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I have always liked math, but have never been great or very good at it. I graduated from a US public school gifted program, and a top 30 US university. The bigg

I have always liked math, but have never been great or very good at it. I graduated from a US public school gifted program, and a top 30 US university. The biggest problem I always had with math education is they didn’t teach the “why” nearly enough. I was lousy at memorizing formulas/methods (esp. in calculus), so I often found myself dead in the water during exams.

zsmi 5y ago
I used to suffer from the lack of "why" too until one day it dawned on me that there really isn't a why, or maybe, the why is always the same: for engineers doing classical physics solving problems by hand, it's convenient to define it that way. No more, no less. When I was a math tutor in school, everyone would complain about lack of why. This was my approach to address the issue. I would describe a problem relevant to the class, say something like find the angle between two vectors, and ask them to think of a general way to do it. Usually this require some help from me but the solution was always "theirs". Then we would play with their solution: find corner cases, figure out which operations broke, develop a couple of theorems with it. Then I would show them the dot product, and a couple of its tricks (a.k.a. theorems), and they would get a good appreciation for why the dot product works the ways it does. And that's the why and the reason it stuck. It's useful way to solve a common problem. There is no magic or deep philosophical truth. The problem is this teaching style works with a dialog to a few number of students. I am not sure how well it would work in a lecture setting. It was also not so easy on me as I would be the person who had to spot the problems in their definition, basically on the spot, and lead them to it on their own without revealing the answer. With basic things like a dot product it's pretty easy to know where to look (i.e. is their definition commutative, pretty much never) and finding issues and nudging them too it wasn't too hard. But it's likely hard to scale things like this on many metrics.
- iammisc 5y ago
  The problem with asking teachers to think on the spot is that teachers in America are especially dumb: https://www.forbes.com/sites/nataliewexler/2019/03/13/why-so-many-aspiring-teachers-cant-pass-a-licensing-test--and-why-it-matters/?sh=64db8b0f321a https://www.forbes.com/sites/nataliewexler/2019/03/13/why-so... My mother was a public school teacher. She passed her test on the first try, which was extremely rare at her school. She was taught in India, and became an American teacher as an older woman. She was an outlier. She graduated from a major public teaching program in our state of California (CSU Fullerton). Most teachers fail their tests on the first try. This is not advanced calculus... this is elementary mathematics and reading. The issue is that accolades in the education department do not translate to results. The state creates a monopoly on teaching via its certification process, with means education departments are guaranteed funding regardless of their results, given that once you have the credential, you're considered 'equal' to other teachers. Despite my mother doing very well on her tests, she was fired and not granted tenure because she didn't follow her principal's teaching methodology. Even though her students were passing their own exams at higher rates and doing better (And of course my mother taught two boys who went on to make careers in science and mathematics), she was fired for not following the failing methodology. This is what is wrong with American schooling. It's a race to the bottom and the teachers have no idea what's actual achievement because they've never achieved basic elementary schooling, much less seen it in others. Here are example questions: https://www.mometrix.com/academy/praxis-math-practice-test/ https://www.mometrix.com/academy/praxis-math-practice-test/ These are not difficult, at all. EDIT: in some ways america is doomed by its own success, in that, for those who can actually do math, reading, and writing, there are significantly more lucrative fields than teaching. Unfortunately, since teachers can't teach those skills, it's really a new aristocracy formed by parents who do know those skills passing them on to children.
  - avmich 5y ago
    So the haves spend money on education of their kids, and have-nots can't do that, relying on public schools which don't perform, and the possibilities gap is widening. Structural problem.
    - iammisc 5y ago
      Absolutely! A major structural problem.
- monkeydreams 5y ago
  > I used to suffer from the lack of "why" too until one day it dawned on me that there really isn't a why, or maybe, the why is always the same: for engineers doing classical physics solving problems by hand, it's convenient to define it that way. No more, no less. 'Why' is shorthand for a description of the context of the operation or method being taught. My senior high-school experience was that my maths teacher just plowed through the curriculum from limits to derivatives to integrals and beyond without explaining what each was used for or why. I managed to scrape a pass in high-school calculus but never grew beyond the most rudimentary of understanding until I was taught by a university lecturer who invested time in explaining the why. The backbone of my mental schema is narrative, so just throwing equations and processes at me does not cause the knowledge to "stick". Teaching me that Gauss had a problem X that he tried to solve by Y, then showing me what he discovered process N, is invaluable to me because it aids my recall.
hhlbf 5y ago
Maths do not have a why.
- horsawlarway 5y ago
  I find this answer to be true, but misguided. In particular, this type of answer was the exact thing that turned me off of math for YEARS. Maths DO have a why - We needed way to describe and model the world around us, and math was a requirement to do that. Now - once the model and rules are put in place - Fine, you can bugger off and be as self-referential and contained as you'd like - "There is no why!"... But lets be clear - there ABSOLUTELY is a why, and the second the world around you no longer matches the model of your maths, we start debating whether or not to throw the thing in the trash and make new rules (see set theory as the classic example...)
- ecshafer 5y ago
  Yes they do. At minimum Mathematics have a why in that it is something that is arrived at logically, and provable. But when you learn mathematics, whether it be algebra or geometry and trigonometry or calculus or discrete, there is another why added, that of real life solutions. Why do we care about geometry? Well it ends up being useful in optics and physics. We like Trigonometry because it can help you build a building or bridge or shoot a cannon and take down said building or bridge. Calculus has innumerable uses in engineering and physics and finance and biology and everywhere.
- vangelis 5y ago
  If maths don't have a why, then programming languages don't have a why. They're both abstractions.
  - iammisc 5y ago
    Programming languages do not have a why. The turing machine is an arbitrary model of computation, as is the lambda calculus, as is horn clauses. The 'why' is deciding which one is most convenient to solve a particular problem. Given that very good computer scientists come up with very different means of solving the same problem, it's clear there's no universal agreement upon which paradigm is better and that there's a lot of subjective reasoning as to why particular models are better.

zsmi 5y ago

I used to suffer from the lack of "why" too until one day it dawned on me that there really isn't a why, or maybe, the why is always the same: for engineers doing classical physics solving problems by hand, it's convenient to define it that way. No more, no less. When I was a math tutor in school, everyone would complain about lack of why. This was my approach to address the issue. I would describe a problem relevant to the class, say something like find the angle between two vectors, and ask them to think of a general way to do it. Usually this require some help from me but the solution was always "theirs". Then we would play with their solution: find corner cases, figure out which operations broke, develop a couple of theorems with it. Then I would show them the dot product, and a couple of its tricks (a.k.a. theorems), and they would get a good appreciation for why the dot product works the ways it does. And that's the why and the reason it stuck. It's useful way to solve a common problem. There is no magic or deep philosophical truth. The problem is this teaching style works with a dialog to a few number of students. I am not sure how well it would work in a lecture setting. It was also not so easy on me as I would be the person who had to spot the problems in their definition, basically on the spot, and lead them to it on their own without revealing the answer. With basic things like a dot product it's pretty easy to know where to look (i.e. is their definition commutative, pretty much never) and finding issues and nudging them too it wasn't too hard. But it's likely hard to scale things like this on many metrics.

iammisc 5y ago
The problem with asking teachers to think on the spot is that teachers in America are especially dumb: https://www.forbes.com/sites/nataliewexler/2019/03/13/why-so-many-aspiring-teachers-cant-pass-a-licensing-test--and-why-it-matters/?sh=64db8b0f321a https://www.forbes.com/sites/nataliewexler/2019/03/13/why-so... My mother was a public school teacher. She passed her test on the first try, which was extremely rare at her school. She was taught in India, and became an American teacher as an older woman. She was an outlier. She graduated from a major public teaching program in our state of California (CSU Fullerton). Most teachers fail their tests on the first try. This is not advanced calculus... this is elementary mathematics and reading. The issue is that accolades in the education department do not translate to results. The state creates a monopoly on teaching via its certification process, with means education departments are guaranteed funding regardless of their results, given that once you have the credential, you're considered 'equal' to other teachers. Despite my mother doing very well on her tests, she was fired and not granted tenure because she didn't follow her principal's teaching methodology. Even though her students were passing their own exams at higher rates and doing better (And of course my mother taught two boys who went on to make careers in science and mathematics), she was fired for not following the failing methodology. This is what is wrong with American schooling. It's a race to the bottom and the teachers have no idea what's actual achievement because they've never achieved basic elementary schooling, much less seen it in others. Here are example questions: https://www.mometrix.com/academy/praxis-math-practice-test/ https://www.mometrix.com/academy/praxis-math-practice-test/ These are not difficult, at all. EDIT: in some ways america is doomed by its own success, in that, for those who can actually do math, reading, and writing, there are significantly more lucrative fields than teaching. Unfortunately, since teachers can't teach those skills, it's really a new aristocracy formed by parents who do know those skills passing them on to children.
- avmich 5y ago
  So the haves spend money on education of their kids, and have-nots can't do that, relying on public schools which don't perform, and the possibilities gap is widening. Structural problem.
  - iammisc 5y ago
    Absolutely! A major structural problem.
monkeydreams 5y ago
> I used to suffer from the lack of "why" too until one day it dawned on me that there really isn't a why, or maybe, the why is always the same: for engineers doing classical physics solving problems by hand, it's convenient to define it that way. No more, no less. 'Why' is shorthand for a description of the context of the operation or method being taught. My senior high-school experience was that my maths teacher just plowed through the curriculum from limits to derivatives to integrals and beyond without explaining what each was used for or why. I managed to scrape a pass in high-school calculus but never grew beyond the most rudimentary of understanding until I was taught by a university lecturer who invested time in explaining the why. The backbone of my mental schema is narrative, so just throwing equations and processes at me does not cause the knowledge to "stick". Teaching me that Gauss had a problem X that he tried to solve by Y, then showing me what he discovered process N, is invaluable to me because it aids my recall.

hhlbf 5y ago

Maths do not have a why.

horsawlarway 5y ago
I find this answer to be true, but misguided. In particular, this type of answer was the exact thing that turned me off of math for YEARS. Maths DO have a why - We needed way to describe and model the world around us, and math was a requirement to do that. Now - once the model and rules are put in place - Fine, you can bugger off and be as self-referential and contained as you'd like - "There is no why!"... But lets be clear - there ABSOLUTELY is a why, and the second the world around you no longer matches the model of your maths, we start debating whether or not to throw the thing in the trash and make new rules (see set theory as the classic example...)
ecshafer 5y ago
Yes they do. At minimum Mathematics have a why in that it is something that is arrived at logically, and provable. But when you learn mathematics, whether it be algebra or geometry and trigonometry or calculus or discrete, there is another why added, that of real life solutions. Why do we care about geometry? Well it ends up being useful in optics and physics. We like Trigonometry because it can help you build a building or bridge or shoot a cannon and take down said building or bridge. Calculus has innumerable uses in engineering and physics and finance and biology and everywhere.
vangelis 5y ago
If maths don't have a why, then programming languages don't have a why. They're both abstractions.
- iammisc 5y ago
  Programming languages do not have a why. The turing machine is an arbitrary model of computation, as is the lambda calculus, as is horn clauses. The 'why' is deciding which one is most convenient to solve a particular problem. Given that very good computer scientists come up with very different means of solving the same problem, it's clear there's no universal agreement upon which paradigm is better and that there's a lot of subjective reasoning as to why particular models are better.