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I recognize the purpose of learning fundamentals but I think we do a great disservice to students learning mathematics by not emphasizing the power of computers

I recognize the purpose of learning fundamentals but I think we do a great disservice to students learning mathematics by not emphasizing the power of computers and calculators in modern arithmetic. Many people with math anxiety see mathematical skills to be the same as mental arithmetic skills. Textbooks still focus things like von Neumann's ability ability to multiply large digit numbers in his head as a proxy for his math proficiency. While impressive, the technologies built on top of von Neumann's work make that kind of skill obsolete. I firmly believe that all brains are good at math but not everyone has had a teacher who will spend the time to teach the cognitive tools required for it. If we stop teaching calculators as "cheating" then a lot more students will be able to focus on the understanding the underlying concepts and will feel less scared about approaching new mathematical problems.

buscoquadnary 4y ago
I'd argue that not all brains are good at math but that they can be trained to be good at math. That's the problem we treat math like it is some innate talent, it's not it just requires certain ways and forms of thinking. Example: I was decent at math in HS, I didn't do very well because I didn't like doing the homework and quite frankly most people good at math aren't going to end up being public school teachers, so the explanations were always lacking. Well I went on learning programming on my own got a job as an SWE, then a couple of years later ran into a situation where I had to learn some new math. It was world's easier this time, because of my work with programming it all jelled immediately and suddenly clicked it all became very logical. I don't believe I suddenly became better at math, but rather I had developed certain thought patterns and ways of thinking that could be applied to mathematics.
- syzar 4y ago
  Not everyone can learn calculus or even basic algebra. A person with a severe intellectual disability isn’t going to be able to learn these subjects. Between severely intellectually disabled and Einstein is a boundary between those who can and those who can’t. I don’t know where the boundary is but it exists.
  - roey2009 4y ago
    A more accurate concept would be the bell curve, and range. Remember, shades of grey.
  - technothrasher 4y ago
    It doesn’t just have to be severe disability. I’m pretty sure I’ve got what is now called NVLD (because my son is diagnosed with it and I share most of his symptoms). It doesn’t cause general intellectual issues, in fact I scored above 130 on the WISC IQ test as a kid. But one of the things it does cause is poor working memory. Trying to keep numbers in my head in order to calculate things is pretty much impossible. Give me calculator and I have no issue with the higher math concepts. But leave me on my own and I’ll give you the wrong answer due to bad calculation every time. Anyway, this all led to me doing very poorly in math at school, and it’s entirely because my brain just isn’t good at it. Luckily these days the educators are smarter and my son has always been allowed a calculator for math class.
    - stonecolddevin 4y ago
      I had no idea this had a name. Sure wish I'd have known this while I was in school.
CivBase 4y ago
Computers and calculators should absolutely be embraced by modern math education. Once someone thoroughly understands long division, there's no sense in forcing them to tediously repeat every division operation by hand. But it's still imperative that students learn conceptually how and why mathematical processes work. Math is different from most other primary school subjects in that it builds on itself year after year. There is a danger to letting computing devices take over before the students fully grasp a math concept. If they fail to learn a concept, moving on with a calculator as a crutch may inhibit them from understanding the next math concept.
- syzar 4y ago
  Students don’t fully grasp concepts at the early stages and they never will. Very few students who master the quadratic formula understand that this formula allows you to factor second degree polynomials. Many students will correctly solve x^2+x+1 = 0 while simultaneously believing that this polynomial doesn’t factor. Here’s a simple problem that tests understanding, A second degree polynomial with leading coefficient 3 has zeros of -1 and 2. Find all the terms of the polynomial. Most students can’t do this. Even most calculus students can’t do it. We teach algorithms like long division and the quadratic formula because they are relatively easy computations to learn but they don’t in any way lead students to fully grasping a concept. It’s only with a certain level of mathematical maturity that one is able to understand the full import of even basic concepts. I can walk into pretty much any first semester calculus class and ask students to write down an example of an equation with no solution. A large majority will fail to do so. It doesn’t occur to them that 0=1 is such an example. They’ll play around with x’s in various complicated looking expressions. Even something as basic and fundamental as the meaning of an equation eludes people at this level even though they have been dealing with equations for years. Well, such is my experience teaching math at a community college for over 20 years.
  - LordDragonfang 4y ago
    >I can walk into pretty much any first semester calculus class and ask students to write down an example of an equation with no solution. A large majority will fail to do so. It doesn’t occur to them that 0=1 is such an example. It's interesting that you picked 0=1 as your example, because I'd argue it stretches the definitions of "equation" and "solution" into semantic triviality. It's more of a falsehood than an "equation", since the two sides are trivially defined as not equal, and there's no variables to "solve". Using that as example exists somewhere between sophistry and pointing out the absurdity that mathematical definitions for terms technically hold even in trivially untrue situations. That's not how normal human communication works, and not recognizing that divide probably goes a long way in explaining the "inability" you see in students. In other words, maybe you should have just used "0x=1" as your example :P
scythe 4y ago
> I think we do a great disservice to students learning mathematics by not emphasizing the power of computers and calculators in modern arithmetic. But all of the students who came to the help desk in college were practically addicted to their overpriced graphing calculators. A decent fraction of the time I would look over their work, they'd be on their last try in WebAssign (may it be cast into the fire) with a scared look on their face, and I would say "you entered this part into the calculator wrong". I knew immediately because I learned to estimate these things. You don't have to even know the multiplication algorithm; you can stick to multiplying the first digit, adding exponents in scientific notation, etc. Learning to work with the calculator instead of behind the calculator will make you use it more effectively. The multiplication table has 45 entries that aren't duplicates or multiples of 1 or 0. Kids memorize longer things all the time. The brain is definitely not the limiting factor. >Textbooks still focus things like von Neumann's ability ability to multiply large digit numbers in his head as a proxy for his math proficiency. I don't recall ever seeing that in a textbook. There is a famous story about JvN summing an infinite series in his head, but very few people can do that or would be expected to. He also estimated the yield of Trinity within a factor of four by just looking at it. One should never compare themself to von Neumann. I honestly believe the biggest factor in math anxiety is osmosis. Their parents are afraid of math, their teachers (often!) are afraid of math, and there's a robust body of evidence in developmental psychology that children inherit the worries of their guardians.
boboralice 4y ago
My husband is terrified of mathematics but he's a great programmer. I'm no genius when it comes to the subject because it's not my passion. I've only used degree-level math a couple of times in the (mumble) years since I graduated and I had to dig the textbooks out and revise it every time. My husband and I went to uni together. I dragged him through the mandatory maths units in the computer science degree by reframing the equations we were working on as "a bunch of steps, just like code" and he got through the unit, but our lecturer was a horrible, impatient person and managed to reinforce his fear of the subject. A lot of people I know have similar stories. Maths and physics seem to attract teachers who think (out loud) that students who don't intuitively "get it" are beneath them.
theteapot 4y ago
> Textbooks still focus things like von Neumann's ability ability to multiply large digit numbers in his head as a proxy for his math proficiency. Really? Which textbooks are these? Done high school, uni. Never heard that, or really much about von Neumann outside a few passing mentions in advanced topics. Sounds like some really strange math textbooks.

buscoquadnary 4y ago

I'd argue that not all brains are good at math but that they can be trained to be good at math. That's the problem we treat math like it is some innate talent, it's not it just requires certain ways and forms of thinking. Example: I was decent at math in HS, I didn't do very well because I didn't like doing the homework and quite frankly most people good at math aren't going to end up being public school teachers, so the explanations were always lacking. Well I went on learning programming on my own got a job as an SWE, then a couple of years later ran into a situation where I had to learn some new math. It was world's easier this time, because of my work with programming it all jelled immediately and suddenly clicked it all became very logical. I don't believe I suddenly became better at math, but rather I had developed certain thought patterns and ways of thinking that could be applied to mathematics.

syzar 4y ago
Not everyone can learn calculus or even basic algebra. A person with a severe intellectual disability isn’t going to be able to learn these subjects. Between severely intellectually disabled and Einstein is a boundary between those who can and those who can’t. I don’t know where the boundary is but it exists.
- roey2009 4y ago
  A more accurate concept would be the bell curve, and range. Remember, shades of grey.
- technothrasher 4y ago
  It doesn’t just have to be severe disability. I’m pretty sure I’ve got what is now called NVLD (because my son is diagnosed with it and I share most of his symptoms). It doesn’t cause general intellectual issues, in fact I scored above 130 on the WISC IQ test as a kid. But one of the things it does cause is poor working memory. Trying to keep numbers in my head in order to calculate things is pretty much impossible. Give me calculator and I have no issue with the higher math concepts. But leave me on my own and I’ll give you the wrong answer due to bad calculation every time. Anyway, this all led to me doing very poorly in math at school, and it’s entirely because my brain just isn’t good at it. Luckily these days the educators are smarter and my son has always been allowed a calculator for math class.
  - stonecolddevin 4y ago
    I had no idea this had a name. Sure wish I'd have known this while I was in school.

CivBase 4y ago

Computers and calculators should absolutely be embraced by modern math education. Once someone thoroughly understands long division, there's no sense in forcing them to tediously repeat every division operation by hand. But it's still imperative that students learn conceptually how and why mathematical processes work. Math is different from most other primary school subjects in that it builds on itself year after year. There is a danger to letting computing devices take over before the students fully grasp a math concept. If they fail to learn a concept, moving on with a calculator as a crutch may inhibit them from understanding the next math concept.

syzar 4y ago
Students don’t fully grasp concepts at the early stages and they never will. Very few students who master the quadratic formula understand that this formula allows you to factor second degree polynomials. Many students will correctly solve x^2+x+1 = 0 while simultaneously believing that this polynomial doesn’t factor. Here’s a simple problem that tests understanding, A second degree polynomial with leading coefficient 3 has zeros of -1 and 2. Find all the terms of the polynomial. Most students can’t do this. Even most calculus students can’t do it. We teach algorithms like long division and the quadratic formula because they are relatively easy computations to learn but they don’t in any way lead students to fully grasping a concept. It’s only with a certain level of mathematical maturity that one is able to understand the full import of even basic concepts. I can walk into pretty much any first semester calculus class and ask students to write down an example of an equation with no solution. A large majority will fail to do so. It doesn’t occur to them that 0=1 is such an example. They’ll play around with x’s in various complicated looking expressions. Even something as basic and fundamental as the meaning of an equation eludes people at this level even though they have been dealing with equations for years. Well, such is my experience teaching math at a community college for over 20 years.
- LordDragonfang 4y ago
  >I can walk into pretty much any first semester calculus class and ask students to write down an example of an equation with no solution. A large majority will fail to do so. It doesn’t occur to them that 0=1 is such an example. It's interesting that you picked 0=1 as your example, because I'd argue it stretches the definitions of "equation" and "solution" into semantic triviality. It's more of a falsehood than an "equation", since the two sides are trivially defined as not equal, and there's no variables to "solve". Using that as example exists somewhere between sophistry and pointing out the absurdity that mathematical definitions for terms technically hold even in trivially untrue situations. That's not how normal human communication works, and not recognizing that divide probably goes a long way in explaining the "inability" you see in students. In other words, maybe you should have just used "0x=1" as your example :P

scythe 4y ago

> I think we do a great disservice to students learning mathematics by not emphasizing the power of computers and calculators in modern arithmetic. But all of the students who came to the help desk in college were practically addicted to their overpriced graphing calculators. A decent fraction of the time I would look over their work, they'd be on their last try in WebAssign (may it be cast into the fire) with a scared look on their face, and I would say "you entered this part into the calculator wrong". I knew immediately because I learned to estimate these things. You don't have to even know the multiplication algorithm; you can stick to multiplying the first digit, adding exponents in scientific notation, etc. Learning to work with the calculator instead of behind the calculator will make you use it more effectively. The multiplication table has 45 entries that aren't duplicates or multiples of 1 or 0. Kids memorize longer things all the time. The brain is definitely not the limiting factor. >Textbooks still focus things like von Neumann's ability ability to multiply large digit numbers in his head as a proxy for his math proficiency. I don't recall ever seeing that in a textbook. There is a famous story about JvN summing an infinite series in his head, but very few people can do that or would be expected to. He also estimated the yield of Trinity within a factor of four by just looking at it. One should never compare themself to von Neumann. I honestly believe the biggest factor in math anxiety is osmosis. Their parents are afraid of math, their teachers (often!) are afraid of math, and there's a robust body of evidence in developmental psychology that children inherit the worries of their guardians.

boboralice 4y ago

My husband is terrified of mathematics but he's a great programmer. I'm no genius when it comes to the subject because it's not my passion. I've only used degree-level math a couple of times in the (mumble) years since I graduated and I had to dig the textbooks out and revise it every time. My husband and I went to uni together. I dragged him through the mandatory maths units in the computer science degree by reframing the equations we were working on as "a bunch of steps, just like code" and he got through the unit, but our lecturer was a horrible, impatient person and managed to reinforce his fear of the subject. A lot of people I know have similar stories. Maths and physics seem to attract teachers who think (out loud) that students who don't intuitively "get it" are beneath them.

theteapot 4y ago

> Textbooks still focus things like von Neumann's ability ability to multiply large digit numbers in his head as a proxy for his math proficiency. Really? Which textbooks are these? Done high school, uni. Never heard that, or really much about von Neumann outside a few passing mentions in advanced topics. Sounds like some really strange math textbooks.