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How do you achieve mathematical maturity?

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There is no one path to mathematical maturity.  Nor is mathematical maturity a skill you can learn on its own.  More to the point, there is no threshold, "beyond here you have mathematical maturity!"

When math educators talk about a student needing to have some level of "Mathematical Maturity," what we mean is that the student has become familiar with the meta-structure of definition -- theorem -- proof on which mathematical discourse is built.  More than any one particular skill, you need to learn to understand how mathematicians think.

You develop this mindset by spending time thinking about mathematics.  As you put time into studying math, you will develop better skills for *how* to study math.  That's mathematical maturity.
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mathematical maturity is like enlightenment, it's not something you seek, but instead a byproduct of disciplining one's mind. If you can snatch the fly from my palm, you will be master, and I student.
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As per personal experience, I think the everlasting procedure goes as follows. Let's call it vicissitudes.

* Study hard and you get some confidence.

* Your confidence is destroyed by some much harder maths. A person may think he has mastered calculus so he is one of the smartest guys in the world but later measure theory make him reason his existence when he starts studying.

* Later your confidence is built up again. Maybe one sees exercises in measure theory relatively more obvious, after studying functional analysis.

* The confidence collapse again. And the loop keeps running.

I think one has to go through such a up and down, to prevent being like people in r/numbertheory...
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taking an abstract algebra course (logic, sets, groups->fields) worked for me.
it encapsulates what a lot of mathematical topics are built upon and has basic and simple proofs.



also beating myself to death over Robert R. Stoll's set theory and logic which was way too advanced for me as a first year :)
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To add to what @ppirilla said, the student must be able to figure out the direction of the study. For example, we start with Real Analysis. The student isn't really aware of how norms work on sets and how we define continuity using this idea. However if the student has taken an Advanced Linear Algebra course, they would be able to relate the norm and inner product right away to the definition of continuity from Calculus.

This intuitive grasp of why we are studying the properties we are studying and what we are trying to build using those properties is immensely important and a part of mathematical maturity in my opinion as a teacher. When I try to teach students something proof related, I see blank faces because they get the idea why the proof works but they don't understand how I was able to think like that. That fog, once removed, is what can be described as mathematical maturity and the only way to get there is do as much Mathematics as possible.
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Finding counterexamples helps build up mathematical maturity.

It gets you thinking about what assumptions were needed and how something failed once you removed that assumption. It also humbles you by how non-intuitive they are.

That break of intuition is the kind of kick in the pants needed to going from an attitude of "it's obvious I don't need to prove it" to regularly thinking about how to prove it.
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When you don't smile to yourself if the sequences of 69 or 420 pop up somewhere in your work.
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Whenever you realize you can actually read Greek.
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One aspect of mathematical maturity is that you have done enough mathematical study that you are able to grasp what things are important in a new setting. You can start to see the forest through the trees. You can anticipate where the narrative is likely to lead and what the roadblocks might be.

As a toy example, if you have studied undergraduate for any length you become accustomed to the general pattern of maths structures in pure maths:

* define an object with structure (e.g. a group)
* define subobjects (subgroups)
* define maps which preserve the structure (group homomorphisms, isomorphisms)
* Build new objects out of old objects (products, quotients, etc.)

Familiarity with this pattern allows you to see where you'd expect it to go even though it might vary a little from setting to setting. For example, you need to be a bit more careful in your definition of isomorphism for a topological space or a ring but once you know that you understand the broader picture even better.

Mathematical maturity (in this sense) means that you can see the way ahead even if it involves some really complicated things in the moment.
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According to Tadashi Tokieda, math is creating mental images. If you are not very mature, you can probably still make sense of Geometry, because it consists of images. Then you go more and more abstract and have to paint mental images of things that aren't actually visualizable.

As to anecdotes: I had a moment of feeling mathematically mature when I read about the fact that SL(2, C) is not compact and thought: Yeah, how could it be, it can transform squares into parallelograms of arbitrary diameter and can therefore not be bounded. This is one case where I just had this mental image which lead to immediate understanding, no proof required. Felt like a real champ that moment.

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