# Tag Archives: precalculus

## 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?

## 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 on The semester in review

## 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.

1 Comment

Filed under Calculus, Math, MATLAB, Problem Solving, Teaching, Technology

## Average velocity

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

$\frac{s(b) - s(a)}{b-a}$

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

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

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

What’s with this topic? Anyone?

Filed under Calculus, Math, Teaching

## Flowchart for data modeling

I just made this flowchart for the Calculus Preparation lesson coming up next Monday, which is on how to model nonlinear data. Click to enlarge:

I <heart> OmniGraffle. It’s one of my top ten e-learning tools for good reason.
The PDF version of this is in my Box.net widget — go to the sidebar and scroll down. Feel free to download and use/share.

Filed under Educational technology, Math, Software

## Age-old questions about freshman math students

Midterms are coming up in a couple of weeks, and while most of the students in my precalculus class are doing reasonably well, some aren’t. Here are some questions I’ve struggled with every time I teach a freshman class, and maybe some of you out there have suggestions. If so, leave them in the comments.

1. How do you impress upon students (freshmen) the importance of coming to office hours? I don’t think I’ve had more than six distinct students visit office hours for help all semester long, and I’d consider this an active semester in terms of office hours. The rest go to the Math Study Center, study tables for football or fraternities, etc. but it does no evident good for a lot of them. I think it would do them good to come see me; but how to convince them of this?
2. How do you convince a student that their purpose for being here, their job, is to be a student? Some of the students don’t come to office hours because they haven’t touched the exercises all semester long, and that’s because they are involved in several different campus activities which are promoted in the name of “getting involved”. The cost to their time budget is unsustainable. How to get them to prioritize time properly?
3. How do you get students to transition from the typical high-school-math mode of “get the answer in the shortest possible time frame” to the college mode of “work hard over an extended period to really understand what you are doing”?

Thoughts?
[Photo by kodama.]