Arthur Benjamin thinks that the current model of the mathematics curriculum — leading from arithmetic to algebra and ultimately to calculus — is flawed and needs to be changed. Watch this 3-minute TED talk for what he thinks ought to be the real summit of the mathematics curriculum:

I am in a great deal of agreement with Prof. Benjamin here. The secondary school math curriculum does indeed seem poised to point students towards calculus. While that is appropriate for some, it is not appropriate for all; while on the other hand, a better knowledge of discrete math, especially probability and statistics, would be appropriate for everybody.

Moreover, Prof. Benjamin did not stress one of the most important selling points for refocusing on discrete math: The mathematical background requirements are a lot lower than they are for calculus. Students currently have to take two years of algebra, at least a semester of trigonometry, and often an entire course in Precalculus on top of all that just to have a fighting change in calculus. And even then it doesn’t always work. Probability and statistics, on the other hand, gets to good ideas, deep ones at that, without nearly so much training.

On the other hand, what about those students who do end up going into science, math, engineering, economics, or other fields requiring calculus? If probability and statistics becomes the summit of the secondary curriculum, then at what point do those kids get the precalculus training they need in order to complete calculus (which I interpret to mean a year of calculus) by the end of their freshman year in college? Would they be having to double up on math courses — statistics on the one hand and precalculus on the other? Would they need to decide that they wanted a STEM-related career early on in high school, and if so, is that good for them?

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?

I’m reading through a number of books and articles related to the scholarship of teaching and learning this summer. One that I read recently was this article (PDF), “Connecting Beliefs with Research on Effective Undergraduate Education” by Ross Miller. There are lots of good points, and teaching tips, in this article. But Miller makes one assertion that doesn’t seem right. He brings up the point, under the general heading of “beliefs”, that “questions arise, both on and off campuses, about whether all students can learn at the college level and whether everyone should attend college” [Miller's emphases]. As to the “should” part of that question, Miller says:

According to Carnevale (2000), from 1998 to 2008, 14.1 million new jobs will require a bachelor’s degree or some form of postsecondary education—more than double those requiring high school level skills or below. Given those data, it makes sense to encourage all students to continue their education past high school. Consistent high expectations for all students to take a challenging high school curriculum and prepare for college (or other postsecondary education) benefit everyone. Our current practices of holding low expectations for many students result in far too many dropouts or graduates unprepared for college, challenging technical careers, and lives as citizens in a diverse democracy.

So, Miller answers, yes — everyone should attend college. But the reasoning seems spurious for a couple of reasons.

• How much of the increasingly common requirement of a bachelor’s degree for new jobs is the result of an existing oversupply of people with bachelor’s degrees? Miller claims that people need to have a postsecondary education because more and more jobs require it. Maybe so. But is that because the jobs themselves inherently use skills developed only through a college education? If so, we have to ask if our higher education system is consistently giving students that kind of education. If not, and if students should get a BA or BS  merely because there are so many people out there with BA’s and BS’s that you have to have one to avoid the appearance of intellectual poverty, then this encourages superficial education at the postsecondary level, and the reasoning here is more mythological than anything and needs to be repudiated.
• As Joanne Jacobs noted back in early 2008 (quoting an article by Paul Barton) it’s not at all settled that the claims about jobs here are even valid. According to that article, only 29% of jobs in 2004 require college credentials, and the percentage is expected to rise only to 31% by 2010 — not exactly a clarion call for all students to matriculate. Also, Barton notes that the wages earned by males with college degrees have slipped, which indicates an oversupply.

College is just not the best choice for every person, and to say that it is merely sets students up for wasting four years of their lives. Some people may have a vocation into a field for which four years of college are a massively inefficient use of time and resources. If you’ve got a vocation to be an electrician, go learn how to be an electrician. If it’s to be a stay-at-home mom, then go do that. Both of these vocations can benefit from a college education if the person is inclined to get one, but neither requires a college education. If you want to go to college and then do those things, fine; but let nobody say that you should go to college, irrespective of your life situation.

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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We’ve finally made it to final exams week in the second semester of what seemed like the longest academic year ever. I thought I would bump this old post from December 11, 2005 (original with comments here) to give props and encouragement to all the students out there who are getting ready for their exams.

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(Inspired by seeing so many students on AIM tonight studying for finals, which for us start tomorrow.)

Dear Lord:
Let those who are filling the library right now with their bodies and their thoughts
Study hard, but also eventually rest.
Let them realize that success on their exams comes
Not from pulling allnighters
Not from cramming
Not from losing sleep
But as the sweet fruits of a long semester
Of diligence, patience, humility, and sweat
Of losing themselves in the laborious doing
That comes when a long-held dream is finally pursued.
Let them know that their final exams not only measure their knowledge
But also, in the ending of the term, show how faithful You have been to them.
They know more now than they did in August.
They are better students, better stewards, of Your blessing of intellect.
Their thoughts are more like Your thoughts.
And no matter what happens, this cannot be taken away.
In that, let them rest
And tomorrow, Tuesday, and Wednesday, let them learn and be satisfied.
In Your Name: Amen.

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.

The two biggest players in the learning management system world, Blackboard and Angel, will soon be one company, since Blackboard has purchased Angel Learning, Inc. for $95 million.  From a superficial reading of the press release, it appears that Blackboard thinks of itself as having a more technologically innovative product, whereas Angel has a better track record with customers — and Blackboard has the money to pull off the purchase.

I can’t verify any of those claims, but I can say that we switched from Blackboard to Angel at my college a few years ago due to a general dissatisfaction with the quality of the product compared to the price we were paying. I don’t recall Blackboard as being particularly innovative, although admittedly that was 4-5 years ago. Angel has not been much of an improvement, and I’ve blogged before about the maddening UI design decisions that Angel has made. In going from Blackboard to Angel, we basically traded one set of deeply flawed LMS technology for another.

And now we have the situation where the current sub-par LMS technology maker is being bought out by the previous equally-but-differently-subpar LMS technology maker. So who knows what exactly we, the users at my college, are going to end up with. The best-case scenario is that we would get the best of both technologies. There are some things that Angel does  pretty well, well enough at least that I am no longer finding myself forced to roll my own LMS at Wikispaces just to retain my sanity. We shall see.

In the meanwhile, Jon Mott has some excellent thoughts about life post-LMS. I think he’s right that the basic problem isn’t the implementation of the technology (although, as I’ve noted, there are some big problems there with Angel and probably with Blackboard as well) but rather the paradigm on which the technology is based. It makes me wonder if the real LMS that best suits the modern college or university is already out there, in the form of previously-released tools that just need to be cobbled together rather than an expensive proprietary software package that tries to emulate those tools.

In a comment on an earlier post, I said I would try to blog about Flat World Knowledge and their business model soon. Here’s a 20-minute video that goes over this business model which allows textbooks to be free but still provides compensation to authors.

Again: Free textbooks can be done; it just requires a different approach than the one we’re used to.

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.