Phenomenal Math

When you read and really get motivated 

Some excerpts from the book "Living Proof : Stories of Resilience Along the Mathematical Journey " - 

" Playing the violin is hard, hitting a baseball is hard, and learning a second language is hard. What seems to make mathematics different from playing the violin or learning Chinese is that the struggle to play violin doesn’t make people feel defeated and dumb. Somehow, when we encounter difficulties in mathematics, our natural tendency is to retreat, to think it’s too hard, we’re not smart enough, or we’re not “math people.’’ We allow ourselves to be defeated by the difficulty. We understand that learning to play the violin requires making many, many hours of horrible screeching sounds, that learning to speak Chinese means making error after error and not being understood. But, somehow, when it comes to mathematics, we fear making mistakes. We imagine that there are “math people’’ to whom it is all transparent and, if it doesn’t come to us immediately, we must not be one of them. 

There are no such people. People who succeed in mathematics, like people who learn a musical instrument or a new language, spend a lot of time not understanding and feeling frustration. The path to understanding in mathematics necessarily involves, in the words of Steve Klee (4), being “willing to struggle.’’ It is strange that people do not understand this about mathematics when it is commonplace in essentially every other field of human endeavor

There is something that, I think, is slightly different about the learning of mathematics. After the long, hard struggle for understanding what often happens is a crystalline perfection of understanding. The resulting knowledge is so obvious, so natural, that we don’t understand our previous difficulty "

Some dangerous myths about mathematics -

There are dangerous myths in mathematics. One of them is that there exist “math people,’’ people to whom it all comes easily and is obvious. People who study the theory of learning are discovering that grit and persistence in the face of difficulty are much more important than any inherent talent in learning mathematics. Simply believing that study and struggle are more important to learning than innate ability leads (through productive study and struggle) to more learning and more understanding. There are no “math people,’’ mathematical thinking is a fundamental part of every human’s intellectual capacity. 

The people we label “good at math’’ are simply those who have taken the time and trouble to engage the struggle more deeply than others. The myth of “talent’’ has pernicious consequences. Math teachers look for “talent’’ and encourage it. Unfortunately, and this is not a phenomenon unique to mathematics, we tend to see that “talent’’ in people who look like us. This has the effect of erecting barriers to entrance into mathematics among populations of people who don’t look like the majority of mathematicians: women, people of color, people with disabilities. No one wants these barriers to be in place. But, they exist, and they make it difficult for many people to access mathematics. Teachers of mathematics can also, often unintentionally, wreak profound damage on students by explicitly, or implicitly, conveying an expectation of non-success. The flip side of this is that teachers can also, again often unintentionally, have a profound positive impact with a simple, kindly encouraging remark. 

We can create - 

We can create a mathematical world where demoralizing, punishing struggle is not necessary. It will always be necessary for people to struggle within their own minds to master mathematics, but we need to teach our students to see, the power and glory of mathematical struggle. Using reason and logic and one’s own mind one can achieve mathematical insight—this is the most awesome of our intellectual capabilities, it’s part of the essence of being human. 

We should talk about - 

Math should be difficult, as should any worthwhile endeavor. But it should not be crippling. The ability to succeed in a mathematical program should not be hindered by a person’s gender, race, sexuality, upbringing, culture, socio-economic status, educational background, or any other attribute. Yes, math is difficult. We should talk about what makes it difficult. But we should also acknowledge the various biases and prejudices that people bring to their study of math that compound its difficulty. By making an effort to understand what we have in common and what makes our experiences different, our hope is that our community will become more inclusive while making the struggle more bearable—perhaps even more fun. 

When it just feels hard - 

Many students who find mathematics hard at some point were students for whom topics in high school came easily. Of course, we don’t mean all topics, as even practicing mathematicians have favorite topics and, well, less favorite topics. One of the editors of the book, David Taylor, still hates factoring polynomials; he was never really good at it, prefers never to do it, and just doesn’t like it at all, but working with matrices that have over 1,600 entries and working with their powers to find long-term steady states in Monopoly comes more easily, and honestly is more fun.

 For most, there’s a time when the content being studied becomes hard and it can be challenging to overcome that. The feeling of understanding everything in class turning into a feeling of not understanding anything can be a shock. What follows are some stories from people who have gone through that exact scenario at some point in their studies; many of these people are currently mathematics professors. If they can struggle with material and make it through to become a professor of the subject themselves, you can too.

Some questions and some interesting answers!

*) https://math.stackexchange.com/questions/528089/show-that-if-a-subseteq-b-then-inf-b-leq-inf-a-leq-sup-a-leq-sup-b?rq=1 

*) https://math.stackexchange.com/questions/392129/proof-that-inf-a-sup-a 

*) https://math.stackexchange.com/questions/674589/prove-n2-diverges-to-negative-infinity 

*) https://math.stackexchange.com/questions/677817/how-to-properly-construct-an-epsilon-n-proof 

*) https://math.stackexchange.com/questions/3291505/a-and-a2-have-same-characteristic-polynomial 

*) Nice one - https://math.stackexchange.com/questions/123712/epsilon-rm-delta-proof-of-the-discontinuity-of-dirichlets-function?rq=1 

Things I plan to read (hope I read) - 

Monkey Saddle

Mathematical visualization

Wired bacteria from nature's power grid

 learning to think well and nurturing curiosity

Apollonian Gasket

But what is a Fourier series? From heat flow to circle drawings 

Attitude towards intuition in calculus

How to reach all students!

Early career academics/ teaching

DISCLAIMER: All the contents of the site are views my own. It has many links which I borrowed from many websites, I am not trying to claim they are mine. This is not for any sort of promotion. The aim of the website to make a collection of resources. To make viewers have a feel of beauty and effectiveness of mathematics and with a hope that this mathematical odyssey becomes a lifelong endeavor.