How does knowing more about mathematics help one's teaching of lower level course, such as calculus?

Question

A question has been asked about why great mathematicians are not necessarily great teachers. On the other hand, I am wondering if knowing more mathematics actually helps with one's teaching of lower level courses in mathematics. For example, I believe that a good student with bachelor's degree in mathematics should have sufficient knowledge to teach calculus. However, how does having a master's or doctorate degree help one's teaching in calculus, if any at all?

I am teaching calculus now and I do not understand commutative algebra; I took a course on commutative algebra long time ago; I did poorly in the course and now I could hardly recall anything from this course. If I invest substantial among of time studying this subject well now, will it help me in my calculus course in any sense?

Answer 1

I will speak purely anecdotally.

However, how does having a master's or doctorate degree help one's teaching in calculus, if any at all?

It isn't the degree per se that helps, but rather the process of having to learn, relearn, and reformulate gargantuan amounts of mathematics of all types that helps. Do I use either my PhD research or my current research in teaching non-major freshman calculus? Not at all. (I do get to mention my current research in linear algebra, so it is possible, of course.) But what I do use on a daily basis is the practice that comes from intense advanced study in seeing every side of a mathematical problem.

To give just one example, when teaching Riemann sums (often without using this phrase), one can just do left and right hand sums, or maybe trapezoid or midpoint if you are ambitious. Fine. What happens when a student decides to pick some left and some right hand endpoints? What happens when a student wants to pick only integer points? What about the student who insists that on the interval $[0,2]$ with $n=2$ you should use $f(0),f(1),f(2)$ with equal weight? Answering these questions beyond "the book says that is wrong" requires experience, and the ability to try to see what the many possibilities are from all sides.

Now, naturally you can obtain this experience in a multitude of ways. I have colleagues in other departments without a PhD whose deep understanding of the material they teach is unparalleled. And getting a doctorate doesn't automatically mean you magically look at calculus (or lower-level courses - I find that precalc is actually conceptually a bigger target because it roams so far and wide) better.

But, on the whole, I find that significant advanced study in mathematics makes me a much better calculus teacher than I would have been otherwise. Don't worry about commutative algebra; do worry about finding ways to stretch yourself mathematically early and often - whether through formal means or informal.

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Can't resist:

If I invest substantial among of time studying [commutative algebra] well now, will it help me in my calculus course in any sense?

If you want to talk commutative algebra and calculus, take aside the two students who really want to do deep AI/machine learning/big data/whatever, and tell them about dual numbers, automatic differentiation, and tangents. Hard to find parts of math that don't influence each other in some way - but of course you don't tell this story to 99% of students! Anyway, the point isn't the specific math content, it's the process.

Answer 2

I'll give an answer by analogy. When you are a kid, you should be exposed to playing with other kids or participating in activities such as team sports. It's natural to ask, "Does playing soccer at seven years old help a child become a better adult?" And the answer seems to be that, in general, yes it does help. Not because of soccer per se, but because it is important for the child to learn how to interact with others, work with others in a cooperative environment, learn about the importance of practice and hard work, and learn how to listen to their coach to improve their skills. Of course, that's not what the child notices. Hopefully, they're just having fun and focusing on the immediate task in front of them.

And it doesn't mean that if a kid plays soccer, then they'll automatically be better at those bigger goals than a kid who doesn't play soccer, but on average playing soccer will be a positive influence. Those experiences tend to accumulate and help shape who we become as adults.

I think studying advanced math and getting a Ph.D. work the same way. Most of the time is hopefully spent enjoying the opportunity to learn mathematics and focus on the problem in front of us. At the same time, we have an opportunity to learn a plethora of skills in the process. Among them are skills that help with our teaching - the ability to think about problems in greater depth, to question which assumptions are really needed, to better understand the essence of an argument or a calculation and to hopefully improve our ability to communicate mathematical insights to others. Again, not all Ph.D. recipients acquire all of these skills, nor is it the only possible route to acquire them, but on average, math Ph.D.'s are much more skilled in these areas than people whose math education stopped much earlier.

Documenting exactly how an advanced degree can make you a better teacher is non-trivial. It's not clear the extent to which the experienced process (which takes years to complete) can be summarized succinctly. But experience shows that those with a Ph.D. in mathematics (or related fields) are generally - but again not universally - much more prepared to teach the content required for a math course such as calculus.

I think it's much more debatable if math Ph.D. recipients are better at the other aspects of teaching calculus (how they deliver the content) than others. And it's debatable how much relative importance good presentation of the material has compared to a good choice of content. But without a strong understanding of the content, no amount of brilliant presentation can help a teacher who either makes a poor choice of how to present specific content or, worse, presents that material incorrectly because they lack a full understanding of the relevant content and the proper perspective to view it in.

Answer 3

The title of the question asks how knowing "more about math" can help in teaching calculus, while the question itself asks specifically about the math learned towards a masters degree or a doctorate. I am going to focus on the "more about math" part, regardless of the stage at which it was acquired, because the math that one person learned in grad school could be learned as an undergraduate by someone else.

A solid understanding of analysis of all kinds (such as real analysis, complex analysis, and Fourier analysis) gives an instructor in a calculus course a thorough mastery of the intricacies behind the often confusing topic of infinite series. It means the instructor understands what makes all those convergence tests work and really understands the math behind boundary behavior of power series ("at the endpoints") or forming power series expansions of the same function around different points. This can help in the construction of examples for the class that emphasize a certain issue and it means the instructor should be up to the challenge of answering almost any non-routine question calculus students may ask about infinite series.

Of course a calculus instructor who has not taken a lot of analysis courses could teach himself/herself why all the convergence tests for series work, but I think the experience of using series in more advanced coursework gives you a perspective on the value of this topic and how it gets used later that someone whose analysis knowledge ends at calculus doesn't have. It's kind of like taking high school French from someone who just knows high school French very well compared to taking high school French from someone who is actually fluent in French (assuming that both instructors actually want to teach French!). Maybe on most days the students won't notice a difference, but as soon as someone asks about something going even slightly beyond the regular curriculum, the instructor with more limited knowledge will be at a loss to answer the questions or point out connections that more mathematically experienced instructors know well.

Since you bring up commutative algebra in your question, let me give an example of how one topic from that could help an instructor understand calculus better: completions of commutative rings include as a special case the rings of formal power series $\mathbf R[[x]]$ and $\mathbf R[[x,y]]$. Some calculations with power series really work if you use just a formal manipulation of the terms, while others are genuinely numerical (convergence issues are essential for the result to even mean anything). When I was first learning calculus, I found the idea of division of power series (e.g., the ratio $\sin x/\cos x = (x - x^3/3! + \ldots)/(1 - x^2/2! + \ldots)$ near $x = 0$) a bit puzzling. Knowing about both analytic functions in complex analysis and formal power series made me much more comfortable with division of power series. (For the record, my understanding of formal power series actually came from my reading about $p$-adic analysis rather than commutative algebra.)

Knowing some number theory and complex analysis gave me a much clearer idea of what is going on with integration of rational functions in general, which if you look in a calculus textbook appears to be a large number of disconnected different cases, of which we may teach only 1 or 2 to avoid excessive hand calculation. If a student were to ask me for some intuition behind decompositions like $1/(x^3+x) = 1/x - x/(x^2+1)$, I can point out a numerical analogy: $7/(3 \cdot 5)$ can be broken up into some number of thirds and some number of fifths: $7/15 = 2/3 - 1/5$. The principle is that when you have a denominator that is a product of "relatively prime parts", you should expect to be able to break it up into a sum with those separate parts as their own denominator. It works for denominators of fractions just as much as it works for denominators of rational functions. I don't know what other people would say if a student asked for intuition behind the partial fraction decomposition process: it just works?

Knowing some algebraic geometry lets me tell students why implicit differentiation has a surprising practical application that they use without even realizing it: the security in ATM cards and other smart cards is based on elliptic curve cryptography (ECC), and the math behind ECC involves being able to compute tangent lines to points on curves like $y^2 = x^3 - x$. (Technically, ECC uses elliptic curves over finite fields rather than over the real numbers, and the equations look more like $y^2 + xy = x^3 + x + 1$ because of issues involving fields of characteristic $2$, but the basic ideas for finding tangent lines are still largely the same as over the real numbers.) Algebraic geometry also lets me know the motivation behind the otherwise peculiar $\tan(t/2)$-substitution in calculus, which turns integrals of rational functions of $\sin x$ and $\cos x$ into integrals of rational functions.

Answer 4

There's a general point I can't see explicitly in the other answers: knowing more maths (and generally having spent time knowing/thinking about maths) helps you have a bigger picture. A lot of maths starts to fit together better as you know more for longer.

As a (not very good) analogy, suppose someone left a sandwich on a bench, and an ant found it. The ant could teach all its ant friends how to reach the sandwich up one of the legs of the bench, and that is arguably enough. But it can't guess that there are three other legs they could climb up.

Having a sense of the bigger picture doesn't automatically make you better at teaching. But it does make it easier to be good at teaching. It means you have a better idea which bits are important, which aspects come up elsewhere in maths, how to reverse-engineer suitable exam questions...

Answer 5

I believe that a good student with bachelor's degree in mathematics should have sufficient knowledge to teach calculus.

I agree with this. I work with a few colleagues who came to teach at the community college after working in the high school, and their credential is a Master's of Science for Teachers (MST). They have never taken graduate (theory) courses in algebra, analysis, geometry, etc, but they are excellent and dedicated teachers/colleagues -- attributes I chalk up to their teaching experience. This experience speaks a lot to me about their potential, more so than a graduate degree.

If I invest substantial among of time studying this subject well now, will it help me in my calculus course in any sense?

In short -- no. While it is good to have a depth of knowledge in a topic so you can engage students when they ask something "out of the box" (like the three students asking me about fractional derivatives in the past two weeks alone), its use has its limits, and the law of diminishing returns applies. It is a reminder that a teacher is not there to simply have knowledge to spew forth, but to help students take on new information. I believe a "good teacher" should be able to take something just out of reach of their own understanding and still help a student learn it. [Of course, they should be willing to learn it alongside that student.]

...how does having a master's or doctorate degree help one's teaching in calculus, if any at all

Simply put, a graduate degree is just a minimum qualification adopted by institutions of higher ed for their faculty. It (and a good curriculum vitae) will get you in the door for an interview, and then you must show your skills as an educator. If this is hard to hear, consider the perennial student complaint "when am I ever going to use this". The answer for them is the same as for the math PhD who wants to teach beginning algebra: "Perhaps you won't."

Answer 6

An aspect that does not show up much in other's answer is: exercise & test design. To construct an exercise, it is very often quite useful to have more advanced knowledge, either to be able to choose the values correctly, or to know what will echo later courses.

Some example:

• knowing finite fields helps understanding that the coincidence between equality of polynomials and equality of their associated polynomial functions is not obvious, and needs a proof (it does not make it so much easy to convey, though),
• having learned quite some differential geometry, I know that constructing a smooth function that is zero on $(-\infty,0]$ and positive on $(0,+\infty)$ is not just for practice, but turns out to be very useful (much) later on,
• when I give a family of vectors for students to decide whether it is free, and whether it is generating, I am happy to have a set of available tools (e.g. dimension theory) that helps me knowing some of the answers in advance, and checking theirs,
• Knowing the partial fraction decomposition of a rational function helps designing exercises on sums, integrals, and systems of linear equation that will put these subject closer together.

Additional examples on the already discussed fact that knowing more gives you better perspective (and warns you from some misconception, and help you answer student's questions):

• knowing Weierstrass continuous nowhere differentiable functions helps me understand better the difference between continuous and differentiable functions; so does knowing that generic continuous functions are nowhere differentiable,
• knowing Taylor series helps understanding derivatives, in particular proving the formula $(fg)'=f'g+fg'$ is much more transparent¹ when one defines $f'(x)$ as the number (when it exists) such that $f(x+h)=f(x)+f'(x)h+o_{h\to 0}(h)$,
• knowing that the antiderivatives of $x\mapsto e^{x^2}$ cannot be expressed with usual functions helps warning students that finding explicit anti-derivatives (or solutions of differential equations) is not always possible, and helps explaining why we define new function names, and what math is really about.

¹ Added in edit. I was asked by email to give more details on this point, as well doing it here. With the first-order Taylor series definition of the derivative, whenever $f,g$ are differentiable at $x$ we get \begin{align*} (fg)(x+h) &= \big(f(x) + f'(x) h +o(h)\big) \big(g(x)+g'(x)h+o(h)\big) \\ &= f(x)g(x) + \big(f'(x)g(x)+f(x)g'(x)\big)h + f'(x)g'(x)h^2 + o(h) \\ &= (fg)(x) + (f'g+fg')(x) h + o(h) \end{align*} so that $fg$ is differentiable at $x$, with the derivative as we know it. This is much more transparent than the usual trick of computing the ratio by adding and subtracting some quantity in order to factorize, as the cross products $f'g$ and $fg'$ appear naturally when developping, and the product $f'g'$ that student might think should be the answer appears clearly, but as a second-order term. (Of course, beware that the higher-order Taylor formulas do not entail high differentiability, a misconception that this approach could unfortunately enforce.)

Answer 7

There are four ways an advanced knowledge of mathematics can help, in rough order of importance: Mastery, understanding student thinking, context and content and tricks. However, general teaching skill also matters, and I would guess that it does not reliably increase along with mathematics studies, or at least that the effect is small.

Mastery

All other things being equal, someone with more experience with difficult mathematics is likely to have mastered lots of easy mathematics. Even when the particular subject is new to them, they have learned how to easily learn a new mathematical idea or abstraction.

Mastery gives confidence, speeds up preparation and makes it easier to find mistakes. It is all around helpful.

Understanding student thinking

You need to master several ways of thinking about something as simple as multiplication, not to speak of more advanced mathematics. It helps to have developed mathematical thinking to be able to understand these different ways of seeing (say) multiplication, and also thereby being able to understand non-standard solutions and solving methods of students. The same is true of common misunderstandings and explaining what is the problem, or maybe giving a spontaneous counterexample of exercise that illustrates the different ways of thinking.

In particular, you need to do this on the fly, in front of a classroom. It is not easy!

Context

Having learned higher mathematics gives context for the basics. This helps in making the content meaningful and in finding examples and applications, which can be made into exercises or project work.

Tricks and content

Sometimes, though rarely, the specific techniques or content one has worked on perfectly line up with the teaching. I doubt this is very common, and the more advanced mathematics one is teaching, the more likely it is. Applied research and courses for students of relevant subjects might also increase the incidence.

Answer 8

My personal impression across many fields is that a certain amount of extra subject matter can help (especially with "top high school" classes like AP Calculus or AP Chemistry or AP Bio). but that by and large, the issues in teaching and learning intro topics like first year college calc, chem, physics, bio, and through diffyQ and engine math are MUCH more about the basics AND about PEDAGOGY as opposed to being further advanced in your field. I have seen in chem where the salty old (barely passed a Ph.D. and with the grace of his adisor) had way more empathy and ability to teach freshman chem (e.g. analogy of wheels to car for limiting reactant) versus the shiny research profs. Plus he knew stoich and equilibrium problems inside out.

So, net, net: I don't think advanced education really helps you be a better teacher. Often it can even be a negative (look at the peeps here wanting to "do something interesting" rather than horsing up the troops.)

Really if you even have a math UNDERGRAD (as opposed to ed) , you are well, well equipped to teach calculus. After that, really concentrate on being a salty, salty TEACHER. Not a mathematician.