Elementary vs Foundational

Euclid’s proof that there are infinitely many primes is simple and ancient. This proof is given early in any course on number theory, and even then most students would have seen it before taking such a course.

There are also many other proofs of the infinitude of primes that use more sophisticated arguments. For example, here is such a proof by Paul Erdős. Another proof shows that there must be infinitely many primes because the sum of the reciprocals of the primes diverges. There’s even a proof that uses topology.

When I first saw one of these proofs, I wondered whether they were circular. When you use advanced math to prove something elementary, there’s a chance you could use a result that depends on the very thing you’re trying to prove. The proofs are not circular as far as I know, and this is curious: the fact that there are infinitely many primes is elementary but not foundational. It’s elementary in that it is presented early on and it builds on very little. But it is not foundational. You don’t continue to use it to prove more things, at least not right away. You can develop a great deal of number theory without using the fact that there are infinitely many primes.

The Fundamental Theorem of Algebra is an example in the other direction, something that is foundational but not elementary. It’s stated and used in high school algebra texts but the usual proof depends on Liouville’s theorem from complex analysis.

It’s helpful to distinguish which things are elementary and which are foundational when you’re learning something new so you can emphasize the most important things. But without some guidance, you can’t know what will be foundational until later.

The notion of what is foundational, however, is conventional. It has to do with the order in which things are presented and proved, and sometimes this changes. Sometimes in hindsight we realize that the development could be simplified by changing the order, considering something foundational that wasn’t before. One example is Cauchy’s theorem. It’s now foundational in complex analysis: textbooks prove it as soon as possible then use it to prove things for the rest of course. But historically, Cauchy’s theorem came after many of the results it is now used to prove.

Related: Advanced or just obscure?

On replacing calculus with statistics

Russ Roberts had this to say about the proposal to replacing the calculus requirement with statistics for students.

Statistics is in many ways much more useful for most students than calculus. The problem is, to teach it well is extraordinarily difficult. It’s very easy to teach a horrible statistics class where you spit back the definitions of mean and median. But you become dangerous because you think you know something about data when in fact it’s kind of subtle.

A little knowledge is a dangerous thing, more so for statistics than calculus.

This reminds me of a quote by Stephen Senn:

Statistics: A subject which most statisticians find difficult but in which nearly all physicians are expert.

Related: Elementary statistics book recommendation

 

 

Least understood bit of basic math

Logarithms may be the least understood topic in basic math. In my experience, if an otherwise math-savvy person is missing something elementary, it’s usually logarithms.

For example, I have had conversations with people with advanced technical degrees where I’ve had to explain that logs in all bases are proportional to each other. For example, if one thing is proportional to the natural log of another, the former is also proportional to the log base 10 or log base anything else of the latter [1].

I’ve also noticed that quite often there’s a question on the front page of math.stackexchange of the form “How do I solve …” and the solution is invariably “take logarithms of both sides.” This seems to be a secret technique.

I suspect that more people understood logarithms when they had to use slide rules. A slide rule is essentially two sticks with log-scale markings. By moving one relative to the other, you’re adding lengths, which means adding logs, which does multiplication. If you do that for a while, it seems you’d have to get a feel for logs.

Log tables also make logs more tangible. At first it seems there’s no skill required to use a  table, but you often have to exercise a little bit of understanding. Because of the limitations of space, tables can’t be big enough to let you directly look up everything. You have to learn how to handle orders of magnitude and how to interpolate.

If the first time you see logs is when it’s time to learn to differentiate them, you have to learn two things at once. And that’s too much for many students. They make mistakes, such as assuming logs are linear functions, that they would not make if they had an intuitive feel for what they’re working with.

Maybe schools could have a retro math week each year where students can’t use calculators and have to use log tables and slide rules. I don’t think it would do as much good to just make tables or slide rules a topic in the curriculum. It’s when you have to use these things to accomplish something else, when they are not simply an isolated forgettable topic of their own, that the ideas sink in.

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[1] That is, loga(x) = loga(b) logb(x). This says loga(b) is the proportionality constant for converting between logs in base a and b. To prove the equation, raise a to the power of both sides.

To memorize this equation, notice the up-and-down pattern of the bases and arguments: a up to x = a up to b down to b up to x. The right side squeezes an up and down b in between a and x.

 

Bottom-up exposition

I wish more authors followed this philosophy:

The approach I have taken here is to try to move always from the particular to the general, following through the steps of the abstraction process until the abstract concept emerges naturally. … at the finish it would be quite appropriate for the reader to feel that (s)he had just arrived at the subject, rather than reached the end of the story.

From the preface here.

When books start at the most abstract point, I feel like saying to the author “Thank you for the answer, but what was the question?”

Differentiating bananas and co-bananas

I saw a tweet this morning from Patrick Honner pointing to a blog post asking how you might teach derivatives of sines and cosines differently.

One thing I think deserves more emphasis is that “co” in cosine etc. stands for “complement” as in complementary angles. The cosine of an angle is the sine of the complementary angle. For any function f(x), its complement is the function f(π/2 – x).

When memorizing a table of trig functions and their derivatives, students notice a pattern. You can turn one formula into another by replacing every function with its co-function and adding a negative sign on one side. For example,

(d/dx) tan(x) = sec2(x)

and so

(d/dx) cot(x) = – csc2(x)

In words, the derivative of tangent is secant squared, and the derivative of cotangent is negative cosecant squared.

The explanation of this pattern has nothing to do with trig functions per se. It’s just the chain rule applied to f(π/2 – x).

(d/dx) f(π/2 – x) = – f‘(π/2 – x).

Suppose you have some function banana(x) and its derivative is kiwi(x). Then the cobanana function is banana(π/2 – x), the cokiwi function is kiwi((π/2 – x), and the derivative of cobanana(x) is –cokiwi(x). In trig-like notation

(d/dx) ban(x) = kiw(x)

implies

(d/dx) cob(x) = – cok(x).

Now what is unique to sines and cosines is that the second derivative gives you the negative of what you started with. That is, the sine and cosine functions satisfy the differential equation y” = –y. That doesn’t necessarily happen with bananas and kiwis. If the derivative of banana is kiwi, that doesn’t imply that the derivative of kiwi is negative banana. If the derivative of kiwi is negative banana, then kiwis and bananas must be linear combinations of sines and cosines because all solutions to y” = –y have the form a sin(x) + b cos(x).

Footnote: Authors are divided over whether the cokiwi function should be abbreviated cok or ckw.

Related post: How many trig functions are there?

Dual polyhedra for kids

Here are a dodecahedron (left) and icosahedron (right) made from Zometool pieces.

dodecahedron and icosahedron

These figures are duals of each other:  If you put a vertex in the middle of each face of one of the shapes, and connect all the new vertices, you get the other shape. You could use these as a tangible way to introduce duality to kids.

There are lots of patterns that kids might discover for themselves. The dodecahedron has 12 faces and 20 vertices; the icosahedron has 20 faces and 12 vertices. At each vertex of the dodecahedron 3 five-sided faces come together; at each vertex of the icosahedron 5 three-sided faces come together.

The two polyhedra have the same number of edges. You can see this by taking one shape apart to make the other. A more sophisticated explanation is that Euler’s theorem says that V + F = E + 2. When you swap the roles of V and F, V+F doesn’t change, so E cannot change.

Here’s a hint on making an icosahedron with Zometool. Stick the red struts with the pentagonal ends into every pentagonal hole on one of the balls. Now if you connect each of the outer balls to each other, you have an icosahedron. You can leave the red pieces inside, or you can use a few of them as a temporary scaffolding to get started, then remove them.

If you do leave the red pieces inside, it’s hard to put the last few pieces in place because the shape is so rigid.

icosahedron with struts to its center

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How long will there be computer science departments?

The first computer scientists resided in math departments. When universities began to form computer science departments, there was some discussion over how long computer science departments would exist. Some thought that after a few years, computer science departments would have served their purpose and computer science would be absorbed into other departments that applied it.

It looks like computer science departments are here to stay, but that doesn’t mean that there are not territorial disputes. If other departments are not satisfied with the education their students are getting from the computer science department, they will start teaching their own computer science classes. This is happening now, to different extents in different places.

Some institutions have departments of bioinformatics. Will they always? Or will “bioinformatics” simply be “biology” in a few years?

Statisticians sometimes have their own departments, sometimes reside in mathematics departments, and sometimes are scattered to the four winds with de facto statisticians working in departments of education, political science, etc. It would be interesting to see which of these three options grows in the wake of “big data.” A fourth possibility is the formation of “data science” departments, essentially statistics departments with more respect for machine learning and with better marketing.

No doubt computer science, bioinformatics, and statistics will be hot areas for years to come, but the scope of academic departments by these names will change. At different institutions they may grow, shrink, or even disappear.

Academic departments argue that because their subject is important, their department is important. And any cut to their departmental budget is framed as a cut to the budget for their subject. But neither of these is necessarily true. Matt Briggs wrote about this yesterday in regard to philosophy. He argues that philosophy is important but that philosophy departments are not. He quotes Peter Kreeft:

Philosophy was not a “department” to its founders. They would have regarded the expression “philosophy department” as absurd as “love department.”

Love is important, but it doesn’t need to be a department. In fact, it’s so important that the idea of quarantining it to a department is absurd.

Computer science and statistics departments may shrink as their subjects diffuse throughout the academy. Their departments may not go away, but they may become more theoretical and more specialized. Already most statistics education takes place outside of statistics departments, and the same may be true of computer science soon if it isn’t already.

How long can you think about a problem?

The main difficulty I’ve seen in tutoring math is that many students panic if they don’t see what to do within five seconds of reading a problem, maybe two seconds for some. A good high school math student may be able to stare at a problem for fifteen seconds without panicking. I suppose students have been trained implicitly to expect to see the next step immediately. Years of rote drill will do that to you.

A good undergraduate math student can think about a problem for a few minutes before getting nervous. A grad student may be able to think about a problem for an hour at a time. Before Andrew Wiles proved Fermat’s Last Theorem, he thought about the problem for seven years.

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Crayons to Computers

I saw a daycare named “Crayons to Computers” recently. I assume the implication is that crayons are basic and computers are advanced. Programming a computer is more advanced than writing with crayons, but surely their clients don’t learn to program computers. They probably play video games, which takes considerably less dexterity and mental skill than using crayons.

So maybe the place should be named “Computers to Crayons.” Or if they really wanted to offer advanced instruction, perhaps they could be “Crayons to Clarinets.” Learning to play clarinet takes much more work than learning to play a computer game.

Easiest and hardest classes to teach

I’ve taught a variety of math classes, and statistics has been the hardest to teach. The thing I find most challenging is coming up with homework problems. Most exercises are either blatantly artificial or extremely tedious. It’s hard to find moderately realistic problems that don’t take too long to work out.

The course I’ve found easiest to teach has been differential equations. The course has a flat structure: there’s a list of techniques to cover, all roughly the same level of difficulty. There are no deep analytic or philosophical issues to skirt around as there are in statistics. And it’s not hard to come up with practical applications that can be worked out fairly easily.

Related post: Impure math

It's not what you cover

Walter Lewin on teaching physics:

What counts, I found, is not what you cover but what you uncover. Covering subjects in a class can be a boring exercise, and students feel it. Uncovering the laws of physics and making them see through the equations, on the other hand, demonstrates the process of discovery, with all its newness and excitement, and students love being part of it.

From For the Love of Physics

What do colleges sell?

Universities are starting to give away their content online, while they still charge tens of thousands of dollars a year to attend. Just what are they selling? Credentials, accountability, and feedback.

Some people are asking why go to college when you can download a college education via iTunes U.

First, you would have no credentials when you’re done.

Second, you almost certainly would not put in the same amount of work as a college student without someone to pace you through the material and to provide external motivation. You’d be less likely to struggle through anything you found difficult or uninteresting.

Third, you’d have no feedback to know whether you’re really learning what you think you’re learning.

The people that I hear gush about online education opportunities are well-educated, successful, and ambitious. They may be less concerned about credentials either because they are intrinsically motivated or because they already have enough credentials. And because of their ambition, they need less accountability. They may need less feedback or are resourceful enough to seek out alternative channels for feedback, such as online forums. Resources such iTunes U and The Teaching Company are a godsend to such people. But that doesn’t mean that a typical teenager would make as much of the same opportunities.

Cartoon guide to the uninteresting

If you’re not interested in a subject, do cartoons make it more palatable?

My guess is that cartoons may help keep your attention if you’re moderately interested in a subject. If you’re fascinated by something, cartoons get in the way. And if you’re not interested at all, cartoons don’t help. The cartoons may help in the sweet spot in between.

No Starch Press has given me review copies of several of their Manga Guide books. The first three were guides to the universe, physics, and relativity. I’ve reviewed these here and here. Recently they sent a copy of the newest book in the series, The Manga Guide to Biochemistry.

I’m much more interested in physics than biology, so I thought this would be a good test: Would a manga book make it more interesting to read about something I’m not very interested in studying? Apparently not. It didn’t seem that the entertaining format created much of an on-ramp to unfamiliar material.

It seemed like the information density of the book was erratic. Material I was familiar with was discussed in light dialog, then came a slab of chemical equations. Reading the book felt like having a casual conversation with a lawyer who periodically interrupts and asks you to read a contract.

Someone more interested in biochemistry would probably enjoy the book. Please understand that the title of this post refers to the fact that I find biochemistry uninteresting, not the book. If I had to study a biochemistry book, the Manga Guide to Biochemistry might be my first choice. At times I’ve found biochemistry interesting in small doses, describing a specific problem. But it would be nearly impossible for me to read a book on it cover to cover.

O’Reilly’s “Head First” series is similar to the Manga guide series, though the former has more content and less entertainment. I enjoyed the first Head First book I read, Head First HTML with XHTML & CSS. Maybe I enjoyed it because the subject matter was in the sweet spot, a topic I was moderately interested in. The cartoons and humor helped me stick with a dry subject.

When I tried another Head First book, I was hoping for more that same push to keep going through tedious content. The books clearly had the same template though with different content. What was interesting the first time was annoying the second time, like hearing someone tell a joke you just heard. So at least for me, the Head First gimmick lost some of its effectiveness after the first book.

When are we ever going to use this?

“When are we ever going to use this?” What a great question! This is a teachable moment. Too bad most teachers blow it. Instead of seizing the opportunity, they reprimand the student for asking. At least that was my experience.

Why would someone not explain how their subject is used? Often because they don’t know. Or they don’t know how to articulate what they do know. But teachers are supposed to know things and be good at articulating them. That’s their job.

Sometimes the student asking how a subject is going to be used is just a lazy whiner. He’s not asking a sincere question, and he will not find a sincere answer satisfying. But maybe the student is genuinely curious. Or maybe there’s at least a drop of curiosity in the whiner. Or maybe someone else sincerely has the question that the whiner insincerely asked.

I am not saying that content needs to be more practical. Attempts at being more “practical” have often been shortsighted. Many subjects that have been discarded as impractical are actually quite practical. We’ve just grown impatient, unwilling to wait for long-term benefits. I’m saying that more teachers should know and articulate the value of what they’re teaching.

It’s more difficult to convey the value of things that are not immediately useful, but it’s also more important.

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