# How to convey the meaning of “mathematical maturity”?

Some university-level courses have no specific prerequisites, yet are mathematically involved to the extent that someone with little to no experience in math will probably find themselves in over their heads. Such courses are often cross-listed and appeal to students in disciplines outside of math like philosophy, computer science, economics, psychology, etc. For example, certain courses in logic or decision theory often fit this bill.

One standard way of "warning" the students about the mathematical nature of the course is by saying something like, "The only prerequisite is mathematical maturity."

To me, this seems like an apt characterization: they're going to be asked to prove things on the homework, to absorb formal definitions fairly rapidly, etc. But just because I have an intuitive grasp on the meaning of "mathematical maturity" doesn't mean they do. I can't help but feel that this warning falls short with some and goes too far with others. I know from experience that some students underestimate this warning, and remain in the course despite, e.g., having no idea what it means to rigourously prove something. Unsurprisingly, these students tend to struggle greatly. On the other hand, I like to view such courses as good alternatives to more traditional courses like, say, calculus, for helping to strengthen and deepen general mathematical facility. It would be a shame if some students who would otherwise find the course quite useful overestimate the meaning of "mathematical maturity" and shy away.

How can I adequately convey to students the meaning of "mathematical maturity"? In some sense, it's one of those, "If you don't know it, you haven't got it" type things, but I'd like to have something more tangible than this, perhaps a method of assessment or a representative class of examples I could offer to students who are unsure.

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Perhaps add a 'required reading' addendum? Something like Hardy's Apology would be appropriate –  Daniel Littlewood Apr 7 at 19:48
@DanielLittlewood, perhaps better something like "The Book of Proof" by Hammack. –  vonbrand Apr 7 at 19:53
I'm afraid it won't be possible without some explicit assessment tool because of the Dunning–Kruger effect. However, I suspect that such a tool could be very simple, like "we expect a student to know what XYZ is and to solve this with a proof equivalent to that". –  dtldarek Apr 7 at 23:22
old joke: Mathematics is really Psychology. Psychology is really Biology. Biology is really Chemistry. Chemistry is really Physics. Physics is really Mathematics. –  Mike Jones Aug 24 at 14:31

This question is tough to answer since I do not know the course you have in mind; I am imagining something that involves proof-writing, possibly naive set theory, and, more generally, the ability to write convincing arguments and criticize unconvincing ones.

With regard to:

a representative class of examples I could offer to students who are unsure.

I might use the course listing to link to a site with a few sample problems from early on in the course, a bit more about the relevant background, and your contact information in case they want to discuss the choice with you one on one. If you want example content for a proof-based course, I might use:

1. Prove in two different ways: The sum of the first $n$ odd numbers is $n^2$.

2. Prove in two different ways: The number of diagonals in a regular $n$-gon ($n > 2$) is $\frac{n(n-1)}{2}$.

3. Prove in two different ways: $N$ a finite set with $n>0$ elements; then its power set has $2^n$ elements.

If you can understand all of the language above, then this is probably an appropriate course for you. If you can carry out most of the proofs above (at least one per question) then you will be fine.

Alternatively, consider the following examples:

1. If $a$ and $b$ are even, then $a+b$ is even. [Proof: Since $a$ is even, we can write $a = 2n$ for some integer $n$; similarly, we can write $b = 2m$ for some integer $m$. Adding the two together, we find that $a+b = 2n+2m = 2(n+m)$. Since $n+m$ is an integer, the sum is two times an integer, i.e., the sum is even. Therefore, $a+b$ is even as desired. Q.E.D.]

2. Follow-up A: Prove that if $a$ and $b$ are odd, then $a+b$ is even.

3. Follow-up B: Prove that if $a$ is even and $b$ is odd, then $a+b$ is odd.

4. Follow-up C: Prove that if $a$ is odd and $b$ is even, then $a+b$ is odd. (Can you use Follow-up B?)

5. Suppose $a$ and $b$ are rational. Prove that $a + b$ is rational. [Insert teacher proof.]

6. Follow-up X: Suppose $a$ and $b$ are irrational. Must $a+b$ be irrational?

7. Follow-up Y: Suppose $a$ is rational and $b$ is irrational. Prove that $a + b$ is irrational.

8. Follow-up Z: Suppose $a$ is irrational and $b$ is rational. Prove that $a + b$ is irrational.

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I had a small moment of panic when I misread 'prove in $N$ different ways'. –  NiloCK Jun 17 at 17:08

Some reasonable phrases are

• For an economics class: Prerequisite: some familiarity with the fundamental theorem of calculus
• For mathematical logic: Required background: understanding one proof of the Pythagorean theorem
• For decision theory: This course is appropriate for students who know why the digits test works for divisibility by 3

Most students who have any inclination to rigorous analysis will have seen these theorems before they look at this new class. Then they can judge for themselves.

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I have a BS in Math and wrote a thesis on Polish spaces. I have no idea what the "digits test" is. I imagine it is something along the lines of "if the last digit is divisible by 3, then the number is divisible by 3". But why would elementary number theory be a prerequisite for a course on the foundations of statistics? –  nomen Apr 8 at 2:23
@nomen Surely it is: "If the sum of the digits is divisible by 3, then the number is divisible by 3." The test you describe fails for 13, 23, 43, 53 ... –  Benjamin Dickman Apr 8 at 5:07
@BenjaminDickman: Fair enough, you're right. Hopefully you see my point as well. At the very least, the test for maturity should be relevant to the subject. –  nomen Apr 8 at 23:15
@Nomen, your point about relevance to decision theory is good. Do you have a better suggestion? It would be relevant but long-winded to say: This course is appropriate for students who know why flipping a coin more often is more likely to result in over 49% heads. –  Matt F. Apr 10 at 14:42
@MattF: Why not something like "use calculus to minimize functions" or even the Riesz representation theorem, if that's not scary enough. It depends on what the goal is, and where you're starting. Some classes might end up covering some "harder" real analysis. Some classes might just assume the results are true. Etc –  nomen Apr 10 at 15:12

I think you answered it youself when you state the need of "proving something rigurously" and "absorb definitions fairly rapidly." You might even give examples of the kinds of proof (formality) required, or (as @MattF's answer says) give some examples of results they should understand the proof of, and some definitions they should understand the why behind them.

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In my personal experience, "mathematical maturity" began when I began understanding how mathematical concepts interact with each other.

"Anyone" can memorize concepts. Understanding "how they fit together" is the real sign of maturity.

So tell students that they have "mathematical maturity" when they can understand the interrelationships between key mathematical concepts, and not just the concepts themselves.

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