I'm studying math right now and I have that problem. We're just being vomited theorems and propositions in class instead of making us think. There's not a single subject dedicated to learning the process of thinking in maths. So I think we're learning the wrong (the hard) way.
Math education: It’s not about numbers, it’s about learning how to think
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Re: Math education: It’s not about numbers, it’s about learning how to think
#42Programming needs to be taught alongside Algebra I. Especially in a language like Haskell or Scheme where algebraic refactoring of type signatures looks like normal algebra notation.
Learning multiple different highly abstract concepts at the same time does not seem like a solution to the difficulty of learning a single highly abstract concept.
Re: Math education: It’s not about numbers, it’s about learning how to think
#43Maybe I'm wrong, but I have always believed that if you want people to be good at math, it's their first years of education which are important, not the last ones. In other worlds, push for STEM should be present in kindergartens and elementary schools. By the time people go to high school it is to late. I never had any problems with math until I went to university, so I was merely a passive observer of everyday stru…
To this point, I have been doing a plenty of 1on1 lessons with kids trying to catch up with Math in high school/college/matura exams and one this I have noticed is that things which are problematic to these people are things which were covered quite well already in elementary school.
Later the curriculum only expands on these problems so it's even harder and harder to catch up.
I have been working with college freshmen struggling with basic linear equations and such. While the profession is marginally unrelated to Math (e.g. Graphical Designer), I still wonder to this day how people can have 20-30-40 years and not know how they could (with a calculator in their hand!!) calculate the price after the discount. How can someone feel ok with this?
Re: Math education: It’s not about numbers, it’s about learning how to think
#44Earlier quoted context omitted.
I think the more pervasive problem in schools is a lack of mastery in basic foundational skills before moving on to more advanced topics. Too many students lack the ability to perform basic algebra. I mean simple stuff like simplification, factoring, working with exponents and even fractions. Faced with a typical problem in calculus, of course they're going to flounder and make errors when each calculus problem invol…
> No amount of explanation and reasoning will help the students and instructors get around the fact that they just have to spend the time to develop fluency with the basics. Conversely, no amount of memorization and drilling will provide students with the intuition necessary to solve problems just beyond the grasp of naive symbolically-driven proof search algorithms. In a typical non-proof-based calculus sequence, tr…
I agree, you need both. I am not optimistic about the state of mathematics education. So many things have to "go right" for a student to master mathematics.
Re: Math education: It’s not about numbers, it’s about learning how to think
#45Learning "how to think" is just one part of it. The other part - the one that makes it much more difficult for many, if not most, people to learn math - especially the more abstract branches of it - is learning to think about math specifically . The reason is that mathematics creates its own universe of concepts and ideas, and this universe, all these notions are so different from what we have to deal with every day…
> Learning "how to think" is just one part of it. The other part [...] is learning to think about math specifically. Do you feel like you were taught how to think about math specifically, or how to think at all? Like probably most people here on HN, I love math and programming, but I'm struggling to recall whether and how I was taught to think in school. We went through a series of problems, and talked about the synt…
I think that people are rarely taught how to think; this basic skill usually comes (rather quickly) in the process of "learning by example" - part of which is struggling with concrete problems - whether it is in mathematics, in programming, or in life in general.
Re: Math education: It’s not about numbers, it’s about learning how to think
#46Maybe I'm wrong, but I have always believed that if you want people to be good at math, it's their first years of education which are important, not the last ones. In other worlds, push for STEM should be present in kindergartens and elementary schools. By the time people go to high school it is to late. I never had any problems with math until I went to university, so I was merely a passive observer of everyday stru…
Not sure it's exactly the early years. But I think there's zero point to moving past a topic until you have mastery & can think WITH it (not just, about it, or hack through it on paper). And since the early years are often where that fails, they need to be rethought.
I remember listening to an episode of The Science Show podcast where the guest was arguing (based on their experience) that chemistry and physics concepts should be taught to young children as it's actually easier to teach it to them. With the stated reason being that if they're taught it at a later age, as they are now, they've already built a view of the world where those concepts are somewhat unintuitive. If they learn the concepts earlier they are more easily absorbed.
I found their point fairly compelling, and I wonder if it applies to mathematics as well.
The episode was http://www.abc.net.au/radionational/programs/scienceshow/you...
Re: Math education: It’s not about numbers, it’s about learning how to think
#47I wish school curricula would embrace that "learning how to think" bit.
With the sole exception of Geometry, every single math class I took in middle and high school was an absolutely miserable time of rote memorization and soul-crushing "do this same problem 100 times" busy work. Geometry, meanwhile, taught me about proofs and theorems v. postulates and actually using logical reasoning. Unsurprisingly, Geometry was the one and only math class I ever actually enjoyed.
Re: Math education: It’s not about numbers, it’s about learning how to think
#48As my old boss once said, "never confuse mathematics with mere arithmetic ."
Yet, - arithmetics (a.k.a. Number Theory) is a vast branch of modern pure mathematics; - numeric calculations, i.e. how to do them efficiently, is a very important part of the (applied) mathematics.
The parent commenter's old boss certainly did not mean Number Theory by "arithmetic".
Re: Math education: It’s not about numbers, it’s about learning how to think
#49Maybe I'm wrong, but I have always believed that if you want people to be good at math, it's their first years of education which are important, not the last ones. In other worlds, push for STEM should be present in kindergartens and elementary schools. By the time people go to high school it is to late. I never had any problems with math until I went to university, so I was merely a passive observer of everyday stru…
> I never had any problems with math until I went to university
So, here's the thing. Did I have no problems with math until uni because I'm smart? Because I have a specific type of brain? Because of my upbringing? Because I had very good education in my early years? Some combination of the above?
I struggle to believe that people can't do algebra. I am convinced that with some thought and help they'd find it trivial. However, is that my own experience overriding my ability to take their perspective as true? I dismiss "I can't" as "I don't think I can", but who am I to judge that better than them?
Re: Math education: It’s not about numbers, it’s about learning how to think
#50Maybe I'm wrong, but I have always believed that if you want people to be good at math, it's their first years of education which are important, not the last ones. In other worlds, push for STEM should be present in kindergartens and elementary schools. By the time people go to high school it is to late. I never had any problems with math until I went to university, so I was merely a passive observer of everyday stru…
> I never had any problems with math until I went to university So, here's the thing. Did I have no problems with math until uni because I'm smart? Because I have a specific type of brain? Because of my upbringing? Because I had very good education in my early years? Some combination of the above? I struggle to believe that people can't do algebra. I am convinced that with some thought and help they'd find it trivial…
I saw what looked like this sort of thing with my daughter. She struggled in her high-school algebra 1 class, but whenever I would sit down with her and she would work out problems, she had no trouble (but she also had no confidence that she could do it).. She ended up having to repeat the class, which ruined her confidence even more, and she hated it even more the second time through.
At the start of middle school, her teachers were recommending accelerated math classes, but she never ended up doing that, because she stopped liking math and stopped trying. Why? Maybe because it was unpopular or she didn't like her teacher... I don't know. In any case, she went from kinda-liking math to completely hating it over the couple of years, even though when I was there, she seemed to understand it just fine.