# Tag Archives: Calculus

29 March 2011 · 4:25 pm

## Five questions I haven’t been able to answer yet about the inverted classroom

Between the Salman Khan TED talk I posted yesterday and several talks I saw at the ICTCM a couple of weeks ago, it seems like the inverted classroom idea is picking up some steam. I’m eager myself to do more with it. But I have to admit there are at least five questions that I have about this method, the answers to which I haven’t figured out yet.

1. How do you get students on board with this idea who are convinced that if the teacher isn’t lecturing, the teacher isn’t teaching? For that matter, how do you get ANYBODY on board who are similarly convinced?

Because not all students are convinced the inverted classroom approach is a good idea or that it even makes sense. Like I said before, the single biggest point of resistance to the inverted classroom in my experience is that vocal group of students who think that no lecture = no teaching. You have to convince that group that what’s important is what (and whether) they are learning, as opposed to my choices for instructional modes, but how?

2. Which is better: To make your own videos for the course, or to use another person’s videos even if they are of a better technical or pedagogical quality? (Or can the two be effectively mixed?)

There’s actually a bigger question behind this, and it’s the one people always ask when I talk about the inverted classroom: How much time is this going to take me? On the one hand, I can use Khan Academy or iTunesU stuff just off the rack and save myself a ton of time. On the other hand, I run the risk of appearing lazy to my students (maybe that really would be being lazy) or not connecting with them, or using pre-made materials that don’t suit my audience. I spend 6-12 hours a week just on the MATLAB class’ screencasts and would love (LOVE) to have a suitable off-the-shelf resource to use instead. But how would students respond, both emotionally and pedagogically?

3. Can the inverted classroom be employed in a class on a targeted basis — that is, for one or a handful of topics — or does it really only work on an all-or-nothing basis where the entire course is inverted?

I’ve tried the former approach, to teach least-squares solution methods in linear algebra and to do precalculus review in calculus. In the linear algebra class it was successful; in calculus it was a massive flop. On some level I’m beginning to think that you have to go all in with the inverted classroom or students will not feel the accountability for getting the out-of-class work done. At the very least, it seems that the inverted portions of the class have to be very distinct from the others — with their own grading structure and so on. But I don’t know.

4. Does the inverted classroom model fit in situations where you have multiple sections of the same course running simultaneously?

For example, if a university has 10 sections of calculus running in the Fall, is it feasible — or smart — for one instructor to run her class inverted while the other nine don’t? Would it need to be, again, an all-or-nothing situation where either everybody inverts or nobody does, in order to really work? I could definitely see me teaching one or two sections of calculus in the inverted mode, with a colleague teaching two other sections in traditional mode, and students who fall under the heading described in question #1 would wonder how they managed to sign up for such a cockamamie way of “teaching” the subject, and demand a transfer or something. When there’s only one section, or one prof teaching all sections of a class, this doesn’t come up. But that’s a relatively small portion of the full-time equivalent student population in a math department.

5. At what point does an inverted classroom course become a hybrid course?

This matters for some instructors who teach in institutions where hybrid, fully online, and traditional courses have different fee structures, office hours expectations, and so on. This question raises ugly institutional assumptions about student learning in general. For example, I had a Twitter exchange recently with a community college prof whose institution mandates that a certain percentage of the content must be “delivered” in the classroom before it becomes a “hybrid” course. So, the purpose of the classroom is to deliver content? What happens if the students don’t “get” the content in class? Has the content been “delivered”? That’s a very 1950′s-era understanding of what education is supposedly about. But it’s also the reality of the workplaces of a lot of people interested in this idea, so you have to think about it.

Got any ideas on these questions?

20 Comments

Filed under Education, Inverted classroom, Life in academia, Teaching

Tagged as Calculus, Classroom, Education, Inverted classroom, khan academy, Linear algebra, matlab, Salman Khan, student

16 December 2010 · 2:30 pm

## 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.

4 Comments

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

Tagged as Calculus, Critical thinking, High School Math, integral, Math, Numerical integration, Population, student

29 November 2010 · 9:00 am

## 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.

11 Comments

Filed under Calculus, Critical thinking, Education, Higher ed, Math, Peer instruction, Problem Solving, Teaching

Tagged as Calculus, Education, Math, mathematics, Peer instruction, Problem Solving

15 November 2010 · 9:17 pm

## Technology FAIL day

This morning as I was driving in to work, I got to thinking: Could I teach my courses without all the technology I use? As in, just me, my students, and a chalk/whiteboard with chalk/markers? As I pulled in to the college, I thought: Sure I could. It just wouldn’t be as good or fun without the tech.

Little did I know, today would be centered around living that theory out:

• I planned a Keynote presentation with clicker questions to teach the section on antiderivatives in Calculus. As soon as I tried to get the clickers going, I realized the little USB receiver wasn’t working. Turns out, updating Mac OS X to v10.6.5 breaks the software that runs the receiver. Clicker questions for this morning: Out the window. Hopefully I’ll find a useable laptop for tomorrow, when I’m using even more clicker questions.
• Also in calculus, the laptop inexplicably went into presenter mode when I tried to give the presentation without clicker questions. Most of the time when I try to get it into presenter mode, I can’t do it. This time I couldn’t make it stop.
• The Twitter client on my laptop got stuck in some kind of strange mode such that clicking on anything made it go to Expose.
• I lost the network connection to our department printer halfway through the day.
• GMail went down.

Fortunately everything I had planned could be done without any technology aside from the whiteboard. But when the technology doesn’t work, I have to improvise, and sometimes that works well and sometimes not. In calculus, I just had to revert back to what is often called the “interactive lecture”, which means just a regular lecture where you hope the students ask questions, and it was about as engaging as that sounds.

I do believe I can teach without all this technology, but the kind of teaching I do with the technology is, I think, more inherently engaging and meaningful for students. I ask better questions, interact more freely with students, and highlight the coherence and the big ideas of the material more adeptly with the technology in place. So when the tech fails on me, things seem odd and out of place and contrived. Students pick up on that. Maybe I’m simply addicted to the tech, but I don’t like teaching without it, and my classes aren’t nearly at the same level without it.

3 Comments

Filed under Educational technology, Life in academia, Math, Profhacks, Teaching, Technology

Tagged as Calculus, Classroom response systems, Clickers, ed tech, Educational technology, Math, Technology, twitter

5 September 2010 · 1:37 pm

## 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.

Comments Off

Filed under Calculus, Education, Educational technology, Math, Peer instruction, Screencasts, Teaching

Tagged as Calculus, Education, Function (mathematics), High School Math, Math, Screencast, stewart, youtube

21 August 2010 · 6:50 pm

## 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.

Comments Off

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

Tagged as Calculus, Calculus III, Contour map, Math, matlab, Multivariable calculus, Screencast, video, visualization, youtube

17 August 2010 · 4:01 pm

## 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.

Now, however, problems are completely different and cannot be so easily encapsulated. I can again pull an example from my Dad’s work history. During the last decade of his career, the Houston Oilers NFL franchise moved to Tennessee. Dad was employed by the Nashville Electric Service and the problem he was handed was: Design the power grid for the new Oilers stadium. This problem has some similarities with designing the navigational circuitry for a space capsule, but there are major differences as well because this was a civic project as well as a technical one. How do we make the power supply lines work with the existing road and building configurations? What about surrounding businesses and the impact that the design will have on them? How do we make Bud Adams happy with what we’ve done? The problem quickly overruns any simple categorization, and it required that Dad not only use skills other than those he learned in his (very rigorous!) EE curriculum at Texas Tech University, but also to learn new skills on the fly and to work with other non-engineers who have more in the way of those skills than he had. Also, the methods use to solve the problem were radically different. You can’t design a power grid that large using hand tools; you have to use computers, and computers need alternative representations of the models underlying the design. And the methods themselves lead to new problems.

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.

6 Comments

Filed under Calculus, Education, Educational technology, Engineering, Engineering education, Higher ed, Math, Problem Solving, Teaching, Technology

Tagged as Bud Adams, Calculus, Education, Engineering, Engineering education, Innovation, Math, mathematics, Newton, Technology

13 August 2010 · 1:21 pm

## 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.

Comments Off

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

Tagged as Calculus, camtasia, Math, matlab, Multivariable calculus, Screencast, screencasting, Technology

8 August 2010 · 12:47 pm

## Calculus and conceptual frameworks

Image via http://www.flickr.com/photos/loopzilla/

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?

9 Comments

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

Tagged as Calculus, Conceptual framework, Education, learning, math education, mathematics, precalculus

15 May 2010 · 11:41 am

## The semester in review

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.

Comments Off

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

Tagged as Add new tag, Calculus, Education, Linear algebra, Math, mathematics, matlab, precalculus, research

• ### Robert Talbert

Here's my personal website.

• ### Subscribe

• I'm also part of the blogging team at:
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 13, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9566676616668701, "perplexity_flag": "middle"}