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A Neat Argument For the Uniqueness of $e^x$

August 09, 2018

In my advanced Calculus 1 class I teach a quick unit on differential equations. We don’t have the tools to solve them since we haven’t done integrals, but I talk about what differential equations are and how you can check whether you have a solution.

And then I spend a day in lab discussing exponential growth, and how the differential equation $y’ = ry$ implies that $y = Ce^{rt}$ for some constants $C$ and $r$. I’ve been telling my students that while it’s easy to check that this is a solution, we don’t have the tools to prove it’s the only family of solutions.

But today thanks to reddit, I discovered that that isn’t quite true. You can prove that $Ce^x$ is the only solution to this differential equation with a simple argument.

Suppose $f(x)$ is a function that satisfies $y’ = r y$, that is, suppose $f’(x) = r f(x)$. Then consider the derivative of $f(x) e^{-rx}$. By the product rule, we have

Thus we see that $f(x)/e^{rx}$ must be a constant; and thus $f(x) = C e^{rx}$. So this family of solutions is unique.


Working Backwards

August 06, 2018

I teach a lot of students who are still learning the basics of proof-writing. My calculus students are seeing their first college math, and often my number theory class is the first really proof-heavy class that a lot of the students take. So I spend a lot of time helping students figure out how to write good proofs. The single best piece advice I’ve come up with is to get comfortable working backwards.

Math students are generally comfortable working forwards. At the beginning of our education this is the only way of working we have: we’re given a problem, like “add these two numbers together”, and we use some algorithm to work out the answer. I like the way Jordan Ellenberg describes this:

[U]ntil algebra shows up, you’re doing numerical computations in a straightforwardly algorithmic way. You dump some numbers into the addition box, or the multiplication box, or even, in traditionally minded schools, the long-division box, you turn the crank, and you report what comes out the other side. Algebra is different. It’s computation backward.

But even once we get to high-school algebra, we don’t really tend to feel like we’re working backwards. In practice we develop an algorithm for solving equations and turn the crank; instead of the addition box or the long-division box we now have the quadratic-equation box. This effect is strong enough that when I give my calculus students problems where I explicitly tell them to guess and check1, they feel very confused and want me to tell them what steps to follow to finish the problem.

But this wanting to know “the steps” is a trap—it leads to treating problems like a black-box request for an algorithm, rather than thinking about what’s actually going on. And as soon as problems require any sort of creative engagement, the search for steps fails.

This first hits students hard when they learn to do integrals. When I taught Calculus 2, I spent a couple lab periods having my students do integral worksheets while I answered questions and gave advice. Fairly frequently, they would spend five or six minutes staring at an integral, before giving up and asking me how to start; I sometimes pointed out that we only had four or five things to try, and if they’d just tried all of them they’d have found one that worked already. But they were uncomfortable with the idea that there wasn’t one correct thing to try.

But all of this becomes especially important when you start doing proofs, because in proofs there is no straightforward algorithm—and often you can’t even really work forward.

Working forward in proofs can be quite useful, to be fair. Given a set of hypotheses, you can start listing off things that the hypotheses obviously imply. Especially in the context of a class, you can often see “okay, if I know these three things, it looks like I should try applying that theorem and see what I get.” But this has really serious limits. You can’t plausibly write down all the implications of your hypotheses.

Instead, you need some idea of where to go. So you should start by looking at the goal, the thing you want to prove. Figure out what you could know that would be enough to finish the proof.

If you can see how to do that, then great, you win. If not, now go look at the hypotheses, and figure out what you can reasonably and easily conclude from them. Can you see how to get from there to the things you wanted?

You develop a sort of push-pull dynamic: work backwards from the goal, then forwards from the premises, then backwards from the goal again. And hopefully, eventually things will meet in the middle.

This seems pretty mundane. But a lot of students are comfortable working forward, and deeply uncomfortable working backwards. They can draw conclusions, but are bad at staying focused on the ultimate purpose of whatever they’re doing. So just prompting them to think about their goals in the middle of proofs can be really helpful.

I do this a lot while lecturing. In the middle of a proof, stop and ask the class to remind me what I’m actually trying to do. They find this surprisingly hard. Sometimes they even struggle to remember what the actual theorem we’re trying to prove is, despite the fact that it’s still written on the board; it’s easy to get lost in the weeds of whatever you’re doing right now and forget the broader context. But without that context, everything you’re doing is kind of pointless, and it’s really difficult to decide what to do next.

And that idea of the importance of context, of focusing on your actual goals, is just as true outside of math class as it is inside. I see this a lot when I give feedback on people’s writing and speaking: they will keep saying things, and the things will be correct, but they won’t do a good job of saying things that are relevant to their goals and message, and of telling us why the things are relevant.

But you can also see this in a lot of bad planning and management. If you lose track of why you’re doing what you’re doing, you’re much less likely to actually achieve your goals. You’ve forgotten what they are, so if you meet them it’s essentially by luck!

This is one of the dangers, for instance, of getting to reliant on management-by-metric. Originally you create a metric to measure how well you’re achieving some goal. But over time, people forget about the goal and remember the metric—and do things that improve the metric, but in ways that don’t advance the original goal.

Staying focused on your actual goal, and working backwards, is really helpful for learning to write proofs. So if you’re teaching people how to write proofs, this is worth explaining explicitly, and then actively training. Keep asking people why they’re doing what they’re doing, and how it gets them closer to the conclusion they want to prove!

But it’s also a really transferable skill, one that can help you in almost all aspects of life. Another example of general thinking skills that studying math helps develop, but are helpful everywhere.

Do you have any suggestions for how to help students get comfortable working backwards? Any other tips for teaching students to write proofs more fluently? Please share them in the comments!

  1. This mostly comes up in the context of inverse functions. One of my favorite questions in Stewart is: Let $f(x) = x^5+x^3+1$. What is $f^{-1}(3)$? You can’t find an explicit formula for $f^{-1}$, but you don’t need to: just plugging in numbers makes it clear the answer is 1. 


Asking the Right Question

July 19, 2018

I’ve been poking around on math reddit lately, and in particular /r/askmath and /r/learnmath. And one thing I’ve really noticed is that many of the posters are really bad at asking questions.

Learning to ask good questions is really important, for a couple reasons. First and most obviously, it’s easier to get help with things and learn things if you can ask better questions. Second, and maybe more importantly, framing questions well is a lot of what makes you a good mathematician.

The most boring way to ask a bad question is just to not include enough information. Sometimes this is just laziness (“I don’t understand how to do this problem, please help”). And I’ve definitely seen questions asked that are thin disguises over “I don’t want to do my homework; can someone do it for me?”

But more often, badly-phrased questions result from deep confusion on the part of the asker. If they understood the material well enough to ask their question clearly and correctly, they wouldn’t need to ask it in the first place.

A lot of math is less about answering questions than about figuring out exactly what question you should be asking, and how to make it precise. We tend to sweep this under the rug a bit when teaching, in a way that I suspect probably leads to a certain amount of confusion.

When we teach, we often ask a question, and then demonstrate a tool to answer it, without necessarily stopping to explain why that question is a good one, or how people settled on asking exactly that question. And this often leaves our students with the sense that what they’re doing doesn’t really mean anything.

This is a major reason students fall back on figuring out what “type of problem” they’re working on, and then following “the steps” to get the answer. They see math as a sort of opaque box, and a question asks them to perform the correct magical ritual to get the answer. Because if the words you’re using—and your questions—don’t have a meaning, that’s all you can really do.

And that’s how you get questions like “how do I find solutions to $f(x) = \sin(x)+ \tfrac{1}{2}$.” I can tell what the original question probably was. But because the student doesn’t really understand what a “function” is, they ask a question that is, read literally, completely nonsensical.

There’s a different type of bad question that comes up: questions of the form “prove to me that this definition is correct”. Or, relatedly: “what is the true meaning of this ambiguous statement”.

I saw one question asking whether $-3$ was a prime number. And that’s actually a tricky question, but for a basically dumb reason. In elementary number theory, we usually define prime numbers to be positive integers. That’s part of the definition, so $3$ is prime and $-3$ is not. Why? Because we said so.

And then when you start doing algebraic number theory, you define primes as a factorization property in a ring, and you see that $-3$ is in fact a prime in the ring of integers. So is $-3$ a prime? Yes, by the definition of prime elements in a ring. Apparently the same question, two different answers—determined by the definition we happened to pick at the time.

Another great example of this problem is questions about order of operations. What’s the value of $6/2 \cdot 3$? It could be 1; it could be 9. I can’t tell you what the original author meant. I can only tell you that they’re a sloppy writer.

This happens in a lot of cases where we’re just discussing a notational convention. There are no underlying facts that make notation right or wrong; screwing up order of operations isn’t really the same type of error as adding numbers incorrectly. It’s just an aid to communication1.

My favorite type of badly written question does this exactly right. Sometimes, you see a question that’s basically “this one thing feels kind of like this other thing, but I can’t tell you how. Can you tell me?”

These people are doing good math. They’re noticing a pattern, and trying to put it into words. They’re maybe not quite there, and sometimes it’s hard to answer the question clearly. But it shows great instincts.

And this is how math tends to actually get done! When we teach, we tend to define terms, then state theorems about them, and then prove the theorems. But this is exactly backwards from how math is often actually done. First we understand what’s going on; then we figure out what the rule is and write it down; and finally understand what conditions are important and give those conditions names. That is, we formulate a proof, then state the theorem, and then define the terms.

These questions are working on step 1. I want to encourage them.

Have any thoughts about questions students ask? Any good techniques for helping your students write better questions? I’d love to hear about them in the comments below!

  1. Aids to communication are useful and important. Math can be hard enough when it’s written well and clearly! But as a reader, if you can tell the writer isn’t following your notational conventions, you shouldn’t hold him to them. The goal of a reader is to figure out what the writer actually meant. And the goal of a writer is to make this as easy as possible. 


Why Word Problems?

June 27, 2018

Why do we assign word problems in math classes?

They aren’t easy. My students usually think that word problems are one of the most difficult and traumatic things I ask them to do, and I doubt I’m alone in this. They take a large chunk of class time, and put a lot of stress on our students. We should only teach them if the benefits justify this effort.

They aren’t really fun. Textbook writers try to make their word problems fun, but this generally winds up looking silly and transparent. Even when the whimsy is more appealing than cringeworthy, students don’t actually enjoy them. So “fun” is a bad reason.

And word problems aren’t really applications of math. I have never in my life needed to figure out how quickly the level of water is dropping in an inverted conical water tank. I don’t think I’ve ever heard of an inverted conical water tank, outside of calculus textbooks1. This just isn’t a real, useful application of calculus.

And yet, I do think word problems are important for teaching people how to apply math to the real world. Not because the problems are a good example of applying math to the real world; they’re too obviously contrived to set up a particular type of math problem. But word problems are very useful for developing a specific skill: the skill of translation. And the fact that we as instructors don’t always understand what word problems are for means it’s much harder for us to successfully teach this skill to our students.

Mathematical theorems, on their own, don’t tell us anything about the world. Theorems are always true when the hypotheses are satisfied, but the hypotheses themselves are pure mathematical abstractions and not real-world objects2. So in the most direct sense, you can’t actually apply math to the world.

And yet, we use math to study the world all the time. We do this more or less by a three step process.

  1. Take the problem and translate it mathematical language by writing down equations that describe it. That is, we take a question about the world and turn it into a math question about mathematical objects.
  2. Apply our mathematical tools to this mathematical description.
  3. Take the mathematical answer we got and translate it back into a statement about the world.

For our purposes, step 2 is boring. It’s the same as doing regular non-word math problems. So the interesting parts of a word problem, the parts that can justify teaching them, show up in step 1 and step 3, where we translate the problem into math and back again. And these steps are interesting both pedagogically and philosophically.

Pedagogically, the translation steps are what makes the problem a word problem; but they are also the steps we need to master if we want to use math to understand anything else. Word problems teach us to represent ideas in a mathematical language.

At an elementary level, this is a straightforward exercise in exchanging English words for math equations. We ask something like

Johnny has three apples, and Susie has five apples. How many apples do they have together?

And our students have to figure out that this means they should take the numbers three and five and add them together. This is important, but it’s really an exercise in key words, and recognizing which formula from class to use.

Word problems can be much more sophisticated. In related rates problems, for instance, we have to figure out what the question is asking (“how fast is the water level dropping”) and translate that into math (“what is the derivative of \(h\)?”). We likewise have to translate the facts the problem gives us, like “when the water depth is three feet”, into equations like “\(h=3\)”. Then we do math, and at the end we get a number, and then we have to figure out what that says about the original problem.

The hard part here is figuring out what the problem is asking and how to talk about it. In my experience, when students fail at word problems, it’s not because they do the actual math incorrectly. Rather, they write down what’s essentially a random equation, and then do math to it. They’re missing the crucial skill of using the math to describe the world.

Also undervalued is the final step, of looking at the number and interpreting what it says about the world. Most students won’t blanch at telling me that a person was running a fifty miles per hour, or that a box has a volume of negative ten cubic meters. This is exactly what happens when you treat math as a numerical black box, rather than saying something about the world: there’s no way to look at a number and see whether it makes any sense.

So when we’re teaching word problems, it’s really important to focus on these steps. I tell my students I give them generous partial credit for noticing that their answers make no sense; and of course most of the points on a word problem aren’t for actually solving the system of equations, or whatever. This is also why middle schoolers often get assigned word problems that have extra or missing information that they need to identify: the goal isn’t to make them do the math, but to engage with why they’re doing math.

But all too often, we lose sight of this as instructors, and treat word problems as a sort of transparent skin on top of a math problem that will somehow make it “fun”, rather than as an important skill in their own right.

Philosophically, this layer of translation between a word problem and a mathematical expression answers the question of why math is so unreasonably effective at describing and understanding the world. It also answers the question of why it’s sometimes so ineffective.

Math is completely accurate in describing exactly one thing: math. Euclid’s theorems apply perfectly to lines and circles, but the world doesn’t contain any lines and circles; it contains imperfect approximations of lines and circles. (Or perhaps more accurately, it contains objects that are imperfectly approximated by lines and circles).

If we want to use math to study the world, these imperfect approximations are important. We build a mathematical model of the thing we care about, and then that model is a mathematical object and so we can study it with math. But it’s important to remember that our math applies to the model, not to the real thing in the world.

If our model describes the world really well—as it does in many cases in physics, for instance—then this isn’t really a problem; your math tells you about the model, and your model tells you about the world. But when the model diverges from the real world, your math stops being helpful.

Many of the worst failures of science and prediction come from people doing good math on a bad model. You can see this in election predictions, where most modelers, like Sam Wang, treated results in separate states as more independent than they were. You can see this in the financial crisis of 2008, where most banks had relied on models that assumed house prices couldn’t all drop at the same time. And you can see this in the replication crisis, largely caused by researchers using statistical tools and tests while violating the assumptions that make them work.

Math is a useful tool for studying a lot of things. But you have to remember that no matter how good your math is, your conclusions about the world are only as good as your model. And your model is always a translation, a description of the world, not the world itself. Your model is always going to lose something. The trick is to find a model that’s good enough, and loses little enough, to do what you want it to do.

  1. Disposable paper cups are the closest I can think of, but they aren’t really water tanks. And I’m still never trying to relate the speed water flows out of the cup to the rate at which the water level changes inside it. 

  2. I’d like to dodge arguments about whether mathematical objects “really exist” in this essay. There are a few different things you can mean by the claim that an ideal mathematical circle exists, with varying degrees of plausibility; but no one seriously claims that they have actually picked up an ideal circle with their hands. 


Course Goals

June 20, 2018

When I started teaching my own courses, one of the less obvious things I had to figure out was how to structure my syllabi. One section that all the syllabi I looked at contained—and all the advice I read mentioned—was the section on “course goals.”

Honestly, I couldn’t see the point. The goal of a calculus course is to, um, learn calculus? But I was supposed to write a course goals section, so write one I did. It was basically boilerplate.

In this course we will master the details of differential calculus, and pursue some advanced topics and applications. We will also develop technical writing skills that allow clear communication of sophisticated ideas, and learn about some technological tools such as Mathematica useful to mathematical and scientific projects.

We will cover limits, continuity, derivatives, modelling, and applications.

That isn’t completely contentless; you can see the emphasis I put on communication and writing skills in my math classes. But honestly, that’s a secondary goal at best. If my students finish calculus 1 with an understanding of what derivatives mean, how to compute them, and how to use them, I’m happy. The goal of calculus 1 is to prepare students for calculus 2. (And the goal of calculus 2 is to prepare students for multivariable calculus, et cetera).

When I first taught an upper-division course, the “course goals” section became possibly more confusing. There is a lot more room for flexibility in designing a number theory course than there is in designing a calculus course: there’s no “next course” that it’s supposed to prepare students for, at least in our department, so I could cover basically whatever I wanted.

So fleshing out the topics list was substantially more important. I needed to decide what topics to include in my course. By the goal was, more or less, “learn some number theory”. Or maybe even just “learn some math, and this happens to be a course in number theory, but the goal is just to have you learn some more advanced math topics.” And that’s not something it really makes sense to write in the syllabus.

But I finally got it last summer. Our department reorganized our senior comprehensive project to involve interdisciplinary half-courses called Math 400, and I got to teach one of the first two we ever offered, which became my course on cryptology.

I was a little nervous about preparing this course. It was very different from any course I’ve taught, and even very different from any course our department had taught, so there weren’t a lot of resources to draw on. And it wasn’t even similar to any courses I’ve taken, so I knew I was very much making it up as I went along. At the same time, I really wanted it to be a good and positive experience for our majors. I wanted to give them something they could feel excited by and proud of for their senior capstone project.

Luckily for me, Occidental’s Center for Teaching Excellence (which is, itself, excellent) held a workshop over the summer to assist us professors with developing new courses. We formed a working group of four professors to brainstorm ideas for our courses and get feedback on our ideas and our syllabi.

Working with this group, the focus on course goals made sense.

I had fallen victim to a common problem in planning and designing processes: I had forgotten that you can’t make good plans unless you know what you’re trying to achieve. If you have goals, you can look at what you’re doing, see whether that achieves your goals, and think about ways to meet your goals better.

Without goals, you’re just doing things. You can’t tell whether they’re good things or bad things or bad things, because you don’t know what good and bad mean.

For core service courses like calculus, this isn’t much of a problem. The goals are standardized: prepare for the next course in the sequence. And the course itself is standardized. I assign homework and give tests, just like every other calculus course. The goals matter, but I don’t have to think about them too much.

With more interesting courses like my cryptology seminar, I had a lot more decisions to make. I had to put some thought into what I actually wanted my students to get out of it, and then I had to think about what course design decisions would actually accomplish those goals. Because there were more choices I could make, there were a lot more things to think about.

The other thing I realized is that most courses are more like the seminar than they are like calculus. A course on English literature, or the history of Islam, or the psychology of gender, doesn’t exist to check a box and prepare students for the next course in the sequence. It isn’t a standardized offering. And thus it demands individualized thinking about what role it does play in the curriculum—what its purpose is, what students should learn from it and assignments will actually further those goals.

(And even a basic calculus class has more of this going on than I’d realized; there’s a lot that all the calculus courses have in common, but also a lot of choices to make. Within my department, there’s wide variance in things like how much weight is put on homework assignments versus tests, or which applications are the most interesting and important to study. We still need to think about the goals of our courses if we want to teach the best courses we can.)

If we want to make good decisions about how to structure our courses, we really do need to think about what we’re trying to accomplish. What do we want our students to take away? What do we want them to learn, and experience? These are important questions, and they help us decide what’s worth spending time on, and what we can skip past.

But why is it so important to put this in the syllabus? One reason, I suppose, is that we can. If we’ve put in the work to decide what our goals are, we might as well share those. And forcing ourselves to write the goals in the syllabus imposes some discipline, and insures that we do put this thought in.

Publishing our course goals helps us interact with our colleagues and share information and expertise more easily. It’s hard to give advice to someone unless you know what they’re trying to do; certainly, in the course development workshop I went to, a lot of the discussion was about what types of assignments would actually achieve the goals I’d set out.

But most importantly, telling our students what the goals of our courses are helps them be better students. None of our students can ever remember every fact and memorize every skill we mention in our courses. They have to prioritize. And one of the most frustrating ways to underperform in a course is to work very hard on things that just aren’t important, while neglecting the most vital parts of the material.

Our students want to focus their study and their practice on the parts that are really important. But precisely because they are students, and not experts, it’s hard for them to tell which parts those are; part of our role, as instructors, is to help them figure out where to focus their efforts.

If we share our course goals, we can do a better job of designing our courses, and our students can do a better job of developing genuine mastery and understanding. The idea felt silly when I first heard of it. But more and more, I realized that the “course goals” section of the syllabus really is critical.