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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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When I learned the rule for differentiating a power, I actually understood why it worked. I'm not sure if it's the case for everybody. If it's not the case for everybody, I have a possible idea but it's only my theory. It's to change the entire education system to teach a whole lot less and have people discover things on their own based on what they were taught. That way, students will become smarter and know how to check things themselves with their smartness in an engineering job and not just put blind trust into rules they were taught always working. They may later encounter new and different formula that either are actually wrong or are liable to be misunderstood so I think it's good to get them into the habit of checking things on their own. Also, some people if they do just get taught how to do it without understanding and then later get an engineering job, even if the boss of that job knows that those formula are mathematically correct, that person might still refuse to put blind trust into using formulae they don't know how to prove are mathematically correct because they don't know that other people aren't making a mistake in thinking they're correct and will have to be respected for refusing to blindly apply formulae they don't know how to prove in the job.

Before that change is made, it will have to be really well researched exactly how the whole system and job market should change where a lot of knowledge moves to work place specific training. Finland has a really good education system so the Finish National Agency for Education might be able to help them with that research. Here's my possible idea which may need researching whether or not to do. I think a lot of students when they get taught that much end up learning less because the curriculum is too fast for them to follow so once we remove the need to be taught so much, they might end up learning more.

Here are some basics of how Polish notation works. To denote a sum of two expressions, you write + followed by each of those expressions but with spaces in between. Polish notation also expresses natural numbers in standard decimal notation with no spaces between the digits but spaces on both sides of the decimal notation. We could use a modified Polish notation that I invented. We could use ∅ to denote 0, S to denote the successor operation, and + to denote addition where once you're constructed a notation for a natural number, you can stick an S onto the beginning to denote its successor but without a space in between. Also, once you're constructed two notations for a natural number, you can use + followed by one of those expressions and then the other expression with no spaces in between to denote their sum. After learning just a few basics like the number 0 denoted ∅, the successor operation denoted S, and the addition operation denoted + and write it in Polish notation, the students can be left to do almost all play with very little teaching. Then they might discover what they're interested in discovering based on those basics and become really smart, but they will also have the right idea of how natural numbers work because there's no dispute about how they work. Also by using Polish notation, the notation is already unambiguous and already has no brackets so the're no need to learn PEDMAS and the time can instead be used to teach something else.

Also, we could switch to senary notation but wait until way later to teach it when they can actually learn it more efficiently. They could be taught that the senary notation of a number is the string of characters that describes the method of getting to that number by starting from 0 and then applying a series of operations each of which is of the form of left multiplying by 6 then left adding a number from 0 to 5. So the senary notation for 95 would be +SSSSS∅×SSSSSS∅+SSS∅×SSSSSS∅+SS∅×SSSSSS∅∅. I think senary is the base that people should use because it makes it easy for students to teach themselves how to find the quotient and remainder of any division problem because it has such a simple single digit multiplication table because every natural number from 1 to half of 6 is a factor of 6. They probably should not even be given the single digit multiplication table and instead be left to completely teach themselves how to find the quotient and remainder in senary, and they will realize that some of the steps are to mentally compute the product of two single digit numbers all on their own without ever having seen a multiplication table. Here, the problem will be considered entirely a problem about pure number theory because they have not yet been taught how to compute a division problem between natural numbers and give the answer as a non whole number in mixed fraction notation. Maybe way later after they've got losts of experience performing calculations in senary notation, then they could be taught that

  • the character 0 is short hand for +∅×SSSSSS∅
  • 1 is short hand for +S∅×SSSSSS∅
  • 2 is short hand for +SS∅×SSSSSS∅
  • 3 is short hand for +SSS∅×SSSSSS∅
  • 4 is short hand for +SSSS∅×SSSSSS∅ and
  • 5 is short hand for +SSSSS∅×SSSSSS∅

Then the compact senary notation for 95 would be 532∅. I know the digits appear in backwards order from conventional senary notation but I think it's worth it for the new school kids in order to define a digit as an operation and have it proceed the operand. If they're introduced to it too early, they might forget how they're really performing calculations in senary and adopt an autopilot method of performing them that they find so intuitive and can't figure out how to break down further. Also by convention, those digits should strictly be used to denote an operation of multiplying by 6 then adding a number and not to denote an operation of multiplying by 10 and then adding a number because it might be confusing having multiple meanings. The convention should be that you use decimal when writing numbers the normal way and senary when using digits to denote operations in modified Polish notation. Then the switch to senary will occur at the same time as the switch to modified Polish notation. If somebody wants to use another base like quinary, they should write it as a long string of characters which really is a valid notation for that number instead of using a digit to represent a different operation than it really means. So it's correct to denote 95 as +∅×SSSSS∅+SSSS∅×SSSSS∅+SSS∅×SSSSS∅∅ but it's not correct to denote 95 as 043∅.

Later when they're a lot smarter and can learn it better, maybe they could be introduced to the negative numbers and even later to all the real numbers, but probably should not be left to discover properties all on their own because they might get the wrong idea of how real numbers work. Instead, the teacher could construct the dyadic rational numbers, the numbers that have a terminating notation in binary except that they don't actually introduce the binary decimal expansion, and then the rest of the real numbers from those Dedekind cuts of the dyadic rational numbers where the lower part has no maximal element nor does the higher part have a minimal element. According to https://nrich.maths.org/2550, students work well with halves.

By the time they start taking calculus and already know how the real numbers work, I think they can be taught to teach themselves Calculus. It might actually be better for them to teach themselves because they might become smarter in the long run as a result. They should probably just about once a month be given some homework problems and be given a whole month to complete them so that they will have more than enough time to teach themselves Calculus.

They could be told just a few basic laws such as $\forall a \in \mathbb{R}\forall b \in \mathbb{R}\forall c \in \mathbb{R}$, if $a$ is positive, then $a^{b + c} = a^b \times a^c$ and exponentation to a natural number exponent is iterated multiplication where 1 is the empty product and exponentation is continuous. The criterion that exponentation is continuous allows you to evaluate a positive number raised to an irrational power. They could also be taught that a logarithm is the inverse of a left exponentation and a root is the inverse of a right exponentation such as taking the third power. They could be taught using a student centered approach until they understand the basics and then guided to figure out other properties all on their own without being told what they are such as $\forall a \in \mathbb{R}\forall b \in \mathbb{R}\forall c \in \mathbb{R}$, if $a$ is positive, then $(a^b)^c = a^{b \times c}$. After they get taught what a derivative is, then they could be told that $e$ is defined to be the number such that $\frac{d}{dx}e^x = e^x$. Then they could be left to teach themselves how to differentiate any function that can be gotten from the functions they were already introduced to.

Maybe schools could get students to teach themselves how to differentiate any function that can be gotten from the functions they know how to differentiate by giving them one homework problem for every 3 school days. For example, their first homework problem could be to differentiate $e^{e^x}$. Then they'll teach themselves the rule for differentiating a composition. Next, they could be asked to differentiate $\ln x$, then they will teach themselves the rule for differentiating an inverse. Next, they could be given the problem of differentiating $x^{2.5}$ and $2^x$ and told that it will not be for 6 school days that they are given more homework problems. Then they'll figure out that they can do that by expressing $x^{2.5}$ as $e^{\ln x \times 2.5}$ and $2^x$ as $e^{\ln 2 \times x}$. 6 days later, they could be asked to differentiate $(x^2 + 1)^{x^3}$. Then they'll teach themselves that for any binary operation, if you know how to differentiate any expression of the form of that binary operation on two seperate operands where one operand of that binary operation is constant and the other operand is a function you already know how to differentiate, then you use the rule for differentiating the expression treating one operand as constant and then the rule for differenting while treating the other operand as constant and then add the two results to get the derivative of the whole expression. 3 days later, they could be asked to differentiate a product and they'll see that they can do it by using the general rule for differentiating a binary operation. Next, they could be asked to differentiate $\log_x2$. Then they'll figure out to use the general binary operation rule in reverse. Then they'll figure out that $\frac{d}{dx}x^{\log_x2} = \frac{d}{dx}2 = 0$ but also $\frac{d}{dx}x^{\log_x2} = x^{\log_x2}\ln x(\frac{d}{dx}\log_{x}2) + \log_{x}2(x^{\log_{x}2 - 1})$ so $\frac{d}{dx}x^{\log_x2} = x^{\log_x2}\ln x(\frac{d}{dx}\log_{x}2) + \log_{x}2(x^{\log_{x}2 - 1}) = 0$. Then they can use that equation to solve for $\frac{d}{dx}\log_x2$.

The article Teaching Is not Learning — The Guided Discovery Approach for Learning seems to support my idea that discovery learning could work so well.

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