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?

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.

You too can own a massive house if you sell enough calculus books.

There’s a new, five-story, 18000 square foot, $24 million house in Toronto that is built of curves and glass and boasts its own professional-quality concert hall. The owner? Not a billionaire financier, head of state, movie or sports star, or anything of the sort — it’s James Stewart, author of the Stewart Calculus franchise of books.

From the Wall Street Journal article:

As visitors descend into the house, the fins disappear and the views widen. On the first floor, push a button and a 24-foot wall of glass windows vanishes into the floor, opening the pool area to the outside. Curves are everywhere, down to the custom door handles and light fixtures. The architects are even working with Steinway to create a coordinating piano. [...]

An hour before five friends arrived for dinner, Mr. Stewart ambled around his kitchen, marinating some pork tenderloin chunks and tossing chopped leeks, red peppers and corn into a deep soup pot to simmer. He laid some ready-made sushi on a large red platter and then leaned back against a green-hued quartz countertop to relax.

Mr. Stewart say he isn’t overwhelmed by his home. “I just enjoy wandering around it,” he says. “Even now I’m still discovering details, and I’ve lived here for more than a year.”

Go to the article and look at the slideshow for more. It’s indeed a beautiful home (in a way it reminds me of St. Procopius Abbey near Chicago, which I visited last year).  I’m certainly not going to be down on Prof. Stewart for building his dream home, for which he apparently saved up money for 60 years. But it certainly destroys the old idea that professors never make money off of textbooks they write. And it also makes you wonder, if you recently spent $150 on a Stewart Calculus book, what part of that house you have a legitimate claim to. If you’re a Stewart Calculus book owner, I’d say you have a right to stop in at his place for sushi unannounced at any time.

A proposition for Prof. Stewart: Now that you’ve built your dream home and established your legacy, take all of your calculus books and make them available as free PDF downloads under a Creative Commons license, so students who are spending down to their last dime on textbooks can have a shot at saving for their dream houses, too.

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.

Spreadsheets are one of the  most underrated tools available for doing and learning mathematics, especially calculus. At my college we include spreadsheets as a central tool for our Calculus I course and use them every chance we get. But as with all technology, there is the possibility of encountering a seemingly inexplicable glitch when using them even in a very tame situation.

Here’s one I encountered this week when setting up a spreadsheet to do an average velocity/instantaneous velocity problem. We started with a falling object whose position from the start point at time t is given in the following table:

The eventual goal is to compute the average velocity from t=2 to t=3, then t=2.5 and t=3, then t=2.9 and t=3, and so on, finally estimating the instantaneous velocity right at t=3. Actually, this is where the glitches started. The “25″ in the third cell was supposed to be a “20″ because I wanted to position data to be fit, exactly, by the function , which we were going to use to generate the position data not in the table. I caught this after an initial edit but decided it would be more interesting to proceed with the 25 there, then use my spreadsheet to get a power function trendline through the data, and go from there.

Go from there I did, and here’s the chart with the trendline:

Pretty close to , right? Well… not exactly. Here’s what we get when we use the trendline formula to generate the position data for value of time approaching t=3 from the left, and then the average velocity from those times to t=3:

What’s supposed to happen is that the position values approach the position attained by the book at t=3, and the average velocities stabilize toward a single number which represents the instantaneous velocity at t=3. But the little deviation in the t=2 position from the original table (25 instead of 20) throws the trendline off so that the position as time approaches 3 overshoots the actual position at t=3, and so we end up with average velocities that are spinning out of control. (-1781 m/sec is roughly 4000 miles per hour, for reference, and the direction is wrong to boot).

But, you say, this is no surprise, because the mistake in the original table had the position off by 5 meters from where you intended. But what’s funny is that if you go back and make the mistake in the original table smaller, even a lot smaller, you encounter the same effect. If you change the 25 in the t=2 cell to 20.1 — that’s just a difference of less than 4 inches from the intended position of 20 meters — the trendline changes to , and here’s what you get as time approaches 3:

As we close in on t=3, we still get the object rocketing upward!

What we learn here is that if you use a trendline for calculations, you really shouldn’t mix data from the trendline with data from the table which produced the trendline. In fact, the original trendline created using s(2) = 25 would predict s(3) ≈ 46.813, and when that value is used instead of 45 in the average velocity calculations you see the averages stabilize, although slowly, towards something like 31.2 m/sec.

But even then, if you took that trendline formula and found using the Power Rule (as we typically end up doing later in the course to tie algebraic diferentiation rules back to table calculations), you get , which is not what the average velocities are approaching in the table.

So spreadsheets are useful tools for learning mathematics, but for the kind of infinitesimal, close-up work that we have to do with calculus, error propagation becomes a viable course topic as the students are learning about limits.

Note: I used Numbers 09 for the charts and trendlines. I think Excel uses the same algorithm to produce power function trendlines as does Numbers, so this isn’t an Apple vs. Microsoft issue.

I got an email from a fellow edu-blogger a couple of days ago asking for my input on the subject of academic rigor. Specifically this person asked:

Is the quest for more rigor an issue for you? Is it good, bad, meaningless? What does rigorous teaching look like in your classroom?

I hope she doesn’t mind my sharing the answer, because after writing it I thought it’d make a good blog post. I said:

For me, “rigor” in the context of intellectual work refers to thoroughness, carefulness, and right understanding of the material being learned. Rigor is to academic work what careful practice and nuanced performance is to musical performance, and what intense and committed play is to athletic performance. When we talk about a “rigorous course” in something, it’s a course that examines details, insists on diligent and scrupulous study and performance, and doesn’t settle for a mild or  informal contact with the key ideas.

Example: A rigorous course in geometry goes beyond just memorization of formulas, applications to simple geometric exercises, and “hand-waving” attempts at proof. Instead, such a course treats details as important, the ability to explain on a deep level the truth of formulas and results as a key goal for students, and sets a high bar for the exactitude of mathematical arguments. Euclid’s “Elements” for example is the prototype of the rigorous treatment of geometry. It’s not a difficult work to understand, necessarily; in fact one of the enduring qualities of the Elements is the clarity and precision of not only each individual proposition but also in how the overall collection of propositions fits together. By contrast, many modern books on geometry are highly non-rigorous, omitting details, putting theorems out of order, and defining a proof as a “reasonable explanation” only.

Is rigor good? It depends on the audience and the goals of the class. When I teach a geometry course for junior and senior Math Education majors, rigor is of the utmost importance because I want those pre-service teachers to go into their classrooms with tough, precise minds for the sake of their students. If I were to teach a geometry class for fifth-graders, on the other hand, I think rigor would obscure the subject, and I would depend a lot more on intuitive explanations and perhaps constructivist techniques for discovering key ideas in geometry and save rigorous proofs for another day. Similarly, when I teach calculus at my college, the audience is about 50% business majors, and so we designed the course not to cover much theory. This is not a rigorous treatment of calculus, but it is more effective for the students than if we included the epsilon-delta proofs and what not.

The quest for more rigor is most important in the post-calculus courses I teach (geometry, abstract algebra, and introduction to proof). These are subject areas where precision and detail-orientation are essential for a complete understanding of the material. Students are not allowed to give examples when a proof is called for, and I nitpick every little thing in their proofs up to and including the choice of punctuation and prepositions. [If any of the five who took this course from me this past semester are reading this, feel free to chime in with an "Amen." - RT.] At the calculus level and below, I lay off on the theory but the rigor in the course comes from getting details of mechanical calculations right. And this is a big issue, because students in high school are generally taught only to produce a correct answer, not a clear and detailed solution. I am on a mission to make sure students can not only get right answers but also communicate their methods in a clear and audience-appropriate way, and that’s what “rigor” looks like there.

[After-the-fact note: To clarify, in calculus I insist on details in mechanical calculations but also on the details of processes and in paying attention to nuances in solving application-style problems. For example, students know that if you just set and solve for , that this doesn't give you an inflection point; and in an optimization problem you can't just find the critical number of the model function, you must also test it with the First or Second Derivative Test to see if it really yields a maximum. Or at least, they don't complain when they forget to do it and I take off points!]

I have two kids, ages 3 and 5. (There’s a third one on the way in three weeks, but that’s another story!) I’m pretty rigorous with them, too — when the 5-year old says “Mimi comed to our house this weekend” I correct her grammar, and she gets it right the next time. You have to do it in a gentle way, but getting details right now will help them get the more complicated things right later. If I were to project myself out of higher ed and into the K-12 sphere I could see my teaching being “rigorous” in that kind of way — insisting that kids get the details right and not gloss over things, but doing it in a lovingly persistent way. I wish more K-12 teachers would do this, though, because it’s obvious from my freshmen in the last 4-5 years that this isn’t happening (or at least it’s not sticking).

[Final note: That last sentence isn't a slam on either my freshmen, who were really quite excellent this year in calculus, or their teachers. It's an observation, and I stand by it. I can show you their work at the beginning of the semester if you don't believe me. Why this kind of "rigor" is not sticking with them is something I can't fully explain because I don't know what was going on with them in high school. Is it them? Is it their teachers? Is it the system? Is it the preponderance of standardized testing, which makes rigor more or less irrelevant? Comment!]

Average velocity is another one of those basic calculus (really pre-calculus) topics that, like difference quotients, leave me at a loss for why students have such a hard time with them. There’s a very simple and common-sense definition, namely that the average velocity of an object with position s(t) from t = a to t = b is

(See? It’s just distance = rate * time solved for “rate”.) There are examples in the book and examples on the internet ad infinitum of how to calculate average velocities, and all of these are simple numerical calculations with absolutely no algebra involved. You have to know how to plug numbers into a function and then do basic arithmetic on your calculator. That’s all.

But students get so turned around. They calculate only the position at time t=b. They add up the positions at t=a and t=b and divide by 2 (“average”). They add in the numerator or denominator (or both). They get the fraction upside-down. And so on. Not all students of course, but many of them — a lot more of them than there should be. And in my calculus classes, it’s certainly not for lack of training data; we’ve done it in lecture, in group activities, in online videos, you name it.

With difference quotients, I can sort of understand where the difficulties might come from — it’s the algebra. But there’s no algebra at all in an average velocity calculation, and even if you struggle to get the concept, can’t you just memorize the formula for the time being? I try always to see student difficulties from the student’s point of view and remember that I was in their shoes once too, but honestly, I am finding it really hard to know where such a consistent mass misunderstanding of this particular idea comes from.

What’s with this topic? Anyone?