4 ms·

I remember reading a book in high school and realizing there could be other ways to prove things that I had been taught only one way. One that particularly stoo

I remember reading a book in high school and realizing there could be other ways to prove things that I had been taught only one way. One that particularly stood out later was using a rotating fishtank to prove the pythagorean theorum. A good friend of mine was so delighted by the example I gave him a copy of the book I found it in. https://press.princeton.edu/books/paperback/9780691154565/the-mathematical-mechanic https://press.princeton.edu/books/paperback/9780691154565/th...) The relevant section is available as a pdf here: http://www.personal.psu.edu/mxl48/Welcome_files/Sample.pdf http://www.personal.psu.edu/mxl48/Welcome_files/Sample.pdf

hypertexthero 4y ago
This brings to mind the visual solution to calculating triangle area in James Somers post “I should have loved biology”: > In his “Mathematician’s Lament,” Paul Lockhart describes how school cheapens mathematics by robbing us of the questions. We’re not just asked, hey, how much of the triangle takes up the box? > That’s a puzzle we might delight in. (If you drop a vertical from the top of the triangle, you end up with two rectangles cut in half; you discover that the area inside the triangle is equal to the area outside.) —https://jsomers.net/i-should-have-loved-biology/ https://jsomers.net/i-should-have-loved-biology/
- Waterluvian 4y ago
  I vividly remember math class one year boring me to death so bad that I distracted myself with my own puzzles like this. It was when I discovered the Fibonacci sequence inside Pascal’s triangle. I didn’t think this was a new discovery but it was new to me and it felt like lightning. I think that might have been an early glimpse of my later discovery that all my best learning would be done outside school.
  - mananaysiempre 4y ago
    > [A]ll my best learning would be done outside school. Choose one: - Experience of discovery and survival of curiosity to adulthood; - Set of job-relevant skills well defined by names of subjects; - Standardized testing and easily comparable grades. (In my admittedly limited teaching experience.) I would guess that the last point will always get chosen, because it’s bureaucracy-friendly, and a bureaucracy makes the choice. But one of my most bizarre experiences is (some) HN readers being quite vocal about their support for it as well, where I haven’t seen it be anything but harmful. The bullshit admission process at US colleges might be to blame—I’m really not sure. References: Lockhart’s “Lament”[1], of course, for describing the feelings that (good) teachers have on this subject; Quinn’s “Revolution in mathematics”[2], as a more clinical analysis of how the bureaucracy won and got to basically redefine what “mathematics” even means for the majority of the population (in a way that’s as hopelessly obsolete as it is intensely harmful to the subject proper). The point shouldn’t be specific to mathematics, but it’s what I have the references for. [1] https://www.maa.org/external_archive/devlin/devlin_03_08.html https://www.maa.org/external_archive/devlin/devlin_03_08.htm... [2] http://www.ams.org/notices/201201/rtx120100031p.pdf http://www.ams.org/notices/201201/rtx120100031p.pdf
- Tao3300 4y ago
  I hated math for most of my childhood. I tested into an advanced track and had to be sequestered into lower level courses in the next higher grade do to a lack of effort. Then when I hit college and had Discrete and Calculus, I found out I loved it and wound up minoring in math. Though my arithmetic is still slow and my trig has major gaps in it due to school math just sucking in general.
coliveira 4y ago
The nice thing about mathematics is that for every true statement there are infinitely many proofs. Granted, some are just variations of others, but there many ways to reach the same point.
- carapace 4y ago
  > for every true statement there are infinitely many proofs No. There are true statements which cannot be proven. For example: "This statement cannot be proven." (Technically it's truth value is neither true nor false. It is an imaginary Boolean value.)
  - coliveira 4y ago
    So this statement is not true, you cannot prove its value. But I understand what you mean, let's just talk about provable statements.
steppi 4y ago
This is a really great book. It’s very accessible but the insights can also be appreciated by a mathematically sophisticated audience. I’m particularly fond of Chapter 11 on understanding complex analytic functions and the part in Chapter 2 that gives a very clear explanation why the determinant formula gives the (signed) volume of the parallelepiped determined by the column vectors of a matrix.
Al0neStar 4y ago
2023 edition: https://press.princeton.edu/books/paperback/9780691242057/the-mathematical-mechanic https://press.princeton.edu/books/paperback/9780691242057/th...

hypertexthero 4y ago

This brings to mind the visual solution to calculating triangle area in James Somers post “I should have loved biology”: > In his “Mathematician’s Lament,” Paul Lockhart describes how school cheapens mathematics by robbing us of the questions. We’re not just asked, hey, how much of the triangle takes up the box? > That’s a puzzle we might delight in. (If you drop a vertical from the top of the triangle, you end up with two rectangles cut in half; you discover that the area inside the triangle is equal to the area outside.) —https://jsomers.net/i-should-have-loved-biology/ https://jsomers.net/i-should-have-loved-biology/

Waterluvian 4y ago
I vividly remember math class one year boring me to death so bad that I distracted myself with my own puzzles like this. It was when I discovered the Fibonacci sequence inside Pascal’s triangle. I didn’t think this was a new discovery but it was new to me and it felt like lightning. I think that might have been an early glimpse of my later discovery that all my best learning would be done outside school.
- mananaysiempre 4y ago
  > [A]ll my best learning would be done outside school. Choose one: - Experience of discovery and survival of curiosity to adulthood; - Set of job-relevant skills well defined by names of subjects; - Standardized testing and easily comparable grades. (In my admittedly limited teaching experience.) I would guess that the last point will always get chosen, because it’s bureaucracy-friendly, and a bureaucracy makes the choice. But one of my most bizarre experiences is (some) HN readers being quite vocal about their support for it as well, where I haven’t seen it be anything but harmful. The bullshit admission process at US colleges might be to blame—I’m really not sure. References: Lockhart’s “Lament”[1], of course, for describing the feelings that (good) teachers have on this subject; Quinn’s “Revolution in mathematics”[2], as a more clinical analysis of how the bureaucracy won and got to basically redefine what “mathematics” even means for the majority of the population (in a way that’s as hopelessly obsolete as it is intensely harmful to the subject proper). The point shouldn’t be specific to mathematics, but it’s what I have the references for. [1] https://www.maa.org/external_archive/devlin/devlin_03_08.html https://www.maa.org/external_archive/devlin/devlin_03_08.htm... [2] http://www.ams.org/notices/201201/rtx120100031p.pdf http://www.ams.org/notices/201201/rtx120100031p.pdf
Tao3300 4y ago
I hated math for most of my childhood. I tested into an advanced track and had to be sequestered into lower level courses in the next higher grade do to a lack of effort. Then when I hit college and had Discrete and Calculus, I found out I loved it and wound up minoring in math. Though my arithmetic is still slow and my trig has major gaps in it due to school math just sucking in general.

coliveira 4y ago

The nice thing about mathematics is that for every true statement there are infinitely many proofs. Granted, some are just variations of others, but there many ways to reach the same point.

carapace 4y ago
> for every true statement there are infinitely many proofs No. There are true statements which cannot be proven. For example: "This statement cannot be proven." (Technically it's truth value is neither true nor false. It is an imaginary Boolean value.)
- coliveira 4y ago
  So this statement is not true, you cannot prove its value. But I understand what you mean, let's just talk about provable statements.

steppi 4y ago

This is a really great book. It’s very accessible but the insights can also be appreciated by a mathematically sophisticated audience. I’m particularly fond of Chapter 11 on understanding complex analytic functions and the part in Chapter 2 that gives a very clear explanation why the determinant formula gives the (signed) volume of the parallelepiped determined by the column vectors of a matrix.

Al0neStar 4y ago

2023 edition: https://press.princeton.edu/books/paperback/9780691242057/the-mathematical-mechanic https://press.princeton.edu/books/paperback/9780691242057/th...