Nov 26, 2016

>I don't know how you think that school systems, especially in the west, place more emphasis on "how" than "why".

I've taught people at college level, and have seen way too many cases of people knowing how to perform the steps to, say, solve an equation, find a derivative or take an integral without understanding what the operations actually mean, or why bother with all this stuff in the first place.

It's not the students' fault; the concerns I am trying to voice have been articulated by many over the recent years, and are best summed up here:

Oct 07, 2016

You are essentially saying - the way I think of math is how it should be learned. Sorry but if you're serious, you will find a way to explain your mathematical intuition.

The problem is detailed in this wonderful essay by Paul Lockdhart

Jun 10, 2016

Step 1: Read Lockhart's Lament:

Step 2: Download the Book of Proof: You read through it and do all the odd numbered exercises (the solutions are at the end of the book).

Step 3: Get a book called Real Mathematical Analysis by Charles Pugh and you work through that and attempt as many problems as you can, with a view not to rush through it, but to expand your mind through each problem.

Step 4: Pick any of these books that interest you the most and do the same:

- Calculus by Spivak

- Algebra: Chapter 0 by Paolo Aluffi

- Linear Algebra Done Right by Axler

By then you should have enough mathematical maturity to know what to do next.

May 05, 2016

I would check out the book "Free to Learn" by Peter Gray - he goes into why "unschooling" or democratic schooling can be effective. It really changed my outlook on education. I had previously read "A Mathemetician's Lament"[0], an essay by Paul Lockhart that made me start to question the value of regular schooling.

Jan 30, 2016

Do we really want to "reduce such a beautiful and meaningful art form to something so mindless and trivial[?]; no culture could be so cruel to its children as to deprive them of such a natural,satisfying means of human expression. How absurd!" - Lockhart [1]. He was talking about math, but I am sure the same results would occur with computer science "education" if done in high schools.

"One of the best ways to stifle the growth of an idea is to enshrine it in an educational curriculum." - Hal Abelson[2]

[1] [2]

Jan 26, 2016

I have never met a mathematician who feels that two-column proofs is the right way to teach high school geometry. Considering that little math taught in high school beyond basic algebra uses anything close to 'absolute rigor,' introducing rigorous proofs should be left for college when one actually has a reason to learn proofs for advanced mathematics (set theory, analysis, etc).

Geometry is taught correctly when proofs are used as a tool to convey a deep intuition about a mathematical pattern. Check out Lockhart's book [1] if you want to see what it's like when done right, though there are many more examples. Two-column proofs are simply a tool for the lazy/unknowledgeable teachers to fill a geometry class.

I am aware of Lamport's work, and it's a specific tool for a specific subfield in which there is a plethora of false results. Ignoring the fact that most of theoretical computer science research and most of math research more does not fall into the category that Lamport is critical of (distributed computing), these temporary issues about rigor in academic publishing should have no effect on high school pedagogy. Instead, we should listen to the world's finest math teachers, who pretty much all agree that two-column proofs are awful. A quote from Lockhart [2]:

> Geometry class is by far the most mentally and emotionally destructive component of the entire K-12 mathematics curriculum. Other math courses may hide the beautiful bird, or put it in a cage, but in geometry class it is openly and cruelly tortured. What is happening is the systematic undermining of the student’s intuition. A proof, that is, a mathematical argument, is a work of fiction, a poem. Its goal is to satisfy. A beautiful proof should explain, and it should explain clearly, deeply, and elegantly. A well-written, well-crafted argument should feel like a splash of cool water, and be a beacon of light— it should refresh the spirit and illuminate the mind. And it should be charming. There is nothing charming about what passes for proof in geometry class. Students are presented a rigid and dogmatic format in which their so-called “proofs” are to be conducted— a format as unnecessary and inappropriate as insisting that children who wish to plant a garden refer to their flowers by genus and species.

[1]: [2]:

Jan 26, 2016

I love these insights, they have strong echoes of "A Mathematicians Lament" by Paul Lockhart, another excellent essay on the topic. [PDF]