If it doesn't interest you, don't read it!

Question

As an algebraist, if I see an analysis question on Mathoverflow that falls outside my expertise, I probably won't read it in details, and if I do read it, I certainly won't judge its "level", because I don't consider myself an expert in analysis. I would assume most people act similarly. However, post a question on the topic of Mathematics Education, and suddenly everyone becomes an expert in Mathematics Education! Questions in Mathematics Education could be very subtle. Just because someone teaches mathematics, it doesn't make them experts in Mathematics Education. Probably many of those who become a Mathematics Education expert as soon as they see a question on that topic, don't have any interest in Mathematics Education, have never faced the challenges of teaching elementary mathematics, or have not thought about possible solutions to such challenges.

Yesterday I posted a question tagged Mathematics Education. It was quickly closed. It was judged to be more appropriate for Stack Exchange Mathematics Educators. One person commented that it was not a research level question. Another person started acting up and making irrelevant and off topic comments. Interestingly, nobody attempted to answer the question itself! These people did not have any knowledge on the topic of the question. In fact, the topic is a research level question in Mathematics Education and the very fact that someone says it is not, shows they have no information about the topic. I like to assume that if a question does not interest someone, they can just skip it without feeling any urge for making hasty comments under the question.

The question that I asked was related to Mathematical Maturity. If you try to search the literature in Mathematics Education for Mathematical Maturity you will not find much. Of course you will find the Wikipedia page and a few blogs related to it, but not many research papers. In fact, I only know of an article named Developing Mathematical Maturity by Lynn Steen (1983) in: The Future of College Mathematics (pp. 99-110), Springer, New York, NY. But the topic is important enough that Terrence Tao and Bill Thurston have written about it. While the phrase Mathematical Maturity does not directly appear in their writings, one can argue that Tao's blog post There is more to mathematics than rigour and proofs and Thurston's On proof and progress in Mathematics address Mathematical Maturity, among other things.

Why is Mathematical Maturity important? We often expect our Calculus III students to be "better" than our Pre-Calculus students in learning mathematics. One can argue that by "better" we mean mathematically more mature. Consequently, one way to see whether our Pre-Calculus to Calculus III sequence is working properly or not, is to look at students' development of Mathematical Maturity from Pre-Calculus to Calculus III. Of course, one may say if students passed their Calculus courses, it means they learned, so why do we need to monitor their development of Mathematical Maturity? But those of you who know something about teaching Calculus, know that just because students pass a course doesn't mean they learned it. I submit that a successful Calculus program is one that develops students' Mathematical Maturity by as much as possible.

While there is no universal definition for Mathematical Maturity, mathematicians have been using the phrase. The situation reminds me of notions such as naturality or moduli space that were used by mathematicians before they had a formal definition. In the case of naturality, MacLane and Eilenberg captured its definition by introducing the notion of natural transformations of functors. Therefore, being able to provide a universally agreeable definition of Mathematical Maturity is a research level question in Mathematical Education, although, that was not what my post was about. Another challenge is to have tools for measuring the development of Mathematical Maturity from Pre-Calculus to Calculus III. But who can address these questions about Mathematical Maturity? I would like to argue that one has to be a mathematician to know what Mathematical Maturity means. Terrence Tao would know more about Mathematical Maturity. Bill Thurston would know more about Mathematical Maturity. Mathematicians are the ones who know about Mathematical Maturity. And this is why my question is not more appropriate for Stack Exchange Mathematics Educators, but appropriate for Mathoverflow, for mathematicians to respond to it (of course, only if the topic interests them). To my point, I recommend reading the report How Do Mathematicians Describe Mathematical Maturity? by Kristen Lew, who is a mathematics educator. What do mathematics educators do when they want to research Mathematical Maturity? They interview mathematicians!! Interestingly, she found that none of the four applied mathematicians whom she interviewed had used or knew of the phrase Mathematical Maturity.

I would like to repost my question on Mathoverflow (which I deleted after it was closed), but I don't want it to get closed again! I am posting here in meta, with enough explanation for why I think my question should be on Mathoverflow rather than Mathematics Educators, to see if there is enough supporting opinion to keep my question open, if I repost. I don't understand the "anger" toward such questions that some people exhibit! I find it very unwelcoming and I think this is an issue that Mathoverflow personnel should address. Make the environment more welcoming.

Answer 1

Might I point out that "Mathematical maturity" is not a mathematical object, subject to rigorous definition, so it probably shouldn't be argued about by comparing to actual mathematical objects that may or may not have had a rigorous definition before being studied. It's a sociological/psychological/behavioural phenomenon presumably best examined and discussed based on case studies (in case it's not clear, I think I disagree with both fedja and Mahdi, here!) To answer your question about consensus, it's something that has moved over time, influenced also by external factors like the presence of math.SE.

In the early days of MO, when it was smaller and the community mostly knew each other either personally or by reputation, then talking about widely disparate questions was ok, in the manner that talking with one's close colleagues at afternoon tea about this that or anything vaguely intersecting the profession is ok. But now MO operates at scale, and allowing all kinds of random questions seems to be viewed as a risk to keeping the noise/signal ratio low. People who are rather famous, like the late Bill Thurston, have asked very "soft" questions in the past, not least because they are known as somebody who thinks deeply about such stuff and written about it to benefit mathematics as a profession outside MO and/or have even formally published such writing. But, and I say this with the utmost respect, more generic mathematicians like us don't have the social capital to get away with asking such a soft question. Is it fair? Maybe not. But it forces people who want to ask a "super-soft" question to work to make the question really tight and high-quality.

So I think it in-principle possible that a question such as your might have survived on MO, at least some point in the past, under ideal circumstances. However, the question really does feel like a question about designing mathematical assessment in an education context, so in it's current form it doesn't feel like an MO question. To counter this negative posture, I watched a talk by Tim Gowers at the recent Fields symposium on YouTube, where he touched on matters that included the curious parallels between what an undergrad struggles to grasp in introductory analysis, and what was difficult for some automatic formal proving software that he was involved in designing. The talk was rather clearly a "sociological" talk, but it also described a fairly ambitious loose research program in trying to quantify what "difficulty" and "centrality" meant in mathematics research, via proof assistants, based on an essay by Akshay Venkatesh about the future of computers in maths research. So you have very famous people, in this instance, talking about big vision matters. I can't say what I would do had I seen your question before closing/deletion, but I hope this is food for thought.

Answer 2

The consensus about the definition of an "MO appropriate question" is currently that it is something way more restrictive than "anything that mathematicians care about and may be the best people to ask", so I am not surprised that vague and controversial questions like yours may receive all sort of treatment from a closure in under 5 minutes to a long lively discussion lasting for weeks. Yours just fell into the first category. Why? You sort of answered it yourself in the passage

What do mathematics educators do when they want to research Mathematical Maturity? They interview mathematicians!! Interestingly, she found that none of the four applied mathematicians whom she interviewed had used or knew of the phrase Mathematical Maturity.

The educators (IMHO) are too obsessed with various metrics (from standardized test scores to "mathematical maturity") when discussing the mathematics development. As a mathematician I view it very differently. You just build your toolbox and become proficient with your tools. That's where the craftsmanship starts and that's where it ends.

If it were about carpentry, I would say that some people carry only a hacksaw and an axe, some other add nails, glue, measuring tapes, hammers, screws, etc. to the set while yet others build a whole factory with their own source of electricity, and so on. Also in this metaphor there are people who can build a whole house with nothing but an axe and there are people who cannot make a simple bench without heavy machinery. The tools and the skills accumulate with time and, while there are some connections (like you cannot efficiently use nails without a hammer or a hammer gets new applications when nails are also around), the order in which people acquire them beyond the first basic set is determined by their particular circumstances more than by any inherent logic, the primary driver for acquiring new tools and skills being the tasks that are hard or impossible to do with the current ones (and you often have a choice between using a new tool and using a new skill).

What would I call "mathematical maturity" if I were to use this word for anything? Probably my answer would be "understanding one's abilities, limitations, and the possibilities for further development reasonably clearly" (so it is a meta-craft notion for me).

I have no idea whether this is what you had in mind or not, but let me attract your attention to one more important thing. In mathematics we do not try to figure out what one might possibly mean by a certain combination of words that is commonly used without an assigned meaning (in fact we try to never use words without unambiguous assigned meanings in mathematical texts). Instead, we often assign (more or less arbitrarily) names to objects, tricks, ideas, etc. that turn out to be ubiquitous once their underlying meaning/structure becomes crystal clear but too long to explain in full every time they are mentioned. So, the words "mathematical maturity" would appear in our language after we notice something common in various instances of student mathematical development, extract that common thing, understand what exactly it is and what it is responsible for, and then decide that it is worth having a name and that "mathematical maturity" is a combination of words that we want to use (though, in all honesty, the probability that we would end up calling it "shmarky plaft" is higher). Thus the definition normally arises in mathematics before the notion, not after it.

What's the point? It is that your phrase

I submit that a successful Calculus program is one that develops students' Mathematical Maturity by as much as possible.

sounds totally meaningless, because you don't even yet know what you are talking about when saying "mathematical maturity", assume (for no apparent reason) that it takes values in a linearly ordered set and the set of all development states is linearly rather than only partially ordered with respect to it, provide no way to measure it, and, above all, offer it as the single objective function by which to measure "success", while calculus programs serve a whole multitude of conflicting goals (resulting in tool vs. skill, breadth vs. depth, rigor vs. intuition, relevance to an external subject vs. internal logic, and many other dilemmas that one faces even when one tries to teach a single class, forget about planning the whole sequence). It is exactly such proclamations that make many working and teaching mathematicians very skeptical about mathematical education as a discipline up to denying its very existence as a meaningful science (which is sort of a pity because there are some good ideas and even some sound practical advice there, as far as I can judge).

IMHO, what we should really do is not to invent various new "measures of success" and argue about their exact meanings, applicability, and importance, but just run small scale experiments to see which techniques do facilitate learning and which do not and share the resulting conclusions. If some technique really works, it will eventually catch and spread by itself. So you want to develop "mathematical maturity" in students and to demonstrate its importance? Fine. Then I don't even care what you mean by it in the most general sense. Just give me one simple example of its instance I should pay attention to and some easy to implement ideas for how to run the class to develop it and I'll give it a try, but I'll judge whether it works or not by seeing how such change influences the values of my objective functions. If they go up, yeah, I'll want to know more about your ideas, but if they go down, I'd rather stick to my old ways.

Well, you wanted a serious response from someone who can (or, at least, could) qualify as a research mathematician with teaching experience, so I tried to provide one.