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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. $\endgroup$ – Wrzlprmft Dec 21 '16 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)... $\endgroup$ – mweiss Dec 21 '16 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. $\endgroup$ – mweiss Dec 21 '16 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. $\endgroup$ – Ben Crowell Dec 22 '16 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. $\endgroup$ – Daniel R. Collins Dec 22 '16 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... $\endgroup$ – Jon Bannon Dec 22 '16 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. $\endgroup$ – user1527 Dec 27 '16 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. $\endgroup$ – Benoît Kloeckner Jan 3 '17 at 11:18
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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. $\endgroup$ – Jon Bannon Apr 19 '17 at 21:46
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    $\begingroup$ This isn't an answer. This would have been more appropriate as a comment. $\endgroup$ – Ben Crowell Apr 20 '17 at 0:37
  • $\begingroup$ @BenCrowell, OK, I addressed this in my answer. $\endgroup$ – Mikhail Katz Apr 20 '17 at 7:06
  • $\begingroup$ @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. $\endgroup$ – Mikhail Katz Apr 20 '17 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. $\endgroup$ – Jon Bannon Apr 27 '17 at 23:13

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