# 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?

### 9 responses to “Calculus and conceptual frameworks”

1. I think there is a major mismatch between the frameworks we often create in earlier math classes, and what the students run into in calculus.

There is plenty of opportunity to develop rules, laws, formulae, rather than just present them. But that requires earlier courses that “waste” time dwelling on where the math comes from, instead of just on how to get the right answer.

We are further squeezed by the newer standardized test driven standards movement, where multiple choice is a way of life.

Jonathan

• jd2718 – good comment! Your middle paragraph about “plenty of opportunity to develop…” seems to indicate that what?… opportunity is being wasted? Really, ideas and relationships should be illustrated and derived in all Mathematics courses. Those need to happen somewhere, either in the textbooks, or in lecture, ideally in both.

2. Justin

I have to disagree with your statement that “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.” As often as not, such students have a wrong framework that they’ve developed over many years of hacking and trial-and-error programming. When they’re presented with a proper framework in an intro CS class, they have more trouble than inexperienced students because the inexperienced students are blank slates and don’t have years worth of bad habits that prevent them from approaching programing properly. I know a number of schools whose CS departments won’t give AP credit because they want to make sure that students go through a (maybe accelerated) course that will present the material using the right frameworks.

3. @Justin, I think my main point here is that students are not blank slates. Even those with zero programming experience have some kind of conceptual framework about computing they bring to bear on a class. In the MATLAB class, for instance, students’ conception of variable declaration was strongly inherited from their experiences with graphing calculators and spreadsheets — if they declared x := 3, then y := x.^2, and then changed x to 4, they expected the number stored in y to be 16. Getting students to the right conceptual framework in this basic concept was a big hurdle. Students always bring something to a new concept, even if it’s not old, bad habits in implementing that concept.

Which isn’t to say your point isn’t any good. Certainly kids who have been coding for a long time have some things in their conceptual framework that are right and some that are way off.

4. Justin

I don’t mean to hijack this thread by talking more about CS than math, but there are definite parallels here. As an example, assignment statements are actually a complex and advanced topic and are fraught with peril, as your example shows. Students are actually better off not being introduced to assignment statements until they have the proper conceptual framework to understand when and how to use them. Unfortunately, most common programming languages are built around assignments, and as a result, they get introduced to students before they are ready. John Backus, lead designer of Fortran touches on this issue in his Turing Award lecture: http://www.stanford.edu/class/cs242/readings/backus.pdf

5. Great post, Robert. You’ve challenged me to (a) remember how I introduced the derivative at a point the last time I taught calculus, which is coming up on six years now and (b) reflect on why I introduced it the way that I did.

The key for me, I think, was not to start with Example 1 (calculate the slope of a tangent line given an algebraic representation of a function) but to start with Example 2 (calculate an instantaneous rate of change given a graphical representation of a function). And instead of going with “the rate at which a charge on a capacitor is discharged” (whatever the hell that means), I went with an application that students could more easily get their heads around–the velocity of someone doing a belly flop off a diving board at the moment of impact.

Why? Because most students learn inductively. If you want them to develop a conceptual framework, you need to start with multiple concrete examples. As they work through those examples, students will start to see the patterns and from those patterns they might just create a reasonable mental model of what’s going on. Starting with something as abstract as finding the slope of a tangent line (geometry = abstract) using an algebraic representation of a function (algebra = abstract) doesn’t work well when the students haven’t abstracted their mental models yet. Sure, we mathematicians like to go from the abstract (slope of a tangent line) to the concrete (impact velocity of a belly flopper), but students don’t work that way in general.

This is an also an example that might best be explored during class and not before class. Students probably shouldn’t read through the belly flop example, because they’ll read too fast, skipping to the end point (the actual impact velocity) when they need to be paying attention to the process. You’re on target here when you say that students will focus on final answers over process too often. I’d rather walk them through this example step-by-step, making them sweat through the process a little.

Here’s part of how I would handle that example: “If it takes you two seconds to hit the water and you’re 10 meters up when you start your dive, your average velocity is 5 meters per second. Is that how fast you’re going when you hit the water? No, because you’re moving more slowly at the start and more quickly at the end. The 5 meters per second is average rate, not the faster rate of change you experience at the end of the dive.” That’s me attempting to connect something very intuitive (you speed up as you fall) and visual (you can see it in the graph of position over time!) to something more conceptual that needs to be a part of the students’ mental models (rates of change change over time).

Only after the students have worked through the belly flop example (which, by the way, has a naturally motivating question at the end of it: How hard are you going to hit that water?) do I move them to the more abstract problem of finding slopes of tangent lines.

6. Great comment, Derek, and I agree totally. I think I’ve complained before here about Stewart’s approach to basically every important concept in his book — namely, the concept is presented in the most context-free way possible (e.g. finding the slope of a line instead of the speed of a diver) and then eventually makes its way to something realistic and familiar. Some people do actually learn this way, for example people who have Ph.D.’s in mathematics! But most people taking calculus learn in completely the opposite way. The best thing I can say about Stewart’s book is that it’s easy to remix, so it’s not hard if I tell students to read example 3 first, then example 2, then example 1. In fact I do this on a regular basis — but it is a little unnerving to warn students NOT to read the book in the order in which it was written.

As to your point about doing things in class rather than outside of class, I agree, because this is a clear case of not acquiring factual knowledge but assimilating conceptual knowledge, which is best done with live supervision by an instructor. However, I would say that one important portion of this lesson could easily be farmed out for a reading assignment: The calculation of average velocity. This is a pre-calculus idea and very easy to rehearse and master prior to the main lesson on instantaneous velocities. Then when they come in for the lesson, they can ramp pretty quickly up to the main conceptual idea using repetitions of that mechanical calculation they were supposed to practice.

7. Have you looked at Briggs/ Cochran’s calculus book from Pearson? It’s a breath of fresh air compared to the major texts out there now in the calculus arena.

I so agree with you in regard to precalculus review being more of a conceptual review into calculus. But the textbook authors probably just edit portions from the their precalc texts and stick it in the front.

Re: new conceptual frameworks – just very hard for getting adopters on board with new ways of presenting material in a textbook. I’ve experienced it firsthand as an author of a precalc book that discusses functions first and introduces concepts via applications.

8. I love the post! I have been asking the following questions to college graduates for about 20 years.

1. Did you pass at least one full year of college level calculus?

If the answer is yes, then I ask the follow-up question.

2. Can you write for me on a piece of paper the fundamental equation for a derivative? Can you show me where it comes from?

I’ve asked about 180 people question #2 over twenty years. How many do you think were able to show the equation for the slope of a tangent line? How many do you think can even say it is the slope of a function at a point? Don’t you think that after taking one to two years of calculus it is reasonable to expect that students should be able to remember this simple idea for the rest of their lives?

People pay thousands of dollars for these calculus courses. And, I have to know, what is the goal of teaching calculus courses at the college level?

I can only assume the goal has nothing to do with actually teaching the students calculus.

What is the point of teaching 180 people one to two years of calculus if less than 10 are able to answer question #2? Other than overcharging for tuition, what do you think you are accomplishing at college? Do you think I am exaggerating my survey results?

And, for those of you teaching calculus, how can you really change your fundamental approach so that you are not failing so miserably in the future. My grade for those of you teaching college calculus? Well, what grade would you give a student who got only 10 right out of 180 questions?