Math education: you’re doing it wrong

Recent discussion about the problems with our educational system reminded me about the story of why my friend J almost didn’t make it into his high school honors math class. Now, to clarify, J is easily one of the smartest people I know. But he is also a smartass, and thirteen-year-old J was certainly no different.

On the entrance exam for his honors math class, several of the problems asked you to fill in the next number in the sequence, such as: 2, 4, 8, 16, _?_. Obviously, whoever wrote the exam wanted you to complete that sequence with “32,” because the pattern they’re thinking of is powers of 2. For n = 1, 2, 3, 4, 5, the formula  2n = 2, 4, 8, 16, 32. But J didn’t write “32.” He wrote “π.”

When his teacher marked that problem wrong (as well as all of the other sequence questions, which J had answered in similar fashion), J explained that there are literally an infinite number of numbers that could complete that sequence, because there are an infinite number of curves which go through the points (1, 2), (2, 4), (3, 8), and (4, 16). Sure, he said, one of those curves is the obvious one which also goes through (5, 32), but you can also derive a curve which goes through (5, π). He showed her an example:

As you can see if you try plugging in the numbers 1, 2, 3, 4, and 5, to the equation above, you get the sequence 2, 4, 8, 16, π. Here are the two curves plotted on a graph, both the “correct” curve and J’s smartass curve (hat tip to the mathematician at www.askamathematician.com for graphing this for me in Mathematica):

Anyway, after thirteen-year-old J explained the math behind his unconventional, but admittedly accurate, answer to the original problem, his teacher replied, “Oh come on, you knew what it was asking for!” and refused to give him any credit. I can’t think of a better illustration of the triumph of the stick-to-the-book method of teaching over kids’ innate creativity… or of the triumph of math education over actual math skills.

The Transplant Problem

In this week’s video, I field a question about a tricky dilemma in moral philosophy: if you had to kill one innocent person to save five people, should you do it?

“Is there an answer?” Searching for the meaning of life in The Hitchhiker’s Guide to the Galaxy.

(posted at 3 Quarks Daily)

The Austrian philosopher Ludwig Wittgenstein gets credit for pointing out that many classic philosophical conundrums are unsolvable not because they are so profound, but because they are incoherent. Instead of trying to solve such questions, he argued, we should try to dissolvethem, by demonstrating how they misuse words and investigating the confusion that motivated the question in the first place.

But with all due respect to Wittgenstein, my favorite example of the “dissolving questions” strategy comes from Douglas Adams’ The Hitchhiker’s Guide to the Galaxy, which contains a cheeky and unforgettable dissolution of which I’m sure Wittgenstein himself would have been proud:  A race of hyper-intelligent, pan-dimensional beings builds a supercomputer named Deep Thought, so that they can ask it the question that has preoccupied philosophers for millions of years: “What is the answer to life, the universe, and everything?”

After seven and a half million years of computation, Deep Thought finally announces the answer: Forty-two. In response to the programmers’ howls of disappointment and confusion, Deep Thought rather patiently points out that the reason his answer doesn’t make any sense is because their original question didn’t make any sense either. As I’ve written before, questions like this one, or the very similar “What is the meaning of life?” question, seem to be committing a basic category error: life isn’t the kind of thing to which the word “meaning” or “answer” applies.

But in this article I want to take my analysis a little further than that.

Read the rest, at 3 Quarks Daily.

RS#34: Why do people listen to celebrities’ opinions?

Episode #34 of the Rationally Speaking podcast is all about celebrities giving their opinion on topics they don’t know much about: why do they get invited to opine on those topics at all, and why are people influenced by them? Even people who are experts in a technical field often give misleading viewpoints when they’re offered a platform to talk about fields other than their own. Massimo and I talk about some examples, describe some relevant psychological studies on influence, and still somehow manage to work in our standard bickering about philosophy.

http://www.rationallyspeakingpodcast.org/show/rs34-celebrities-and-the-damage-they-can-do.html

The D.I.Y. way of getting a probability estimate from your doctor

One frustrating thing about dealing with doctors is that they tend to be unwilling or unable to talk about probabilities. I run into this problem in particular when they’ve told me there is “a chance” of something, like a chance of a complication of a procedure, or a chance of transmitting an infection, or a chance of an illness lasting past some time threshold, and so on. Whenever I’ve pressed them to try to tell me approximately how much of a chance there is, they’ve told me something to the effect of, “It varies” or “I can’t say.” I sometimes tell them, look, I know you’re not going to have exact numbers for me, but I just want to know if we’re talking more like 50% or, you know, 1%? Still, they balk.

My interpretation is that this happens due to a combination of (1) people not having a good intuitive sense of how to estimate probabilities and (2) doctors not wanting to be held liable for making me a “promise” – perhaps they’re concerned that if they give me a low estimate and it happens anyway, then I’ll get angry or sue them or something.

So I wanted to share a useful tip from my friend, the mathematician who blogs at www.askamathematician.com, who was about to have his wisdom teeth removed and was trying unsuccessfully to get his surgeon to tell him the approximate risks of various possible complications from surgery. He discovered that you can actually get a percentage out of your doctor if you’re willing to just construct it yourself:

Friend: “I’ve heard that it’s possible to end up with permanent numbness in your mouth or lip after this surgery… what’s the chance of that happening?”

Surgeon: “It’s pretty low.”

Friend: “About how low? Are we talking, like five percent? Or only a fraction of one percent?”

Surgeon: “I really can’t say.”

Friend: “Okay, well… how many of these surgeries have you done?”

Surgeon: “About four thousand.”

Friend: “How many of your patients have had permanent numbness?”

Surgeon: “Two.”

Friend: “Ah, okay. So, about one twentieth of one percent.”

Surgeon: “I really can’t give you a percentage.”

Why learn geography?

Over the weekend I met up with a group of friends who host an education-themed discussion salon. Some of them are teachers, others just interested in the subject, and all of them are very smart. (It also turned out, though we didn’t plan it this way, that they’re all quantitatively-minded, specializing in math, computer science, or statistics. This is nice because it means that whenever someone proposes an idea, someone else will inevitably say, “Let’s see, how would we test that?” and we end up in a discussion of control groups and confounding factors.)

Afterwards, I pinged Jesse on Gchat to hash out one of the more interesting questions that came up during the salon: Should students have to memorize geographical facts? Our conversation, edited somewhat for clarity, is below.

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Julia: Last night at the education salon we were talking about whether students should have to memorize geography – you know, identifying countries on a map, knowing capital cities, etc. It’s definitely one of those things where, when news articles are lamenting how ignorant Americans are, they cite polls in which (e.g.) 2/3 of Americans can’t identify Iraq on a map. But even though that triggers this knee-jerk “What a travesty!” response, on second thought I’m not convinced it’s such a bad thing.

Jesse: Well, not knowing where things are makes it tougher to notice regional patterns. You’re not going to pick up on the common features that form a shared culture in the South if you hear about something that happened in a particular state and you don’t know whether it’s in the South or Midwest.

Julia: But it’s so easy to look up. If people hear about some place mentioned in the news, they can literally just google it, right?

Jesse: I expect they often won’t… though I’m surprisingly ok with that.

Julia: Yeah. One thing we talked about in the salon was how the most effective thing for schools to instill in their students are meta-skills: instead of making them memorize vocabulary lists, get them in the habit of looking up words they don’t know. And instead of making them memorize maps, get them in the habit of looking up unfamiliar place names.

The deeper issue here, though, is how much relevant knowledge you actually get about a situation or event, from knowing where it’s located relative to other places. Take the Iraq case. There are certainly relevant facts you need to know about Iraq in order to understand world politics: it’s in the Middle East, it’s Shiite Muslim, it has oil, etc. But how much additional important knowledge do you get by being able to locate it on a map?

Jesse: I’m sure we could come up with some elaborate example in which it’s essential to understand geographical features – access to water, relation to mountains, etc. But the fact that it’s a stretch to think of examples indicates to me that those cases are rare enough to warrant just looking up places on a map as needed. The kind of useful knowledge you were describing in the Iraq case sounds like it can be picked up through history classes, current events, simple interaction in society, etc.

Julia: That’s what I was envisioning, yeah. My general principle with education is that it’s always better to learn “motivated” facts than unmotivated facts. By which I mean: if it’s clear to you why the fact you’re learning is important or useful, then you’re going to be more interested, more willing to learn, and more likely to remember it. So the ideal way of learning geography, in my opinion, is simply on an as-needed basis, contextually, in other classes.

For example, if you’re learning about the Roman Empire, you need to learn what regions it covered in order to appreciate what a huge undertaking it was, and in order to understand the spread of Roman infrastructure and ideas. Or if you’re learning about WWII, you need to know which countries bordered each other, because it’s relevant to understanding the war. But you wouldn’t take a separate geography unit in which you’re memorizing maps.

Jesse: I’m trying to think of other times we encourage rote memorization. Just thinking ‘aloud’ – take the multiplication tables. Yes, they’re easily calculated or looked up, but we consider it valuable to learn them by heart. That’s because we use multiplication at that scale (through 12×12) so much that it’s impractical to look up answers all the time. To what extent does that apply to geography?

…I would say, a very small extent.

Julia: Although… there is one point someone made which I think might be a good one: That having a visual framework in which to store information is a really effective way of remembering it. So, if you hear something about a civil war in Burma, and you can place that on your mental map of the world, you’re more likely to remember that knowledge than if you didn’t have the mental map.

Jesse: Ah, this makes me think of your ‘Memory Palaces‘ post

Julia: Yes! That’s what I thought of too.

Jesse: I think that’s correct, but is it worth it learning the entire map ahead of time? Especially for places that are likely to come up in discussion and in the news, a map will form through interaction with the news and the urge to look it up. I don’t think it’s worth the prep time – considering how unlikely it is to be remembered without use, I suspect the time could be better spent doing other things.

That said, I have to say that video games like Medieval: Total War and Rome: Total War taught me more about those geographic regions than any class. If we ever decided that the map-facts are important, games are the way I’d do it.

Julia: Interesting… what is it about the game format that works so well?

Jesse: I think it’s just the motivated learning – there’s a reason to care which city is Milan vs. Venice. When a message pops up that your army in Venice is under attack, you care where that is in relation to the rest of the region.

Beware of the Granfalloon

In this week’s video I discuss my new favorite word — “Granfalloon” — and how identifying yourself with a particular group can distort your thinking.

Visualizing data with lines, blocks, and roller coasters

Randall Munroe's infographic on radiation dose levels (Click to enlarge)

I’m a huge fan of clever ways of visualizing data, especially when there’s something challenging about the data in question. For example, if it contains more than three important dimensions and therefore can’t be easily graphed with the typical representations (e.g., position on x-axis, position on y-axis, color of dot). Or if it contains a few huge outliers which distort the scale of the data.

This recent infographic in Scientific American by my friend (and co-blogger, at Rationally Speaking) Lena Groeger is a great example of the latter. The challenge in displaying relative levels of radioactivity is that there are a few outliers (e.g., Chernobyl) which are so many times higher than the rest of the data that when you try to graph them on the same scale, you end up with the outlier at one end and then all the rest of the data clumped together in an indeterminate mass at the other end.

Randall Munroe over at the webcomic XKCD came up with a pretty good, inventive solution that relies on our intuitive sense of area, rather than length. Each successive grid represents only one small block of the next grid, which is how he manages to cram the entire skewed scale into one page. It’s cool, but I don’t think it works that intuitively. We have to consciously keep in mind the reminder of how big each grid is relative to the next, and it’s easy to lose your grip on the relative scales involved.

However, one of the benefits of online infographics as opposed to print is that you don’t have to fit the whole image in view at once. Lena and her colleagues created a long, leisurely scale that has the space at one end to show the differences between various low levels of radiation dose, below 100,000 micro-Sieverts… and then it hits you with a sense of relative magnitude as you have to scroll down, down, down, until you get to Chernobyl at 6 million micro-Sieverts.

It reminded me of one of my all-time favorite data visualizations: over one hundred years of housing prices, transformed into a first-person perspective roller coaster ride. There are a number of wonderful things about this design choice. For one thing, it works on a visceral level: reaching unprecedented heights actually makes you feel giddy, and sudden steep declines are a little scary.

I also love the way it captures the most recent housing bubble — as you keep climbing higher, and higher, and higher, and higher, and higher, the repetitive climb starts to feel relaxing, and you even forget that you’re on a roller coaster. You forget, in other words, that you’re not going to keep going up forever. And that moment at the end, when the coaster pauses and you turn around to look down at how far away the ground is (this video stops right before the 2008 crash) — shiver. Just perfect.

RS#33: New Dilemmas in Bioethics

During the taping of Rationally Speaking episode #33, at NECSS 2011. (Photo credit: Brian Gregory)

Episode #33 of the Rationally Speaking podcast is out: “New Dilemmas in Bioethics.”  This is the one Massimo and I recorded live at the 2011 Northeast Conference on Science and Skepticism. We discuss bioethics with two special guests: Jacob Appel, doctor, author, lawyer and bioethicist; and Jennifer Michael Hecht, poet and historian of science. Topics covered included: Should parents be allowed to select the gender and sexual orientation of their babies? Should pharmacists and physicians be allowed to refuse to provide treatments that violate their own religious or ethical principles? And when is assisted suicide acceptable?

One of the interesting things about this episode was the strikingly different approaches Jacob and Jennifer used when considering bioethical issues — Jacob clearly has pretty utilitarian inclinations, so the guiding principle behind his answers was “What would the expected positive and negative effects of this policy be?” He also has a relatively libertarian approach to bioethical policy, which I think grows naturally out of his utilitarianism — in general, allowing people the freedom to make their own choices will maximize utility (though of course you can find plenty of exceptions; I don’t mean to imply that Jacob’s worldview is that absolute).

Jennifer, meanwhile, had a much more deontological (rule-based) approach to ethics: she appears to judge some things as wrong not necessarily because they reduce overall utility, but because they’re inherently distasteful or because they violate a principle that she holds sacrosanct. (I’m interpreting their respective views, of course, so let me add the disclaimer that I can’t guarantee they’d agree with these characterizations).

I suspect most people are closer to Jennifer’s worldview, but Jacob’s is much more aligned with mine.

What is 0^0? And is math true, or just useful?

When you hear mathematicians talk about “searching” for a proof or having “discovered” a new theorem, the implication is that math is something that exists out there in the world, like nature, and that we gradually learn more about it. In other words, mathematical questions are objectively true or false, independent of us, and it’s up to us to discover the answer. That’s a very popular way to think about math, and a very intuitive one.

The alternate view, however, is that math is something we invent, and that math has the form it does because we decided that form would be useful to us, not because we discovered it to be true. Skeptical? Consider imaginary numbers: The square root of X is the number which, when you square it, yields X. And there’s no real number which, when you square it, yields -1. But mathematicians realized centuries ago that it would be useful to be able to use square roots of negative numbers in their formulas, so they decided to define an imaginary number, “i,” to mean “the square root of -1.” So this seems like a clear example in which a mathematical concept was invented, rather than discovered, and in which our system of math has a certain form simply because we decided it would be useful to define it that way, not because that’s how things “really are.”

This is too large of a debate to resolve in one blog post, but I do want to bring up one interesting case study I came across that points in favor of the “math is invented” side of the debate. My friends over at the popular blog Ask a Mathematician, Ask a Physicist did a great post a while ago addressing one of their readers’ questions: What is 0^0?

The reason this question is a head-scratcher is that our rules about how exponents work seem to yield two contradictory answers. On the one hand, we have a rule that zero raised to any power equals zero. But on the other hand, we have a rule that anything raised to the power of zero equals one. So which is it? Does 0^0 = 0 or does 0^0 = 1?

Well, I asked Google and according to their super-official calculator, the answer is unambiguous:

Indeed, the Mathematician at AAMAAP confirms, mathematicians in practice act as if 0^0 = 1. But why? Because it’s more convenient, basically. If we let 0^0=0, there are certain important theorems, like the Binomial Theorem, that would need to be rewritten in more complicated and clunky ways. Note that it’s not even the case that letting 0^0=0 would contradict our theorems (if so, we could perhaps view that as a disproof of the statement 0^0=0). It’s just that it would make our theorems less elegant. Says the mathematician:

“There are some further reasons why using 0^0 = 1 is preferable, but they boil down to that choice being more useful than the alternative choices, leading to simpler theorems, or feeling more “natural” to mathematicians. The choice is not “right”, it is merely nice.”