My initial motivation in asking the questions in my last post was to provide some structure for what a calculus class should look like. Is the role of calculus really to make sure that all freshman know how to compute integrals by hand, and can use this to find the volume and surface area of a solid of revolution? That rising second years can go through mindless algebraic manipulations to arrive at an answer to a question that was asked without any context or applicability? Moreover, that they need to be able to do so by hand? Quick, find the antiderivative of $\sqrt{3-2x-x^2}$, no computers. Who gives a crap? How many times will that function, or one like it, actually show up in the lifetime of my student? I get paid to do math, and the only time I've ever cared about the answers to these algebraic questions was when I was taking or teaching a calculus course.

Looking back, I'm a little ashamed that the problem above was actually one I assigned as homework for my students this semester. What good does it do to have them work this integral by hand? Surely the answer depends on the students, right? My students going away from the sciences have gotten nothing out of this assignment. My students staying in the sciences have spent another... 5 minutes? 30?... practicing their algebra skills. Hurray‽

Look, I'm not saying algebra isn't useful. But I really do think it is over-emphasized in calculus courses (the ones I've encountered). My calc II course covered: techniques of integration, improper integrals, arc length and surface area of revolution, parametric curves, polar curves, multiple integrals, and infinite sequences and series. Lots of these are wonderfully fascinating and fun topics. Parametric and polar curves force students to start thinking about curves differently, to start re-interpreting functions and points in the plane. Infinite sequences and series are so unlike anything that's ever come before that they can't help but be interesting. Add up infinitely many things that get infinitely small, and produce an infinite value? Or a finite one? How the hell does that work? What does it even mean?

If not a single one of my students can do the integral mentioned above, I'm not sure that I'd feel bad about my semester. If not a significant portion of my students could tell me what was new about a parametric or polar curve, I think I'd be fairly upset (admittedly, now I'm a little afraid to ask).

My problem with my course is that algebraic manipulations are essentially given precedence over conceptual understanding. This is done by making exams that test algebraic nonsense, forcing instructors to make sure their students can do the algebraic nonsense, taking up valuable time that could be better used in other ways. With less time spent on algebra, more time could be spend on concepts, and more time could be spent on improving student's writing. The time spent on concepts makes it a more mathematically interesting and, I'd argue, worthwhile course. Whatever negative impact on my students is brought on by slightly less algebra practice will be more than compensated for by having them spend time thinking about ideas. More time spent teaching students how to write mathematics will force them to think more precisely about what they are saying and how to say it. How could this not be advantageous to students, whether they will be taking more math or not? Unless, of course, thinking about these ideas and how to write them is not aligned with the goals of my university. If the role of higher education isn't something that thinks this would be better, then I'm getting off this boat before it sinks.

So why not go for it? Spend drastically less time on techniques of integration and series convergence tests. Force students to turn in well-written assignments. Spend class time talking about what well-written math is. Spend time letting students play with computers to get a feel for parametric and polar curves. Computers will be faster and more accurate in plotting curves than any student ever will be. Give students time to plot lots of curves, changing parameters to see the effect on the graph. Make students look up resources online to learn how to find arc lengths or surface areas, instead of just telling them the formula (or even deriving it) and having them memorize it. Spend class time talking about how to read the resources they find, and how to evaluate them for quality.

My advisor pointed out, during our conversation mention in my last post, that there are some issues with trying new things. At the lowest level: who gets to teach such a course, and where does their funding come from? But I think better concerns that he mentioned are two that bring me back to some of my earlier questions.

- It is hard to test conceptual understanding. It is much easier to test if a student can do some algebra. With several hundred students going through calculus every semester, this is a practical concern. How does a university handle scale?
- Institutional inertia would have to be overcome. Change is hard. We do this because this is how we've done this for a while now,
~~and it is clearly the best way to do things~~~~and it's going ok~~~~and nobody has complained~~.

These questions brought me back, full circle, to my initial desire for some formal guidelines for courses. I think it would be hasty to change all of our calculus courses around right away. I think a better solution involves some experimenting. Make a new course, make grad students apply to teach it, arguing for what they want to do differently, and require the instructor to report back on how it went. I think having formal guidelines for courses would aid the experimentation, because it gives an objective measure for comparing a newly designed course with a more traditional one.

I know that much of what I'm saying is oriented toward the way things are set up in my department. How is your department set up differently, and what does that mean about how calculus courses are structured?

Thoughts? Objections? (non-spammy) Links?

## 24 comments:

I have three questions, which I hope are of the so-stupid-they're-actually-cunningly-insightful Zen master variety.

1) To what extent do you believe you "understood" the calculus after taking a calculus class for the first time?

2) To what extent do you believe you "understand" the calculus at this point?

3) What has happened in the interim?

The kinds of things you are suggesting in both the posts in this miniseries are being done, and done well, at small colleges where there is less inertia, less red tape, and less territorial instinct among faculty.

Especially at liberal arts colleges where the value of writing, critical thinking, and problem solving are (at least supposed to be) infused throughout every course, it is very easy to create over-arching goals for the entire mathematics curriculum and then define the place of calculus in context. Ironically the most future-ready models of higher education are at the liberal arts colleges, which are the oldest forms of higher education.

The MAA has done some work in the past on this; see especially their "Lean and Lively Calculus" stuff from 10 or so years ago. As for putting problem solving at center stage in calculus rather than decontextualized calculation, MAA has done and still does do a lot of work in this area. There are lots of instructional strategies that keep algebraic calculations an important part of the course without making them the focus, such as the use of Mastery exams and online homework. There's lots to check out that has already been done here.

Good luck! You're asking the right questions, I think.

1. I know now that I didn't know as much at the time as I thought I did. I know that my profs did a great thing by designing the course around spending time reading the text.

2. Fairly well?

3. A decade of math classes, 5 of them also spent teaching calculus?

So... how do I set up my class to get a reasonable amount of understanding in the first year? And how much does it matter that my students actually understand the calculus? I guess I should find some numbers about how many of my students actually go on to be math majors, or science majors, and how much calculus is used in those science courses. And for my students that end up not going into math or science... they're better off for having spend a year learning calculus because...?

I'm sorry if I'm being dense.

Thanks for the pointers @RobertTalbert. Definitely useful, and giving me some optimism about liberal arts schools (my undergrad background, and what I've always thought would be my professor future). Oddly enough... I actually have that MAA "lean and lively" report (or so) on my bookshelf. Time to dust that off.

This will likely appear as a trackback soon, but let me just say that my thoughts (all 1488 words of them) are now over at Partially Ordered Thoughts.

It's not about being dense, it's about separating what you know now from what they need to know. Just because you didn't really have the handle on it then that you thought you did, your real understanding didn't come within the context of the single semester course anyway, but in letting it sink in and applying it elsewhere.

Nice post @Mitch. Everybody head over and read that one too.

@unapologetic so... I'm still confused. I definitely don't expect my students to have the same understanding, at the end of the semester, that I have. I know my understanding has been built up over time. I think my point is that what seems to be the case (at my school, anyway) is that we only care (mostly) that our first year students to have a symbol-pushing understanding of calculus, and I don't see the good in that at all.

Great post, Nick. I'm in agreement with much of what you say. One thing I was struck by during much of the semester was how the exam writing didn't have nearly the same emphasis on application and conceptual understanding that even the textbook had. This kind of surprised me, but it seems if we had just remained truer to the textbook we would have actually ended up placing a much greater emphasis on concepts rather than symbol-pushing. Were we just showing deference to tradition in our exam-writing? Maybe it's just easier to write exam questions that look more or less like last year's exams...

I'm concerned that a lot of the problem has to do with the culture of mathematicians in general and grad students in particular. We really need to take a poll to figure out what mathematicians think "conceptual understanding" even means. There are at least a few instructors for calculus who have a genuine disdain for application questions. Honestly, I think this is sad. Application questions, in my opinion, force students to gain what I think is truly a "conceptual understanding" of calculus. If there's one thing I hope my students learned about calculus, it's that calculus is all about modeling the real world using functions. From this point of view, every exam should have been entirely application questions. In reality, none of the exams came close to this.

From my point of view, we should be able to change a few things without seriously altering the current structure of things, simply by talking about how we write exams. But I have a pretty limited perspective on things. It's good to hear someone else's thoughts.

@Jameson I think one of the reasons some mathematicians have issues with application problems is that they take the students more time. It's hard to test on all of a course's content if you give students application questions and expect them to have enough time to solve them. Take-home components of tests are part of the solution, but there's way too much cheating to rely too heavily on them. I think more people need to explore mastery exams as an option here. You don't need to put a bunch of basic differentiation and integration questions on the midterms if you're giving the students mastery exams on those topics. However, with a large class and no testing center, this can be horribly time-consuming.

The other reason I can see people not liking to give students application questions is that it's harder to teach them to think about the concepts and applications in the way they need to be successful than it is to teach them to manipulate symbols. For too many, it's about what's easiest for the instructor rather than what's most beneficial for the students. Of course, part of this is the reward structure the institutions use, as it rarely incentivizes doing things that are best for student learning.

Thanks for more things to think about @Jameson and @Mitch. I'll have to look into this concept of mastery exams a bit more, they're not something I'm familiar with.

Perhaps I'm about to contradict myself (I'm sure I should analyze my own thoughts more carefully), but I think I'm one of the instructors who doesn't care so much about applications. My initial reaction is... put applications in the class they apply to :) Physics applications in physics class, econ/business there... I guess this creates a pre-req nightmare (you have to have non-applications calculus before you can take this particular physics course...). And perhaps that's why we have 2 courses at UVA, one slightly more geared to applications and one less so (oh, and those engineers have their own...).

And I certainly don't want to knock the applicability of calculus to real world problems (though recently some folks have been arguing that prob/stats is more useful for more people these days...).

But I have zero problem with 'math for math sake'. No other department is going to get students thinking about functions, just because you can. No other department is going to ask what it actually means to add up infinitely many numbers.

Maybe the first-year calculus sequence is not where math for math sake comes in. Perhaps these are the topics for the further math courses. I don't know. But first-year calc is the last chance for math for math sake for many students.

I think I'm not sure what else to say right now, which probably means it is time to stop, and do some more thinking. Clearly I'm pretty un-sure about a lot of this. Thanks, @all, for being around to help me think about it more.

I've never been fond of teaching applications in calculus, but I've come to realize that it's important for most students to understand

whywe're having them learn something (and this is usually an application). Most of our students in service courses are not mathematics majors, and so math for math's sake is hard for them to grasp. I think we should incorporate a certain amount of that. However, when we care about what they can do, being able to use the mathematics really needs to be paramount.PART I:

I agree with some of what you say, but I have to take issue with many of your claims.

First of all, I think performance of integration technique problems by hand is probably a bit over-emphasized in calculus classes. But only a bit. My main concern with your post is that you seem to be setting up this false (in my opinion) dichotomy between symbolic fluency (what you call "mindless algebraic manipulations") and explanatory discourse (what you call "conceptual understanding"). It's not at all clear to me that these two are somehow in opposition to each other. I would argue that they are complementary and reinforce each other.

Let's start with the example you give: "Quick, find the antiderivative of $\sqrt{3-2x-x^2}$, no computers. Who gives a crap? How many times will that function, or one like it, actually show up in the lifetime of my student?"

Oh, I don't know... maybe if they ever needed to find the area of a circle with center (-1, 0) and radius 2, and somehow forgot the formula for area of a circle, or (god forbid), took the initiative to use calculus to verify that the usual geometry formula gives the correct answer. Of course, maybe circles and semicircles don't "show up" very often in the lifetime of our students. I highly doubt that.

Interestingly, I decided to try my hand at that integral, by 3 methods: (1) by hand, using only what's in my noggin; (2) using tables; (3) using a CAS.

The results may shock you. I found the solutions the fastest by method (2), next fastest by method (1), and I was never able to get the answer by (3)! I was only able to verify that (3) gave the correct answer after a lot of "mindless algebraic manipulations".

By method (2), it's a no-brainer. Complete the square, see that it's sqrt(a^2 - u^2), where a = 2, u = x + 1, and look it up in a table of integrals. You don't even have to worry about substitutions since du = dx. I had the answer in 30 seconds flat: (x+1)(sqrt(3-2x-x^2))/2 + 2arcsin((x+1)/2).

Now method (1) took a bit longer, but still worked (here, I "erased my brain" of my previous work). Mostly because you're essentially re-deriving the formula from the table of integrals. (BTW, did you ever wonder how they got those integrals in the first place?) You complete the square as before, then do the trig substitution x = 2sin(theta). It's a little work, but nothing to write home about. Took me about 2 minutes.

Now, I have Scientific Workplace, which uses MuPad, and when I asked it to find the antiderivative, it gave me:

(1/2)(x)(sqrt(3-2x-x^2) - (2i)(ln(ix + sqrt(3-2x-x^2) + i)) + (1/2)(sqrt(3-2x-x^2).

Hmmmmm...

Logarithms and complex numbers! My lord! I can only imagine the hapless calc II student pondering this! I looked up in the help menu... no help. I tried all the ways I could to "simplify" or "rewrite" the expression, in various combinations of commands. I simply could not get rid of those dang logs and i's. So our hapless calc II student (who may not even be very familiar with complex numbers) would be totally stuck. Now, maybe there is a combination of commands that could simplify this, I don't know. My point is, I have a PhD in math and spent 15 minutes trying, and I couldn't get succeed, so we shouldn't count on our student to find it.

I WAS able to verify that this is an antiderivative. It turns out (of course) that all the i's cancel out, leaving the appropriate real-valued expression. I'm still not sure exactly what type of algorithm would generate a log for this particular function. It's an interesting exercise to think about.

My point is, the example you gave is neither (a) completely irrelevant in the real world, nor (b) automatically solvable by someone with a CAS but no knowledge of using integration techniques or tables of integrals.

PART II:

"Looking back, I'm a little ashamed that the problem above was actually one I assigned as homework for my students this semester. What good does it do to have them work this integral by hand?"

Well, let's see. How about: (1) it reinforces their algebra skills (and my experience teaching calculus is that, even at elite universities, roughly half to 2/3 of students are poor in their algebra skills); (2) it reinforces their knowledge and appreciation of trig identities and trigonometry in general; (3) it shows them how the formula in the table of integrals was derived; (4) in case they are using a CAS that doesn't immediately give them the answer they're looking for (see above), it gives them a way to quickly solve it.

"If not a single one of my students can do the integral mentioned above, I'm not sure that I'd feel bad about my semester."

I WOULD.

"My problem with my course is that algebraic manipulations are essentially given precedence over conceptual understanding. This is done by making exams that test algebraic nonsense, forcing instructors to make sure their students can do the algebraic nonsense, taking up valuable time that could be better used in other ways."

Again, I think your positing a false dichotomy here. It is true that "algebraic manipulations" should not be taught AT THE EXPENSE OF conceptual understanding, but neither should conceptual understand be taught AT THE EXPENSE OF SYMBOLIC FLUENCY. I DO NOT consider these algebraic manipulations to be "nonsense". What you really mean I think is "trivial", i.e. "these calculations are trivial for someone who has achieved the conceptual understanding I have". Which may be true. But the manipulations themselves may be far from trivial.

To take an example, consider partial fraction decomposition. On the one hand, this is a purely rote algorithmic process. However, the EXISTENCE of the algorithm itself is an important CONCEPTUAL fact, not just in calculus or diff eqs, but its version applied to rational functions over a general field is important in other areas.

"Make students look up resources online to learn how to find arc lengths or surface areas, instead of just telling them the formula (or even deriving it) and having them memorize it."

Yes, god forbid they should understand how to DERIVE the formulas for surface area or volume of revolution by Riemann sums, an important application of integration that requires a sophisticated conceptual understanding. Better just to have them mindlessly "look it up online".

PART III:

"Spend drastically less time on techniques of integration and series convergence tests."

I read at a blog that referred to this post (partially ordered thoughts, I think) that they thought it would be better not to teach those "arcane tests for convergence". You know, like the comparison test, absolute convergence, alternating series test, p-series, geometric series, ratio and root tests, those ones that you never, ever see again, and have absolutely no conceptual significance in later courses or applications.

"So why not go for it? Spend drastically less time on techniques of integration and series convergence tests. Force students to turn in well-written assignments. Spend class time talking about what well-written math is. Spend time letting students play with computers to get a feel for parametric and polar curves. Computers will be faster and more accurate in plotting curves than any student ever will be."

These are all good ideas. My concern is that an over-reliance on abstract conceptual understanding and sophisticated computational software, combined with an under-reliance or jettisoning of traditional paper-and-pencil symbolic fluency will lead to a generation of students who will have achieved a conceptual understanding devoid of any concrete substance or intuition, and who will be helpless to judge the validity of results they get from computers, or unable to do anything if their computer tools fail them.

@Anonymous thank you very much for your long response(s). I very much appreciate the time you've given me and this discussion.

Part A (not lining up with your parts)

On my integral example, you say: "maybe if they ever needed to find the area of a circle with center (-1, 0) and radius 2, and somehow forgot the formula for area of a circle, or (god forbid), took the initiative to use calculus to verify that the usual geometry formula gives the correct answer.". Let me just say I'm not too worried about the first situation. As for the second, why should a geometric verification use an example that requires completing the square? I'd hazard a guess that a student with such initiative would be perfectly happy to use the unit circle. And if they're just verifying that formal symbol games work out, a table of antiderivatives seems sufficient.

"Now, I have Scientific Workplace, which uses MuPad". I hope you didn't pay much for it. Wolfram|Alpha, which is free, seems to do fine here. But let's not argue specific tools.

"Well, let's see. How about: (1) it reinforces their algebra skills (and my experience teaching calculus is that, even at elite universities, roughly half to 2/3 of students are poor in their algebra skills); (2) it reinforces their knowledge and appreciation of trig identities and trigonometry in general; (3) it shows them how the formula in the table of integrals was derived; (4) in case they are using a CAS that doesn't immediately give them the answer they're looking for (see above), it gives them a way to quickly solve it.". Ok... (1) I don't argue that algebra skills are lacking. I just question making that a priority. (2) Perhaps. If a student told me they appreciated such an example as a trig reinforcement, I'd probably think they were joking. (3) Top of the list of their priorities, I'm sure. (4) (see above). And I'd like to try to help students be able to overcome whatever difficulties they have with a CAS, by themselves, even if they've never used some particular system.

You mention the example of partial fraction decompositions. I agree it's a useful tool. In my classes (I'm willing to accept as much blame as anybody wants to place on me here) it never seems like the point of them comes out. We spend time on the tedium of finding the pfd of expressions, instead of emphasizing "When would it be good to do this? Why would we do it? How does it make life better? What understanding about a function does knowing its pfd give?"

Part B

"Yes, god forbid they should understand how to DERIVE the formulas for surface area or volume of revolution by Riemann sums, an important application of integration that requires a sophisticated conceptual understanding. Better just to have them mindlessly "look it up online"." I would absolutely love it if my class was set up to get students to understand how to derive the formula. If it gets mentioned at all in class, besides "feel free to read the book", it is gone over quickly and then drowned in a sea of algebra in examples before rushing on to the next topic (actually, I'm a little proud of the explanation I give, but that's the subject for another day). "Mindlessly" isn't exactly what I have in mind for looking things up online. Even for just looking up the formula to apply in a particular example, students will have to be able to appropriately test the reliability of information they find. Did a student just look at a single page? How can we rely on that page? Is the answer obtained by using some stray formula found online reasonable? I think my intention was to have students go through these questions to gain an understanding of the derivation of the formula. Have them compare explanations found on several pages and synthesize the findings. Compare with the book and see which is easier to read. Use the easier to read source as a way to work on understanding the harder source, making it easier to read the harder source next time.

Later on, in Part III... series tests are all well and good. Coming up with goofy examples to see if students can manipulate limits is what bothers me. We should spend more time on why the tests work, the meaning of what's going on. Leave the algebra to the computers. They're better at it.

You worry that some of the things I seem to be proposing "will lead to a generation of students who will have achieved a conceptual understanding devoid of any concrete substance or intuition". I think a conceptual understanding should do wonders for intuition, and don't see how formal symbol manipulation leads to too much intuition. I'm not sure what to say about concrete substance, at this point.

I'd probably argue that among students in calculus class, most will be allowed (or, at least, there's no reason they shouldn't be allowed) to use the tools available to them should an integral arise. I still don't have an understanding of how frequently that is. As a math grad, they seem to have arisen for me when I was teaching them. In more applied sciences, they may come up, and I expect computers are welcome (I have no experience, so I have no idea what I'm talking about). Outside of that, I'd much rather that my students decades from now look back on calculus class as an enchanting exploration into the infinite, rather than a forgotten algebraic exercise.

Anyway, I don't know what I'm talking about. I've got no experience outside of the classes I've taken or taught.

Thanks for pushing me in my thoughts. I honestly do appreciate it.

(PART I)

Nick,

Thanks for responding to my comments. No, I don't think you "don't know what you're talking about". Everyone will have their own feelings about something like this, based on their unique experiences. Please keep in mind, most of my strongly held opinions on this are based on teaching and/or TAing many classes (at least 2-3 each of precalc, calc I and II, vector calc, and lower-division ODEs and PDEs).

Regarding your comments about my noting the integral is the area of a semicircle. My point wasn't that this particular situation (a student forgetting the formula for area of a circle, and using calculus to find it) would likely happen. I was trying to counter your claim that an integral of such algebraic complexity would never come up in the lifetime of a student. Certainly finding the area of a circle is a trivial task, and I would hope they wouldn't use calculus to solve it, but my point was that other problems will require calculus, and using calculus to solve such problems is not trivial. If finding the area of a circle produces integrals that are not at all obvious, that require techniques of integration, then other more difficult problems will produce even more difficult integrals. These integrals are not academic exercises. They pop up all over the place in physics, engineering, geometry, differential equations, etc. Heck, just finding the arc length of an ellipse produces an integral that can't even be "solved"!!

I think it might be important to point out that calculus was invented primarily to solve problems in geometry (pure math), physics, and astronomy, and its applications to the "softer" sciences (chemistry, biology, psychology, sociology, economics) only came decades or centuries later. One of the problems with calculus courses may be that we are trying to find "one size fits all". At Santa Barbara when I was a grad student, we had two calculus sequences -- the traditional one (for math, physics, engineering majors) and a symbolically watered down, but conceptually driven sequence for "life and social sciences". The latter seemed to work well -- after all, how many psychology or biology majors will ever need to integrate by parts or use partial fractions? Not many. But I think it's a disservice not to teach these techniques to "hard science" majors, esp. math and physics majors.

So back to those "hard science" majors taking the traditional calc sequence. It's my opinion (and again, an opinion) that students in these hard majors should be expected to be far more symbolically fluent than other majors taking calculus, simply because these disciplines are more symbolically complicated. Which brings me to a question: If these "mindless algebraic nonsense" techniques are really so mindless, if they're really so rote, so meaningless, requiring so little conceptual understanding, and just plug-and-chug, why do so many students have so much difficulty mastering them? If calc II is tested by giving a lot algebraic computation problems on exams, they should breeze through easily. But they don't.

My experience with calc II is that the class typically separates fairly quickly into two groups -- a smaller group (maybe 1/3 of the class) who quickly understand the applications, master the techniques with practice and seem bored after a while, and a much larger group (about 2/3 of the class) who struggle and struggle with the problems, never understanding the derivations of the formulas and never being able to complete problems without getting lost in algebra or calculus mistakes.

(PART II)

And this is what I meant earlier about symbolic fluency and conceptual understanding going hand in hand, not in opposition to each other. Relying on technology to find answers is fine for students who already have a solid grasp, but can become a crutch otherwise.

Let me give some examples. I taught calculus last semester and this semester. I had a significant number of students who could not graph functions like

y = sin(2x) + 3, or

y = 4e^(-x), or

y = x^3 - x, or

y = sqrt(1 - x^2)

without a graphing calculator. They literally didn't know where to start. Several of them used a table, like x = 0, 1, 2, 3, -1, -2, -3, and y = ... Worse, they had no intuition about graphing functions: they didn't understand and were unable to handle or explain translations, compressions, and reflections. They didn't even know the basic shape and properties of the graphs of fundamental functions like exponentials, logs, trig funcs, polynomials, and rationals. Some of them even reached for their calculator to graph y = x^3. Seriously. Worse, they didn't seem to have an intuition about functions -- behavior at +/- infinity, oscillations, periodicity, relative growth rates, asymptotes, intercepts, etc. I can only suspect that this is because they have never been asked to graph functions without a calculator before.

Another disturbing trend I've found in calculus classes is an increasing inability to handle mildly complicated algebraic steps. They balk at any expression involving combining square roots or rational functions. They make mistakes like log(a + b) = log(a) + log(b), which is far more than a simple algebra mistake, but belies a profound lack of conceptual understanding of the logarithm function. Some don't understand fractional exponents or negative exponents, or at least can't use them to simplify expressions.

And this goes all the way down to basic classes. I've taught elementary algebra, and there are students who could explain to me the conceptual understanding of fractions, but still took 2/5 + 3/4 = 5/9. At my undergrad alma mater, a California St. Univ. school, something like HALF of all incoming students require remedial math work. This means, SIXTH, SEVENTH, and EIGHTH grade level stuff. Something is not succeeding earlier in their classes.

Graphing functions by hand, doing algebra by hand, simplifying complicated algebraic expressions, doing rote problems with trig identities, logarithms, and so on is now considered passe, a waste of time, after all we have computers that can do all these things for us. But I think part of gaining an intuition and conceptual understanding of fractions is to do fractions by hand. Same for precalc stuff, all the way up to Laplace transforms. I haven't done scientific studies to verify this, of course. But it's based on my experience. If you can't do certain things by pencil and paper, your conceptual understand will get stuck up eventually. You might understand calc II without learning all those integration techniques, but I bet there are few people who succeed in upper-division PDEs, e.g. who can't do those techniques cold.

I think (esp. for the "hard" sciences) we should expect both from them -- frankly, if they have a great deal of trouble with integration by parts or partial fraction decomposition, if they don't pick it up fairly quickly and easily, it's debatable whether they're in the right major. But we should also expect the thinking part. Is it too much to ask? The ironic thing is, whenever I try to introduce the conceptual part to classes like calculus, they balk, saying it's not "real math" and complaining that it's way too difficult for them, that I'm expecting too much. I guess we can't win!

I think there's a bit of talking past each other going on here. Let me take graphing as an example, because I think the recent discussion is looking for the same result but viewing it differently. The anonymous poster sees benefit in having students sketch graphs of functions by hand. For some students, this is the best approach. They benefit from the hands-on, tactile aspects of it. For other students, the opportunity to explore with a graphing utility is far more powerful. This enables us to help students build up intuition about the behavior of functions through

carefully designed learning activities. Turning them loose with WAlpha is a recipe for disaster. However, having them do a few graphs, record certain behaviors, and then make predictions about the behavior in other situations before graphing them builds conceptual and computational understanding. With the old-style, drill and kill approach to graphing, students just memorize a list of steps to carry out without any idea why they're doing them or what they're supposed to take away. I think we can all agree this is sub-optimal.Partial fractions has come up here a couple of times. I agree, this is an absolutely fundamental technique. However, what's important for students to be able to do? To me, they need to be able to tell me what the decomposition should look like (precise denominators and the form of the numerators). From there, a computer can solve the system of equations. In reality, they'll usually just have a CAS find the decomposition. However, I want them to see how it comes about because it teaches them something about the behavior of the function in question. Knowing what the decomposition looks like helps them know if it's something that might help them out or if they should look elsewhere.

It's not about removing techniques from the curriculum (although I would like to give trig substitution the old heave-ho most days); it's about figuring out what is important for students to leave our classes knowing.

One final thought: Nick's recent comments about looking up online, in a book, etc. are hitting upon an important point: information literacy. Most of us (myself included) do not do a good job of helping our students develop information literacy skills. I learned that this semester through the writing assignments and project I gave my upper-division applied combinatorics students. I think it's high time we think about how we can teach information literacy skills across the curriculum.

"This enables us to help students build up intuition about the behavior of functions through carefully designed learning activities. Turning them loose with WAlpha is a recipe for disaster. However, having them do a few graphs, record certain behaviors, and then make predictions about the behavior in other situations before graphing them builds conceptual and computational understanding."

I agree completely, and I'd be the first to admit that with such "carefully designed learning activities", student understanding in a classroom with technology would inevitably be better than without. But I have a feeling that

in the high schools at least, calculators are not used in such a structured way, and more often than not become a short-cut for students to do computations and graphing. It does not help that they are oftennevergiven an exam in which they are not allowed to use a calculator.Of course, I have heard the opposite is true, that studies show that classrooms with technology always outperform those without. All I can say is that my own experience doesn't bear this out. Over the past 20 years, technology has been increasingly allowed in the classroom, and over the past 20 years, I've seen students' ability to do basic tasks without technology gradually but consistently deteriorate. I know that doesn't prove causation, but it's hard for me not to suspect so.

I feel that if technology is used in the classroom in the positive way you suggest, then students would necessarily gain the skills to perform relatively simple to moderately complex algebra, trig, and graphing problems by hand with little or no difficulty. Since a significant number of them obviously cannot do this by the time they reach precalc or calculus, the only possible conclusion I can come to is that technology is being used as a crutch in the high schools, not as an effective learning tool.

I was helping a family friend with precalc several weeks ago. She had to add something like

2/5 + 1/7.

Instead of doing it by hand (or in her head), she did the following:

2/5 --(convert to decimal) --> 0.4 (store); 1/7 --(convert to decimal) --> 0.1428571429 (store) --> add stored values (0.5428571429) --(convert to fraction) --> 19/35.

The calculator gave the exact fraction because it stored the values as fractions and added them as fractions.

"With the old-style, drill and kill approach to graphing, students just memorize a list of steps to carry out without any idea why they're doing them or what they're supposed to take away. I think we can all agree this is sub-optimal."

I hope you're not talking about curve-sketching in calc I.

Didn't someone say "science without religion is lame; religion without science is blind"? How about "rote mechanical algorithms without conceptual motivation and understanding is lame; conceptual motivation and understanding without rote mechanical algorithms is blind"?

I was thinking primarily of the way curve sketching is taught in precalculus (since I've taught it twice since I last taught Calculus I). However, my students are really good at algorithmic things and not so good at conceptual things if you don't push them. I think even calculus students are inclined to memorize the steps the book gives for curve sketching and blindly apply them. They know what a positive second derivative means for the shape of the graph because it's part of the algorithm. However, if you ask them a more conceptual question, they have fits. For instance, students have lots of issues if given the graphs of the first and second derivatives of a function and asked to draw conclusions about the graph of the function. Typical curve sketching problems in calculus also suffer from the problem of taking derivatives. To make a good curve sketching problem, we tend to make up functions with messy derivatives. If students make mistakes here, then they have a horrific time graphing. There's a lot of great stuff that calculus-based curve sketching brings together. However, we have to focus on making sure the students understand those concepts and don't blindly memorize steps.

Hi, Mitch (and Nick). Sorry if I sound like I have an axe to grind. I don't mean to come off that way. But I see what in my mind is a disturbing attitude, and I care about how (I think) it's affecting students.

Maybe you've have better students than I've had, I don't know. I've taught at community colleges, a small state school, and at UC Santa Barbara. A pretty wide swath. They all have similar problems, obviously not as bad at one vs. the other.

"I think even calculus students are inclined to memorize the steps the book gives for curve sketching and blindly apply them. They know what a positive second derivative means for the shape of the graph because it's part of the algorithm."

I'm not sure what you're expecting... should they be expected to PROVE the I/D test, 1st/2nd derivative tests, etc.? Just the step you're talking about (pos. 2nd deriv => concave up) requires:

a. knowing what a 2nd derivative is.

b. understanding the graphical interpretation of concave up.

c. understanding the conceptual relationship between 2nd deriv. and concavity.

I don't consider putting those 3 pieces of knowledge together in a step of curve-sketching just blindly following steps. It HAS to be more -- I actually list all the steps of curve-sketching on exams, don't make them memorize anything, and many still can't do it.

Again, if curve-sketching in calculus is so rote, so meaningless, such a blind application,

why do the vast majority of students struggle and struggle with it?"However, if you ask them a more conceptual question, they have fits. For instance, students have lots of issues if given the graphs of the first and second derivatives of a function and asked to draw conclusions about the graph of the function."

Yes, I've seen that a lot. All I can say is my experience has been different from yours, in the following sense. Almost all calculus students I've had who can curve-sketch well have no trouble with these problems (relating graphs of f, f', f'', etc.) Or equiv., if they can't do what you're saying, then they can't curve sketch either. Another way of saying this is that I don't view curve-sketching as less "conceptual" than the type of problem you're describing. The only difference is knowing how to read the graph of a function, which would seem to me a much lower-level skill than curve-sketching itself.

"Typical curve sketching problems in calculus also suffer from the problem of taking derivatives. To make a good curve sketching problem, we tend to make up functions with messy derivatives. If students make mistakes here, then they have a horrific time graphing."

Apparently we've reached our point of fundamental disagreement. I think it's quite reasonable to expect calculus students to be able to compute by hand, derivatives (and integrals) of reasonably (not horrificly) complex functions consistently and correctly. They may not actually compute them by hand in the real world, but they should at least be able to do so, if asked. Consistently.

The reason they can't do so is because they can't do algebra, they can't do trig, they can't even do basic arithmetic sometimes. And why is that? Because the same reasoning was used earlier in high school -- there's no need to expect students to practice complicated algebraic computations by hand, or do lots of trig problems and identities by hand, or or do mildly annoying arithmetic (like adding 2/7 + 1/5) by hand. That's not demonstrating understanding, that's just memorizing rote mechanics. And that's why they fall apart when asked to compute the derivative of anything more complicated than f(x) = 3sin(x).

Again, sorry if I come off with an axe to grind. I think we just have a fundamental disagreement, and I think I've said my piece.

@Anonymous @Mitch thanks for continuing this conversation. I truly do intend to get back in to it. I'd like to continue thinking about it, and chime in if I can put some thoughts together into something coherent.

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