What's Wrong with High School Math

I am an occasional contributor to the local weekly "The Jewish Post", and last year Bernie (the editor) published an article I wrote about high school math. Usually he reposts my articles on his website, but I noticed that this one doesn't appear. So I thought I'd put it up on my blog. I think it's a pretty good article.

What's Wrong with High School Math

I used to be the math guy on Channel 11. It was twenty years ago, and community access TV was a platform for kooks of all stripes to do their thing in front of a TV audience, and my “thing” was math. I started my show because I didn’t like the way math was taught in university, and I thought I could show by example that there was a better way to do things. Instead of teaching math as a set of rules to be followed in order to get correct answers, I wanted to show math as a way of looking at the world so that things made sense. In its three years on the air, my show had at least some success in that my presentations seemed to resonate with quite a few viewers, and not only for their quirkiness; on the other hand, I’m quite sure I had zero impact on the way math was taught in university.

Last year I went back to university to certify as a high school teacher. I didn’t last long: after repeatedly arguing with my professors, I was unceremoniously drummed out of the program after only ten weeks. But in that short time, I was appalled by what I saw going on in the schools. I know there has been a lot of public debate recently over “back to the basics” and the lack of basic literacy in math, but my issues with the system are a little different. This is article is about what I saw.

I am not so much interested in the nuts-and-bolts problem of instilling skills of manipulation in very young children. I am more concerned about what I might call the spiritual consequences of the way math is taught in the senior years.  To understand what I mean, let’s look at the reasons why math is supposedly important:

1. It teaches you how to think.

2. It provides important skills needed in everyday life.

3. “In today’s high-tech global economy”…well, you know the rest.

I’m going to go way out on a limb here and say these are bad reasons to teach math in high school. Let’s put aside for the moment the 5% of graduates who will go on to careers in the technical fields: not because I am willing to concede that they are well-served by the present system, but because the problem of high-tech education is too vast to deal with in this short article. I want to talk now about the 95% of students who won’t  become engineers or scientists. Why do they need to learn how to factor polynomials? Is it worth the cost in human suffering? Because surely only the scourge of acne can rival math as a cause of suffering among teenagers. How do we justify it?

We routinely justify it by reciting the three reasons listed above, but I find them very hard to take seriously. Does anyone seriously believe that mathematical reasoning is of any use in working out solutions to the ordinary problems of daily life: relationships, jobs, happiness or whatever? Even such iconic problems as rent-vs-buy, or how fast to pay down your mortgage…those are lifestyle choices that people will inevitably make for reasons that have very little to do with the the textbook “present-value” calculations that they may be taught in school. No, there is a fourth reason why we teach math, an unspoken reason:

4. We teach math because when we were young, we suffered through it: then, as we grew older, we validated that suffering by convincing ourselves in retrospect that it was “good for us”. And if it was good for us, it will be good for our children.

The pervasiveness of this attitude explains everything that is wrong with math teaching in high school. It explains why you’re not supposed to enjoy math, and it explains why it is alright to forget everything you learned the day after the final exam.

Mostly, it explains why you need to memorize algorithms to get the right answer even if you don’t know what you’re doing. Because the big, soul-crushing lesson students learn from high school math is that you will only succede if you follow directions. If you try to think for yourself, to ask why you need to do what you’ve been told, you will surely fail. 

Yes, high school math does teach you how to think. That’s what scares the s*** out of me.

The Foundations of Quantum Mechanics

Shortly after I started this blog two and a half years ago, I came up with the idea of Quantum Siphoning, a causal mechanism to explain the "collapse of the wave function". At that time I was planning a trip to Eastern Canada for later in the summer, and so I wrote letters to some two hundred physics professors, enclosing a link to my article and asking them if they would be interested in meeting with me to discuss it. I received a positive response from Prof. John Sipe of the University of Toronto, who was good enough to offer me a full hour of his time.

Professor Sipe obviously moves in circles where he regularly has the opportunity to discuss physics at the highest level with all kinds of brilliant people, so it would be presumptuous of me to think I could have made much of an impression on him. But the meeting was very exciting for me, and I was very much on my game. I think I held up my end of the discussion to the fullest extent of my abilities, and I have to say that Professor Sipe gave me his fullest attention for the whole hour. We touched on the possibility of him taking me on as a grad student, but he told me that although he publishes work in "Foundations" (that's what they call the philosophical underpinning s of Quantum Mechanics), he does not accept students in that field. So basically we spent the hour talking physics.

I told him that I was interested in developing natural explanations for the major phenomena which were traditionally called upon to justify the present-day Copenhagen-inspired interpretation of quantum mechanics. I listed what I considered to be the six most influential of these phenomena:

1. The black-body spectrum.
2. The photo-electric effect.
3. The Compton effect.
4. The discrete clicks in the Geiger counter.
5. The flecks of silver on a photographic plate exposed to weak light.
6. The straight-line paths observed in cloud chambers.

What these phenomena share in common is that they all purport to illustrate situations where the physics is completely described by differential equations (what we commonly call "wave equations") but the visible manifestations of these phenomena are inexplicable without ascribing some sort of particle-like behavior to the waves. In several of these instances, that behavior is best characterized as what we call the "collapse of the wave function". Whereas a differential equation by its very nature describes a disturbance as propagating in a natural and continuous manner through time and space, the experiments seem to show the sudden and random occurence of discontinuities in these otherwise well-behaved functions.

I think I am correct in claiming that in the late 1920's, when the great struggles were taking place in Copenhagen, Goettingen, Berlin and elsewhere to try and come to terms with the new quantum mechanics of Schroedinger and Heisenberg, that any explanation would have had to deal with each of these six subjects. Taken as a whole, they constituted the litmus test for any proposed paradigm. And the most prominent victim of that litmus test was of course the wave theory of light.

I told Professor Sipe that my goal was to re-habilitate the wave theory of light by showing how it was compatible with all six experiments in my list. And that was what we talked about for an hour. I'll tell you more about our meeting when we return.

Statistics of Ice Cream Cones

The other day my student brought me a homework problem from First-Year Stats at U of Winnipeg. Here is the problem:

1b). An ice cream shop offers an end of season special Fall deal for a fixed cost, where you can choose two of twenty flavors, two of eight toppings, and one of four cones. How many different ice cream treat combinations are available?

In the preamble to the problem sheet, the prof asks the students to show what formula they are using, but she doesn't ask them to state any assumptions about what is being sought. Where do you begin with a problem like this? You are getting two scoops of ice cream, two toppings, and one cone. What makes this kind of problem challenging is that you have to decide if you care what order the choices are made. Since the prof doesn't tell you whether or not you should care, you have to decide based on the real-life interpretation of the problem.

Is it a different order if you get chocolate with peppermint sprinkles and butterscotch with chocolate sauce, versus chocolate with chocolate sauce and butterscotch with peppermint sprinkles? You tell me. Is it different if you have black cherry on the bottom and cappucino on the top, or the other way around? Or would both of those count as the same order? Your guess is as good as mine.

Whether or not you choose to count those permutations as different or the same makes all the difference in the world as to how you would do the calculation. But the professor doesn't want you to think about that. She wants you to pick a formula and show your work, which means show how you plug the numbers into the formula. That's what passes for "education" in today's univeristies.

More Fun With Statistics

The other day I wrote about a homework problem in first-year Stats that, according to my thinking, illustrates much that is wrong with the way math is taught. A few days ago I met my student again, and she had more homework. Let's have a look at it.

The assignment begins with the following general instructions:


"Complete the following questions. Show all your work. Be sure to include the formula as part of showing your work. Please be neat and clearly state your answers. Please state your answer in a complete sentence and interpret the result. Remember to add a title page and to staple the assignment together. Follow the instructions for each question on how many decimal places to carry your final answer."

It's good to show your work. But I find it disturbing that the professor assumes that a formula must be part of your "|work". What if you figure out the problem by pure logic? How do you reduce that to a formula? I certainly can't.

What's going on here is that the professor is revealing her true philosophy of math education: learning math consists of memorizing a bunch of formulas and learning how to recognize which formula to use on which problem. Her instructions don't make much sense otherwise. And the problem she gave last week about the woman picking random shoes out of her closet...well, the whole problem doesn't make any sense unless you interpret it as a case of "match-the-numbers-the-correct-formula".

Today's assignment includes more of the same. To be sure, most of the problems are OK, but a few of them are objectionable. I have issues with the following three:

1a). An industrial plant will randomly select six machines from an assembly line containing 20 machines for a quality control check. How many ways are there to do this?

1b). An ice cream shop offers an end of season special Fall deal for a fixed cost, where you can choose two of twenty flavors, two of eight toppings, and one of four cones. How many different ice cream treat combinations are available?

4. On a 15-item true-false test, where a true item is as likely to appear as a false item, what is the probability of getting 10 true items on the test?

It's actually the middle problem that I find most objectionable, so let's set that aside for the time being. My complaints about the other two items are perhaps mere quibbles, but I find them to be revealing of an unhealthy attitude. Let's talk about the industrial plant first.

The statistics of quality control is an important and practical issue of which there is much to be learned. My problem with this question is that it has nothing to do with the quality control. It is true that there are so-and-so-many ways of picking six machines out of twenty, but I cannot for the life of me imagine any practical situation where you would care how many such choices there are. The beauty of a good real-world math problem is that it illustrates an interesting connection between math and the real world: and this question, while at first pretending to deal with quality control, turns out in the end to be nothing more than a case of plug-the-numbers-into-the formula.

My issue with the third question is similar, but in this case it is the pointless awkwardness of the question which stands out. There is a very important and interesting question which could be asked here, and in fact it is answered with the same formula which is intended for this item. But the professor goes into great contortions to avoid asking the interesting, practical question. Instead she asks an artificial and contrived question which is of no general interest but happens to be answerable by using the same formula as the interesting question.

In case you haven't figured it out yet, here is the practical and interesting question which the professor fails to ask:

"On a true-false test with 15 questions, what is your chance of getting ten out of fifteen by simply guessing?"
 
If you can go back and re-read the professor's question I think you can see how she asks something theoretical and esoteric that really misses the point that ought to be made.

By the way, it should also be noted that what is significant here is not really the chance of scoring exactly ten out of fifteen, but the chance of scoring ten or better, which is a somewhat more invovled calculation. These are the things that are important in math, and ought to be talked about. But they become irrelevant if your purpose is simply to plug-the-correct-numbers-into-the-formulas.

Which brings us to the middle question, the one about the ice-cream stand. Let's take this one up when we return.

Handicapper-General at Work

In "Welcome to the Monkey House", Kurt Vonnegut writes about a future where all men are finally equal, thanks to the unrelenting efforts of the United States Handicapper-General. It seems the Handicapper General was at work in Sweden this week, as we read what happened to the head cook in a school cafeteria who was reined in for daring to prepare better food for her students than the other cooks in neighboring schools.

My readers will know that last year I was expelled from the Teacher Certification Program at the University of Winnipeg. While I was in the program I learned a lot about what is wrong with the system. One aspect is the enforced mediocrity similar to what we read about in Sweden. I have a small example in front of me. I am looking at a Unit Plan I prepared for teaching Grade Nine Static Electricity. I was docked five marks because my work failed to satisfy the following criterion:

"Could this Unit Plan be used by a Substitute Teacher for 2 weeks?"

Apparently the fact that I know more about electricity than the average substitute drawn from the random pool should not be a factor in how I teach. Presumably if I had stacked my lesson plans with two weeks worth of worksheets on Ohms Law I would have gotten full marks for this line item.

I knew when I wrote my lesson plans that a typical substitute would not be capable of what I could do in class, but I refused to limit myself according to the philosophy of the tin-pot Handicapper-Generals who make up the Education Faculty at the U of W. I gladly sacrificed the five marks. This apparently enraged the professors, because I was given a failing grade (almost unheard of in Education) on the next set of lesson plans I handed in. I think I'll have more to say about that later.

Of Bumblebees and Orange Trees

The world of mathematics is populated by strange creatures like bumblebees who while normally flying in straight lines at constant velocity, are able to change or even reverse direction instantaneously. We have men in rowboats who proceed along a river at constant velocities and drop their hats in the water when they pass under a bridge. And recently I wrote about a spider capable of instantly calculating the optimum trajectory along the walls and ceilings between two arbitrary points in a room.

I do not object to these magical creatures even though their abilities and behavior are strange or even absurd when compared to real spiders and bumblebees. Why then do I object to a woman who decides to randomly "sample" her shoe collection when packing her suitcase for a business trip?

In mathematics, we use spiders and bumblebees as shorthand symbols to represent certain ideal behavior which is graphically suggested by the creatures we use as stand-ins. There is a certain charm to a math puzzle which is composed using such symbols, and there is little doubt as to what is intended. The homework problem we looked at the other day is an entirely different kettle of fish. It represents everything that is wrong with math teaching in today's schools, be they high schools or universities.

The school system pays lip service to the idea that they want students to learn to think and understand, but in practise the system demands that the student memorize instructions and follow algorithms. The homework problem I discussed the other day shows both of these hypocrisies in their fullest form: first, the lip service to the idea that the student should think about what is about what is going on...namely, a woman randomly choosing two pairs of shoes for a trip...and then, the turnaround where the student is told what formula to use for the calculation...."sampling with replacement", taking into account the order of selection.

I have already discussed the absurdity of calculating the answer based on which order the shoes are selected. If you pick the pumps first and the boots second or vice versa, you still end up with the same two pairs of shoes in your suitcase. If you want to create a word problem where the student is required to take into account the order of selection, then there has to be a practical reason within the word problem why it should matter. You don't write a problem where the order doesn't matter, and then tell the student to use the formula where it does.

And then there is the point of "sampling with replacement". Sampling with replacement means you put the first pair of shoes back in the closet before you choose the second pair. Now, how are you going to pack your suitcase if you do that? It just doesn't make sense. She picks a pair of shoes, puts it right back in the closet, shuffles the boxes around so they are again totally randomized and then picks another pair. How does she end up with two pairs of shoes in her suitcase? Maybe their are women who pack their bags that way, but I don't know how you can do any kind of mathematical calculation.

If I thought I was picking on one example of a bad teacher who made a silly mistake in a homework assignment, I wouldn't be writing this article. I've singled it out because this goes on all the time, and it's typical of everything that's wrong with math teaching.

(Oh, and I forgot to mention the orange trees.)

How To Lie About Statistics

I left you the other day with a homework problem one of my students brought me. I am often dismayed by what goes on in the education system, but this little item sums it all up pretty nicely for me. I'll let you read the problem, and see if it bothers you in any way. Then I'll tell you what's wrong with it.

Here is the problem:

"You're going to a two-day conference and can't decide what shoes to pack. You own 5 pairs of flats, 7 pairs of heels, and 8 pairs of boots. To save time you decide to randomly pick your shoes. If you sample two pairs of shoes, one at a time, with replacement, what is the probability you will get a pair of heels and a pair of boots in that order"

I always like word problems in math. The nice thing about word problems is they don't (or shouldn't) tell you how what formula to use: you have to understand what's going on and put two and two together for yourself. And of course, that's why a lot of students don't like word problems. It's not that they don't like to think: it's that the education system has never really encouraged them to think for themselves. The system is all about teaching you the rules for how to solve specific problems. If you are a good student, and learn the rules, you will be able to solve the problems. Within this paradigm, it is completely irrelevant whether you understand what you are doing. The system puts a premium on the ability to follow instructions.

Except when it comes to word problems. That is where a small complication creeps in: you have to interpret the problem and decide which formula to use in solving it. Students who are conditioned to learning by rules are uncomfortable with this, and so the teachers cater to them. They give a whole worksheet of word problems all based on the same formula, so you don't really have to think about what's going on; you just have to pick out the relevant numbers. It's fundamentally dishonest, because it pretends to encourage students to interpret math realistically, whereas it actually ends up being all about plugging numbers into formulas.

That's all very well for me to make these claims, but what is it about this particular problem that I find so offensive? Well, let's read it over again. I was okay about the woman needing to pick to random pairs of shoes, although it really is a very awkward proposition. We all know how to pick a random card out of a deck, or a handful of scrabble tiles from a bag. But how do you pick a random pair of shoes out of a closet? Do you close your eyes and reach in, fumbling around on your hands and knees? It really doesn't make a convincing scenario, but for the sake of the math, we can try to work around it. We might imagine that all her shoes are in boxes, and the boxes are lined up in a row on her shelf, in random order. I don't know any woman who stores her shoes that way, but let it be.

The red flags start to fly when the jargon words appear: we are told the woman samples two pairs of shoes. How the hell do you sample a pair of shoes? Do you take a bite out of the heel? It doesn't make any sense. Unless...you ignore the whole business about the woman going on the trip and you flip through your math book for a formula that relates to something about "sampling". I don't know any such formulas. I understand the idea of probabilities, and I can figure out how to calculate them in all kinds of situations, but I really don't know any formulas about sampling. So I'm starting to get annoyed.

Then it gets worse: we are told that the woman samples her shoes "one at a time, with replacement". What does this even mean? Surely she selects her shoes two at a time...in pairs, that is. She "randomly" chooses a pair of boots, and then a pair of flats or whatever. Surely we don't expect her to take a left heel and then a right flat. Why do they tell us she samples them one at a time?

And what does it mean when they tell us she samples them "with replacement"? She is going on a trip. She takes two pairs of shoes from her closet. What could "replacement" mean in that context?

It took me some time but all at once I saw what was going on. The students were given a formula for "sampling with replacement", and the professor puts those code words into the problem statement so the students will know which formula to use. The whole story about the woman who needs to pack for a trip has nothing to do with anything. The clincher is the last line of the problem, where it asks "what are the chances she will get a pair of heels and a pair of boots....in that order?"

What difference does the order make if she's packing for a trip? All you care about is what she ends up with. It's true that in probability and statistics, sometimes the order matters and sometimes it doesn't. But the beauty of math is that when something matters, it matters for a reason. In this case it doesn't matter because once you have your two pairs of shoes, it doesn't matter what order you packed them. So to write up a math problem where it the order doesn't matter, and then at the very end to tell the student to use the formula for when it does matter....well, that's a total perversion of everything that math is supposed to be all about.

It's a perversion because it tells the student in no uncertain terms: don't try and think about what the problem means. Don't try to make sense of what is going on. Just use the formula you were taught in class. Otherwise you will fail.

I hope you see why I don't like this problem. But I'm not done yet. There is one more outrageous aspect to this question that I haven't yet explained, although it's been mentioned in passing. Do you know what it is? I'll let you think about it....
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..,.did you figure out what's wrong about "sampling with replacement"? I'll take up this topic when we return.

A funny homework assignment

I've been having a bit of a dry spell on the physics over the last six months. In that time, I've started three new topics and gotten bogged down on each, unable to bring any of them to conclusion. First I started a series on induction motors. I wasn't really expecting to run into problems here. I have a couple of pretty good insights into this topic, and I thought it would make for some good articles. But as I got into it, I ran into a couple of surprises. First, in the process of carefully describing the field distributions in the rotor and the stator, I noticed an apparent error: according to the relative phases of the currents, the motor seemed to be working mainly by force of magnetic repulsion rather than attraction. I had never heard it described this way, and it seemed bizarre. But I couldn't find any other way of making things add up. I believe in the end I was correct about the repulsion business.

But that's not where I really got into trouble. While attempting to analyze the importance of the air gap, I noticed that when I drew the magnetic flux lines, the seemed to go around the rotor bars instead of through them. But if this were the case, how would you generate the very strong IxB forces needed to turn the motor? Try as I might, I couldn't explain this, and I still don't know the answer.

The next problem that I couldn't solve came from quantum mechanics. I figured out that I can analyze the physics pretty well when there's only one electron, but things get very dicey when there are two. I can break down some very simple cases, and I can do some cool things with approximations, but the fundamental essence of the physics remains elusive to me. I thought I was going to be able to solve a problem discussed by Feynmann in Vol. 3 of the Lectures: the scattering of two electrons from each other. It's basically the quantum mechanical version of the billiard ball collision, which is of course pretty much the starting point of classical mechanics. So it would be nice to really understand it. It's the spin states of the electrons that makes this problem especially significant, and I had recently been reviewing some pretty cool stuff about the basis states of some very simple two-electron systems, and I thought I ought to be able to apply this to the scattering problem. I butted my head against the wall for a few weeks and in the end I couldn't do it. It's still out there, and I'd like to figure it out one day.

The physics was going so bad I thought maybe I needed a change of pace, so I took up an old math problem: the question of the solvability of the fifth-degree equation. I knew I had some pretty good insights on this one, and I thought maybe if I forced myself to blog it out, I would be able to finally put all the pieces together. In fact, I made some progress, and I think I'm almost there, but I still can't put it all together. In fact, I came across this very good website recently according to which I seem to be at a very similar stage of thinking as was Lagrange, some forty years before Galois. So while my insights appear to be fairly sound, and quite original in the context of the way things are taught in university, the fact remains that I don't appear to have come up with anything the Lagrange didn't already know way back then. I'd still like to be able to write up the whole story in a way that puts things in perspective for a modern audience, but as with my other stalled topics...I'm stalled on this one too.

So what have I got for you today? Well, I've been getting into the math tutoring recently, and one of my students brought in a fascinating item from first-year stats. I call it fascinating because in my opinion, this little homework question sums up everything that is wrong with the education system today. I'm going to write out the question for you and see if you can figure out why I have a problem with it. After you've had a chance to think it over, we'll take it up when I return. Here is the question.

"You're going to a two-day conference and can't decide what shoes to pack. You own 5 pairs of flats, 7 pairs of heels, and 8 pairs of boots. To save time you decide to randomly pick your shoes. If you sample two pairs of shoes, one at a time, with replacement, what is the probability you will get a pair of heels and a pair of boots in that order"

Emily, my student, is taking first year stats at the University of Winnipeg, the same university that kicked me out of the Teacher Certification program last year. When she showed me this homework question, I almost couldn't believe my eyes. It is appalling to me on some very fundamental levels, which I promise to take up when we return. In the meantime, I wonder if my readers can figure out just what it is that I object to in this item?