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If it is true that we first learn formalism...how to do things that we don't understand, should we regard teaching students mathematics as programming dumb machines with formal rules (to the greatest extent possible) and allow them to eventually incorporate meaning?

I am thinking of this as the passage from "M" mode to "I" mode in Godel Escher Bach.

This seems to be what we are saying when we set specific learning goals for students to be able to do, like find the derivative of $x^{3}$...

Question: Is there mathematics education research, perhaps in cognitive load theory, that indicates that the above approach is superior to concept-based teaching, especially at the undergraduate level. Particularly, that mechanics should precede concepts that organize them?

The reason I ask this is that our colleagues in physics and engineering departments (see my first link) want to see students proficient in mechanical computation...and cannot understand why two semesters of calculus are needed to train students to do this. We, though, as mathematics professors, try to emphasize and train reasoning and concept. Some of us (like myself) trying very hard to get students to move past mechanical symbol pushing toward metacognition.

The basic starting assumptions of the physics faculty and mathematics faculty seem to differ. They say that students should "at least be able to find integrals" etc. (we all agree), whereas it is likely the case that trying to teach conceptual thinking increases cognitive load of weaker students so much that they cannot handle the symbol pushing as well as if it were emphasized WITHOUT the meaning. Which basic approach is better?

It is certainly the case that a differentiated approach will work, so I am not asking about that. What I am asking is whether or not doing what students and physics and engineering departments (and Keith Devlin in the above link) seem to want: blind and correct manipulation of symbols before concepts that organize such calculations, is more sound than teaching students organizing principles that rudder such calculations with meaning first. (To me, the answer seems obvious...if you are taking unjustified steps then you will make serious errors due to being very dumb. I could be wrong, though...hence this question!)

EDIT: I think my original version of this may not be as accurate as I wanted. The question is, perhaps, more about whether moving past simplified (perhaps even oversimplified) intuition in calculus classes creates so much cognitive load that we'd be better off sticking with a course that looks pretty much like this:

Differential calculus:

  1. The derivative is the slope of a tangent line. Look, instantaneous velocity is an example. Limit means that if x gets really really close to a then f(x) gets arbitrarily close to its limit. (So, don't get into the problems of what we mean by arbitrarily.) In these velocity problems, we end up dividing by zero, so we do algebra to get rid of this problem. (Very little discussion of why or emphasis on definition of derivative beyond saying "change in y over change in x" and talking about "infinitesimal changes" like an 18th century mathematician or physicist.)

  2. Here are lots of nifty formulas for computing derivatives. Let's practice the daylights out of them. Sometimes this is hard, so we implicitly differentiate...so let's do that to death, too!

  3. There are higher derivatives. The second one describes concavity, which acceleration is an example of.

  4. Maxima and minima happen at endpoints or at the tops and bottoms of hills. We don't care so much about anything tricky...let's do a slew of optimization problems.

  5. Talk about linear approximation. Do a bunch of mechanical problems that ensure students can do such problems. Maybe even require them to be able to explain with a picture why this works. (No, that's REALLY pushing it...produces too much cognitive load.)

  6. Maybe do some differential equations. Teach them to write $y=e^{kx}$ if they see $\frac{dy}{dx}=ky$ and to plug in initial conditions. Very little discussion of, or emphasis on, the fact that $y=e^{x}$ is a fixed point for differentiation and such things...

Integral calculus:

  1. The derivative of the area function is the original function (hand wave hand wave)...here's how to take integrals using antiderivatives...practice to death with u substitutions and integration by parts.

  2. Talk about slicing and Riemann sums and do very basic examples. Don't make too much of a big deal about knowing how to model something by appropriately employing Riemann sums (appropriately slicing things, for example, along directions where things are well-behaved and can be considered locally constant)...just do a bunch of examples so students can model later experience on these. Don't worry about general conceptual development that will allow the use of integrals in any new situation so much as getting the student to be able to recognize that "If the force doesn't vary with x then you can just do force times distance, but if the force varies with x then you have to integrate". Or, we could aim to get the students to recognize that "When you are working with finite things we can add, but when we move to continuous things we have to trade the sum for an integral"...without emphasizing the nuts and bolts of Riemann sums and various subtleties.

I'm trying to tease out what the difference in emphasis is, here. There is surely a spectrum, here. Experienced mathematicians teaching calculus probably are guilty of trying to move their students toward thinking about calculus in a way that is preparatory for later, more modern, mathematics...whereas physicists want us to churn out great 18th century mathematicians...I don't know, but I'd really like to.

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    $\begingroup$ As a physicists I disagree with your physicists. $\endgroup$ Commented Dec 21, 2016 at 15:02
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    $\begingroup$ I think the weight of the evidence is that formal or mechanical computational skills are best retained if they are built on a substrate of conceptual understanding. However, it is important to distinguish between 2 different types of "conceptual understanding". Many mathematicians think that you can't "understand" something like a derivative unless you have a precise definition and a careful existence proof. I do not think that is helpful for beginners; rather, the kind of "conceptual understanding" that seems to be most important for beginners is more informal and heuristic. (cont'd)... $\endgroup$ Commented Dec 21, 2016 at 15:26
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    $\begingroup$ ... In that sense the optimum sequence for learning would be something like (informal conceptual) --> (formal, skill-based proficiency) --> (rigorous conceptual). Although to be honest I think that most really deep learning is not "monotonic" with respect to this order but rather involves tacking back and forth among these three phases. $\endgroup$ Commented Dec 21, 2016 at 15:27
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    $\begingroup$ @DanielR.Collins: instructors who do more rote-memorization presentations have higher passing rates in our classes (but students less prepared for a later real math course) Or less prepared for later physics and engineering courses. There is evidence that this does really happen: academia.stackexchange.com/a/75968/1482 . See the quote from the Braga paper. $\endgroup$ Commented Dec 22, 2016 at 14:07
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    $\begingroup$ @BenCrowell: That's a great reference ("teachers can either engage in real teaching or in teaching-to-the-test"), exactly my experience here (esp. with standardized, multiple-choice final exam regimes); thanks so much for pointing that out. $\endgroup$ Commented Dec 22, 2016 at 15:00

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I'm primarily a physicist, but I also teach first-semester freshman calc once in a while. Your characterization of a cultural divide between physicists and mathematicians on this subject does not seem at all accurate to me. If anything, I think the characterizations should be reversed, at least on the average -- but it would only be an average, because different teachers are different.

Here is a dialog that I have had many, many times with my physics students in my office hours:

student: I'm supposed to find the maximum power. How do I do that?

me: Think back to your calculus.

student: Oh, I can do it using calculus? OK, so ... what do I do?

me: It's a function, and we're trying to find an extremum...

student: Oh, an extremum! So I set it equal to zero.

me: Set what equal to zero?

student: The power?

me: Er, if you were running a business, and you wanted to maximize your profits, would you set your profits equal to zero?

student: Oh, no. Huh. So what do I do?

At this point, I prompt the student to sketch a function with a maximum and draw a tangent line at the peak. Then they remember that they should be taking the derivative and setting it to zero. The point of relating this dialog, which I've had dozens of times over the years, is that my students' problems are almost never with computing things using calculus. The problems are with recognizing when they need to use their calculus, and applying it in a way that shows conceptual understanding.

our colleagues in physics and engineering departments (see my first link) want to see students proficient in mechanical computation...and cannot understand why two semesters of calculus are needed to train students to do this.

As described above, this remark is very hard to reconcile with my experience. I would also point out that in freshman calc and freshman physics classes these days, biology majors are about as numerous as engineering majors. (This varies, e.g., here in California, the UC system requires calc-based physics for bio majors, but the Cal State system doesn't. I teach at a community college, so we serve both populations.) The real problem IMO is that the content of second-semester calc is utterly irrelevant to the biologists. They are simply never going to integrate using a trig substitution or evaluate a Taylor series.

I think it's important for calculus students to understand why calculus works, not just how to do computations. However, I find that many of my colleagues who teach calc imagine that the "why" is very narrow. In the case of a derivative, they seem to think of "why" understanding as absorbing the definition of the limit, being able to do epsilon-delta proofs, and applying those skills to computing derivatives, before learning to do derivatives using computational rules. "Why" knowledge should also mean the kind of knowledge lacked by the student in the dialog above. And it should mean things like understanding why the Leibniz notation makes sense, interpreting differentials as small changes, being able to explain why the chain rule makes sense in terms of dimensional analysis, and being able to sketch the graph of the derivative of a function given a graph of the function.

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    $\begingroup$ There is a spectrum of opinion on the things found in your last paragraph, and I think this spectrum is what I'm asking about... $\endgroup$ Commented Dec 22, 2016 at 12:14
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    $\begingroup$ @Jon I think there is a context dependency that affects the ability to recall and apply knowledge. The teacher ought to take some responsibility for showing the students how to apply in the current course what they've learned in another. I think this goes for physics teachers as well as for mathematics and other teachers. $\endgroup$ Commented Dec 27, 2016 at 17:56
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    $\begingroup$ Depending on what exactly is contained in "second-semester calc", I might actually think that it should be made relevant to biologists, see my answer to another question: matheducators.stackexchange.com/questions/2060/… At least first-order differential equation, exact integration of simple rational functions with parameters, partial derivatives and first-order Taylor formulas are quite relevant to many biological model, and the innumeracy (or should I say the acalculy?) in biology is actually an issue. $\endgroup$ Commented Jan 3, 2017 at 11:18
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    $\begingroup$ @Ben Crowell: I must have been in a hurry when I commented above, because I was actually rather happy to see your answer. I should clarify that I am not in the camp that believes "why" knowledge has much to do with epsilon-delta proofs and such, but more that "why" knowledge is about the basic coherence of ideas of what the derivative means in context. This takes time to teach, often via encountering the concepts in various contexts in order to see the common features. The physics colleagues I'm referring to seem to complain that it is not so hard to teach someone to do nx^{n-1}... $\endgroup$ Commented Oct 10, 2019 at 14:06
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    $\begingroup$ @Ben Crowell: I should also clarify that I was an undergraduate physics major, and so find the attitude of some of these colleagues a little surprising...since as a physics major I remember thinking about this coherence and context as something that took a lot of time and contemplation. Labeling pictures and understanding meaning was very close to seeing how the ideas are "physical". If math colleagues also want to get epsilon-delta intuition in there, it seems that this should take MORE time, not less...which is what surprises me about this programming suggestion! $\endgroup$ Commented Oct 10, 2019 at 14:08
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With regard to your comment "talking about 'infinitesimal changes' like an 18th century mathematician or physicist". The implication of this comment is that we should not teach students this way, because this way of teaching is outdated. However, today we can combine intuition and rigor in a course in infinitesimal calculus; see this article. Infinitesimals are no longer things of the 18th century, but rather the cutting edge of both mathematical research and mathematics education.

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    $\begingroup$ I am not saying that rigorous infinitesimals are in any way wrong, I am saying that the way we teach students, typically, to think about them does not come to grips with subtleties suggested in the linked article. We introduce them in a way that an 18th century mathematician or physicist may have thought about them, not how Robinson thought about them. This was my point. $\endgroup$ Commented Apr 19, 2017 at 21:46
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    $\begingroup$ This isn't an answer. This would have been more appropriate as a comment. $\endgroup$ Commented Apr 20, 2017 at 0:37
  • $\begingroup$ @BenCrowell, OK, I addressed this in my answer. $\endgroup$ Commented Apr 20, 2017 at 7:06
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    $\begingroup$ @JonBannon, I think of our course as an ordinary calculus course, as opposed to honors calculus based on epsilon-delta exclusively. Our experience indicates that the students find the material accessible and do not find it difficult "to come to grip with" as you put it. $\endgroup$ Commented Apr 20, 2017 at 7:08
  • $\begingroup$ @Mikhail Katz: Let's put it this way, many of the freshmen many of us teach would glaze over at the definition of hyperreal numbers and for them the takeaway would be that there are infinitely small positive real numbers. Serious math students can probably handle what you suggest, but many students would not do well with it...the same weaker students, though, would not be amenable to the completeness axiom or any construction of the reals from the rationals. $\endgroup$ Commented Apr 27, 2017 at 23:13
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There is evidence that both a computational and conceptual approach are needed: https://www.jstor.org/stable/3482237

The paper of Sfard linked to does seem to agree that the scale must tip first toward computation, in the beginning, but with a view toward conceptualizing.

The paper aims to uncover why mathematics is hard for many to learn, and makes the interesting claim that process and concept are prerequisites of one another...which is clearly problematic. Unsurprisingly, "programming" students with procedures and very few concepts won't work for the reasons we expect (students end up memorizing many things with few relations between them) and overemphasizing abstract concepts without sufficient work with procedures amounts to talking about things completely divorced from student experience...and this is no good, either.

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  • $\begingroup$ It should be noted that the linked article (from Educational Studies in Mathematics, about seeing math concepts as either objects or procedures) is pretty much entirely theoretical, and doesn't have any hard study data one way or the other. $\endgroup$ Commented Aug 8, 2024 at 3:54
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I'm compelled to add a quote that's taken residence in my head for a long time. Looking at the question title "Should we 'program' calculus students, like the physicists seem to want us to?", and OP's further motivation for this question:

The reason I ask this is that our colleagues in physics and engineering departments (see my first link) want to see students proficient in mechanical computation... We, though, as mathematics professors, try to emphasize and train reasoning and concept. Some of us (like myself) trying very hard to get students to move past mechanical symbol pushing toward metacognition.

I'll point out that more than one commentator to the question was surprised by this, and opined that's not how their local engineering faculty would see things. In that vein, here's the observation that made a big impact on me. From Stein/Barcellos, Calculus and Analytic Geometry, 5E, "To the Instructor", p. xxii (1992):

At the Tulane conference on "Lean and Lively Calculus" in 1986 we heard the engineers say, "Teach the concepts. We'll take care of the applications." Steve Whitaker, in the engineering department at Davis, advised us, "Emphasize proofs, because the ideas that go into the proofs are often the ideas that go into the applications." Oddly, mathematicians suggest that we emphasize applications, and the applied people suggest that we emphasize concepts. We have tried to strike a reasonable balance that gives the instructor flexibility to move in either direction.

So in short, the premise of the present question is at best not established. I know I don't feel like calculation precedes understanding for myself, and I don't see any evidence in the link to the Devlin blog for it generally. Moreover, we'd need a wider survey of physics/engineering faculty to see if they mostly feel the way this question describes them; at least for Stein/Barcellos at the Tulane conference, they seem to have somewhat bigger/better evidence that the case may be reversed.

I'd like to think that the search for evidence-based best ordering of pedagogy would be useful, although (somewhat separately) it's too often the case that any findings are very local, subject to many variables, and not replicable large-scale, to really hope for great results here.

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    $\begingroup$ This is very interesting, Daniel. In fact, I was wondering if the phenomenon I was seeing was local to my institution. $\endgroup$ Commented Aug 8, 2024 at 11:45
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That's a good but very tough question. It seems to me that by only explaining what the student is required to do, or by correctly explaining what and incorrectly explaining why it is that way, the student is stimulated to figure out for himself or herself the 'why' bit.

Spike Milligan's father told him lies quite a bit when he was a child, and he turned out a genius who was very good at figuring things out for himself.

"You cannot teach a man anything, you can only help him find it within himself." -'Galileo

"If you teach a man anything, he will never learn." ― George Bernard Shaw

I suspect that a very narrow view of understanding is being taken when you say that someone is able to solve an equation correctly but doesn't understand what he or she is doing. As I see it, some sort of understanding is needed, otherwise the answer will not be correct, though it might not be the type of understanding we think is desirable.

Maybe the teacher should make the student figure out the meaning of what he or she is doing by putting challenging problems to the student that can't be solved without the supposedly superior/desired type of understanding.

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