4 ms·

Computers and calculators should absolutely be embraced by modern math education. Once someone thoroughly understands long division, there's no sense in forcing

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
  - syzar 4y ago
    0x is the same thing as 0 so it appears my example is a good one in that you yourself don’t fully understand the concepts involved. This isn’t pejorative. Suppose I said solve x = x+1 You then subtract x from both sides and end up with 0 = 1 Then you conclude that the original equation has no solution. I’m guessing that you wouldn’t realize that the reason we conclude that the original equation has no solution is because the two equations x = x + 1 and 0 = 1 have the same solution set since adding the opposite of x to both sides is a solution set preserving operation. It transforms a given equation into a new equation with the same solutions and clearly 0=1 has no solution. That is, 0=1 is a perfectly valid equation. The larger point, that is missed by people, is that an equation in essence is asking for one to find the instances when two expressions are equal. To find an example of an equation with no solution just find two expressions that are never equal to each other.
    - LordDragonfang 4y ago
      >The larger point, that is missed by people, is that an equation in essence is asking for one to find the instances when two expressions are equal. Respectfully, you've got this backwards. An equation, by definition, is an assertion that two expressions are equal. 0=1 is a logically consistent assertion, but it happens to be false. Most students will intuitively have trouble with the idea that you want them to make a false statement, even if they don't realize that, because their whole schooling has taught them the opposite. The issue is precisely that we are teaching those students that that "an equation in essence is asking for one to find the instances when two expressions are equal". Mathematical statements don't "ask" anything, they simply are. That's a pedagogical definition, not a mathematical one, and by teaching students that, you're teaching them how to pass a math test rather than teaching them math. And there's no blame on you for that, since you're paid to teach students to pass math tests. But framing it that way doesn't teach them math, it teaches them how to guess the teacher's password[1]. It's a focus on getting an answer rather than understanding the actual axioms. So of course students don't come up with an equation with nothing to solve, because you've taught them equations are things that only exist as things with unknowns to solve. It might be obvious to someone who already is extremely well versed in mathematics that 0=1 is "an equation without a solution". But it's unfair to expect students who don't already have that answer to derive it, because they're working off of the wrong axioms. It's a communication failure, not a mathematical one. [1]https://www.lesswrong.com/posts/NMoLJuDJEms7Ku9XS/guessing-the-teacher-s-password https://www.lesswrong.com/posts/NMoLJuDJEms7Ku9XS/guessing-t...
- knaik94 4y ago
  Actually the quadratic formula is a great example of exactly what I mean. I was taught it as a way to solve for roots of a second degree polynomial when "completing the square" didn't work. The terms in the quadratic formula are defined as the coefficients in the polynomial and the answer is written in the form "x = +-___". Looking back at that chapter in math, that along with the rational roots test for finding zeros in a higher order polynomial were genuine wastes of time. I don't mean the learning the intuition, I mean a question asking to list every single possible root given a second or higher order polynomial. I was taught that using the quadratic formula to find roots is "cheating" when I tried using it before we had covered it in class. What exactly does "completing the square" test for other than your skills at mental arithmetic? The reason why many student believe that polynomial doesn't factor is because teachers do a lot of hand waving when it comes to explaining what it means to have no rational roots of a polynomial. Few teachers will take the time to teach the foundations of the cartesian coordinate system and how complex solutions don't map easily on the typical plane of rational numbers. All students learn is if there's an "i" the answer is "no solution". Zeros being the solutions of functions is a question on the finding roots of a polynomial chapter in basically every single high school algebra 2 class. It's a prerequisite to learning how to graph second order polynomial functions. Many students learn and forget how to do it before reaching calculus, let alone college. I genuinely haven't done long division in the last decade. I struggled to help my younger cousin with it recently and had to relearn myself because it's such a useless algorithm in the age of computers. Certain multiples and powers I remember, because of how often I come across the numbers, but in general I will choose a calculator every time. I would even choose a calculator to double check my own work with a paper and pencil. At this point what is the value in doing the work by hand? In many cases a decimal to the hundredths is required as well. When I hear the question, write down an example of an equation with no solution, my intuition and experience doesn't lead me to writing an incorrect equation. It leads me to think about writing a polynomial function that I know will have complex roots because I was taught the answer to that is "no solution", or writing a system of equation where x is a specific number while at the same time having an equation where that specific number can't be a part of the domain. More fundamentally, a system of equation with no solution is one where the two lines that are graphed are parallel. I have to admit, my experience is a little biased as I was placed in accelerated math since elementary school. It wasn't difficult for me or my peers. My math class senior year was fundamentals of multivariable calc and linear algebra as a senior in high school, having finished AP calc bc the year before. I was far from the only one in that situation, there were at least 60 of us that year, some seniors and some juniors. I can't say I have experience teaching a full class but I have been tutoring high schoolers in math for over 8 years. Many of my students have also been in accelerated math, but not all of them. I don't think anyone tested out of multivariable calculus, but I did have a friend who tested out of linear algebra at reputable universities. I know that I have some time and experience left before I feel confident in my own mathematical maturity, but I'd like to imagine I'm somewhat good at math. At the very least I wouldn't consider myself bad at math, even though I still feel like I am at the early stages of learning in specific branches of math.
jrgoff 4y ago
I'm curious what value you consider understanding long division to hold? In my mind it has some value in training how to follow an algorithm and obviously has some value in being able to divide arbitrary integers (and beyond). But of course as you point out there's not much point in forcing long division when there are ubiquitous computers available to do the job better. And while I think following an algorithm carefully can be a valuable skill, if that is the primary goal then I'm not sure teaching long division is a particularly good way to go about it. I agree with your general point that math curriculum builds on itself and if students fall behind it can make things worse and worse for future classes. But in general I think, at least when I was in school, there was probably too much focus on algorithms and not enough on conceptual understanding. I think part of the reason is that algorithms historically were necessary to know before calculators were ubiquitous, and also I think algorithms are much easier to teach and evaluate so there's probably a tendency to focus on them for that reason.
- CivBase 4y ago
  It's not necessarily the algorithm that matters. There are a few different algorithms you can follow for division and a few ways you can accurately represent the results. For example, kids these day commonly learn something called the "lattice method" for multiplication. It's a different algorithm from what I learned, but that doesn't necessarily matter. The point is not for students to learn how to follow an algorithm (or master doing it quickly), but that they learn why the algorithm works and how to interpret the result.
  - knaik94 4y ago
    Honestly, I hadn't come across the lattice method until you mentioned it. I would strongly argue that it hinders learning because it's hard to draw an association between the numbers you are writing down and how the lattice method is set up. It is equivalent to long division after you recognize that the lattice method removes the shifting to the right as you multiply by the next digit. With the standard long multiplication method, you recognize that what you were really doing is splitting the multiplier by powers of 10 once you learn the distributive property. The lattice method is actually weirder because you don't move by increasing powers of 10, you go in the reverse direction. You still have to carry and you still add up numbers in column/power of 10 order, starting with the smallest power at the very end. If anything, the lattice method represents everything about the common core math approach that many parents and teachers dislike. The algorithm matters and I argue that once students can naturally derive the algorithm on their own, there's no place for human calculations. Computers and calculators should be used. We're no longer living in a society where students will have to worry about "what if you don't have a calculator on you". It seems as weird as memorizing unit conversions to me. What's interesting is that the purpose of implementing the common core standard was exactly what I am arguing for. A shift away from algorithmic memorization to a number sense way of doing math. The failure lies with the teachers not being able to teach well much more than students not learning.

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
- syzar 4y ago
  0x is the same thing as 0 so it appears my example is a good one in that you yourself don’t fully understand the concepts involved. This isn’t pejorative. Suppose I said solve x = x+1 You then subtract x from both sides and end up with 0 = 1 Then you conclude that the original equation has no solution. I’m guessing that you wouldn’t realize that the reason we conclude that the original equation has no solution is because the two equations x = x + 1 and 0 = 1 have the same solution set since adding the opposite of x to both sides is a solution set preserving operation. It transforms a given equation into a new equation with the same solutions and clearly 0=1 has no solution. That is, 0=1 is a perfectly valid equation. The larger point, that is missed by people, is that an equation in essence is asking for one to find the instances when two expressions are equal. To find an example of an equation with no solution just find two expressions that are never equal to each other.
  - LordDragonfang 4y ago
    >The larger point, that is missed by people, is that an equation in essence is asking for one to find the instances when two expressions are equal. Respectfully, you've got this backwards. An equation, by definition, is an assertion that two expressions are equal. 0=1 is a logically consistent assertion, but it happens to be false. Most students will intuitively have trouble with the idea that you want them to make a false statement, even if they don't realize that, because their whole schooling has taught them the opposite. The issue is precisely that we are teaching those students that that "an equation in essence is asking for one to find the instances when two expressions are equal". Mathematical statements don't "ask" anything, they simply are. That's a pedagogical definition, not a mathematical one, and by teaching students that, you're teaching them how to pass a math test rather than teaching them math. And there's no blame on you for that, since you're paid to teach students to pass math tests. But framing it that way doesn't teach them math, it teaches them how to guess the teacher's password[1]. It's a focus on getting an answer rather than understanding the actual axioms. So of course students don't come up with an equation with nothing to solve, because you've taught them equations are things that only exist as things with unknowns to solve. It might be obvious to someone who already is extremely well versed in mathematics that 0=1 is "an equation without a solution". But it's unfair to expect students who don't already have that answer to derive it, because they're working off of the wrong axioms. It's a communication failure, not a mathematical one. [1]https://www.lesswrong.com/posts/NMoLJuDJEms7Ku9XS/guessing-the-teacher-s-password https://www.lesswrong.com/posts/NMoLJuDJEms7Ku9XS/guessing-t...
knaik94 4y ago
Actually the quadratic formula is a great example of exactly what I mean. I was taught it as a way to solve for roots of a second degree polynomial when "completing the square" didn't work. The terms in the quadratic formula are defined as the coefficients in the polynomial and the answer is written in the form "x = +-___". Looking back at that chapter in math, that along with the rational roots test for finding zeros in a higher order polynomial were genuine wastes of time. I don't mean the learning the intuition, I mean a question asking to list every single possible root given a second or higher order polynomial. I was taught that using the quadratic formula to find roots is "cheating" when I tried using it before we had covered it in class. What exactly does "completing the square" test for other than your skills at mental arithmetic? The reason why many student believe that polynomial doesn't factor is because teachers do a lot of hand waving when it comes to explaining what it means to have no rational roots of a polynomial. Few teachers will take the time to teach the foundations of the cartesian coordinate system and how complex solutions don't map easily on the typical plane of rational numbers. All students learn is if there's an "i" the answer is "no solution". Zeros being the solutions of functions is a question on the finding roots of a polynomial chapter in basically every single high school algebra 2 class. It's a prerequisite to learning how to graph second order polynomial functions. Many students learn and forget how to do it before reaching calculus, let alone college. I genuinely haven't done long division in the last decade. I struggled to help my younger cousin with it recently and had to relearn myself because it's such a useless algorithm in the age of computers. Certain multiples and powers I remember, because of how often I come across the numbers, but in general I will choose a calculator every time. I would even choose a calculator to double check my own work with a paper and pencil. At this point what is the value in doing the work by hand? In many cases a decimal to the hundredths is required as well. When I hear the question, write down an example of an equation with no solution, my intuition and experience doesn't lead me to writing an incorrect equation. It leads me to think about writing a polynomial function that I know will have complex roots because I was taught the answer to that is "no solution", or writing a system of equation where x is a specific number while at the same time having an equation where that specific number can't be a part of the domain. More fundamentally, a system of equation with no solution is one where the two lines that are graphed are parallel. I have to admit, my experience is a little biased as I was placed in accelerated math since elementary school. It wasn't difficult for me or my peers. My math class senior year was fundamentals of multivariable calc and linear algebra as a senior in high school, having finished AP calc bc the year before. I was far from the only one in that situation, there were at least 60 of us that year, some seniors and some juniors. I can't say I have experience teaching a full class but I have been tutoring high schoolers in math for over 8 years. Many of my students have also been in accelerated math, but not all of them. I don't think anyone tested out of multivariable calculus, but I did have a friend who tested out of linear algebra at reputable universities. I know that I have some time and experience left before I feel confident in my own mathematical maturity, but I'd like to imagine I'm somewhat good at math. At the very least I wouldn't consider myself bad at math, even though I still feel like I am at the early stages of learning in specific branches of math.

jrgoff 4y ago

I'm curious what value you consider understanding long division to hold? In my mind it has some value in training how to follow an algorithm and obviously has some value in being able to divide arbitrary integers (and beyond). But of course as you point out there's not much point in forcing long division when there are ubiquitous computers available to do the job better. And while I think following an algorithm carefully can be a valuable skill, if that is the primary goal then I'm not sure teaching long division is a particularly good way to go about it. I agree with your general point that math curriculum builds on itself and if students fall behind it can make things worse and worse for future classes. But in general I think, at least when I was in school, there was probably too much focus on algorithms and not enough on conceptual understanding. I think part of the reason is that algorithms historically were necessary to know before calculators were ubiquitous, and also I think algorithms are much easier to teach and evaluate so there's probably a tendency to focus on them for that reason.

CivBase 4y ago
It's not necessarily the algorithm that matters. There are a few different algorithms you can follow for division and a few ways you can accurately represent the results. For example, kids these day commonly learn something called the "lattice method" for multiplication. It's a different algorithm from what I learned, but that doesn't necessarily matter. The point is not for students to learn how to follow an algorithm (or master doing it quickly), but that they learn why the algorithm works and how to interpret the result.
- knaik94 4y ago
  Honestly, I hadn't come across the lattice method until you mentioned it. I would strongly argue that it hinders learning because it's hard to draw an association between the numbers you are writing down and how the lattice method is set up. It is equivalent to long division after you recognize that the lattice method removes the shifting to the right as you multiply by the next digit. With the standard long multiplication method, you recognize that what you were really doing is splitting the multiplier by powers of 10 once you learn the distributive property. The lattice method is actually weirder because you don't move by increasing powers of 10, you go in the reverse direction. You still have to carry and you still add up numbers in column/power of 10 order, starting with the smallest power at the very end. If anything, the lattice method represents everything about the common core math approach that many parents and teachers dislike. The algorithm matters and I argue that once students can naturally derive the algorithm on their own, there's no place for human calculations. Computers and calculators should be used. We're no longer living in a society where students will have to worry about "what if you don't have a calculator on you". It seems as weird as memorizing unit conversions to me. What's interesting is that the purpose of implementing the common core standard was exactly what I am arguing for. A shift away from algorithmic memorization to a number sense way of doing math. The failure lies with the teachers not being able to teach well much more than students not learning.