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Real understanding of concepts has been replaced with step by step methods that allow you to arrive at a solution without needing to understand what you are doi

Real understanding of concepts has been replaced with step by step methods that allow you to arrive at a solution without needing to understand what you are doing. Isn’t this how pre-undergraduate math education has been for the entire history of mass education? How else would you describe everything from how to multiply two numbers to simplifying fractions to the chain rule?

jjk166 3y ago
Well you kind of have to take addition on faith, but everything else is pretty built up - subtraction is adding negatives, multiplication is repeated addition, division is repeated subtraction, fractions are division, decimals are fractions, and so on and so forth. Maybe some explanations were a little more handwavy than others, and certainly some algorithms could be better explained (took me quite some time to grok long division), but I can't think of anything that was just "trust me it works."
- bradleyjg 3y ago
  Do you need to understand that addition is commutative, or even what commutative means, to use the schoolboy multiplication algorithm? How is that anything other than a “step by step methods that allow[s] you to arrive at a solution without needing to understand what you are doing?”
  - dotnet00 3y ago
    The point is that math is not typically taught as just memorizing the schoolboy multiplication algorithm without being given an understanding that addition is commutative and what commutative means.
    - bradleyjg 3y ago
      I challenge you to select a random high schooler and report back on his or her knowledge of commutativity.
      - peteradio 3y ago
        If you ask them if A+B=B+A I'm sure they'd tell you yes for the most part. Quizzing on vocabulary is hardly the point.
        bradleyjg 3y ago
        And be able to explain why that must be true for schoolboy multiplication to work. Come on, this argument is ridiculous. There was no golden age of mass mathematics education.
        jjk166 3y ago
        That is moving the goal posts to a different stadium. I don't think I could explain that now, there's no reason for a child to be required to do so.
        bradleyjg 3y ago
        Real understanding of concepts has been replaced with step by step methods that allow you to arrive at a solution without needing to understand what you are doing. This is exactly what happens with multiplication (and simplify fractions and taking derivatives). That’s been exactly my claim all along. Virtually no 16 year olds know why the procedure they were taught to multiply 123 x 456 produces the correct answer. Not today, not 20 years ago, not 50 years ago. They know how to execute the algorithm. That’s it.
        dotnet00 3y ago
        "This argument is ridiculous" only because you keep shifting goalposts and refusing to actually back up anything you're saying while just repeating your personal belief that mathematics has always been rote memorization of algorithms. If it was that way, there would not be so much variation between performance in math around the world, as they'd all just be having kids memorize the same stuff.
- Jensson 3y ago
  > Well you kind of have to take addition on faith No you don't, addition is taught by making kids count up sums, not by giving them magic rules to follow. Addition has by far the strongest connection to reality and that connection gets hammered into every child. Basically everyone knows how to add by counting on fingers.
  - jjk166 3y ago
    You can see that it works, but only because you have defined it to work that way. There is no explanation of why it works based on more fundamental principles. With a much more advanced math education you can derive addition from more fundamental axioms, but that is way beyond what a child should ever be required to learn.
- ChainOfFools 3y ago
  > trust me it works I can think of a couple. limits at infinity are thoroughly hand wavy, at least at the level they're introduced in high school or lower division college calculus. And I would say this even projects backwards into geometry, and even the Blurred overloaded operations of rational numbers and simple fractions. the notion of being able to add up an infinity of nothing and get a quantifiable something is just sort of taken for granted without elaboration. The equality operator ( = ) is also presented as an implicit expression of truth when introduced, when it clearly admits to shades of true at best. 1*1 = 1 = 1^1 = 1^2 is one kind of true and a bunch of kinds of false. and although at a young age kids may not be sensitive to this, letting it go without explanation or curious challenge can be the source of inarticulate senses of doubt about what's really going on when they get introduced to higher levels of math.
tennisflyi 3y ago
> Isn’t this how pre-undergraduate math education has been for the entire history of mass education? Yes. Not sure why you're in the negative ATM.

jjk166 3y ago

Well you kind of have to take addition on faith, but everything else is pretty built up - subtraction is adding negatives, multiplication is repeated addition, division is repeated subtraction, fractions are division, decimals are fractions, and so on and so forth. Maybe some explanations were a little more handwavy than others, and certainly some algorithms could be better explained (took me quite some time to grok long division), but I can't think of anything that was just "trust me it works."

bradleyjg 3y ago
Do you need to understand that addition is commutative, or even what commutative means, to use the schoolboy multiplication algorithm? How is that anything other than a “step by step methods that allow[s] you to arrive at a solution without needing to understand what you are doing?”
- dotnet00 3y ago
  The point is that math is not typically taught as just memorizing the schoolboy multiplication algorithm without being given an understanding that addition is commutative and what commutative means.
  - bradleyjg 3y ago
    I challenge you to select a random high schooler and report back on his or her knowledge of commutativity.
    - peteradio 3y ago
      If you ask them if A+B=B+A I'm sure they'd tell you yes for the most part. Quizzing on vocabulary is hardly the point.
      - bradleyjg 3y ago
        And be able to explain why that must be true for schoolboy multiplication to work. Come on, this argument is ridiculous. There was no golden age of mass mathematics education.
        jjk166 3y ago
        That is moving the goal posts to a different stadium. I don't think I could explain that now, there's no reason for a child to be required to do so.
        bradleyjg 3y ago
        Real understanding of concepts has been replaced with step by step methods that allow you to arrive at a solution without needing to understand what you are doing. This is exactly what happens with multiplication (and simplify fractions and taking derivatives). That’s been exactly my claim all along. Virtually no 16 year olds know why the procedure they were taught to multiply 123 x 456 produces the correct answer. Not today, not 20 years ago, not 50 years ago. They know how to execute the algorithm. That’s it.
        dotnet00 3y ago
        "This argument is ridiculous" only because you keep shifting goalposts and refusing to actually back up anything you're saying while just repeating your personal belief that mathematics has always been rote memorization of algorithms. If it was that way, there would not be so much variation between performance in math around the world, as they'd all just be having kids memorize the same stuff.
Jensson 3y ago
> Well you kind of have to take addition on faith No you don't, addition is taught by making kids count up sums, not by giving them magic rules to follow. Addition has by far the strongest connection to reality and that connection gets hammered into every child. Basically everyone knows how to add by counting on fingers.
- jjk166 3y ago
  You can see that it works, but only because you have defined it to work that way. There is no explanation of why it works based on more fundamental principles. With a much more advanced math education you can derive addition from more fundamental axioms, but that is way beyond what a child should ever be required to learn.
ChainOfFools 3y ago
> trust me it works I can think of a couple. limits at infinity are thoroughly hand wavy, at least at the level they're introduced in high school or lower division college calculus. And I would say this even projects backwards into geometry, and even the Blurred overloaded operations of rational numbers and simple fractions. the notion of being able to add up an infinity of nothing and get a quantifiable something is just sort of taken for granted without elaboration. The equality operator ( = ) is also presented as an implicit expression of truth when introduced, when it clearly admits to shades of true at best. 1*1 = 1 = 1^1 = 1^2 is one kind of true and a bunch of kinds of false. and although at a young age kids may not be sensitive to this, letting it go without explanation or curious challenge can be the source of inarticulate senses of doubt about what's really going on when they get introduced to higher levels of math.

tennisflyi 3y ago

> Isn’t this how pre-undergraduate math education has been for the entire history of mass education? Yes. Not sure why you're in the negative ATM.