Tag Archives: Calculus

This week in screencasting: Making 3D plots in MATLAB

I’ve just started on a binge of screencast-making that will probably continue throughout the fall. Some of these screencasts will support one of my colleagues who is teaching Calculus III this semester; this is our first attempt at making the course MATLAB-centric, and most of the students are alums of the MATLAB course from the spring. So those screencasts will be on topics where MATLAB can be used in multivariable calculus. Other screencasts will be for my two sections of calculus and will focus both on technology training and on additional calculus examples that we don’t have time for in class. Still others will be just random topics that I would like to contribute for the greater good.

Here are the first two. It’s a two-part series on plotting two-variable functions in MATLAB. Each is about 10 minutes long.

Part of the reason I’m doing all this, too, is to force myself to master Camtasia:Mac, which is a program I enjoy but don’t fully understand. Hopefully the production value will improve with use. You’ll probably notice that I discovered the Dynamics Processor effect between the first and second screencasts, as the sound quality of Part 2 is way better than that of Part 1. I’d appreciate any constructive feedback from podcasting/screencasting or Camtasia experts out there.

I’m going to be housing all these screencasts at my newly-created YouTube channel if you’d like to subscribe. And if I manage to do more than one or two a week, I’ll put the “greatest hits” up here on the blog.

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Filed under Calculus, Camtasia, Screencasts, Teaching, Technology, Textbook-free

Calculus and conceptual frameworks

I was having a conversation recently with a colleague who might be teaching a section of our intro programming course this fall. In sharing my experiences about teaching programming from the MATLAB course, I mentioned that the thing that is really hard about teaching programming is that students often lack a conceptual framework for what they’re learning. That is, they lack a mental structure into which they can place the topics and concepts they’re learning and then see those ideas in their proper place and relationship to each other. Expert learners — like some students who are taking an intro programming course but have been coding since they were 6 years old — have this framework, and the course is a breeze. Others, possibly a large majority of students in a class, have never done any kind of programming, and they will be incapable of really learning programming until they build a conceptual framework to  handle it. And it’s the prof’s job to help them build it.

Afterwards, I thought, this is why teaching intro programming is harder than teaching calculus. Because students who make it all the way into a college calculus surely have a well-developed conceptual framework for mathematics and can understand where the topics and methods in calculus should fit. Right? Hello?

It then hit me just how wrong I was. Students coming into calculus, even if they’ve had the course before in high school, are not guaranteed to have anything like an appropriate conceptual framework for calculus. Some students may have no conceptual framework at all for calculus — they’ll be like intro programming students who have never coded — and so when they see calculus concepts, they’ll revert back to their conceptual frameworks built in prior math courses, which might be robust and might not be. But even then, students may have multiple, mutually contradictory frameworks for mathematics generally owing to different pedagogies, curricula, or experiences with math in the past.

Take, for example, the typical first contact that calculus students get with actual calculus in the Stewart textbook: The tangent problem. The very first example of section 2.1 is a prototype of this problem, and it reads: Find an equation of the tangent line to the parabola y = x^2 at the point P(1,1). What follows is the usual initial solution: (1) pick a point Q near (1,1), (2) calculate the slope of the secant line, (3) move Q closer to P and recalculate, and then (4) repeat until the differences between successive approximations dips below some tolerance level.

What is a student going to do with this example? The ideal case — what we think of as a proper conceptual handling of the ideas in the example — would be that the student focuses on the nature of the problem (I am trying to find the slope of a tangent line to a graph at a point), the data involved in the problem (I am given the formula for the function and the point where the tangent line goes), and most importantly the motivation for the problem and why we need something new (I’ve never had to calculate the slope of a line given only one point on it). As the student reads the problem, framed properly in this way, s/he learns: I can find the slope of a tangent line using successive approximations of secant lines, if the difference in approximations dips below a certain tolerance level. The student is then ready for example 2 of this section, which is an application to finding the rate at which a charge on a capacitor is discharged. Importantly, there is no formula for the function in example 2, just a graph.

But the problem is that most students adopt a conceptual framework that worked for them in their earlier courses, which can be summarized as: Math is about getting right answers to the odd-numbered exercises in the book. Students using this framework will approach the tangent problem by first homing in on the first available mathematical notation in the example to get cues for what equation to set up. That notation in this case is:

m_{PQ} = \frac{x^2 - 1}{x-1}

Then, in the line below, a specific value of x (1.5) is plugged in. Great! they might think, I’ve got a formula and I just plug a number into it, and I get the right answer: 2.5. But then, reading down a bit further, there are insinuations that the right answer is not 2.5. Stewart says, “…the closer x is to 1…it appears from the tables, the closer m_{PQ} is to 2. This suggests that the slope of the tangent line t should be m = 2.” The student with this framework must then be pretty dismayed. What’s this about “it appears” the answer is 2? Is it 2, or isn’t it? What happened to my 2.5? What’s going on? And then they get to example 2, which has no formula in it at all, and at that point any sane person with this framework would give up.

It’s also worth noting that the Stewart book — and many other standard calculus books — do not introduce this tangent line idea until after a lengthy precalculus review chapter, and that chapter typically looks just like what students saw in their Precalculus courses. These treatments do not attempt to be a ramp-up into calculus, and presages of the concepts of calculus are not present. If prior courses didn’t train students on good conceptual frameworks, then this review material actually makes matters worse when it comes time to really learn calculus. They will know how to plug numbers and expressions into a function, but when the disruptively different math of calculus appears, there’s nowhere to put it, except in the plug-and-chug bin that all prior math has gone into.

So it’s extremely important that students going into calculus get a proper conceptual framework for what to do with the material once they see it. Whose responsibility is that? Everybody’s, starting with…

  • the instructor. The instructor of a calculus class has to be very deliberate and forthright in bending all elements of the course towards the construction of a framework that will support the massive amount of material that will come in a calculus class. This includes telling students that they need a conceptual framework that works, and informing them that perhaps their previous frameworks were not designed to manage the load that’s coming. The instructor also must be relentless in helping students put new material in its proper place and relationship to prior material.
  • But here the textbooks can help, too, by suggesting the framework to be used; it’s certainly better than not specifying the framework at all but just serving up topic after topic as non sequiturs.
  • Finally, students have to work at constructing a framework as well; and they should be held accountable not only for their mastery of micro-level calculus topics like the Chain Rule but also their ability to put two or more concepts in relation to each other and to use prior knowledge on novel tasks.

What are your experiences with helping students (in calculus or otherwise) build useable conceptual frameworks for what they are learning? Any tools (like mindmapping software), assessment methods, or other teaching techniques you’d care to share?

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Filed under Calculus, Critical thinking, Education, Educational technology, Math, Problem Solving, Teaching, Technology, Textbooks

The semester in review

Plot of the vector field f(x,y) = (-y,x).

Image via Wikipedia

I’ve made it to the end of another semester. Classes ended on Friday, and we have final exams this coming week. It’s been a long and full semester, as you can see by the relative lack of posting going on here since around October. How did things go?

Well, first of all I had a record course load this time around — four different courses, one of which was the MATLAB course that was brand new and outside my main discipline; plus an independent study that was more like an undergraduate research project, and so it required almost as much prep time from me as a regular course.

The Functions and Models class (formerly known as Pre-calculus) has been one of my favorites to teach here, and this class was no exception. We do precalculus a bit differently here, focusing on using functions as data modeling tools, so the main meat of the course is simply looking at data and asking, Are the data linear? If not, are they best fit by a logarithmic, exponential, or power function? Or a polynomial? And what should be the degree of that polynomial? And so on. I enjoy this class because it’s primed for the kind of studio teaching that I’ve come to enjoy. I just bring in some data I’ve found, or which the students have collected, and we play with the data. And these are mainly students who, by virtue of having placed below calculus on our placement exam, have been used to a dry, lecture-oriented math environment, and it’s very cool to see them light up and have fun with math for a change. It was a small class (seven students) and we had fun and learned a lot.

The Calculus class was challenging, as you can tell from my boxplots posts (first post, second post). The grades in the class were nowhere near where I wanted them to be, nor for the students (I hope). I think every instructor is going to have a class every now and then where this happens, and the challenge is to find the lesson to learn and then learn them. If you read those two boxplots posts, you can see some of the lessons and information that I’ve gleaned, and in the fall when I teach two sections of this course there could be some significant changes with respect to getting more active work into the class and more passive work outside the class.

Linear Algebra was a delight. This year we increased the credit load of this class from three hours to four, and the extra hour a week has really transformed what we can do with the course. I had a big class of 15 students (that’s big for us), many of whom are as sharp as you’ll find among undergraduates, and all of whom possess a keen sense of humor and a strong work ethic that makes learning a difficult subject quite doable. I’ll be posting later about their application projects and poster session, which were both terrific.

Computer Tools for Problem Solving (aka the MATLAB course) was a tale of two halves of the semester. The first half of the semester was quite a struggle — against a relatively low comfort level around technology with the students and against the students’ expectations for my teaching. But I tried to listen to the students, giving them weekly questionnaires about how the class is going, and engaging in an ongoing dialogue about what we could be doing better. We made some changes to the course on the fly that didn’t dumb the course down but which made the learning objectives and expectations a lot clearer, and they responded extremely well. By the end of the course, I daresay they were having fun with MATLAB. And more importantly, I was receiving reports from my colleagues that those students were using MATLAB spontaneously to do tasks in those courses. That was the main goal of the course for me — get students to the point where they are comfortable and fluent enough with MATLAB that they’ll pull it up and use it effectively without being told to do so. There are some changes I need to make to next year’s offering of the course, but I’m glad to see that the students were able to come out of the course doing what I wanted them to do.

The independent study on finite fields and applications was quite a trip. Andrew Newman, the young man doing the study with me, is one of the brightest young mathematicians with whom I’ve worked in my whole career, and he took on the project with both hands from the very beginning. The idea was to read through parts of Mullen and Mummert to get basic background in finite field theory; then narrow down his reading to a particular application; then dive in deep to that application. Washington’s book on elliptic curves ended up being the primary text, though, and Andrew ended up studying elliptic curve cryptography and the Diffie-Hellman decision problem. Every independent study has a creative project requirement attached, and his was to implement the decision problem in Sage. He’s currently writing up a paper on his research and we hope to get it published in Mathematics Exchange. (Disclaimer: I’m on the editorial board of Math Exchange.) In the middle of the semester, Andrew found out that he’d been accepted into the summer REU on mathematical cryptology at Northern Kentucky University/University of Cincinnati, and he’ll be heading out there in a few weeks to study (probably) multivariate public-key systems for the summer. I’m extremely proud of Andrew and what he’s been able to do this semester — he certainly knows a lot more about finite fields and elliptic curve crypto than I do now.

In between all the teaching, here are some other things I was able to do:

  • Went to the ICTCM in Chicago and presented a couple of papers. Here’s the Prezi for the MATLAB course presentation. Both of those papers are currently being written up for publication in the conference proceedings.
  • Helped with hosting the Indiana MAA spring meetings at our place, and I finished up my three-year term as Student Activities Coordinator by putting together this year’s Indiana College Mathematics Competition.
  • Did a little consulting work, which I can’t really talk about thanks to the NDA I signed.
  • I got a new Macbook Pro thanks to my college’s generous technology grant system. Of course Apple refreshed the Macbook Pro lineup mere weeks later, but them’s the breaks.
  • I’m sure there’s more, but I’ve got finals on the brain right now.

In another post I’ll talk about what’s coming up for me this summer and look ahead to the fall.

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Filed under Abstract algebra, Calculus, Inverted classroom, Life in academia, Linear algebra, Math, MATLAB, Personal, Teaching, Vocation

Boxplots: Curiouser and curiouser

The calculus class took their third (and last) hour-long assessment yesterday. In the spirit of data analytics ala the previous post here, I made boxplots for the different sections of the test (Conceptual Knowledge (CK), Computation (C), and Problem Solving (PS)) as well as the overall scores. Here are the boxplots for this assessment — put side-by-side with the boxplots for the same sections on the previous assessments. “A2” and “A3” mean Assessments 2 and 3.

Obviously there is still a great deal of improvement to be had here — the fact that the class average is still below passing remains unacceptable to me — but there have been some definite gains, particularly in the conceptual knowledge department.

What changed between Assessment 2 and Assessment 3? At least three things:

  • The content changed. Assessment 2 was over derivative rules and applications; Assessment 3 covered integration.
  • The way I treated the content in class changed. Based on the results of Assessment 2, I realized I needed to make conceptual work a much greater part of the class meetings. Previously the class meetings had been about half lecture, with time set aside to work “problems” — meaning, exercises, such as “find the critical numbers of y = xe^{-x}. Those are not really problems that assess conceptual knowledge. So I began to fold in more group work problems that ask students to reason from something other than a calculation. I stressed these problems from the textbook more in class. I tried to include more such problems in WeBWorK assignments — though there are precious few of them to be had.
  • The level of lip service I gave to conceptual problems went up hugely. Every day I was reminding the students of the low scores on Conceptual Knowledge on the test and that the simplest way to boost their grades in the class would be to improve their conceptual knowledge. I did not let their attention leave this issue.

Somewhere in a combination of these three things we have the real reason those scores went up. I tend to think the first point had little to do with it. Integration doesn’t seem inherently any easier to understand conceptually than differentiation, particularly at this stage in the course when differentiation is relatively familiar and integration is brand new. So I think that simply doing more conceptual problems in class and stressing the importance of conceptual knowledge in class were the main cause of the improvements.

Quite interestingly, the students’ scores on computation also improved — despite the reduced presence of computation in class because of the ramped-up levels of conceptual problems. We did fewer computational problems on the board and in group work, and yet their performance on raw computation improved! Again, I don’t think integration is easier than differentiation at this stage in the course, so I don’t think this improvement was because the material got easier. Maybe the last test put the fear of God into them and they started working outside of class more. I don’t know. But this does indicate to me that skill in computation is not strictly proportional to the amount of computation I do, or anybody else does, in class.

To overgeneralize for a second: Increased repetition on conceptual problems improves performance on those problems dramatically, while the corresponding reduction in time spent on computational exercises not only does not harm students’ performance on computation but might actually have something to do with improving it. If we math teachers can understand the implications of this possibility (or at least understand the extent to which this statement is true) we might be on to something big.

The scores on problem solving went two different directions. On the one hand, the median went up; but on the other hand the mean went down. And the middle 50% didn’t get any better on the top end and got worse on the bottom end. I’m still parsing that out. It could be the content itself this time; most of the actual problems in integration tend to take place near the end of the chapter, after the Fundamental Theorem and u-substitution, so the kinds of problems in this section were less than a week old for these students. But quite possibly the improvement in conceptual knowledge brought the median up on problem solving, despite the newness of the problems. Or maybe the differences aren’t even statistically significant.

What I take away from this is that if you want students to do well on non-routine problems, those problems have to occupy a central place in the class, and they have to be done not outside of class where there’s no domain expert to guide the students through them but in class. And likewise, we need not worry so much that we are “wasting precious class time” on group work on conceptual problems at the expense of individual computation skill. Students might do just fine on that stuff regardless, perhaps even better if they have enhanced conceptual understanding to support their computational skills.

It all goes back to support the inverted classroom model which I’ve been using in the MATLAB course, and now I’m wondering about its potential in calculus as well.

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Filed under Calculus, Critical thinking, Education, Inverted classroom, Math, MATLAB, Problem Solving, Teaching

The case of the curious boxplots

I just graded my second hour-long assessment for the Calculus class (yes, I do teach other courses besides MATLAB). I break these assessments up into three sections: Concept Knowledge, where students have to reason from verbal, graphical, or numerical information (24/100 points); Computations, where students do basic context-free symbol-crunching (26/100 points); and Problem Solving, consisting of problems that combine conceptual knowledge and computation (50/100 points). Here’s the Assessment itself. (There was a problem with the very last item — the function doesn’t have an inflection point — but we fixed it and students got extra time because of it.)

Unfortunately the students as a whole did quite poorly. The class average was around a 51%. As has been my practice this semester, I turn to data analysis whenever things go really badly to try and find out what might have happened. I made boxplots for each of the three sections and for the overall scores. The red bars inside the boxplots are the averages for each.

I think there’s some very interesting information in here.

The first thing I noticed was how similar the Calculations and Problem Solving distributions were. Typically students will do significantly better on Calculations than anything else, and the Problem Solving and Concept Knowledge distributions will mirror each other. But this time Calculations and Problem Solving appear to be the same.

But then you ask: Where’s the median in boxplots for these two distributions? The median shows up nicely in the first and fourth plot, but doesn’t appear in the middle two. Well, it turns out that for Calculations, the median and the 75th percentile are equal; while for Problem Solving, the median and the 25th percentile are equal! The middle half of each distribution is between 40 and 65% on each section, but the Calculation middle half is totally top-heavy while the Problem Solving middle half is totally bottom-heavy. Shocking — I guess.

So, clearly conceptual knowledge in general — the ability to reason and draw conclusions from non-computational methods — is a huge concern. That over 75% of the class is scoring less than 60% on a fairly routine conceptual problem is unacceptable. Issues with conceptual knowledge carry over to problem solving. Notice that the average on Conceptual Knowledge is roughly equal to the median on Problem Solving. And problem solving is the main purpose of having students take the course in the first place.

Computation was not as much of an issue for these students because they get tons of repetition with it (although it looks like they could use some more) via WeBWorK problems, which are overwhelmingly oriented towards context-free algebraic calculations. But what kind of repetition and supervised practice do they get with conceptual problems? We do a lot of group work, but it’s not graded. There is still a considerable amount of lecturing going on during the class period as well, and there is not an expectation that when I throw out a conceptual question to the class that it is supposed to be answered by everybody. Students do not spend nearly as much time working on conceptual problems and longer-form contextual problems as they do basic, context-free computation problems.

This has got to change in the class, both for right now — so I don’t end up failing 2/3 of my class — and for the future, so the next class will be better equipped to do calculus taught at a college level. I’m talking with the students tomorrow about the short term. As for the long term, two things come to mind that can help.

  • Clickers. Derek Bruff mentioned this in a Twitter conversation, and I think he’s right — clickers can elicit serious work on conceptual questions and alert me to how students are doing with these kinds of questions before the assessment hits and it’s too late to do anything proactive about it. I’ve been meaning to take the plunge and start using clickers and this might be the right, um, stimulus for it.
  • Inverted classroom. I’m so enthusiastic about how well the inverted classroom model has worked in the MATLAB course that I find myself projecting that model onto everything. But I do think that this model would provide students with the repetition and accountability they need on conceptual work, as well as give me the information about how they’re doing that I need. Set up some podcasts for course lectures for students to listen/watch outside of class; assign WeBWorK to assess the routine computational problems (which would be no change from what we’re doing now); and spend every class doing a graded in-class activity on a conceptual or problem-solving activity. That would take some work and a considerable amount of sales pitching to get students to buy into it, but I think I like what it might become.

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Filed under Calculus, Clickers, Critical thinking, Inverted classroom, Math, Teaching, Technology

MATLAB as a handout creator

One of the fringe benefits of having immersed myself in MATLAB for the last year (in preparation for teaching the Computer Tools for Problem Solving course) is that I’ve learned that MATLAB is an excellent all-purpose tool for preparing materials for my math classes. Here’s an example of something I just finished for a class tomorrow that I’m really pleased with.

I was needing to create a sequence of scatterplots of data for a handout in my Functions and Models class. The data are supposed to have varying degrees of linearity — some perfect/almost perfectly linear, some less so, some totally nonlinear — and having different directions, and the students are supposed to look at the data and rank the correlation coefficients in order of smallest to largest. (This is a standard activity in a statistics class as well.)

I could have just made up data with the right shape on Excel or hand-drawn the scatter plots, but whenever I do that, it looks made it up — not with the randomness that a real set of data, even if it’s strongly linear, would have. So instead, I thought I would take a basic linear function and throw varying degrees of noise into it to make it less linear.

I wrote this little function to generate noise:

function n = noise(degree, size)
n = degree*cos(2*pi*rand(1,size));

This just creates a vector of specified length (“size”) centered roughly around 0, and the bigger “degree” is the more wildly the numbers vary. (I’m sure there’s some built-in way to do this in MATLAB, but it probably took less time to write the function than it would have taken for me, the MATLAB neophyte, to look it up.)

Then I just made four linear functions and literally added in the noise for each, as well as a fifth function that was just spitting out 25 random numbers and a sixth that was a pure linear function with no noise. Then plot all of those in a 2×3 subplot. Here’s the code:

x = 1:25;
y_bigneg = -3*x+90 + noise(5,25);
y_smallneg = -5*x + 100 + noise(30,25);
y_bigpos = 3*x + 3 + noise(3,25);
y_smallpos = 3*x + 3 + noise(10,25);
subplot(2,3,1), scatter(x, y_bigpos)
subplot(2,3,2), scatter(x, y_smallneg)
subplot(2,3,3), scatter(x, y_smallpos)
subplot(2,3,4), scatter(x, y_bigneg)
subplot(2,3,5), scatter(x, rand(1,25))
subplot(2,3,6), scatter(x, 90-4*x)

Here’s the result, after going in and adding titles and removing the legends in the Plot Tools window:

That can then be saved as a PDF and embedded into a \LaTeX document or just posted directly to Moodle for students to play with. All of that code above could easily be compacted into one big M-file with some modifications to let the user control the number of points and whatever else.

This is basic stuff, but it’s awfully handy for creating professional-looking documents and graphics for teaching mathematics. That’s an area where I’m finding MATLAB is highly underrated.

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Calculus reform’s next wave

There’s a discussion going on right now in the Project NExT email list about calculus textbooks, the merits/demerits of the Stewart Calculus textbook, and where — if anywhere — the “next wave” of calculus reform is going to come from. I wrote the following post to the group, and I thought it would serve double-duty fairly well as a blog post. So… here it is:

——-

I’d like to add my $0.02 worth to this discussion just because (1) I’m a longtime Stewart Calculus user, having used the first edition (!) when I was an undergrad and having taught out of it for my entire career, and (2) I’m also a fairly consistent critic of Stewart’s calculus and of textbooks in general.

I try to see textbooks from the viewpoints of my students. From that vantage point, I unfortunately find very little to say in favor of Stewart’s franchise of  books, including the current edition, all of the previous five editions, the CCC version (which is almost exactly the same as the non-reform version of the book but with less clarity in its language), or the “Essential” calculus edition. Stewart has a relentlessly formalistic approach to calculus that, while admirable in its rigor, renders it all but impenetrable to students who are not used to such an approach, which is certainly nearly every student I teach and I would imagine a large portion of the entire population of beginning calculus students.

If you don’t believe me, go check out his introductory section on the definite integral (Section 5.2 in the sixth edition). Stewart hopelessly confuses the essentially very simple idea of the definite integral by hitting students with an avalanche of sigma-notation right out of the gate. Or, try the section on exponential functions (1.5), in which Stewart for some reason feels like it’s necessary to explain how it is we can define an exponential function at rational and irrational inputs. This is all well and good, but does the rank-and-file beginning calculus student need to know this stuff, right now?

As a result, I find myself having to tell students NOT to read certain portions of the book, and then remixing and rewriting large parts of the rest of it. But that leads to the ONE thing I can say in the positive sense about Stewart, which I can’t say about many “reform” books: Stewart is what you make it. The book does not force me to teach in a certain way, and if I want to totally ignore certain parts of it and write my own stuff, then this generally doesn’t cause problems down the road. For example, at my college we don’t cover trigonometry in the first semester. In most other books we’ve examined, trig and calculus are inextricable, and so the books are unusable for us. With Stewart, though, given a judicious choice of exercises to omit, you can actually pull off a no-trig Calculus I course with very little extra work on the prof’s part.

I can also say, regarding Stewart CCC, that the ancillary materials are excellent. The big binder of group exercises that comes with the instructor edition is much better than the book itself.

I don’t think that I have yet seen a calculus book that is really fundamentally different from the entire corpus of calculus textbooks, with the possible exception of Hughes & Hallett. They all cover the same topics in the same order, more or less, and in the same ways. If you’re looking for the next wave of calculus reform, therefore, you’ll have to find it outside the confines of a textbook, or at least the textbooks that are currently on the market. Textbooks almost by definition are antithetical to reform. Perhaps real reform will come with the rejection of textbooks as authoritative oracles on the subject in the first place. That could mean designing courses with no centralized information source, or using “inverted classroom” models utilizing online resources like the videos at Khan Academy (http://www.khanacademy.org) or iTunesU, or some combination of these.

Actually, more likely the next wave of reform will be in the form of reconsidering the place of calculus altogether, as the CUPM project did several years ago. Is it perhaps time to think about replacing calculus with a linear combination (pardon the pun) of statistics, discrete math, and linear algebra as the freshman introduction to college mathematics, or at least letting students choose between calculus and this stat/discrete/linear track? Is calculus really the best possible course for freshmen to take? I think that’s a discussion worth having, or reopening.

Enjoy,
Robert Talbert
Franklin College
Peach Dot 1997-1998

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