# Category Archives: Calculus

## Speaking of the inverted classroom

On Wednesday, I gave a talk at Indiana University – Purdue Universty – Indianapolis (IUPUI, for short) to the teaching seminar for math graduate students on the inverted classroom. It was sort of a generalization of the talk I gave on the inverted linear algebra classroom back at the Joint Mathematics Meetings in January. Carl Cowen was in attendance at that talk and invited me to make the 20-minute drive from my house to IUPUI to do something like it, and I was happy to oblige.

Since putting the talk up on Slideshare yesterday morning, it’s gotten over 200 views, 2 favorites, a handful of retweets/Facebook likes, and is currently being highlighted on Slideshare’s Education page. So I thought I would share it here as well. Enjoy and ask questions!

## Coming up in January

Fall Semester 2010 is in the books, and I’m heading into an extended holiday break with the family. Rather than not blog at all for the next couple of weeks, I’ll be posting (possibly auto-posting) some short items that take a look back at the semester just ended — it was a very eventful one from a teaching standpoint — and a look ahead and what’s coming up in 2011.

I’ll start with the look head to January 2011. We have a January term at my school, and thanks to my membership on the Promotion and Tenure Committee — which does all its review work during January — I’ve been exempt from teaching during Winter Term since 2006 when I was elected to the committee. This year I am on a subcommittee with only three files to review, so I have a relatively luxurious amount of time before Spring semester gets cranked up in February. A time, that is, which is immediately gobbled up by the following:

• I’ll be at the Joint Mathematics Meetings in New Orleans from January 6–9. This will be my first trip to the Joint Meetings since 2002, and I’m pretty excited about it. I will be giving two talks, one in the MAA Session on Undergraduate Cryptology (PDF) about my five-day micro-unit on cryptology for freshmen and the other in the MAA Session on Innovative and Effective Ways to Teach Linear Algebra (PDF) on experimenting with the inverted classroom model in linear algebra. Both of those sessions are loaded with interesting-sounding talks, so I hope to attend the entire session. I also hope to catch up with friends I haven’t seen since, well, 2002 — and maybe connect with some new ones. If you’re attending, let me know!
• The second iteration of the MATLAB course is coming up in the spring as well, and I will be doing some significant redesign work on it based on experiences and data from the first iteration. I’m constantly humbled and gratified by the interest and positive responses that the course has generated in the MATLAB community and elsewhere — and by how much interest and attention the course has received. I’ve had a chance to observe and talk to the alumni from the first run of the course during their Calculus III course that used MATLAB significantly, and their usage habits and feedback have given me some ideas for what should be positive changes in the course. I’ll elaborate on that later.
• I am teaching Linear Algebra again in the spring, as I have done for the last 4-5 years, and this year I am targeting that course for a more robust implementation of inverted classroom techniques. A lot of the students in that course will be MATLAB course alumni, so they will be used to all that inversion. But I’ve had enough experience with peer instruction and classroom response system (“clicker”) use on the one hand from this past semester (which I never blogged about, and I’ll try to remedy that) and inverted classroom approaches in MATLAB on the other that Linear Algebra seems well-positioned to benefit from a combination of these approaches. I’ll be sketching out and planning the course in January.
• Like I said, I used a lot of peer instruction and clickers in calculus this semester with great success (I think; at least the students say so). I’m teaching two more sections of calculus in the spring and will be refining my teaching using these tools. But calculus in the spring has a different flavor than calculus in the fall, so we will see how it goes.
• What I’m reading this January: Teaching with Classroom Response Systems by Derek Bruff; Learning to Solve Problems by David Jonassen; The Craft of Research by Booth, Colomb, and Williams; and catching up on a mountain of articles that accumulated during the semester.
• I’m also reading Geometry and Symmetry by Kinsey, Moore, and Prassidis leading up to an MAA review of the book. The “Prassidis” in the author list is Stratos Prassidis, who was my Ph.D. dissertation advisor.

Throw a couple of consulting projects on top of all that, and you’ve got yourself a busy January!

## A problem with “problems”

I have a bone to pick with problems like the following, which is taken from a major university-level calculus textbook. Read it, and see if you can figure out what I mean.

This is located in the latter one-fourth of a review set for the chapter on integration. Its position in the set suggests it is less routine, less rote than one of the early problems. But what’s wrong with this problem is that it’s not a problem at all. It’s an exercise. The difference between the two is enormous. To risk oversimplifying, in an exercise, the person doing the exercise knows exactly what to do at the very beginning to obtain the information being requested. In a problem, the person doesn’t. What makes an exercise an exercise is its familiarity and congruity with prior exercises. What makes a problem a problem is the lack of these things.

The above is not a problem, it is an exercise. Use the Midpoint Rule with six subintervals from 0 to 24. That’s the only part of the statement that you even have to read! The rest of it has absolutely nothing with bees, the rate of their population growth, or the net amount of population growth. A student might be turning this in to an instructor who takes off points for incorrect or missing units, and then you have to think about bees and time. Otherwise, this exercise is pure pseudocontext.

Worst of all, this exercise might correctly assess students’ abilities to execute a numerical integration algorithm, but it doesn’t come close to measuring whether a student understands what an integral is in the first place and why we are even bringing them up. Even if the student realizes an integral should be used, there’s no discussion of how to choose which method and which parameters within the method, or why. Instead, the exercise flatly tells students not only to use an integral, but what method to use and even how many subdivisions. A student can get a 100% correct answer and have no earthly idea what integration has to do with the question.

A simple fix to the problem statement will change this into a problem. Keep the graph the same and change the text to:

The graph below shows the rate at which a population of honeybees was growing, in bees per week. By about how many bees did the population grow after 24 weeks?

This still may not be a full-blown problem yet — and it’s still pretty pseudocontextual, and the student can guess there should be an integral happening because it’s in the review section for the chapter on integration —  but at least now we have to think a lot harder about what to do, and the questions we have to answer are better. How do I get a total change when I’m given a rate? Why can’t I just find the height of the graph at 24? And once we realize that we have to use an integral — and being able to make that realization is one of the main learning objectives of this chapter, or at least it should be — there are more questions. Can I do this with an antiderivative? Can I use geometry in some way? Should I use the Midpoint Rule or some other method? Can I get by with, say, six rectangles? or four? or even two? Why not use 24, or 2400? Is it OK just the guesstimate the area by counting boxes?

I think we who teach calculus and those who write calculus books must do a better job of giving problems to students and not just increasingly complicated exercises. It’s very easy to do so; we just have to give less information and fewer artificial cues to students, and force students to think hard and critically about their tools and how to select the right combination of tools for the job. No doubt, this makes grading harder, but students aren’t going to learn calculus in any real or lasting sense if they don’t grapple with these kinds of problems.

Filed under Calculus, Critical thinking, Math, Problem Solving, Teaching

## What correlates with problem solving skill?

About a year ago, I started partitioning up my Calculus tests into three sections: Concepts, Mechanics, and Problem Solving. The point values for each are 25, 25, and 50 respectively. The Concepts items are intended to be ones where no calculations are to be performed; instead students answer questions, interpret meanings of results, and draw conclusions based only on graphs, tables, or verbal descriptions. The Mechanics items are just straight-up calculations with no context, like “take the derivative of $y = \sqrt{x^2 + 1}$“. The Problem-Solving items are a mix of conceptual and mechanical tasks and can be either instances of things the students have seen before (e.g. optimzation or related rates problems) or some novel situation that is related to, but not identical to, the things they’ve done on homework and so on.

I did this to stress to students that the main goal of taking a calculus class is to learn how to solve problems effectively, and that conceptual mastery and mechanical mastery, while different from and to some extent independent of each other, both flow into mastery of problem-solving like tributaries to a river. It also helps me identify specific areas of improvement; if the class’ Mechanics average is high but the Concepts average is low, it tells me we need to work more on Concepts.

I just gave my third (of four) tests to my two sections of Calculus, and for the first time I started paying attention to the relationships between the scores on each section, and it felt like there were some interesting relationships happening between the sections of the test. So I decided to do not only my usual boxplot analysis of the individual parts but to make three scatter plots, pairing off Mechanics vs. Concepts, Problem Solving vs. Concepts, and Mechanics vs. Problem Solving, and look for trends.

Here’s the plot for Mechanics vs. Concepts:

That r-value of 0.6155 is statistically significant at the 0.01 level. Likewise, here’s Problem Solving vs. Concepts:

The r-value here of 0.5570 is obviously less than the first one, but it’s still statistically significant at the 0.01 level.

But check out the Problem Solving vs. Mechanics plot:

There’s a slight upward trend, but it looks disarrayed; and in fact the r = 0.3911 is significant only at the 0.05 level.

What all this suggests is that there is a stronger relationship between conceptual knowledge and mechanics, and between conceptual knowledge and problem solving skill, than there is between mechanical mastery and problem solving skill. In other words, while there appears to be some positive relationship between the ability simply to calculate and the ability to solve problems that involve calculation (are we clear on the difference between those two things?), the relationship between the ability to answer calculus questions involving no calculation and the ability to solve problems that do involve calculation is stronger — and so is the relationship between no-calculation problems and the ability to calculate, which seems really counterintuitive.

If this relationship holds in general — and I think that it does, and I’m not the only one — then clearly the environment most likely to teach calculus students how to be effective problem solvers is not the classroom primarily focused on computation. A healthy, interacting mixture of conceptual and mechanical work — with a primary emphasis on conceptual understanding — would seem to be what we need instead. The fact that this kind of environment stands in stark contrast to the typical calculus experience (both in the way we run our classes and the pedagogy implied in the books we choose) is something well worth considering.

## This week in screencasting: Optimization-palooza

My calculus class hit optimization problems this week — or it might be better to say the class got hit by optimization problems. These are tough problems because of all their many moving parts, especially the fact that one of those parts is to build the model you plan to optimize. Most of my students have had calculus in high school, but too many calculus courses in high school as well as college focus almost primarily on algorithms for computation and spend little to no time with how to create a model in the first place. Classes that are so structured are doing massive harm to students in a number of ways, but that’s for another post or two.

Careful study of worked-out examples is an essential part of understanding optimization problems (though not the only part, and this alone isn’t sufficient). The textbook has a few of these. The professor can provide more, but class time really isn’t best spent just by having the professor put examples on the board. Class time should also be spent working on optimization problems with the professor there to provide guidance. And since I can’t spend 8-10 class days both working examples and giving students time to work themselves, screencasts on optimization problems have been the obvious solution.

This week I did screencasts for four problems. Here they are (one problem needed two screencasts):

To my students’ great credit, they have embraced YouTube as a great source of help in calculus. They’ve utilized not only these screencasts but many other ones, most of them excellently produced, and now doing a search on YouTube is an essential component of studying for many of them. I think that’s a great approach, obviously.

## This week in screencasting: The polar express

It’s been a little quiet on the screencasting front lately, but in the next couple of weeks my colleague teaching Calculus III will be hitting material for which I volunteered to provide some content: namely, using MATLAB to visualize some of the surfaces and solids used in multiple integration. Yesterday, I finished two of these. The first on is on polar coordinates and polar function plotting in MATLAB:

And the second one is on cylindrical coordinates and plotting two-variable functions in cylindrical coordinates:

MATLAB doesn’t provide a built-in function for plotting in cylindrical coordinates. Instead — and this is either ingenious or annoying depending on how you look at it — to plot something in cylindrical coordinates, you generate all the points you need in cylindrical coordinates and then use the pol2cart function to convert them en masse to cartesian coordinates, then plot the whole thing as usual in cartesian coordinates.

I think this is smart, since by avoiding the use of a specialized function for cylindrical plots and sticking instead to a single command for 3D plotting, you learn one command for all 3D plots and you get to use all the extras available, such as adding a contour plot onto the cylindrical plot. Overloading the pol2cart function so that it can accept and produce the third coordinate makes this all work. Overall I like how MATLAB doesn’t try to make a function for everything but rather creates a well-featured set of relatively simple tools that will do lots of things.

But I can see where some people — especially MATLAB novices — would find all this annoying, since the entire process takes several steps. There’s a workflow diagram for doing this in the screencast, but a better way is to make an M-file that holds all the steps. Here’s the one I flashed briefly at the end of the screencast:


% Script for plotting a cylindrical function.
% Written by Robert Talbert, Ph.D., 10/20/2010

% Theta: Change t1 and t2 to set the starting and ending values for theta.
t1 = 0;
t2 = pi/2;
theta = linspace(t1, t2);

% r: : Change r1 and r2 to set the starting and ending values for theta.
r1 = -5;
r2 = 5;
r = linspace(r1, r2);

% Create meshgrid for inputs:
[theta, r] = meshgrid(theta, r);

% Apply the function to create a matrix of z-values. Change the function to
% match what you want to plot.

z = r*cos(theta);

% Convert to cartesian and plot using mesh:

[x,y,z] = pol2cart(theta, r, z);
mesh(x,y,z)


It would be simple enough to modify this so that it’s a function rather than a script, accepting the arrays theta and r and a function handle for z, and then producing the 3D plot. Or, one could even make an “ez” version where the user just enters a string containing the function s/he wants. If somebody wants to try that out, and you want to share your results, just put the source code in the comments.

The third one in this series will be up later this weekend. It’s on spherical coordinates and it’s pretty much the same process, only using sph2cart instead of pol2cart. There might be a fourth one as well, dealing with some special cases like constant cylindrical/spherical functions (you can’t just say “rho = 5”, because rho has to be a matrix) and how to plot not just the surfaces but the volumes underneath them.

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Filed under Calculus, Engineering, Math, MATLAB, Teaching, Technology

## This week (and last) in screencasting: Functions!

So we started  back to classes this past week, and getting ready has demanded much of my time and blogging capabilities. But I did get some new screencasts done. I finished the series of screencasts I was making for our calculus students to prepare for Mastery Exams, a series of short untimed quizzes over precalculus material that students have to pass with a 100% score. But then I turned around and did some more for my two sections of calculus on functions. There were three of them. The first one covers what a function is, and how we can work with them as formulas:

The second one continues with functions as graphs, tables, and verbal descriptions:

And this third one is all on domain and range:

The reason I made these was because we were doing the first section of the Stewart calculus book in one day of class. If you know this book, you realize this is impossible because there is an enormous amount of stuff crammed into this one section. Two items covered in that section are how to calculate and reduce the difference quotient $\frac{f(a+h) - f(a)}{h}$ and doing word problems. Each of these topics alone can cover multiple class meetings, since many students are historically rusty or just plain bad at manipulating formulas correctly and suffer instantaneous brain-lock when put into the presence of a word problem. So, my thought was to go all Eric Mazur on them and farm out the material that is most likely to be easy review for them as an outside “reading” assignment, and spend the time in class on the stuff that on which they were most likely to need serious help.

Our first class was last Tuesday and the second class wasn’t until Thursday, so I assigned the three videos and three related exercises from the Stewart book for Thursday, along with instructions to email questions on any of this, or post to our Moodle discussion board. I made up some clicker questions that we used to assess their grasp of the material in these videos, and guess what? Many students didn’t have any problems at all with this material, and those who did got their issues straightened out through discussions with other students as part of the clicker activity.

They’ll be assessed in 2 or 3 other ways on this stuff this week to make sure they really have the material down and are not just being shy about not having it. But it looks like using screencasts to motivate student contact with the material outside of class worked fine, at least as effectively as me lecturing over it. And we had more time for the hard stuff that I wouldn’t expect students to be able to handle, not all of them anyway.

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## This week in screencasting: Contour plots in MATLAB

By my count, this past week I produced and posted 22 different screencasts to YouTube! Almost all of those are short instructional videos for our calculus students taking Mastery Exams on precalculus material. But I did make two more MATLAB-oriented screencasts, like last week. These focus on creating contour plots in MATLAB.

Here’s Part 1:

And Part 2:

I found this topic really interesting and fun to screencast about. Contour plots are so useful and simple to understand — anybody who’s ever hiked or camped has probably used one, in the form of a topographical map — and it was fun to explore the eight (!) different commands that MATLAB has for producing them, each command producing a map that fits a different kind of need. There may be even more commands for contour maps that I’m missing.

I probably won’t match this week’s output next week, as I’ll be on the road in Madison, WI on Monday and Tuesday and there are several faculty meetings in the run-up to the start of the semester. But at the very least, I need to go back and do another two-variable function plot screencast because I inexplicably left off surface plots and the EZMESH and EZSURF commands on last week’s screencasts.

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Filed under Calculus, Educational technology, Math, MATLAB, Screencasts, Technology

## Why change how we teach?

Sometimes when I read or hear discussions of innovation or change in teaching mathematics or other STEM disciplines, whether it’s me or somebody else doing the discussing, inevitably there’s the following response:

What do we need all that change for? After all, calculus [or whatever] hasn’t changed that much in 400 years, has it?

I’m not a historian of mathematics, so I can’t say how much calculus has or hasn’t changed since the times of Newton and Leibniz or even Euler. But I can say that the context in which calculus is situated has changed — utterly. And it’s those changes that surround calculus that are forcing the teaching of calculus (any many other STEM subjects) to change –radically.

What are those changes?

First, the practical problems that need to be solved and the methods used to solve them have changed. Not too long ago, practical problems could be neatly compartmentalized and solved using a very small palette of methods. I know some things about those problems from my Dad, who was an electrical engineer for 40 years and was with NASA during the Gemini and Apollo projects. The kinds of problems he’d get were: Design a circuit board for use in the navigational system of the space capsule. While this was a difficult problem that needed trained specialists, it was unambiguous and could be solved with more or less a subset of the average undergraduate electrical engineering curriculum content, plus human ingenuity. And for the most part, the math was done by hand and on slide rules (with a smattering of newfangled mechanical calculators) and the design was done with stuff from a lab — in other words, standard methods and tools for engineers.

So it is with calculus or almost any STEM discipline these days. Students today will not go on to work with simple, cleanly-defined, well-posed problems that fit neatly into a box. Nor will they be always doing things by hand; they will be using technology to solve problems, and this requires both a different way of representing the models (for calculus, think “functions”) they use and the flexibility to anticipate the problems that the methods themselves create. This is not what Newton or Leibniz had in mind, but it is the way things are. Our teaching must therefore change to give students a fighting chance at solving these problems, by emphasizing multiple representations of functions, multiple methods for solution of problems, and attention to the problems created by the methods. And of course, we also must focus on teaching problem-solving itself and on the ability to acquire new skills and information independently, because if so much has changed between 1965 and 1995, we can expect about the same amount of change in progressively shorter time spans in the future.

Also, the people who solve these problems, and what we know about how those people learn, have changed. It seems undeniable that college students are different than they were even 20 years ago, much less 200 years ago. Although they may not be natively fluent in the use of technology, they are certainly steeped in technology, and technology is a primary means for how they interact with the rest of the world. Students in college today bring a different set of values, a different cultural context, and a different outlook to their lives and how they learn. This executive summary of research done by the Pew Research Foundation goes into detail on the characteristics of the Millenial generation, and the full report (PDF, 1.3 Mb) — in addition to our own experiences — highlights the differences in this generation versus previous ones. These folks are not the same people we taught in 1995; we therefore cannot expect to teach them in the same way and expect equal or better results.

We also know a lot more now about how people in general, and Millenials in particular, learn things than we did just a few years ago. We are gradually, but also rapidly, realizing through rigorous education research that there are other methods of teaching out there besides lecture and that these methods work better than lecture does in many situations. Instructors are honing the research findings into usable tools through innovative classroom practices that yield statistically verifiable improvements over more traditional ways of teaching. Even traditional modes of teaching are finding willing and helpful partners in various technological tools that lend themselves well to classroom use and student learning. And that technology is improving in cost, accessibility, and performance at an exponential pace, to the point where it just doesn’t make sense not to use it or think about ways teaching can be improved through its use.

Finally, and perhaps at the root of the first two, the culture in which these problems, methods, people, and even the mathematics itself is situated has changed. Technology drives much of this culture. Millenials are highly connected to each other and the world around them and have little patience — for better or worse — for the usual linear, abstracted, and (let’s face it) slow ways in which calculus and other STEM subjects are usually presented. The countercultural force that tends to discourage kids from getting into STEM disciplines early on is probably stronger today than it has ever been, and it seems foolish to try to fight that force with the way STEM disciplines have been presented to students in the past.

Millenials are interested to a (perhaps) surprising degree in making the world a better place, which means they are a lot more interested in solving problems and helping people than they are with epsilon-delta definitions and deriving integrals from summation rules. The globalized economy and highly-connected world in which we all live has made almost every problem worth solving multidisciplinary. There is a much higher premium now placed on getting a list of viable solutions to a problem within a brief time span, as opposed to a single, perfectly right answer within an unlimited time span (or in the time span of a timed exam).

Even mathematics itself has a different sort of culture now than it did even just ten years ago. We are seeing the emergence of massively collaborative mathematical research via social media, the rise of computational proofs from controversy to standard practice, and computational science taking a central role among the important scientific questions of our time. Calculus may not have changed much but its role in the larger mathematical enterprise has evolved, just in the last 10-15 years.

In short, everything that lends itself to the creation of meaning in the world today — that is, today’s culture — has changed from what it used to be. Even the things that remain essentially unchanged from their previous states, like calculus, must fit into a context that has changed.

All this change presents challenges and opportunities for STEM educators. It’s challenging to go back to calculus, and other STEM disciplines, and think about things like: What are the essential elements of this subject that really need to be taught, as opposed to just the topics we really like? What new facets or topics need to be factored in? What’s the best way to factor those in, so that students are really prepared to function in the world past college? And, maybe most importantly, How do we know our students are really prepared? There’s a temptation to burrow back in to what worked for us, when faced with such daunting challenges, but that really doesn’t help students much — nor does it tap into the possibilities of making our subjects, and our students, richer.

## 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