[jayhoon]

Interestingly, this guide states that the intuitive understanding of maths is only suitable at the school level but not for the university.

In his recently published book "Mathematica: A Secret World of Intuition and Curiosity", David Bessis argues that the intuition is the "secret" of understanding maths at all levels.

Not sure what conclusion to draw from here, but my (rather dated) experience with university maths tells me that the intuition is a powerful tool in developing the understanding of the subject.

[chongli]

This seems like a contradiction but I don’t think it is. What it’s really saying is that experience is a precondition for intuition.

When a high school student looks at a high school math problem they’re drawing on all of their experience in K-12 math to get intuition for how to solve the problem. When they leave high school to study math in undergrad they struggle because their experience is no longer sufficient. They’re faced with a lot more abstract problems and the demands for rigour are much higher. The problems also tend to operate at higher levels on Bloom’s taxonomy [1] than high school math, something with which the average high school student would have little or no experience. It is this unfamiliar territory where intuition is hard to come by.

After gaining more experience (later undergrad and into grad school and beyond) the intuition starts to come back. But it’s fundamentally a different kind of intuition. In high school math it was often a visual/geometric intuition that teachers were trying to build. In higher math it’s an intuition for abstractions and for the tools you need to attack problems. This is really no different from a programmer looking at a problem and saying “I need a hash map and then this problem is trivial.”

[1] https://en.wikipedia.org/wiki/Bloom's_taxonomy

[609venezia]

Possible harmonization of the two ideas: the intuition that we go into math at high school level can help serve us at that level of math. We have some idea of geometry-like objects and 2d-calculus like curves from our everyday life

At university level the objects become more abstract, so the intuition we use in normal daily life may no longer apply. New kinds of intuition may develop but it takes work, including lots of time spent with the formal processes and calculations along with reflection on that time, and the active creation of new metaphors to drive the intuition. For example, I still remember a professor using "Ice-9" (from _Cat's Cradle_) as a metaphor for how proving some local property of a holomorphic function on the complex plane made that property true for its global behavior

[tseid]

I think he's saying here that intuition is sufficient for high school math, but not sufficient for college. That's not to say that it isn't necessary, only that it isn't sufficient.

[Tazerenix]

This is related to Terence Tao's notion of the stages of mathematical rigor.

As Tao puts it, the value of intuition becomes much higher in the post-rigorous stage once you have sufficiently developed your technical skills.

https://terrytao.wordpress.com/career-advice/theres-more-to-...

[alganet]

I read the book.

To me, what it says is "intuition can be honed and it is powerful, but hard to pass along to others". Just that.

Bessis actually mentions examples of how intuition and technique complement each other nicely.

[agumonkey]

That topic fascinates me. As a kid, a lot of topics felt intuitive, and college became a pit of darkness. Makes me wonder what in our brains makes something feel natural, obvious, with that feeling of playfulness and certainty .. while some times you're drowning in a blur.