students to check that their background was sufficient. Sometimes it wasn't, and our conclusion then was pretty much the same as what math profs conclude now: there must be a problem with the way math is being taught in school. Sigh. Perfectly logical mathematicians affirming the consequent. 

There is another thing that bothers some mathematicians. They are worried about Canada's falling rank in international math tests, you know, those PISA tests that have caused so much panic. Some trace the decline back to the introduction of our current elementary math curriculum along with the teaching methods that support it. I don't know if its true that a majority of math profs agree with that viewpoint, but a good many have signed a petition that promotes it, so I assume that plenty actually do believe it. Sigh. Post hoc, ergo propter hoc.

I don't personally think that there is a problem with our PISA rank, but that's a topic for a later discussion. However, in the meantime I would point you to an article by Joanne Jacobs. Take a look at this question:

Did this spark a WTF moment for you like it did for me? Well, what is happening here is that the children are being asked to compute 8 + 5 by splitting the 5 into 2 + 3 as follows:

8 + 5 = 8 + (2 + 3) = (8 + 2) + 3 = 10 + 3

There's no mystery here: As the teacher's feedback says, take 2 from the 5 and add it to the 8. That's what "making 10" out of 8 + 5 means. It's a method for addition that doesn't rely completely on rote memorization, and it is one of the strategies that some think confuses the children and contributed to our reduced PISA score.  

The comments following the Joanne Jacobs post are worth a look. Although there is the expected outrage, at least one person pointed out that "making 10" is one of the strategies taught to the kids in Singapore. And if you have been following the articles about the PISA math test, you know that Singapore ranked much higher than Canada. I find that somewhat thought-provoking.

That's it. Now, if I could just remember where I put that Dale Carnegie book.

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.

Twelve-times

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.