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IME a big part of this misdirection stems from school focusing on the mechanics of math, rather than the intuition of math. In the long run, the mechanics of m

IME a big part of this misdirection stems from school focusing on the mechanics of math, rather than the intuition of math. In the long run, the mechanics of math (how to do long division, or differentiate an expression, or expand a geometric series) are important only insofar as to help us model and predict and analyze the world, intuitively. However, students who do not intuitively see the power of mathematical intuition as a tool for understanding and modeling the world better, think that they are taught the mechanics of math just for the sake of "no phone/calculator/computer available" circumstances.

deleted 3y ago
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butterNaN 3y ago
I can only speak for myself here, but my journey with Maths did start with concrete, practical relationship between concepts and usage. I still remember fascinated by Geometry and Trigonometry, because how it can be used to create graphics and video games is obvious. The more abstract it gets, the more it warrants an introduction with "here's how it's used in real life". I can recall the point when it started becoming mechanical - it was when I started doing Derivatives and Integrations. I was 'just solving puzzles' until I hit a chapter, tucked away way back in the syllabus, almost at the end of the 2 year arc - a chapter about Applied Differential Calculus. I still remember the feeling of "Oh... I get it now" euphoria, with a tinge of sadness - why wasn't this covered early on? Same thing would have happened in Engineering as well, but at the time my teacher was good. They started by explaining the applications of Fourier Transform before we actually got to learn the mechanics of it.
roel_v 3y ago
I've never taught mathematics but I think it's much harder to teach the intuition, and I also think that more people are capable of learning the mechanics than the intuition. So from the point of view of someone designing a methodology and curriculum - you have to design for scale. I have the same thing with learning languages: I despise focusing on the rules of conjugation and word order and rote memorizing rules, exceptions to them and exceptions to exceptions. I much prefer to bootstrap with some vocab and then acquire by immersion, even if that means I get the conjugations wrong in the beginning. But I also find myself in the minority by far, to the extent that pretty much all language teachers I ever had (except my high school French teacher - she was awesome letting me translate MC Solaar lyrics for grades) didn't even really seem to believe that there are other ways of learning other than studying rules.
- SnowdustDev 3y ago
  Wholeheartedly agree about bootstrapping with vocab and learning through immersion. Eventually all the nuances with grammar and rules get understood implicitly with enough practice (at least to the extent I'm satisfied with). For me, it makes language learning a lot more fun and palatable
jameshart 3y ago
One of the problems I find with mathematics education is that we seamlessly morph the introductory numeracy lessons of ‘counting’ and ‘arithmetic’ into the subject of ‘mathematics’, without ever stopping and telling kids ‘okay, now you can count, we’re going to start doing something different called ‘mathematics’.’ Mathematics is about examining things and understanding their essence - what statements can we make about all such things? How can we prove that? How can we use that to figure out other things? But nobody ever tells kids that - they think they’re just in ‘advanced counting’ lessons. I wonder whether a split in curriculum could help - similar to English language/literature. Mathematics needs a ‘numeracy’ program that starts in kindergarten and covers the mechanical ‘how numbers work’ stuff… then a separate mathematics program that starts in middle school and teaches reasoning and proof. Start with geometry.

deleted 3y ago

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butterNaN 3y ago

I can only speak for myself here, but my journey with Maths did start with concrete, practical relationship between concepts and usage. I still remember fascinated by Geometry and Trigonometry, because how it can be used to create graphics and video games is obvious. The more abstract it gets, the more it warrants an introduction with "here's how it's used in real life". I can recall the point when it started becoming mechanical - it was when I started doing Derivatives and Integrations. I was 'just solving puzzles' until I hit a chapter, tucked away way back in the syllabus, almost at the end of the 2 year arc - a chapter about Applied Differential Calculus. I still remember the feeling of "Oh... I get it now" euphoria, with a tinge of sadness - why wasn't this covered early on? Same thing would have happened in Engineering as well, but at the time my teacher was good. They started by explaining the applications of Fourier Transform before we actually got to learn the mechanics of it.

roel_v 3y ago

I've never taught mathematics but I think it's much harder to teach the intuition, and I also think that more people are capable of learning the mechanics than the intuition. So from the point of view of someone designing a methodology and curriculum - you have to design for scale. I have the same thing with learning languages: I despise focusing on the rules of conjugation and word order and rote memorizing rules, exceptions to them and exceptions to exceptions. I much prefer to bootstrap with some vocab and then acquire by immersion, even if that means I get the conjugations wrong in the beginning. But I also find myself in the minority by far, to the extent that pretty much all language teachers I ever had (except my high school French teacher - she was awesome letting me translate MC Solaar lyrics for grades) didn't even really seem to believe that there are other ways of learning other than studying rules.

SnowdustDev 3y ago
Wholeheartedly agree about bootstrapping with vocab and learning through immersion. Eventually all the nuances with grammar and rules get understood implicitly with enough practice (at least to the extent I'm satisfied with). For me, it makes language learning a lot more fun and palatable

jameshart 3y ago

One of the problems I find with mathematics education is that we seamlessly morph the introductory numeracy lessons of ‘counting’ and ‘arithmetic’ into the subject of ‘mathematics’, without ever stopping and telling kids ‘okay, now you can count, we’re going to start doing something different called ‘mathematics’.’ Mathematics is about examining things and understanding their essence - what statements can we make about all such things? How can we prove that? How can we use that to figure out other things? But nobody ever tells kids that - they think they’re just in ‘advanced counting’ lessons. I wonder whether a split in curriculum could help - similar to English language/literature. Mathematics needs a ‘numeracy’ program that starts in kindergarten and covers the mechanical ‘how numbers work’ stuff… then a separate mathematics program that starts in middle school and teaches reasoning and proof. Start with geometry.