Sal interviews the AP Calculus Lead at College Board | AP Calculus AB | Khan Academy

So this is Sal Khan, founder of the Khan Academy, and this is a very exciting Skype call that we're on. I'm with Ben Hedrick, who's the lead for AP Calculus. What do you do at the College Board?

Uh, really anything with AP Calculus and AP Statistics is something that I have a little bit of control over. So for example, for AP Calculus, anything with the curriculum, anything with the assessment is something that I work with. And I'm sure that you're as excited as I am, and several hundred thousand students are around the country for the AP exams that are coming up in roughly a little less than a week.

I thought a fun place to start would be when you guys write the test, what are you trying to assess? What's kind of the spirit of the questions, both on the multiple-choice and the free response section?

So what we're looking for from students is evidence of good conceptual understanding in calculus. Now, we have some things that are going to test procedures, but really we want students to understand what's going on with the calculus rather than just calculating things or putting numbers on paper. There are a lot of thinking questions, a lot of insight, to make sure that whatever good calculus those students have learned in their courses, we're assessing it on the AP exam.

You know, I've done a bunch of example problems for the AP exam, and my sense of it is you sometimes will do some of the minutia work or you'll give the formula, and you're really trying to understand whether students understand the essence of what an integral is, or the essence of a derivative as a rate of change or the slope of a tangent line.

So if you were preparing for this exam, if you're one of these students, what would you suggest to them?

You know, obviously we have a little less than a week for this current test, so what advice would you have for those students? And then just generally as students go through the year, how should they think about it?

Yeah, the advice I would give is depending on how much time the students have to prepare. The advice that I give at the beginning of the year is different from the advice that I give now. So several days before the exam, getting ready for it, first relax. You know, this is a calculus exam. We're not going to be throwing anything at you from left field; it's calculus.

You need to know limits, you need to know derivatives, you need to know integrals. In terms of preparation, make sure you're taking the time to review the basics. Every question that they see is going to hit one of those big topics. So make sure you know your derivative rules, make sure you know your integral rules.

Take a look at the old exams and see how we're asking questions. Look at the patterns and see that we're always asking about understanding what a derivative means, understanding integration, and the idea of accumulation. There are no secrets on the exam, so really stick with what we've done.

Make sure that you're thinking of ways to apply your good knowledge and just review your basics. My experience with the exam is there's a lot of, as you just said, the really mainstream topics, you know, chain rule, fundamental theorem, or theorems of calculus—things like that.

What's your sense of the more, what I would concern, maybe a little bit more of the minutia, some of the more special case derivatives or special case integrals, you know, arc tangent. How much does that play into the exam?

You know, I think so. You've actually answered your own question in asking that—is it the minutia? So really at this point in time, if you're getting ready for the exam, spending time on the minutia is not where you're going to be best served with your time. You should be hitting the major things, and you said it right in the beginning—chain rule.

I mean, any kind of question we do with derivatives, you know, if you're doing maxes or mins, if you're doing tangent lines, if you're doing a related rates problem, they're all derivatives, derivatives, derivatives. If you're having trouble with the chain rule, it doesn't matter how much minutia you spend your time with; the chain rule is going to come up.

So I would say don't sweat the minutia. There might be something on the exam that you haven't reviewed as much as you want to, but it's going to be one single solitary question. Basic derivatives are going to form, well, the basis for all those problems, so that's where your time is best spent.

I definitely get that sense, especially, you know, how do you interpret the first, second, you know, derivatives, concavity—things like that. Those seem to show up a lot, but once again, that's the conceptual idea of derivatives. And on the integration side, I've been doing a bunch of the free response. There seems like, I mean, you know, what happens if you swap the bounds of integration or if you add integrals and things—so a lot of the properties and the conceptual understanding of what an integral is, but you know, not necessarily like the fancy tricks, at least at this point, if you're studying.

Absolutely. There's nothing that we go after the fancy tricks. We're not trying to ask trick questions or play any gotcha moments on it. It's really assessment of calculus, and for the students who've been going through these great courses all year, this is a wonderful opportunity to show us everything that you know and get a really great score on the exam.

And what about calculators? I mean, I was doing some of the free response. At first, part of the free response, knowing your calculator well helps.

Oh, absolutely. The calculator is a tool, and like any tool, you want to make sure that you're applying it appropriately and strategically, which sometimes means using your calculator and using it well and correctly, but other times not using your calculator. I mean, even on the free response section where the calculator is being used, we expect students to show their work.

If you're taking an integral, we want to see that you've actually taken that integral. Now, the calculator will do the work of it, but we want to see the notation that says this is the integral you calculated and this is the answer you obtained. Then do whatever you will with that answer as required by the question. But, you know, good communication through writing, even on the calculator section, is necessary.

And for students taking the BC Exam, above and beyond the core differentiation and integration, a lot of what you learn about parametric equations is actually just an extension of what you learn in differentiation and integration. But probably some of the convergence tests are something to become pretty familiar with—that would be correct?

Yes, and in terms of grading of the exam, you know, some people, to get a five or a high, my understanding is you don't need to answer absolutely every question perfectly. Well, like any test, if you want to do well, you don't have to have perfection on it, and this is a test where students are coming in and it's a three-hour time test—well, a little over three-hour time test.

You know, you can do a lot of really good work and earn a five pretty easily. So I wouldn't tell anyone to, when they're worrying about trying to figure out what magical numbers they need to get in order to get a perfect score on the test or to get a five on the test or whatever it is they're going for, just go in like you would on any test. Do your best on every question; get every point that you can, and hopefully it'll work out fine for you.

What advice would you have for students, especially on the free response where there's multiple sections, and if you know on part A they say, "Hey, what—" that's— I don't— I don't get with that part. What, you know, should they skip to the next few? What should they do?

I actually have very common advice on that one, and that's to make sure that students are trying every part of a free response question. One of the misconceptions students have is exactly what you said: if you hit part A and you're freaking out a little bit because you don't know what part A is about, that's okay. Move on to part B.

There are some questions where the answer for part B depends on part A, but for the majority of the questions, they're separate pieces. So part A, part B, part C, if there's a part D, they might all be asking very different things. And if you have trouble even with part C, that doesn't mean— you know the question gets progressively more difficult. Give part D a try, and a lot of the things that we do give points for are good solid calculus work.

So if you can set up a problem, maybe you don't have time to finish it; maybe you make some mistakes along the way—the answer point is only one little point out of the n—there are other points for the setup and the conceptual understanding of the problem. A lot of students have good conceptual understanding that they could set up and pick up a point here, pick up a point there, but they panic a little bit and they move on to another question and never come back.

And that's okay if you want to move on to another question, but absolutely, I would say if you're having trouble with part A, that's all right. Take a look at part B. One thing that I've observed is a lot of these questions, at first, when you look at them, like, "Oh, wow, this looks like some really, you know, super deep thing," but it really is some core basic calculus ideas and that if you're on the right track, it's actually quite simple.

So, would you say it's fair if a student finds himself doing a very hairy calculation that they might question whether they need to?

I would go beyond might; they should question what they're doing. If you've done something where you feel like you need to prove a new calculus theorem in order to answer it, something has gone terribly, terribly wrong. And also, if you feel like you need to bring in some sort of mathematics you've never seen before in calculus, I mean, as much as I hate to say it, you know, this is calculus. It's limits; it's derivatives; it's integrals.

Even though there's a lot of wealth in that one, if you're doing something beyond that, probably something has gone wrong.

Yeah, well, thanks. I think thousands of AP students are going to really appreciate that. Any other kind of parting words for the coming test?

You know, it's the same thing that I used to tell my students when I was teaching AP Calculus: this is just another test, and the beauty of AP is the teachers have been prepping their students since the beginning of the year. It's one of the few tests where everyone knows what's coming.

So when you sit down and you've got free response questions, you've got multiple choice questions, there really shouldn't be any surprises. This is the stuff they've seen all year; they know what to do. Take a moment, take a breath, take a look at the problem, and get all the points that you can. Show us that good calculus.

Awesome, and I'll throw in a plug—we have tons of resources for the students as well, and we're going to have our office hours on Monday, so super exciting. Thanks a bunch, Ben.

No, thanks for having me.