Projecting onto Projections

The first time I saw the expression , I thought, “Why should that dot product be in there. By the time I saw , I resigned myself to the fact that there was always a dot product in these seemingly random integrals. At some point, I decided that the dot products are in there to turn vectors (or vector fields) into scalar functions—which is something we know how to integrate. More recently, I’ve decided that the purpose of these dot products is to capture the projection of one vector on the other.

For example, if I apply a force to an object, then the work done by that force in moving the object a certain distance in a given direction (denote this shift by ) is . If the force is not constant over some curve parametrized by (), then we compute the work by evaluating the integral since, at any given point, our from above is just the tangent vector to the curve at that point, i.e., .

If you understand multivariable calculus, then you are probably laughing at me. “Duh. Why did it take you so long to figure that out?”

Here is my answer: We (or maybe just I) improperly motivate the dot product. This semester, I’m using Stewart for Multivariable Calculus*. He introduces vectors in a way that seems fairly standard for math texts.

Definition: The dot product of and is .

The great thing about this definition is that it is bloody easy to compute and understand.

Theorem: If and be vectors with angle between them, then .

The beauty here is that you can use the dot product to help compute angles and it is immediately obvious that the dot product of orthogonal vectors is .

*This wasn’t my choice, but rather the choice of my department. Oh, did I mention I got a new job? Indeed I finally gave up on east coast living and moved back to California. I am now in the mathematics department at California State University Fullerton.

I’ve heard that in physics textbooks, they switch the order of the above, i.e., they define the dot product via the cosine formula and then prove the above definition as a theorem. As a mathematician, I always went with the first definition. Now, I am not so sure. What follows is the introduction to the dot product I plan to give to my students (until I come up with something better, anyway*).

*In the comments, please do set me straight about the real purpose of the dot product or how you think it best to introduce it in this context.

Let’s start with two vectors, joined by their tales.

I am interested in how far extends along , so I drop a line perpendicular to from the end of .

At this point, I’m already confused by what would happen if I had tried to see how far goes along , but I decide that I could simply extend and at least draw the following picture:

Awesome, I have a couple of right triangles. And, heck, since they are right triangles that share the angle (let’s call it ) between and , they are similar triangles. Let’s give some names to the important sides.

The comment about similar triangles implies that . Ugh, let’s clear denominators to get . On the other hand, , and so if we multiply by , we get

The moral is that this important quantity——coming from projecting the vectors onto each other, has a very simple reformulation as  which only relies on knowing the original vectors and the angle between them. Since this projection property is so important to us physically, we give a short name to this expression: , and call it the dot product of and .

If is orthogonal to , then the projection should be , which of course it is since the cosine of is .

At this point one can go about proving that the dot product is obtained directly from the components, i.e., without knowing the angle between them. Of course,  there is still the issue of when is obtuse, and it will probably be helpful to cover that case as well. Geometrically it will look a bit different, but the algebra and trig will be almost the same*.

*You do get to use the fact that the cosine of an angle equals the cosine of the supplementary angle!

There is nothing really new here, but I think the ordering is important. Their first impression of the dot product should convey the purpose of the dot product, not just the easiest algorithm for computing it. As it stands, the projection of a vector onto another vector gets a a somewhat token reference at the end of the dot product chapter. As ubiquitous as the idea is throughout the end of the class, it deserves its time in the sun.

A multivariable calculus list

In addition to my calculus course this semester, I also get to teach a multivariable calculus course with only six students. I’ll start with the standard list for those interested in that sort of thing.

Spring 2011 Multivariable Calculus Standards List

Let me admit something, here, in between two documents—less likely to read in here—about teaching this course, now for the third time: I’m a fraud. That’s right, I’m a fake, a charlatan, an impostor. I’ve created a counterfeit course and hustle the students with a dash of hocus-pocus and a sprinkle of hoodwinking. It is only through mathematical guile that my misrepresentations, chicanery and flim-flam go unnoticed. In short, and in the passing Christmas spirit, I am a humbug. This is a physics course. It should be taught be someone proficient in physics, someone with honed intuition about the geometry of abstract mathematical notions like div, grad, curl and all that, someone who sees everything as an application of Stokes’ theorem and has strong feelings about whether it should be written Stokes’ theorem or Stokes’s theorem. About the only thing I bring to the table is that I can teach students to remember that:

and

because curl and cross both start with c, while div and dot both start with d.

Here is the calendar for the course. After it, I’ll explain a little bit of what I’m trying.

Spring 2011 Multivariable Calculus Calendar

There are several big differences here from how I’ve taught this course in the past. First, I am going to try with all my might to get to Stokes’ theorem before the last week. Part of the way I plan to do this is, similar to my calculus class, to cut out most of the stuff on limits and continuity that I usually get bogged down on in the first couple of weeks—am I the only person who finds interesting the pathological examples that make Clairaut’s theorem necessary? I get to teach an extra hour a week to a subset of the class and that stuff will fit perfectly in there. For the science majors, I’m more interested in helping them figure out how to use this stuff and how to develop intuition. Second, I’m skipping Green’s theorem until the end. Yes, it changes the story I normally tell, one that progresses so nicely up the dimension chart, but the trade-off is that I get more time to show them Stokes’ theorem and more time to focus on the physical interpretation.

Speaking of interpretation, you will notice in the calendar  eleven or so ‘Group Activities’. These are stolen from an excellent guide produced by Dray and Manogue at Oregon State as part of their Bridge Project. To work within their framework, I’ve made another structural change that I’d never considered given how I think about the subject. Immediately after finishing triple integration (which, essentially, finishes the first half of the course), we start with vectors (I never start with vectors as most calculus books do) and then I want to get to line integrals and surface integrals as fast as possible. Normally, I mess around with div and curl before getting to integration of vector fields. Instead, I’m going to push out the Divergence theorem—the theorem I always cover in the last 45 minutes of the course—and use this to motivate the definition of div. Then I’ll push out Stokes’ theorem and use this to help motivate the definition of curl. This ought to give me two solid weeks to explore the physical meaning of these theorems as well as to use them to prove some of the standard cool corollaries (like Green’s theorem).

This class will also be the first of my SBG courses to incorporate a final project. If anyone has good suggestions based on experience about how best to incorporate projects into the SBGrading scheme, I would live to hear them. My current system is quite simplistic. The standards for the course are given a 90% weighting for the overall grade—did I mention that midterms and finals now are simply extended assessments whose grades are treated like an arbitrary quiz, just with a lot more standards tested?—and 10% weighting for the project.

A Calculus List

One of the advantages of my job is the incredible scheduling. We finished the fall semester the second week of December and my first class of the spring is next Tuesday! The downside is that this gives me way too much time to plot, scheme, doodle, dabble, think, rethink, and overthink. In the end, I usually settle on a plan that is far too ambitious, pedagogically impossible, philosophically suspect, and utterly indefensible. Thus, I bring you my plan for calculus this semester.

I read the wonderful article, Putting Differentials Back Into Calculus, which argues for using differentials in a way closer to their original creation than the way they are employed in modern textbooks.  As a huge fan of Thompson’s Calculus Made Easy, this suggestion didn’t seem half bad. Considering that only a fifth of my students are math majors and that, for the rest, using calculus outside of their physics class is unlikely, why not make things as easy as possible. I’m going to push to teach this same group next fall in calculus II, so the only teacher I can hurt is myself, right? But then I started thinking: the reason the differential approach will work so well with these students is that they will always be using differentiable functions. What reason is there to mention limits and continuity? These are technical issues that won’t help them at all in understanding calculus or how to apply it in their field of inquiry.

Oh dear, so here I am with essentially two months of material (this includes learning to differentiate any elementary function and using this to solve the standard problems). What will I do for the last month and half? I quickly remembered to add Taylor series because I love teaching that in calculus I. Then I added in the obligatory introduction to antiderivatives and integration. I even sprinkled in some partial differentiation at the end so that I could show the students the totally-awesome-implicit-differentiation-trick that would save them five minutes on the final exam. Grr…still two weeks left. These are precisely the two weeks that I usually spend on limits in the beginning. Now I remember why I always do this. It perfectly fills in the semester calendar. And then it hits me.

Any subject can be made repulsive by presenting it bristling with difficulties.
—Silvanus P. Thompson

Limits sure confuse the heck out of students. Why in the world are we leading calculus off with limits, especially to non-math majors? For the purpose of rigor?

You don’t forbid the use of a watch to every person who does not know how to make one? You don’t object to the musician playing on a violin that he has not himself constructed. You don’t teach the rules of syntax to children until they have already become fluent in the use of speech. It would be equally absurd to require rigid demonstrations to be expounded to beginners in the calculus.
—Silvanus P. Thompson

So I decide I just won’t do it.  No limits for us. We will just do some extra exploratory work. There is a great article on math in medicine we could read together. Okay, time for sleep.

But sleep does not come.

Toss.

Turn.

Toss.

Turn.

All right. All right. I’m up.

Why can’t the limits just die? Why do I feel the compulsion to put them back in? They’re like the tell-tale heart beating under my floor. What will become of my math majors if they don’t see limits? No, I can’t do that to them. What a cruel joke to play: send them to real analysis without having used limits. Back in they go. Of course, those biology majors are going to be completely turned off and once you lose them, they’re gone for good. Argh, out they go. On the other hand, if I get audited by the department, they are sure going to ask questions. With my review coming up, I can’t afford that kind of chatter. Put them back in…

…but later. Huh? Put them back in, but later. Yes, of course. Later. The course description says I need to cover limits, but it doesn’t say when! What if we introduced everything in a reasonable way and, only after the students know what is going on and why any of this is important, then showed them those funky limit do-hickies? Hmm. Interesting. And that is my explanation for the following calendar and skills list:

View this document on Scribd

View this document on Scribd

All Hail Ailles

This post is a sequel of sorts to my tricks of the trade post describing how to quickly compute the standard values of sine and cosine in the first quadrant. This is a very nice little exercise, known by quite a few high school teachers, but unknown to everyone I know personally. This lesson assumes that the student is aware of the basic facts from geometry about 30-60-90 and 45-45-90 right triangles as well as the Pythagorean theorem. If they know the definition of sine and cosine, you can go a little further at the end—this is my target audience. As a good preface to this, you might start with Kate Nowak’s post on these special right triangles. We start with the following rectangle:

Give this to the students (I do it in groups) and ask them find the length of each side and the measure of each angle. Simple arithmetic will take them here:

At this point, perhaps with some nudging, they apply the Pythagorean theorem to get at least two of the missing sides of the inner triangle.

The last side might be too intimidating for your students. If so, suggest they work on the angle problem. Those right triangles on the bottom should look mighty familiar…

My students got stuck here. However, I channeled my help less personality and let them struggle a bit. A few decided to work on the missing side again, but all eventually came around to noticing the angle on the bottom had to be 90 degrees—I must have said, “I don’t know. Why do you think it is a right angle?” half a dozen times that day.

It will not surprise you how few notice that the triangle is a 45-45-90 right triangle. It will surprise you how many don’t see that they can now use the Pythagorean theorem again to get the last side.

At this point, the students should have no problem sorting out the last two angles.

At the end, you might try asking them to compute the sine or cosine of and . Normally, I teach these via the sum angle formulas, but this is a far more intuitive approach.

This rectangle was noticed by high school teacher Doug Ailles and is knows as the Ailles’ rectangle. Not too shabby.

Super-Sized, Circumcised, Circumscribed

Today was supposed to be observation day. At least once per semester since I arrived at Suffolk, I’ve been observed in the classroom by one of the department’s senior faculty. Just so that you don’t get the wrong idea, this isn’t some kinky thing where I prance in front of the examiner in a Speedo. No, outside of Europe, nobody wants to see that kind of nonsense. The faculty actually want to watch how I teach a class.

While I always try to outwardly act as if this is no big deal, the thought of being observed gets me super excited. It gives me a really good excuse to spend that extra hour preparing a special lecture. With that in mind, this observation session came at a pretty awkward time: we just finished the course material and this is review week with the final coming next Tuesday. This is usually when we kick back, tighten up the few topics that are hurting everyone and watch Raiders of the Lost Ark. Instead, though, I decided to review old material by introducing something slightly new.

Enter the Extended Law of Sines

It always surprises me how few people have heard of the extended law of sines. In short, it says that the quantity appearing in the law of sines is equal to the diameter of the circumcircle of the given triangle. If the radius of the circumcircle is , then the theorem is usually written as:

(where and represent the sides and angles of triangle, respectively, with opposite angle , etc.). Aside from giving a geometric interpretation of the terms in the law of sines, the result is also quite useful (I’ll give a fairly involved exercise at the bottom where it is possible to use it).

I wanted the students to prove the result, but I really was not confident in their elementary geometry skills, so I set things up for them. I told them to start from the triangle and draw the circumcircle. Naturally, they didn’t know what I was talking about and wondered if I’d said what they thought I’d said. Planning for this, I started explaining out loud how you physically construct the circumcircle. Eyes glaze. Planning for this, I opened up handy-dandy mathopenref.com to show them what I can’t vividly describe verbally. Next came Geogebra where we found the circumcenter of the triangle and connected it to two of the vertices of the triangle. I now referenced the Central Angle Theorem which, again, nobody knew. Thankfully, Geogebra calculated the angles for us and since these students believe the convenient falsehood that one example proves that something always works, we were golden.

After connecting a few more dots, we were ready to prove the result. Here is the worksheet I gave them to work on in groups. It is nothing fantastic, but even some of the students whose attendance has been…spotty…worked out the answers. The beauty of the last question is that they need to shift gears and use the law of cosines before they can solve the problem.

Oh, and the professor who was coming to observe me? He didn’t show up. Go figure.

 

Exercise: With the notation as above, prove that the triangle is isosceles if and only if

.

Evil SBG

It’s been a while and I really ought to do some reflection on my first true semester of SBG. But, alas, the train ride is short and I wanted to throw something out there. I just finished proctoring the Putnam exam and had a wonderful idea for how to make SBG more evil.

You see, there are times when I really want to add weights to my topics. For instance, there is no way that finding the slope, y-intercept, etc. of a line should be worth as much in a precalculus class as graphing rational functions or using the law of cosines. It isn’t that I don’t think it is as important—it is probably more important in many cases—it is just that knowledge of lines (at that level) is considered remedial material, though for some it isn’t.

So how should I weight things? Should I do it by order of (my) perceived importance? How about by difficulty level—again from my point of view? I realized that while I may think I know the correct weightings, it is unlikely that I actually do. Consequently, I came up with this evil scheme. It isn’t really something serious, but I’m not completely joking either.

At the beginning of the semester, inform the students that each topic (or standard if you prefer) will be weighted by an amount inversely proportional—you’ll have to explain what that means—to the final average class score for that topic (or standard). The topic with the lowest class average score would then be weighted the highest. Those topics which everyone figures out would be weighted the least. This would give students (especially your top tier students) an incentive to attack those topics the whole class is failing. Potentially, students could try to game the system by learning the really hard stuff so well that it doesn’t count for so much. Hah! Now wouldn’t that be something?

Cleaning Out the Gutters

School is starting this week. Hooray!!!

No more summer
Here comes class
Time to get up
Off my chair

The syllabi are written. The schedules are finalized. The students are being informed of their impending doom. Mmmm, it’s good to be alive.

Let’s see, what can I offer you today? First, in case you aren’t checking out the awesome SBGBeginners Wikispace, here is a link to my (always in flux) standards list for precalculus. I would post my standards list for Topics in Finite Mathematics (and maybe I will still), but  I doubt there are too many people teaching a course covering precisely what this course covers. Nonetheless, I should tell you a bit about my experience with producing the grade distribution for that course.

First, my precalculus course is controlled by me. I get to decide what, where, when and how much. So when I decided to make standards comprehension 70% of the grade and to reduce the midterm and final to 10% and 20%, respectively, that was all good. Unfortunately, the other course is not under my control. I had to do some serious negotiating with the course coordinator who is worried that my not grading homework policy will be a disaster. There are four exams in that class and, in the end, they will make up 60% of the grade. I’m making 30% of the grade standards comprehension. Those of you adept at addition will notice the missing 10% and that is the cool/scary/”oh gawd, what am I doing” part. This last bit is called Oral Evaluation. Way back in May, I mentioned my desire to teach only three students, instead of twenty-five. Well, I’m giving it a shot in one of my courses. Every student will sign up three times during the semester to be one of my three interloculars for the day. Their grade will be determined by their preparedness and their performance. More on this story as it develops.

In an effort to waste time, I collected links to videos targeting the standards for my classes (see here for precalculus and here for finite mathematics). Originally, I was going to record the videos myself, but even I can’t justify spending that much time right now on duplicating previous efforts. In the process of creating these lists, I started hanging around on the Art of Problem Solving website. This reminded me of all the time I spent taking the California Math League contests in high school. Man, those were fun. And…

Homeschooling Connection

As I’ve mentioned before, my wife and I are starting homeschooling with our five-year old this fall. Actually, since I happen to believe that

1. Summer vacation should be a time to do things you enjoy
2. School is something you should enjoy

we started a proper curriculum in April and went all summer, forgetting to tell him that he shouldn’t want to learn over the summer. (Mwoo hah hah!!! Kids are so gullible.) So, we’ve covered addition, subtraction and multiplication and are just starting with division. Incidentally, division is the first operation that I think my son finds practical. At the store today, he was constantly telling me how much each member in the family would get of the things we were buying. Anyhow, I was thinking to myself that knowing the four basic operations at his age is pretty good and just think how far I can take him over the next few years. But then, I thought to myself how my biggest complaint with my students is how they are so weak at the basics. Even those who can perform calculations efficiently, can rarely apply that math to solve real-world problems or interesting abstract problems. What if I got my son up to the level of understanding the basic vocabulary and then, instead of striving for breadth, went for depth? What if I started feeding the kid competition style problems and applied problems, not with the goal of getting him into the Math Olympiad, but rather aiming towards mastery of the foundations of arithmetic and mathematical thinking? The idea gives me goosebumps…oh, wait, the air conditioning was set to 63.

One last bit: I made up a few songs to help my son learn the multiplication tables and I think I’ll record them and post them on the blog for posterity. I searched hard for such songs and hated just about everything I found. Mine aren’t really any better, but they don’t annoy me as much and my son sings them all the time.

More Bad Poetry

There once was a teacher who dared
To teach as if all students cared
But his methods all stunk
‘Til he read Think Thank Thunk
Now his class is no longer impaired

Happy New School Year!!!

Pop Quiz

I’m still reading the book used by our department for “Topics in Finite Mathematics” and I found this interesting little snippet from the introduction to the section entitled “The Multiplication Principle”.

Here are some examples of counting problems that are unlike any we have seen up to this point, but that we should be able to answer after studying this section:

How many different Social Security numbers are possible?

How many different telephone numbers can be given the area code 870?

In how many ways can a president and a vice president be selected from among 12 people?

How many ways can the sum of eight be rolled on a pair of dice?

Here is my question for everyone: Which of the above four questions is the most different from the others? If you like, you might rephrase this as: Which of the above four questions doesn’t belong?

My answer differed from my wife’s answer and I think both are valid (but I won’t say what they were yet). I’d love to hear some other responses to this. If you happen to guess the same one I chose—for the same reason—then you win the grand prize for making me feel less alone in the universe. Sadly, the grand prize is merely reciprocity.

Venn do we start?

I’m back after a little break to spend time with family who, coincidentally, showed up a few weeks after the birth of baby #3. In a couple of weeks, I start teaching precalculus again and the pseudo-finite-mathematics class. Last night, while reading a sample problem prepared for the class by the course coordinator, I decided that I don’t draw circles well—especially several that are supposed to intersect—and that, while I understand what Venn diagrams represent and how they are put together, Venn diagrams do not illuminate raw data for me. Inspired by this TED talk of David McCandless, I tried to redraw the problem. Ah, perhaps you would like to see the problem.

A survey of 100 students shows that: 48 take English (E), 49 take History (H), 38 take Spanish (S), 17 take E and H, 15 take E and S, 18 take H and S, and 7 take all three.  How many students:

1. take only S
2. take S and H but not E
3. take only one of those courses
4. take none of those courses

The solution is not difficult to find without the aid of a diagram, but for the type of students in this class, every little picture helps. The book suggests the following Venn diagram will be helpful:

Once the students draws this (by hand, I always seem to be to impatient and screw it up, but maybe the students are better at this) he or she might then argue that the middle region is 7 and that the three regions surrounding the middle are (in clockwise order from the top) 10, 11, and 8. Starting from the left and working clockwise, the three missing regions are then 23, 21 and 12. At this point, they would be able to answer the four questions.

This is a best case scenario, I think. Let’s assume that they are taught to always throw the 7 in the middle. My guess is that many of the students will then toss the numbers 17, 18 and 15 into the three cells around the middle. Why? Well, duh! Those three cells represent E and H, H and S and E and S. Obviously. At this point, I don’t think it matters what they do, they will get the wrong answer. If they get to the next step, however, I’m still not sure they will see how to compute the outer ring. If they get that far, will they really get how to read the picture to answer the questions? I don’t know; I’ve never had to teach Venn diagrams. Perhaps they will find this very easy. On the other hand, maybe they won’t.

An Alternative to the Venn Diagram

Here is an alternative way to solve this problem, one that mathematicians should appreciate for its passing reference to Euler’s formula (Not that one; the other one 😉 ).

First, we draw a triangle. The type is not particularly relevant, only draw it sufficiently regular in order to fit numbers. Label the edges with E, H and S.

We now the label the vertices, edges and faces as naturally as possible. By naturally, I mean as a typical student would do it. The vertices are the numbers given in the problem for the individual subjects, the edges are the numbers given for combinations of two subjects and the middle (face) is for all three subjects.

This is a graphical representation of the given data—and I understand it. This is what the students want to write, so let them write it down. Now, we draw a new triangle. I don’t want to give it a name. If I must, I might call it the companion triangle, but there is probably a better name for it.

How will we label this new companion triangle? Euler’s formula, of course. Half of it, anyway.

Historical Sidenote

Euler proved for any convex polyhedron that the number of vertices plus the number of faces is two more than the number of edges, i.e., V + F = 2 + E.* Traditionally, this is written V – E + F = 2. An interesting thing is that this formula fails in higher dimensions, but the idea of alternating between plus and minus as you increase the dimension of the object is still useful.

* Note that the E in the formula V – E + F = 2 stands for edges, not English. This ambiguity will continue below.

Back to our triangle

With a bit of thought, one sees that to put the “right” data into our new triangle, we simply need to compute V – E + F for the vertices, E – F for the edges and F is still the face. Let me explain what I mean. In our new triangle, we want the E vertex number to represent the number of people who took only English. The original E double counts the numbers coming from the EH and ES edges, so we should subtract those. But this double subtracts things that were on both edges, i.e., are in the middle, so we need to add those back in. So, V – E + F. The EH edge should represent the people who took both English and History and not Spanish. As the original edge number counts both those who took just English and History as well as those who took all three, we need to subtract those in the middle. So E – F. The middle represents those who took all three and so does not change from the original. Let’s compute the new E vertex.

V – E + F

The original E vertex says 48
The edges going out of E (the EH and ES edges) add up to 32, so we subtract them from 48 to get 16.
The face is 7, so we add that to get 23—our new E vertex is 23.

The original S vertex says 38
The edges going out of S (the ES and HS edges) add up to 33, so we subtract them from 38 to get 5.
The face is 7, so we add that to get 12—our new E vertex is 12.

Similarly, we find the H vertex is 21

The original EH edge is 17

Subtracting the face (still 7) gives 10—our new EH edge is 10.

The original ES edge is 15

Subtracting the face (still 7) gives 8—our new EH edge is 8.

Similarly, we find the new HS edge is 11.

Our companion triangle is then:

The student now can see that exactly 12 students take only Spanish, 11 take Spanish and History but not English and 23+21+12 = 56 take exactly one of the courses. The last question is probably the hardest. How many students (of the 100 surveyed) aren’t taking any of the courses? Of course, the answer is to add up all the numbers in the triangle and subtract this from 100 (which gives 100 – 82 92 =18 8).

Is this really better?

Well, it is for me. I can totally see this better without the confusing circles. It is even easier when you’re in the case where there would be two circles in the Venn diagram (now, you would just draw a line segment). If we take it up a dimension, I think it holds up better than Venn diagrams (I can draw a tetrahedron, I can’t draw a Venn diagram with 4 circles). Finally, in my one experiment on a non-math type (my wife), she was able to see the justification for V-E+F pretty easily, but the Venn diagram business stumped her a little.

Waiting for your thoughts…

Annotated Papers

As I mentioned in my last post, I recently had a research paper accepted (yay!). The process of writing a paper is both satisfying (I’m accomplishing something) and frustrating (won’t this something ever end?). Perhaps, though, the most annoying thing is that the structure of papers that describe mathematics is so far removed from the production of the same mathematics. For instance, the reader will only see my most polished proofs, not the three less elegant proofs that actually contained the motivation. I like to keep my proofs short, so I usually pull out steps and turn them into lemmas. Part of the justification is that I (or a reader) can then easily cite just that lemma if that is all one needs. This is vastly superior to making people cite the proof of a theorem. However, this is what you end up with:

1. Preliminaries

Lemma 1 Something completely random at this point holds.

5 sections laters:

6. Main Theorem

Theorem 2 Wow! What an amazing theorem.

Proof: The only thing cooler than the theorem is its proof. And here is where we use Lemma 1.

There is nothing logically wrong with the order. But how can someone read the lemma and have any reaction other that WTF? Perhaps I add a little remark that the proof of Theorem 2 uses the lemma. This is helpful, but it still feels random.

Now, part of the problem is that I am expecting people to read a paper linearly. The better mathematicians don’t do this; they jump to the good part and only if the idea of the proof doesn’t immediately become transparent do they hop around the paper looking for answers. This, incidentally, is why sticking an important lemma in the preliminary section of a paper can be dangerous: the experts are likely to skip it.

Back on Track

I was reading my favorite group theory text this last weekend and noticed how uneven the exposition is. Early on in the book, the sections jump around from one (seemingly) random topic to another, frequently omitting any discussion of why it is being treated this way or where these topics will show up again. Although the terms below won’t mean much to my readers, I think the point will be obvious. At the end of section 2.7, Gorenstein writes:

The class of [Frobenius groups]…is of fundamental importance in the theory of finite groups, and several basic problems that we shall later investigate stem from this class of groups.

Contrast this with section 2.8 where he writes

The class of [Zassenhaus groups]…is a very important and interesting one, which we shall study in detail in Chapter 13…in a Zassenhaus group, the subgroup fixing a letter is always a Frobenius group in its action on the remaining letters.

Notice that in both remarks he talks about their importance, but that only in the second one does he give an explicit connection to anything. At least with Zassenhaus groups, I know that I can look for more info in Chapter 13 and that they always contain a Frobenius group (which is of fundamental importance in the theory of finite groups, although I don’t have a clue why). I have a bit more to say on this subject, but it probably would behoove me not to get into doubly transitive permutation groups.

No, really. Back on Track.

Historically, we have to write mathematics in such a terse unhelpful way because of two considerations: space and time. Journals don’t want to publish twenty pages of my thoughts just to get 3 pages of mathematical progress. Also, readers (i.e., experts) don’t want to spend an hour wading through explanations of my missteps, wrong turns and false proofs just to get one idea they might already know. On the other hand, people with less experience in the field are likely to wonder at just about every step: why? How in the world did you come up with that unnatural monstrosity? This reader would gladly read some exposition that brings a few things together. They’d also love the reassurance that even experts make mistakes; even better, they would love to see how the experts move from false belief, back to ignorance and finally to enlightenment.

Didn’t I say I had an idea?

The system isn’t going to change any time soon and even if it could, I’m not sure it should. However, perhaps a bit of (would Dan Meyer call it) scaffolding would be useful. Imagine if, in addition to downloading my newest paper off of the arXiv, you could also download a screencast of me going through the paper, telling its story, elaborating on difficult sections and giving hints about how I really understand the subject. I’d be willing to try it myself if I thought anyone would watch, but my research is a bit technical and only a handful of people would follow and even less would care.

So, if nobody cares, why are we still reading?

The thing is, this translates to the relationship between teacher and student. Our students read textbooks, refined over the years to be ruthless, efficient and deadly. The story is missing, the context is missing and the connections are missing. The textbooks are a reference, not a teacher. It is then the teacher’s responsibility to add the missing ingredients, to tell the story, to explain how experts actually think about these things and, most importantly, to teach the students how to read (or understand) a subject non-linearly. Mathematics is structured so poorly K-12 partly because we keep treating the learning of mathematics as a well-ordered system and it isn’t (the axiom of choice, notwithstanding).

Okay, I have so much more to say, but I’ve begun to bore myself. I’ll save the rest for my post on story telling.

Next time: Tricks of the Trade (pt. 3): A Symmetry Trick for Integration