This report Frinom the Atlanta Journal-Constitution, citing an article in the June 1 Proceedings of the National Academy of Sciences, says that differences between boys’ and girls’ performance on standardized mathematics tests correlates with the level of gender equity and other socio-cultural factors in the country in which the test was taken.
The study’s co-author says:

“There are countries where the gender disparity in math performance doesn’t exist at either the average or gifted level. These tend to be the same countries that have the greatest gender equality,” article co-author Janet Mertz, an oncology professor at the University of Wisconsin-Madison, said in a university news release.[...]

“If you provide females with more educational opportunities and more job opportunities in fields that require advanced knowledge of math, you’re going to find more women learning and performing very well in mathematics,” Mertz said.

The study goes on to cite the US as a country where there is a relatively high degree of gender equity and hence a relatively equal performance on standardized tests between boys and girls, with more and more girls taking advanced courses in science and math. But, importantly, the study also warns that

“U.S. culture instills in students the belief that math talent is innate; if one is not naturally good at math, there is little one can do to become good at it,” Mertz said. “In some other countries, people more highly value mathematics and view math performance as being largely related to effort.”

This is a point well worth noting. What will it take for the culture in the US to get away from the idea that you’re either born with mathematical ability or born without it — in other words, mathematical predestination?

By now, you’ve probably heard about Wolfram|Alpha, the “computational knowledge engine” that was recently rolled out by the makers of Mathematica. If you haven’t, here’s a good place to start. There is considerable debate among ed-tech people as to exactly what kind of impact Wolfram|Alpha, abbreviated W|A, is going to have in education. For me, W|A is still a little raw and gives back  too many “Wolfram|Alpha isn’t sure what to do with your input” responses when given mathematically legitimate (at least they seem so to me) queries. But the potential is there for W|A to be a game-changing technological advance, doing for quantitative information what Google did for text and web-based information back in the 90’s. (W|A is already its own verb.)

One thing that seems clear is that, with technology available that is free and powerful and hardware-agnostic, technology that previously has ruled the ed-tech roost can’t survive for much longer. I’m thinking particularly of the graphing calculator. These have been a fixture in math education, especially at the pre-college level, for the better part of 20 years. But now here is W|A, which can graph functions, perform symbolic algebra and calculus computations, even solve differential equations and do number theory and statistics and all manner of interesting stuff besides, including but very much not limited to mathematics. In short, it does everything a graphing calculator does. But, importantly: W|A is free, runs on any web-enabled device (including, as I can attest to by experience, an iPod touch), is fast, is portable (see the links I just shared?), and — perhaps most importantly of all –  has an army of developers who are constantly adding new features into the system.

You could spend $150 to get the latest and greatest from Texas Instruments, a handheld device that does what a graphing calculator does — but no more. (Here’s my first-hand take on the NSpire and details on what I see as its demerits.) Or, you could spend a little more than twice that much and get a netbook computer that gives you access to W|A as well as a suite of office tools and more. Computing hardware has become so small and cheap, and online quantitative tools so functional and powerful, that it’s very hard to see how graphing calculators can survive the next 5 years.

If graphing calculators do survive, it will be for one main reason: The AP exams. I was talking with a local high school AP Calculus teacher this week who impressed on me that  she cannot afford to drop graphing calculators and move on to using netbooks or some other more sensible technology because, quite simply, there are questions on the AP Calculus exams that require the use of graphing calculators. Students have to have total fluency with graphing calculators — and not some other, calculator-like technology — in order to do as well as they possibly can on the exam, which is part of this teacher’s professional responsibility. The AP already succeeded in killing the TI-92 calculator — a really good technology for its time, when laptops still weighed 15 pounds and costs thousands of dollars — for no better reason than because it had a QWERTY keyboard. Today, the AP might succeed in keeping W|A and other similiarly useful, perhaps even transformative, technologies out of the hands of students pretty much for the same reasons, which is a real shame and quite backwards-looking.

But then again, I don’t know what the AP folks have in mind. Perhaps there are plans afoot to migrate the AP exams away from dependency on graphing calculators. It certainly wouldn’t take much for the AP folks to write their own lightweight graphing tool that does nothing more than plot functions, find intersection points, shade in areas, and do numerical integration (rarely are graphing calculators used on the AP free-response portion for more than these four things). Make it extremely basic, put it on the web, free for all to use, and provide it on specialized computers for students taking the exam. That way, students can learn how to use technology rather than learn how to use a graphing calculator, and both teachers and students can be freer to choose the extent and type of technology they want to use in their classes. And such a thing would probably have a longer shelf life than any TI calculator for sale or in production.

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I’ve been teaching calculus since 1993, when I first stepped into a Calculus for Engineers classroom at Vanderbilt as a second-year graduate student. It hardly seems possible that this was 16 years ago. I can’t say whether calculus itself has changed that much in that span of time, but it’s definitely the case that my own understanding of how calculus is used by professionals in the real world has developed, from having absolutely no idea how it’s used to learning from contacts and former students doing quantitative work in business amd government; and  as a result, the way I conceive of teaching calculus, and the ways I implement my conceptions, have changed.

When I was first teaching calculus, at a rate of roughly three sections a year as a graduate student and then 3-4 sections a year as a newbie professor:

• I thought that competency in calculus consisted in the ability to think through difficult mechanical calculations. For example, calculating using multiplication by the conjugate was an essential component of learning limits.
• There were certain kinds of problems which I felt were inseparable from a proper understanding of calculus itself: related rates, trigonometric integrals, and a few others.
• I thought nothing of calculus that didn’t involve algebra. I’m not saying I held a low opinion of numerical or graphical calculus problems or concepts; I’m saying I didn’t even have them on my radar screen. I spent no time on them, because I didn’t know they were there.
• Mechanical mastery was the main, and in some cases the sole, criterion for student learning.

Since then, I’ve replaced those criteria/priorities with these:

• I care a lot less about mechanical fluency in algebra and trig, and I care a lot more about whether a student can read a problem for comprehension and then get an optimal solution for it in a reasonable amount of time and using a reasonable method.
• I don’t think twice about jettisoning any of the following topics from a calculus course if they impede the students’ attainment of the previous bullet point: epsilon-delta proofs of limits*, algebraic limits that involve sophisticated algebra tricks that students saw five times three years ago, formal definitions of continuity, related rates problems, calculation of integrals using limits of Riemann sums, and so on. I always want to include these, and I do it if I can afford to do so from the standpoint of managing class time and maximizing student learning. But if they get in the way, out they go.
• I care very much about whether students can do calculus on functions of all shapes and sizes — not only formulas but also tables of data and graphs — and whether students can convert one kind of function to the other, and whether students can judge the relative pros and cons of doing calculus on one kind of function versus another. The vast majority of functions real people encounter are not formulas — they are mostly evenly split between tables and graphs — and it makes no sense to spend 90% of our time in calculus working with formulas if they are so rarely the only option.
• I don’t get bent out of shape if a student struggles with u-substitution and the like; but it drives me up the wall if a student gets the units of a derivative wrong, or doesn’t grasp that a derivative is a rate of change, or doesn’t realize that the primary purpose of calculus is to quantify what we mean by “rate of change”. I guess that means my priorities for student learning are much more about the big picture and the main ideas than they are the minute, party-trick algebra/trig calculations.

Perhaps the story would have been different if I’d remained tasked with teaching calculus to an all-engineer audience. But here, my classes are usually 50% business majors, about 25% biology or chemistry majors, and 15% undecided with only a fraction of the remaining 10% being declared majors in mathematics (which includes students in our 3:2 engineering program). But that’s the story as it is, and I’m sticking to it.

* Technically I never have to omit these, because we don’t do them in our intro Calculus class here.

On Twitter right now I am soliciting thoughts about calculus courses, the topics we cover in them, and the ways in which we cover them. It’s turning out that 140 characters isn’t enough space to frame my question properly, so I’m making this short post to do just that. Here it is:

Suppose that you teach a calculus course that is designed for a general audience (i.e. not just engineers, not just non-engineers, etc.). Normally the course would be structured as a 4-credit hour course, meaning four 50-minute class meetings per week for 14 weeks. Now, suppose that the decision has been made to cut this to TWO credit hours, or 100 minutes of contact time per week for 14 weeks.

Questions: What topics do you remove from the course? What topics do you keep in the course at all costs? And of those topics you keep, do you teach them the same way or differently? If differently, then how would you do it? Finally, would there be anything NEW you’d introduce in the course that would be pertinent for a 2-hour course that wouldn’t show up in a 4-hour version of that course?

Keep Twittering your comments to me at @RobertTalbert, or comment below. I’ll sum them up later.

UPDATE: I also meant to say, feel free to play with the assumptions I am making here. For example, if it’s impossible to think of a 2-hour calculus course, change that to a 3-credit course and see if you can come up with anything.

The last time I taught abstract algebra, I used no textbook but rather my own homemade notes. That went reasonably well, but in doing initial preps for teaching the course again this coming fall I realized my notes needed a serious overhaul; and since I’m playing stay-at-home dad to three kids under 6 this summer, this is looking more like a sabbatical project than something I can get done before August. So last month I set about auditioning textbooks.

I looked at the usual suspects — the excellent book by Joe Gallian which I’ve used before and really liked, Hungerford’s undergraduate text*, Rotman — but in the end,  I went with Abstract Algebra: Theory and Applications by Tom Judson. I would say it’s comparable to Gallian, with a little more flexibility in the topic sequencing and a greater, more integrated treatment of applications to coding theory and cryptography. (This last was something I was really looking for.) There’s even a free companion to the book which incorporates Sage, which I am sorely tempted to use as well because learning Sage has been a pet project of mine.

But what’s really different about this book is that it’s free, licensed under the GNU Free Documentation License. I am having the bookstore prepare print copies for the students — I asked the students if they wanted a print version in addition to the free PDF’s online, and they said “yes” — which the bookstore will sell for a whopping $16.95, just enough to cover the costs of copying and 3-hole punching the 400+ pages of the book. I’m happy because I found a book that really fits my needs; the students are happy because they get a good book too, for a tremendous bang-to-buck ratio.

In the long and contentious comment thread for my post about James Stewart’s new $24M mansion, I suggested that Stewart should consider topping off his impressive (and apparently lucrative) teaching and writing career by making his Calculus book freely available online for anybody who wants it. That suggestion was met with shocked incredulity: “If you had any idea how much work it was to write and maintain a textbook, you’d never consider making it free.” Well, I’m happy to report that hard work and good writing need not necessarily be mutually exclusive with giving it away.

In fact, as more well-written textbooks appear for free online — and there were even more free abstract algebra e-books I did not end up selecting — the commercial market might find itself in trouble.

* Actually, I requested the Hungerford algebra book, complete with a crystal-clear note that I needed to have it in hand by April 10 in order to be able to adopt it in time for our bookstore. To this date I have not received it. Another problem with commercial textbooks: the distribution model for review copies is dreadful. I’m always receiving multiple copies of books I neither need nor am interested in, and not getting the books I do need and am interested in.

All snarks about $24M mansions being funded by calculus textbook sales aside, there is an emerging relationship between calculus and architecture that is really fascinating. Since WordPress.com now allows direct embedding of TED talks, I thought I’d share this talk from architect Greg Lynn on this subject. I ran across this a couple of weeks ago and I’ve been wondering about it ever since. The point about using calculus to change architecture from a “discrete” notion into a “continuous” notion is particularly interesting.

From Joanne Jacobs, a handy interactive certification map for prospective teachers.

Chester Finn and Mike Pitrelli say that education needs more efficiency, not more investment. We could at least give it a try.

Killing the buzz over “21st-Century Skills”: ““The error at the heart of P21 is the idea that skills are all-purpose muscles that, once developed, can be applied to new and unforeseen domains of experience.”

Want a job? Don’t major in business. Instead, major in a “classical” liberal arts major and then take 4-6 math courses on the side (i.e. get a math minor).

Using math to make the perfect pancake.

Albrecht Dürer and the heptagon.

Experiments with MATLAB, a high-school (!) level book on MATLAB from Cleve Moler.

Several apps from Omni Group — OmniWeb, OmniDazzle, OmniDiskSweeper, and OmniObjectMeter — are now freeware. These used to be about $25 apiece. I tried OmniWeb once and thought it was really good, and I’ll probably try it out again. If you’ve got a Mac and some disk space, have at ‘em!

I’ve started reading through Stewart and Tall’s book on algebraic number theory, partly to give myself some fodder for learning Sage and partly because it’s an area of math I’d like to explore. I’m discovering a lot about algebra in the process that I should have known already. For example, I didn’t know until reading this book that the Gaussian integers were invented to study quadratic reciprocity. For me, the Gaussian integers were always just this abstract construction that  Gauss invented evidently for his own amusement (which maybe isn’t too far off from the truth) and which exists primarily so that I would have something to do in abstract algebra class. Here are the Gaussian integers! Now, go and find which ones are units, whether this is a principal ideal domain, and so on. Isn’t this fun?

Well, yes, actually it is fun for me, but that’s because I like abstract nonsense and I like just constructing rings out of nowhere and seeing what works and what doesn’t. But this approach to algebra is a lot harder to convince others to adopt, particularly college math majors whom I teach, most of whom struggle with abstraction. For them, any connection, no matter how tenuous, to the real world is a comfort and a reason to study. Quadratic residues aren’t exactly in the same league as designing airplanes in terms of “real world” utility, but it’s at least something that’s easy enough to understand and explain. Even if you care nothing for real world utility, it’s important to know why something was invented when you are setting about studying it. Otherwise you learn a subject in abstraction and without connections to its roots.

In fact, it seems like a lot of what we take as being canonical in abstract algebra was invented to study number theory. And yet, I have never taken a number theory course, and the number theory that was included in my studies of algebra was added mainly to set up the study of abstract groups and rings, as if to say that number theory exists to make studying algebra easier instead of the other way around as appears to be the case. And it’s not because I had a bad algebra education; I studied under some of the best algebraists around, but I never got the memo that abstract algebra was for something. I learned algebra mainly in isolation for the sole purpose of calculating homotopy groups. Likewise, my entire grad school training was focused on topology, which is supposedly a branch of geometry, but the only course in geometry I have in my background was Mrs. Buttrey’s class at William James Junior High School in the eighth grade — and that didn’t exactly give me the disciplinary perspective I needed to put topology in its proper context. (Even though it was a really good geometry class — thanks Mrs. B!)

I’ve been thinking that my post about the, er, pedagogically challenged way that Stewart Calculus does its examples about instantaneous velocity is really about the idea that you need to make sure that a person learning a new idea has some reason to learn it, before you give it to them in full complexity. Or at least before they’ve finished a course in it. Perhaps this idea extends to all of mathematics and maybe even beyond.

In the Stewart calculus text, which we use here, the first chapter is essentially a precalculus review. The second chapter opens up with a treatment of tangent lines and velocities, with the idea of secant line slopes converging to tangent line slopes and average velocities converging to instantaneous velocities taking center stage.

Calculating average velocity is just a matter of identifying two time values and two position values and then performing two subtractions and a division. It is not complicated. Doing this several times for shorter and shorter time periods is also not complicated, and then using the results to guess the instantaneous velocity is a little complicated but not that bad once you understand the (essentially qualitative, not quantitative) idea behind shrinking the length of the interval to get an instantaneous value out of a sequence of averages.

So I nearly hit the roof when a student came in this morning needing help understanding the Student Solutions Manual for the Stewart text on a problem where you had to find the average velocity of a moving object from 2 seconds to 2.5 seconds. A formula for position is given, . The simple way to do this — the way that works, does not dumb the process down, and yet makes it understandable to the broadest possible audience and therefore sets  up general understanding of the more complicated idea of derivative calculations later — is to calculate , calculate , and then calculate . Fifth-graders do this.

Instead, the Student Solution Manual does it like this:

• Let h represent some positive number.
• Calculate and fully simply the expression .
• Plug in .

This is crazy, absurd, and downright dangerous. It’s as if Stewart, and the person who wrote the manual, really believe that calculus is made up of algebra, and students who are in calculus are uniformly comfortable and skilled with algebra to the point that their way is just as transparent and simple as calculating distance divided by time — as if the algebraic work that ensues when you perform step (2) above were as natural as the concept of velocity itself and students spoke algebra like a first or second language.

Yes, the book’s approach works — and it closely mirrors what’s going to happen later when we want to get an exact value of the instantaneous velocity by letting . But that’s not what students are doing right now. What students are doing is trying to understand the concept of average velocity. It’s not complicated. The complications should come, if at all, on the back end of the subject — where we are trying to make the concept of instantaneous velocity precise through limit calculations — but not on the front end when students are just trying to figure out what’s going on.

In the middle of typing this post out, another student came in, equally confused about the exact same problem. I told him to close his solutions manual. I asked him: What’s the definition of average velocity? He thought about it, and then gave me the right definition. “OK, then,” I said, “How would you get the average velocity from t=2 to t=2.5 here?” And he gave me an exactly right description of the process. The relief on his face was palpable. He understood this concept but the student solutions manual made it appear that he didn’t! How bad is it when you need a manual for the student manual?

Calculus is a really simple subject when you get to its core. I wish the book treated it that way.

Down under, the Australians are going through many of the same arguments about mathematics education that we are here in the US. In this column from The Age, Marty Ross — who holds a PhD in mathematics from Stanford — lambastes the Australian mathematics education community in ways that might seem eerily familiar to those who follow the similar issues in America. Quote:

[H]ere is an exercise from a current Victorian year 9 maths text: a farmer has 2C cows and 3H horses. The exercise is to find the square of the sum of the farmer’s animals.

The Victorian texts are not uniformly that pointless or that bad. But not much is good. Definitions are clumsy, problems are contrived, natural connections and beautiful insights are overlooked. The texts do not reflect a mathematical culture.

It is not just the textbooks. Teachers are poorly trained; the curriculum is moribund, rife with silly, contrived applications; and everywhere there is pointless calculation. And calculators – the cane toads of education.

Is there still proof? Proof is the source of the power of mathematics, the reasoning and the understanding: it’s what holds the discipline together. But it is practically dead. The very little proof that remains is meaningless and ritualised: maths as Latin Mass.

How did it get this bad? Primarily, it results from the failure to involve mathematicians, the people for whom mathematics is their life’s blood. The simple fact is, many of those responsible for mathematics education do not know sufficient mathematics to do the job.

There are lots more “ouch” moments in the article. Ross concludes by saying:

What do I want from a national curriculum? I want a dodecahedron in every classroom, and beautiful diagrams to ponder. I want students to know why there are infinitely many prime numbers, and for them to realise no one knows about twin-primes. I want them to know what the golden mean is, and why it is irrational, and why we care. I want pattern and play and beauty. And I want the times tables.

Is teaching any of the above useful? It is exactly as useful as teaching Harry Potter and Shakespeare.

Mathematicians do mathematics because it is fun and it is beautiful. If the curriculum is not written in that spirit, and if teachers are not trained in that spirit, then we are doomed. We will have yet another generation devoted to gradgrinding students into hating mathematics.

I’ll agree on many of these points — especially why mathematicians are motivated to do mathematics, the criticism about the lack of proof in the math curriculum, and to some extent Ross’ critiques of the mathematical background of mathematics education people. But what do you think — is Ross’ assertion that fun and beauty form the proper basis for a mathematics curriculum really sound? I mean, I’d like all my students to know about the infinitude of primes too, but does that sort of thing make a reasonable organizing principle for an entire curriculum?