News on the math gap
We interrupt these long essays on religious belief to bring you a tidbit of news about math education!
Back at the beginning of the month, the MIT News Office reported on a recent paper by economics professor Glenn Ellison and econ PhD student Ashley Swanson regarding the gender gap in math performance. Their research showed “not only that girls are a small minority of elite high school math students, but also that the prevalence of high-achieving girls in math varies from school to school.” It’s a very interesting result.
This extreme concentration of talent strongly indicates the crucial role that environmental factors, not just innate ability, play in shaping the accomplishments of students. “It’s significant that the top girls are coming from a very, very small subset of schools with strong math programs,” says Ellison. “That suggests most of the girls who could be doing well, aren’t doing well. The thousands and thousands of other schools in the United States must have a lot of talent, too, but it’s not coming out.”
I’m really happy about this work because so much of the dialogue about gender disparity in STEM fields centers around anecdotes, or on assertions of trends which, while convincing, are still assertions. And I think that’s part of why it’s still such a contentious issue. If we’re ever going to reach a consensus about the extent of the problem and what, if anything, should be done about it, we need to roll up our sleeves and crunch some numbers. (And okay, I’ll admit, I’m also happy about this paper because it makes me feel a little more secure about not having done so well on the AMC back in high school.)
There’s a PDF of the paper online, so give that a read if you’re interested in more details.
Dear Prudence
No, not the Beatles’ song, the advice column on Slate.com! I love Emily Yoffe’s advice, and the sense of humor in her columns and videos, but I have to say I found something lacking in the answers she gave regarding this math grad student who wrote to her during a Washington Post chat a few days ago. Here’s the student’s original question and her reply.
Philadelphia, Penn.: I’m a graduate student in mathematics, and my particular area is very abstract. When people ask me what I do, or see me with a textbook and ask what I’m reading, no matter how simplified an explanation I give them, inevitably the person remarks that my area is “way beyond” them or that they’d “never be able to grasp that”. I always want to tell them, “You definitely won’t with that attitude.” To me, all of these concepts are perfectly intuitive, and while I’m certainly aware that not everyone has my capacity for, or interest in, mathematics, I am still annoyed by all of these people putting themselves down to me—did they ever try to understand the subject? At least some of them might find it easier and more interesting than they expected! For some reason, they all seem to take pride in how poor their math ability is, or at the very least they aren’t troubled by it. Should I be harsher and just say “Yes indeed, this is much too hard for you”? How do you suggest I respond to these kind of comments?
Emily Yoffe: A few years ago, in an attempt to help my daughter with her math homework, I enrolled in the elementary school math prep program, Kumon. I scored at the first grade level. Even if I tried, I probably couldn’t truly understand what you’re doing. But I would be interested if you could explain what this math is used for—modeling subprime mortgages? Global warming? Then we’d have something to talk about. So ignore the self-put downs, and don’t add any of your own. Instead think of it as an opportunity to show that what you do is interesting and can—on some level—be grasped.
Later in the chat, someone else wrote in to say exactly what I’d been thinking. If I didn’t know better (I didn’t even know about this chat thing until yesterday), I’d have sworn it was actually me.
“Math? That’s over my head”: I think this writer deserves more of an answer. I’m a woman in physics, and nearly everyone makes a self-deprecating comment when I say so. The point really is this: there is a cultural pride in innumeracy that doesn’t exist for illiteracy—no one will brag about not being able to read, yet feel free to essentially brag about not being good at math. This is not people being candid about their abilities. It actually is a way of dismissing the importance of the field of study by implying that it has no cultural necessity or meaning. There has to be a way of responding to this, but subtly encouraging people to believe that it can be understood (with perhaps some effort—but what doesn’t take effort?). Most people can do math to at least the calculus level with time and effort, not talent. And most people can understand even high mathematical concepts, if not perform them, if they’d get past the mental block. This hurts everybody!
Emily Yoffe: I’m sure most people could change their carburetor with some time and effort, too. Instead of lecturing people about how their innumeracy hurts everyone, understand that people are complimenting you on your impressive skills. So come up with some things to say about your field that can engage even the innumerate, like me, in conversation.
I get what she’s saying here. In terms of polite conversation, certainly it’s inappropriate to lecture and berate someone who was just trying to make small talk with you. And people in math/science fields should certainly work on a simplified, clear description of their job so that they can explain it to people without four hours and a chalkboard. That part, I buy.
What I think Emily doesn’t understand, though, is that it’s quite possible (really, extremely probable) that this math grad student is not doing anything that is useful for anything, in a way that could be explained without four hours and a chalkboard. There’s a whole lot more to math than calculus and statistics. She asks, “But I would be interested if you could explain what this math is used for—modeling subprime mortgages? Global warming?” I can easily imagine the answer is, “Nothing.” Nothing like that, at least. Maybe it’s just used for solving other problems that pretty much only exist in the world of mathematics.
That’s not to say that it will never be used for anything else. There are plenty of theorems proved long ago by mathematicians that were only recently discovered to be useful for describing physical systems. The thing is, that’s not the point. We don’t expect all occupations or fields of study to be “useful” in some way that drastically betters the course of human history. Sometimes, they’re just interesting or enjoyable. Sometimes, it’s not useful in the sense of building a better mousetrap or curing the common cold, but useful in the sense of adding to our understanding of what we are made of and how the universe works. Even if we can’t build tiny computers out of that understanding, it’s good that people are developing it.
My advice to the math grad student would be: tell people how interested you are in your research. Tell them how much you love solving puzzles, and how satisfying it is to complete a proof. Then ask them about their job, and move the conversation along elsewhere.
I was saddened to see that our Dear Prudence doesn’t give any advice to the people approaching math grad students everywhere and saying things like, “Pshew, math! That stuff is way over my head.” My guess is, from the way she answers, that she has on occasion been that person. I would hope that people who make those kind of comments learn two things. First, it’s incredibly other-izing; second, it kills the conversation and puts the burden of recovery on the person you’re speaking to. Those two things make it rude, so you should find some other way to respond.
Let’s deal with the first issue. Comments like “Oh, I could never do that” and the like make it sound as though the mathematician (or scientist) is some kind of alien being imbued with freakish abilities. You may think it sounds like a compliment, but it doesn’t feel like one. It’s usually said with a tone of surprised disdain—it’s not just that you don’t know the math, but also that you would never, ever want to—so it sounds like you’re saying, “You could never be a normal person like me.”
Secondly, when people say things like, “I could never do that”… we already know. We operate under the assumption that most of the people we interact with outside the university do not have graduate degrees in math or science. You are saying something as obvious as, “Occasionally, it is cloudy outside.” Additionally, the natural responses to a statement such as this are either self-aggrandizing, insulting, or patronizing, and that is an awkward position to be put in. We struggle to find a way to respond to your remarks—hence the letter to an advice columnist. Good small talk doesn’t just mean saying the first thing that comes to mind, no matter how inane or impossible it is to answer. If you’re going to kill the conversation, do it in such a way that places the burden on yourself, and transition to a new thing to talk about.
There are bigger issues here—as we’ve said before, they encourage the already-widespread belief that it’s fine to be bad at math—but I thought it was worth pointing out that, in addition, they make for really unpleasant conversation.
Math instincts
This article from the New York Times is somewhat old, but it’s still interesting. Research about how humans (and other animals) understand numbers, in a kind of primitive mathematics, can give us insight on how to teach math to young children.
One research team has found that how readily people rally their approximate number sense is linked over time to success in even the most advanced and abstruse mathematics courses. Other scientists have shown that preschool children are remarkably good at approximating the impact of adding to or subtracting from large groups of items but are poor at translating the approximate into the specific. Taken together, the new research suggests that math teachers might do well to emphasize the power of the ballpark figure, to focus less on arithmetic precision and more on general reckoning.
Try this Flash game to evaluate your own number sense.
Are you here for advice?
We get a significant amount of traffic for searches like
- do you have to be good at math to understand sciences
- bad at math but a good engineer
- smart people who are bad at math
- why do i always do bad at math
- can you be successful if youre bad at maths
because of a few of our old posts on innumeracy and attitudes about it. It seems like people are looking for advice and answers, and our posts don’t really provide them, so I thought I’d give it a shot today.
It appears that many of you Googlers (or Yahooers, or what-have-you) want to know whether it’s possible to be a good engineer or scientist while being bad at math. My short answer is: probably not. My longer answer begins with a question back to you: what do you mean by math? For most things I’d consider engineering or science, you need to have a good handle on geometry, high school algebra, trigonometry, statistics, and calculus. If you’ve taken classes with those names and failed them, a career in the sciences is probably not for you.
I assume most people who would ask this type of question are already in school to become a scientist or engineer, though, and are having second thoughts. Don’t worry, university classes are supposed to be harder than high school, and it’s normal to struggle a bit in the transition. Maybe you’re taking your first math class that includes serious proofs as part of the curriculum. If you’re good at calculations but struggle with proofs, there’s no problem with a career in engineering and plenty of available paths to take in the sciences. You probably won’t end up a string theorist, but we’ll still love you. (I suppose I should also ask, what do you mean by science? If you consider anything ending in -ology to be science, you have a wider range of options, including some that use almost no math.)
Then there are those of you asking why you are bad at math. I don’t know. I would obviously need some more information. Maybe you’re not taking notes or paying attention. I tutored some kids during high school that were failing math for that exact reason. Once I suggested that they bring their notebooks to class and, you know, write things down once in a while, their grades shot right up. Maybe you have a terrible teacher. That’s far from impossible, based on my experience. It’s also a possibility that you just don’t naturally think in the way that math requires you to think. It’s good to try to think that way — don’t write off math from the very start — but if you find that over and over again you just can’t seem to get the hang of it, perhaps math is not for you. That doesn’t mean you’re bad, and it also doesn’t mean that math is bad. Find something to do that you enjoy and excel at, and please, continue to respect people who are good at math — the same way that I respect people who are good oboists, or good swimmers.
It’s certainly possible to be successful and bad at math and/or science. (Just look at Congress!) I would also say that it’s possible to be smart without being good at math, though I’m a little more hesitant. What you mean by “smart” and what you mean by “good at math” plays a big role. I don’t think it’s possible to be smart without being capable of logical analysis. You can be smart without knowing that “integral” is a noun as well as an adjective. There are plenty of brilliant lawyers and insightful columnists and effective managers who never got the hang of l’Hôpital’s rule, and I’d still consider them smart. (I think Gardner’s theory of multiple intelligences doesn’t have much merit as science, but it does offer some insight into what we mean when we talk about being smart. There’s more than one way to do it.)
Any more advice for our search engine visitors? Feel free to leave it in the comments.
Carnival of Mathematics #39
We at It’s the Thought that Counts are happy to be hosting the 39th edition of the Carnival of Mathematics. The carnival is published biweekly and includes a wide variety of articles about math — everything from general-interest posts about math in society to advanced technical proofs. This edition has been plagued by some organizational difficulties — many weren’t aware that we were hosting this time, and the Blog Carnival submission form was down — so if you didn’t get your submission in in time, I’d be happy to hear from you and add things to the carnival. Just email me at a@thoughtcounts.net.
Although I study theoretical computer science and my coauthor Z studies physics, our posts here are more often about math and science policy considerations than about specific topics in our fields. I nevertheless couldn’t resist taking this opportunity to pose a clever math problem in honor of the number 39:
39 people are attending a large, formal dinner, which must of course occur at a single, circular table. The guests, after milling about for a while, sit down to eat. It is then pointed out to them that there are name cards labeling assigned seats, and not a single one has sat in the seat assigned to them. Prove that there is some way to rotate the table so that at least two people are in the correct seats. (update: answer in the comments)
While you’re mulling that problem over, if you’d like to see another with no solution yet posted, head over to Continuous Everywhere but Differentiable Nowhere for one that Sam Shah wrote. It’s based on a problem he found from 1896, and it’s definitely a challenge.
Several of you took the opportunity this time around to teach some math. If you’re looking for advanced and technical math writing, the clear winner this week is Charles Siegel at Rigorous Trivialities with a post on algebraic geometry. He’s written about how to explicitly construct curves of arbitrary degree with specified nodes.
Catsynth offers us a taste of knot theory. Knot theory is one of those areas of math lucky enough to have associated with it lots of cool pictures, some of which you can see in this post. There’s also an introduction to some potentially interesting but unanswered questions about prime knots.
There are even more pretty pictures available at Jon Ingram’s blog, Lessons Taught; Lessons Learnt. In “The Joy of Hex,” Jon shows lots of tilings using rotations of a single hexagonal shape, and poses some interesting questions about relationships between the tilings. He also describes how tilings can be a useful educational tool to show students that not all math looks like algebra.
In other educational news, Larry Ferlazzo points out a new website which gives step-by-step instructions for a wide variety of high school math problem types. He also points out the extensive glossary, useful for students learning English along with math.
If you’re still not convinced that the internet provides many great opportunities for teaching and learning math, check out Maria Andersen’s post on “Teaching from the Online Calculus Trenches.” She showcases the 100 slides she made for a presentation on online teaching software and its promises and limitations.
Speaking of software, there’s a poll going on at Walking Randomly concerning what mathematical software you would use if you could pick whatever you wanted. These competing software packages generate some fierce and probably irrational loyalty among users, so I’m happy to see a growing collection of opinions. If you feel qualified you should definitely weigh in. Otherwise, check it out to see what might be worth learning.
We’ll close with two posts on math in the Olympics. John Cook at The Endeavor gives us a statistical model of Olympic performance by athletes of different genders. He points out that the difference in performance could be the result of a difference in variability across genders rather than a difference in average ability, and gives a quantitative illustration of this possibility.
Finally, while following the Olympic pole vaulting coverage, Xi at 360 notes a silly consequence of the official USATF system for converting between Imperial and metric units. Apparently, the convention is always to round down, which means that by converting back and forth it’s possible to get hilarious results.
That’s all for now. Don’t forget, we’ll continue to update this over the weekend to accommodate new submissions.
Carnival of Mathematics #38
The 38th edition of the Carnival of Mathematics went up at Catsynth on Friday. It includes A’s recent post about typical math teachers’ efforts to make their classes seem relevant to students. There’s a bunch of other neat stuff in the carnival — you should go and check it out! Of particular interest to me was Jon Ingram’s post on winning and losing at Nim. Nim is a very simple game, but there are a lot of really interesting results associated with it, and it’s the foundation for a lot of stuff in combinatorial game theory. His post is easy to understand even if you don’t have a lot of background in math.
We’re looking forward to hosting the 39th edition of the carnival here at It’s the Thought that Counts. To submit your entry, fill out the submission form by August 21.
Why we learn math
There are lots of students who aren’t interested in math, and lots of math teachers who want to motivate them to actually put effort into the subject. Most of this struggle seems to revolve around how useful math is, with kids saying (or at least perceived to be saying) something like “I’ll never need this” and teachers trying to convince them that math is everywhere.
For a variety of reasons, I’ve never been happy about this way of looking at why one learns math. Most importantly, I think, teachers are bound to fail at convincing students of the usefulness of math, even with the most applicable of topics. Part of the reason is that the subjects that apply the math are always taught after the math itself. Physics, economics, and so forth make the need for quadratic equations very clear, but you would never teach someone physics unless they had first mastered quadratic equations. This means that the math a student is currently learning is never being used in their other classes as they learn it. Some students see stuff they learned years before now being used routinely, and they learn to trust that the things they’re being taught now will turn out equally useful in years to come. That’s a healthy attitude, but even with an explicit explanation of the situation, many students will not adopt it.
It’s also worth mentioning that there are in fact topics taught in high school (especially in a school with a strong math curriculum) that aren’t likely to be used in the future by many of the students. Calculus is helpful in a variety of occupations, but far from all. Geometry, barring the straightforward basics, is only helpful in unusual circumstances. Imaginary numbers are unlikely to come up too frequently. A teacher is therefore bound to fail at showing the usefulness of these things directly.
I also think, though, that judging math by its usefulness is missing the point. Why do math teachers need to prove that their material will be vital for daily life in order to make it worth learning? Poetry would never be able to pass that test, nor would history or art. You don’t “use” Shakespeare very frequently. Of course, that’s obviously not the point. We fully recognize that sonnets aren’t “useful,” but we still learn them. We think it makes your life better to have the wider, deeper view of the world that comes with having studied art and literature. We think they’re part of being an educated person. We think that by studying them you build fundamental skills of critical thinking, imagination, and interpersonal relationships that are important, even if the actual material you’re learning is not. These things are all true about math as well, but people don’t think of math that way. They think of it as a prerequisite for other (important) things.
Math should be taught as something you learn because it’s interesting and enlightening. It should of course be mentioned that it’s also useful, but that should never be seen as the only reason why it’s being taught. To do this, it would help to change the math curriculum a bit. Anyone who’s ever taken proof-based math in college knows that mathematicians don’t spend their days doing the stuff that’s taught in high school. Engineers solve a lot more equations than mathematicians do. Mathematicians deal with abstract topics, proving new facts. This kind of open-ended, puzzle-like problem is a lot more fun to do, and a much better way to show students what math is really like, than the computational topics that take up the current standard curriculum.
If math can be taught as something that’s interesting, rather than as something that’s useful, it changes the way students look at it. It becomes the kind of thing that one would expect some people to really, deeply like. It can be fun and exciting. If it’s being done because it helps design bridges, it’s a chore. The application is cool sometimes, but the thing you’re learning never is. Math as useful calculation will never be appreciated by anyone who can’t see themselves going into science, engineering, or accounting, but math as clever puzzles that help us understand the wonder of pure reason can be something people really want to learn.
Gallons per mile
There are lots of very difficult, nuanced issues in politics — issues where two intelligent people could disagree, have an intelligent back and forth for hours, and still come out with totally intact, cogent views on the topic. These are often fundamental questions about the very way our country works. But there’s another kind of issue. There’s the kind of issue where there is just an obvious correct decision, with very little room for intelligent discussion. Sometimes these issues are incredibly important, but more often they are small and just slip by, because only a handful of people is involved in the relevant decision, and they didn’t realize what they were doing. These issues annoy me the most, because, however minor they are, there’s no excuse for failing to do the obviously correct thing.
The way we measure fuel efficiency is one of these dumb things. Using miles per gallon is really misleading. It can make tiny gains seem huge, and huge gains seem tiny. Let’s take two totally hypothetical vehicles. One one hand you have a hybrid car, which gets 40 mpg, and you convert it to a plug-in hybrid, which gets 100 mpg. On the other hand, you have a very inefficient small truck, which gets 10 mpg, and you put in a more efficient engine, pushing it to 15 mpg. It seems like the former improvement is better. It’s a 60 mpg improvement rather than a 5 mpg improvement. It’s a 150% improvement rather than a 50% improvement. Nevertheless, if we assume both vehicles are driven 1000 miles, the hybrid goes from using 25 gallons to 10 gallons, saving 15 gallons, where as the small truck goes from 100 gallons to 67 gallons, saving 33 gallons. The gain from improving the truck’s efficiency is massively better than the gain from improving the car’s.
This is a general mathematical fact. The inefficient vehicles are the ones using lots of fuel, and small changes in their mileage are large percentage changes, so very small mpg changes can save a lot of fuel. The super-efficient cars use very little fuel anyway, so even massive improvements can’t save that much. Consumers, obviously, think about mileage in the units that it’s given to them in, so they value it in an irrational way. (Science Pundit has a great post about this.) It would make a lot of sense to change to gallons per mile (or per 1000 miles) and get consumers thinking more rationally, but I can understand the reluctance. The switch to a new unit takes mental adjustment, and it’ll take a while for consumers to get a good handle on what counts as “good” or “bad” mileage, meaning they’ll probably take efficiency into account less during that unit transition. (Interestingly, it seems that this is already frequently done in many other countries. Sociological Images posts this video that shows mileage in L/100km.)
What really makes no sense is using mpg in regulation. US automobile efficiency is regulated by the CAFE standards, which mandate a minimum average mileage for the fleet of vehicles produced by each manufacturer. The problem is, by using miles per gallon, rather than gallons per mile, the economic incentive is to produce more super-efficient hybrid small cars, whereas much bigger gains could be made through smaller improvements to the worst vehicles. Adding 3 mpg to a hybrid doesn’t cancel out a loss of 3 mpg in a pickup truck, but that’s how the standards work. You could easily pick the new required average to be no more or less stringent than the current one — it would just be more intelligent. If anything, it would help US manufacturers over Asians ones, since it’s the Asian manufacturers that are producing the small, hybrid cars.
There is only one intelligent argument I can think of against this change, which is that these super-efficient cars are the ones that are pioneering technologies that will push down to all vehicles sooner or later. This might be true, but I doubt it’s fundamentally necessary. (It’s easier to put a more advanced engine or a battery in a big vehicle than a small one.) Maybe you can make an argument that this over-counting of gains for small cars is a way of subsidizing the technological innovation behind them. I don’t really buy that, though. A change in the regulatory measures seems obviously good.
More on innumeracy
Given all the comments on my post about people who readily admit to being bad at math, as well as the discussion occurring on various other blogs, I figured it was time to respond to some of what’s been said.
There were some people who expressed skepticism of the phenomenon I was complaining about. These comments (both here and elsewhere) were things like “I always talk about how bad I am at writing” or “I’m an English major, but I know plenty of science.” I have no doubt at all that the incidents cited in the comments really did happen, and they do go against the trend I talked about, but I think they are the exception rather than the rule.
Putting aside for a moment the question of how much knowledge someone should have about any particular field, I want to give some clear support for my assertion that math/science people do know more about the humanities than humanities people know about math and science. I should first be clear about what I’m counting as what. By “humanities” I mean not only literature and fine arts, but also history, social sciences and languages. While there are some arguable cases (economics comes to mind), I think it’s pretty clear that that stuff clearly goes on the humanities side of the divide. When I refer to “sciences,” I mean technical fields in general, including both theoretical and applied math, computer science, engineering, and applications like medicine.
It’s obviously impossible to compare levels of understanding in two different fields. How much calculus do you need in order to equal the amount of knowledge that encompassed by fluency in a foreign language? It doesn’t make any sense to compare these things directly. Still, I believe that we can make the general claim that some incredibly basic, simple science is considered “equivalent” to much more advanced levels of humanities knowledge. Z commented to this effect, using Jeopardy! questions as a proxy. Something a little more quantitative (ha, ha) would of course be preferable.
The best metric I could come up with was simply to look at how much effort was being put into learning material on the other side of the divide, rather than how much material was actually being learned. I decided to look up core curricula at some of the country’s most prestigious universities. These curricula seem as good a proxy as any for what the intellectual class feels a well-educated person should know. The humanities part of the core requirement generally determines how much time a science student has to spend on humanities, while the reverse is true of the science part of the requirement. Of course, many on both sides choose to learn much more than is required, but I think the requirements are a good proxy of what is considered necessary in order to consider yourself well-educated. I tried to vary the colleges I looked at. I chose two schools with a technical focus (MIT and Caltech), three general top universities (Harvard, Princeton, and Yale), and two of the top liberal arts schools (Swarthmore and Williams). Results below: Read more
Being bad at math
If you work or study in a technical field — particularly if you’re in math itself — you get used to a particular type of reaction when you tell someone what you do. It’s far from universal, but frequently the response is something like, “I was never very good at math” or “Math just wasn’t my thing.” You learn to get used to it, but really, I’m sick of it.
It’s not that it’s not true. Probably these people really are bad at math. Probably it was always their worst subject. I’m just tired of no one feeling bad about it. These kinds of sentiments are very common, but imagine how weird it would be if you replaced “math” with “English” or “reading.” Do you think authors, when they tell people they write for a living, ever get told “I was just never very good with words” or “I’m just not a reading person”? They obviously don’t, because it’s not acceptable to be lacking in reading skills. Some people are, but they would never go around saying so, and they usually work very hard to get better. Somehow it’s become dishonorable to admit English is a bad subject for you, but perfectly fine to say the same for math.
It’s really sad that in a technical age, where more and more people are engineers, scientists and computer programmers, we don’t have this deep societal appreciation for math and science. The same thing that makes people freely admit their math skills also affects college curricula. Look at the required curricula at most liberal arts colleges, which proudly proclaim the value of the “well-rounded” education that they give. There are very few math/science/computer/engineering classes, and extremely few math classes in particular. What requirements there are can always be filled by worthless classes. Then look at the curricula for technical, math/science-focused schools. They always have a substantial humanities requirement, and a totally unscientific survey of people I know has found that there tend to be few joke courses to fill those requirements, and that most students don’t take them. Which schools really give the most well-rounded education?
In this day and age, there is no excuse for brushing off math. It’s tough if you’re bad at it, especially since faking competency is a lot harder in math than in the humanities. Nevertheless, brushing it off and not caring is not an acceptable defense mechanism. There’s a part of me that really wants, next time I hear someone say “I was always bad at math” to respond with “Well, I guess you’re just stupid.” It’s obviously not the correct response, but at least it’d move the average in the right direction.
Update: some follow-up comments here.