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Myths of Mathematics Teaching

Daniel J. Velleman

By “myths of mathematics teaching” I mean statements about mathematics teaching that one might reasonably suppose are true, but that are not always true. I am not suggesting that anyone has actually asserted these statements; nor will I argue that there is no truth to them. I am merely claiming that one might be tempted to assume that they are true, but that such an assumption merits some skepticism. Perhaps the best way to explain what I have in mind is to state my first myth.

1. The Reason We Teach Students to Solve Problems of Type $X$ is So That They’ll Know How to Solve Problems of Type $X$

Perhaps no one has ever actually said this, but it sounds so much like a tautology that it may be hard to imagine how it could be false. And of course it is often true; we do want students to know how to solve some of the problems we teach them to solve—for example, how to compute derivatives and integrals and how to find extreme values of functions. But sometimes we have other reasons for teaching students to solve certain kinds of problems.

A good example of this might be related rates problems. Perhaps some people don’t care whether or not students can solve related rates problems, but teaching the topic has other educational benefits. It gives students practice with treating real-world quantities that change over time as unspecified functions of time and interpreting their rates of change as the derivatives of those functions; it gives them practice with the chain rule and implicit differentiation; and it teaches them the technique of taking the derivative of both sides of an equation. This technique is subtle; it can be used only on an equation between functions (not an equation between numbers), and understanding this requires a conceptual understanding of the meaning of the derivative (see Figure 1). I have probably never, in my life as a mathematician, had to solve a related rates problem (except as an exercise in a calculus class). But I have often found it useful to take the derivative of both sides of a functional equation, and that is a technique I learned when I took calculus and had to solve related rates problems.

Graphic without alt text — **Figure 1.**
Calculus books often warn students that in related rates problems, they should not plug in values before differentiating. These figures explain why.

Should we cover related rates in calculus? I’m not going to take a position on that question. There are many worthy topics in calculus; often the semester isn’t long enough to cover all of them, and choices must be made. But my point is that when making the decision about related rates, we shouldn’t simply ask ourselves the question, “Do we want our students to be able to solve related rates problems?” A better question to ask is, “What are the educational benefits of teaching related rates problems?”

2. When Students Have Trouble Learning to Work with a Mathematical Concept, What They’re Having Trouble with is the Mathematical Concept

Again, this sounds so reasonable that one might assume without thinking that it must be true. And of course it is often true—but not always.

My example this time is mathematical induction. Many students have trouble learning to write proofs by induction, and it is natural to assume that they are having trouble understanding the principle of mathematical induction. If they understood this principle, then they’d understand proofs based on the principle, wouldn’t they? To address the problem, we appeal to images that elucidate the principle of induction: if you knock over the first in a row of dominoes and each domino knocks over the next, then all of the dominoes will fall; or, if you step on the first rung of a ladder and you always climb from one rung to the next, then you will reach the top of the ladder. There’s nothing wrong with these images, and they may help students understand mathematical induction.

But when students have trouble with induction proofs, is it always the principle of mathematical induction that they’re having trouble with? Consider the induction step, in which one must prove a statement of the form $\forall n \in \mathbb{N}(P(n) \to P(n+1))$. Typically one proves this by letting $n$ be an arbitrary natural number, assuming $P(n)$, and proving $P(n+1)$. Students are sometimes confused about why we can assume that $P(n)$ is true for an arbitrary $n$. Doesn’t that amount to assuming $\forall n \in \mathbb{N}(P(n))$, which is what we’re supposed to be proving? Some students, confused about this distinction, may even mistakenly state the inductive hypothesis as $\forall n \in \mathbb{N}(P(n))$. It seems to me that students who experience this confusion are probably not confused about the principle of mathematical induction. Rather, they are confused about the logic of universally quantified implications. Proving a universally quantified implication almost always involves making an assumption about a fixed but arbitrary object, and understanding the logic of such proofs requires understanding that this is not the same as making the assumption about all objects. In some cases, what students need help with is this logic and not the principle of mathematical induction.

3. We Shouldn’t Do Things in Class That the Students Might Find Confusing

Confusing our students is a bad thing, right? You might think, therefore, that we shouldn’t do things in class that the students might find confusing. When preparing to teach a certain topic, you might try to find the way of thinking about the topic that will seem most natural to your students, rather than considering how you as a mathematician think about it. For example, in some courses you might choose to favor plausibility arguments over proofs. After all, students find proofs confusing; they seem to accept plausibility arguments more readily. You might even avoid entire topics because students find them confusing. But is that really the best way to deal with confusing topics?

Of course, straying too far out of our students’ comfort zone can lead to disaster—students may simply tune out if they become thoroughly confused. But that doesn’t mean that it is best to stay entirely within the students’ comfort zone. We can’t expect beginning students to think like mathematicians, but that doesn’t mean we shouldn’t nudge them in that direction, even if that risks confusing them. Confusion needs to be managed and responded to, not avoided at all costs.

Presenting something that confuses your students can sometimes create an opportunity to confront and clear up the students’ confusion. Consider, for example, a proof in which a mathematical object is introduced into the discussion with no motivation, as if it were just picked out of the air. Some students are bound to say that they find the proof confusing. They are likely to ask, “How did you get that?”

Students who ask this question are probably not confused about the proof in question; rather, they are confused about the rules of mathematical proof. They don’t realize that their question has two very different interpretations:

(1): How did you justify your conclusion?
(2): How did you think of your argument?

And they don’t realize that a correct proof must answer question (1), but it need not answer question (2). Presenting a potentially confusing proof that prompts this student question could lead to a valuable discussion of the two interpretations of the question and their different roles in proofs. But, of course, that discussion should be followed by an answer to the student’s question!

4. Rigor is an Advanced Topic

There are times when we must teach a topic to students who are not ready for a complete, rigorous treatment of the topic. But does that mean that we should consider rigor to be an advanced topic?

There are certainly ways in which demanding rigor makes mathematical reasoning harder. But there are also ways in which it makes it easier: standards of rigor make it clearer what steps are allowed in a particular mathematical context and what steps aren’t. Rigor makes the rules of the game of mathematics clearer, even if it makes the game harder to win. Sometimes students aren’t ready for a rigorous development of the theory behind a topic they are studying, but they still need to know what steps they are allowed to use when solving their homework problems and what steps are not allowed. Adhering to standards of rigor in class, and demanding that students adhere to them in their homework, can give students this necessary guidance.

Consider, for example, teaching techniques of integration in a calculus course. We cannot expect that students will always choose the right technique for evaluating an integral on their first try. So they must learn to recognize when a technique isn’t working and switch to an alternative technique. Recognizing when a technique isn’t working is much easier when reasoning is held to high standards of rigor; if the steps of an attempted solution cannot be rigorously justified, then an alternative approach must be pursued.

Here’s an experience I have sometimes had when teaching techniques of integration: A student comes to my office with an integral that they have computed incorrectly by using a step that cannot be justified. After pointing out the mistake, I ask the student if they can think of another technique that might work on this integral. Sometimes the answer is yes, and the student is then able to evaluate the integral. If only the student had held their work to a higher standard of rigor in the first place, they could have gotten the right answer on their own!

The advantages of rigor in mathematics instruction are described well by David Gries and Fred Schneider 1, p. 696:

Teaching mathematics through informalism is like driving in a fog. One sees dim figures in the distance, and every once in a while some of them suddenly appear clearly, but usually everything is veiled and mysterious. It’s dangerous to drive in the fog, especially in a strange territory, and one must drive slowly. Even so, one may not always be sure where one is. Teaching rigor and precision, provided it is done without the veil of complexity interfering, burns away the fog, leaving everything crisp and clear and making it possible to drive faster and to enter uncharted lands.

College-level mathematics is full of informal arguments that seem plausible but that, on further analysis, turn out to be flawed. Learning to detect these flaws is one of the greatest challenges of learning mathematics, and doing it when reasoning is kept at an informal level is especially challenging. Indeed, one might argue that it is not rigor that is an advance topic, but rather hand-waving.

5. Before You Can Start Writing a Proof of a Theorem, You Must Understand Why the Theorem is True

Perhaps some mathematicians believe that this is how they write proofs. But I think it is bad advice to give to students who are trying to learn to write proofs.

Would you ever tell students that before they can evaluate an integral, they must see intuitively what the answer is? Of course not. To evaluate an integral, you come up with a step that looks promising—perhaps a substitution, an algebraic reexpression, or an application of integration by parts. Then you write out the details of that step and see where it leads. Similarly, to start writing a proof you just have to come up with a first step.

Of course, when you are writing a proof, it can be very helpful to have an understanding of why the theorem is true. But if you make that a requirement, you risk causing students to be unable to get started on a proof, and getting started is often what gives students the most trouble. It is better to tell students that it is OK to write a step in a proof even if they don’t know where that step will lead.

My worry is that if we tell students that they can’t start writing a proof until they understand why the theorem is true, then the only theorems they will ever be able to prove are those that are so simple that they can see why they are true before they start writing the proof.

6. Conclusion

No doubt some readers—perhaps all readers!—disagree with some of my conclusions. But I hope I have at least convinced readers that some issues that might seem straightforward are deserving of further thought.

References

[1]: David Gries and Fred B. Schneider, Teaching Math More Effectively, Through Calculational Proofs, Amer. Math. Monthly 102 (1995), no. 8, 691–697, DOI 10.2307/2974638. MR1542732,

Show rawAMSref
\bib{gs}{article}{ author={Gries, David}, author={Schneider, Fred B.}, title={Teaching Math More Effectively, Through Calculational Proofs}, journal={Amer. Math. Monthly}, volume={102}, date={1995}, number={8}, pages={691--697}, issn={0002-9890}, review={\MR {1542732}}, doi={10.2307/2974638}, }

Article DOI: 10.1090/noti3088

Credits

Figure 1 is courtesy of Daniel J. Velleman.

Photo of Daniel J. Velleman is courtesy of Henry Amistadi.