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?

Green versus U of W: The gloves are off

Back in January, I told my readers that I had been expelled from the teacher certification program at the University of Winnipeg. Over the next two months, I attempted to fight my expulsion. The University set up a kangaroo court to hear my final appeal, which was rejected in the end.

Last week, I filed a Statement of Claim at the Manitoba Court of Queen's Bench claiming damages from the University of Winnipeg and various individuals who were involved in removing me from the program. I have just learned that the existence of my lawsuit is likely to be announced in the media very soon. So the fight goes public.

People in my situation have tried to sue many universities in the past, and the courts are not in general sympathetic to us. The university will undoubtedly attempt to have my claim dismissed outright before even going to trial, and they might succeed. Anything can happen when you go before a judge. On the other hand, I know what they did to me, and they know what they did, and I can tell you that they don't want to have to defend their actions in either the media or in court.

While I was still fighting my expulsion through the internal process, I was posting information on my other blogsite, "Due Process, Natural Justice and the University of Winnipeg". At that time, I didn't want to give away anything to the university that might compromise my position in any subsequent litigation, so I was only posting information that they already knew. At some point, I decided that it might not be in my best interest to have that information public, so I removed my posts. Tomorrow the fight becomes public, so I have restored all my old posts. To get the whole story, you have to start at the very oldest post and click through them one by one.

And that's where we stand as of today. I think I have a pretty good case against the university, but the judicial process can drag on for years, and even a huge settlement at the end can never make right what was taken away from me. I was a good teacher. I only spent a few weeks in the classroom but I knew that was where I belonged, and the kids knew it too. But my name has been blackened by the University and their collaborators, and now I have virtually no chance of getting back into the teaching profession.

I will be keeping my readers up to date on the fight as it develops from now on. You can follow my exploits on my "Due Process..." blog. In the meantime, I need to support myself for the duration of the fight, and I am therefore accepting private students. I have created a teaching blogsite called "Math with Marty" where I have started posting free on-line lesson modules for the high school math and science curriculum. If you like the on-line lessons, you will love the live in-person version.

 

What is a quotient group?

Group theory is all about permutations. The elements of a group are the various operations whereby you can re-arrange a set of elements. I used the word "elements" twice in that sentence, and I did it on purpose. It's confusing because the word "elements" refers to two different things: you have the "elements" A,B,C and D which we are going to shuffle around, and you have the operations such as "flips" and "rotation" which are the "elements" of the permutation group on the four elements (letters).

There are obvious way of getting around this confusion, but mathematicians have a way of being just a little too clever about these things. They do something which seems brilliant but turns out to be nasty in the end. They figured out that it's unnecessary to talk about the "elements" A,B,C, and D. Everything you need to know about the structure and properties of a group can be worked out by constructing the algebra of flips and rotations (F's and R's). You don't need to think about what is being flipped or rotated. I wrote about this yesterday.

It's all very elegant but it leads to a very awkward situation when you come to deal with the topic of quotient groups. Quotient Groups are something that makes a lot of sense in terms of permutations of elements like A,B,C and D. Let's consider the set of all permutations on these four letters; as we remarked yesterday, there are 24 possible re-shufflings (including the trivial one which leaves the letters as they were). What gets interesting is when we look at the set of functions of A,B,C, and D defined as:

O1 = AB + CD

O2 = AC + BD

O3 = AD + BC

I talked about these three functions last spring when I wrote about solving the fourth degree equation. The interesting thing about them is what happens when you re-shuffle the four letters. We've already noted that there are 24 possible re-shufflings of the letters A,B,C, and D. But you ought to be able to convince yourself that no matter what you do, the result of the re-shufflings is going to simply return the three functions you started with. They might be re-arranged: you might flip O1 and O2, or you might rotate O1 => O2 => O3. Or the thetas might just stay where they were. The overall effect is that from the twenty-four possible permutations of the four elements A,B,C and D, you have generated the six possible permutations of the three elements O1, O2, and O3. 

That's what a quotient group is. You get a quotient group whenever there exists a set of functions which is preserved by the action of the group elements...that is, any permutation on the A's, B's and C's returns the same set of functions (the thetas in my example) that you started with. The funny thing is I don't think you'll find it explained this way anywhere. The way it's traditionally done is by defining something called a normal subgroup and its cosets. You investigate the group action on these sets, and you identify the resulting structure as a "quotient group". You can look up quotient groups and normal subgroups on Wikipedia and you'll see that this is how it's done.

Just how different are these two approaches? I think they're very, very different. For one thing, the "normal subgroup" defined in the traditional approach has four elements, while my functional approach operates on three elements. It's different. Different approaches usually give useful insights, but I can't find my method being used anywhere. 


What disinguishes the two approaches is that I treat a group as a set of operations working on a set of target elements, whereas the sophisticated mathematical approach is to ignore the target elements and define everything in terms of the properties of the operations. The formal development of quotient groups uses this lean, efficient methodology. But at what cost?