Mathematicians Are Chronically Lost and Confused

An anonymous reader writes "Mathematics Ph.D. student Jeremy Kun has an interesting post about how mathematicians approach doing new work and pushing back the boundaries of human knowledge. He says it's immensely important for mathematicians to be comfortable with extended periods of ignorance when working on a new topic. 'The truth is that mathematicians are chronically lost and confused. It's our natural state of being, and I mean that in a good way. ... This is something that has been bred into me after years of studying mathematics. I know how to say, “Well, I understand nothing about anything,” and then constructively answer the question, “What’s next?” Sometimes the answer is to pinpoint one very basic question I don’t understand and try to tackle that first.' He then provides some advice for people learning college level math like calculus or linear algebra: 'I suggest you don't worry too much about verifying every claim and doing every exercise. If it takes you more than 5 or 10 minutes to verify a "trivial" claim in the text, then you can accept it and move on. ... But more often than not you'll find that by the time you revisit a problem you've literally grown so much (mathematically) that it's trivial. What's much more useful is recording what the deep insights are, and storing them for recollection later.'"

Failing as a math teacher

[Anonymous Coward]

All too often I've encountered math teachers who failed to properly explain advanced mathematical concepts because to them it was obvious and trivial.

Gee, thanks.

Re:Failing as a math teacher

[jones_supa]

I have sometimes thought that the best teacher might be another student who has just a moment ago barely (but still correctly) grasped the concept.

Bizarre advice

[AthanasiusKircher]

He then provides some advice for people learning college level math like calculus or linear algebra: 'I suggest you don't worry too much about verifying every claim and doing every exercise. If it takes you more than 5 or 10 minutes to verify a "trivial" claim in the text, then you can accept it and move on. ...

While I agree that one shouldn't waste time questioning every statement you encounter, there's a very ancient and useful tradition in math pedagogy that emphasizes these sorts of things. See, for example, gradually building up geometrical theorems from a few axioms, a la Euclid.

Often, the process of working out complex proofs for yourself is crucial to understanding why things work, not to mention developing and practicing logic skills that are essential in math and elsewhere.

I'm not saying one should waste time trying every exercise or redoing every proof, but some of my greatest insights into the inner workings on math have come from exercises that took me a couple hours to work out or textbook passages I went over a number of times and really dug into how the details worked. If I skipped everything I couldn't do in 5-10 minutes, I doubt I'd ever have developed the more advanced skills and intuitions that would be necessary to see why some results are "trivial."

Any Good Scientist Knows This

[Anonymous Coward]

We must all learn to exist in that exquisitely uncomfortable place where everything we know is always up for reassessment. Otherwise, we miss change, and change is the only constant.

Re:Failing as a math teacher

[techno-vampire]

What's worst is a teacher who defines a new term in a way that only makes sense if you already understand the concepts behind it. As an example, Rudy Rucker once defined a cardinal number (in a book) as, "A number is a cardinal number if it doesn't share its cardinality with any other number." Now, if you know what a cardinal number is, and what "cardinality" means, that's true. If you don't, as most of the readers of that book wouldn't, it's useless.