Permit me to make a tangentially related comment of interest to parents reading this thread: This camp for 11-14 y/o kids: http://www.mathpath.org/ is absolutely excellent. My kid loved it so much they attended three years. Great faculty... John Conway, Francis Su, many others. If you have a math-loving kid of middle-school age, I encourage you to check it out.
Math education: It’s not about numbers, it’s about learning how to think
101–110 of 343 posts
Re: Math education: It’s not about numbers, it’s about learning how to think
#102Is this a US thing? Why would people still think that math is about numbers? Math is about patterns, which got drilled into us by our teachers in primary school. I really don't understand how US education system can fuck up so badly on fundamental subject like math.
I was taught multiplication by rote memorization of the tables. It didn't go well as I'm bad at rote memorization.
People ask me if they see the beauty in math when I'm complaining. I do, but I also see the other kids outside playing while I'm held back from recess until I can tell you what 7*8 equals. Fuck that so much.
Re: Math education: It’s not about numbers, it’s about learning how to think
#103Earlier quoted context omitted.
In theory it shouldn't matter but it does. We brainwash students to think in terms of y being a function of x. y = f(x) If you give a problem with x = f(y) it really confuses people. If I label the horizontal axis y and the vertical axis x in the cartesian plane then confusion ensues. t = b(a) where t is the velocity and a is time That's confusing. I suppose it's analogous to me saying "The male, fluffy, white, large…
There's a nuance here too. in programming, y = f(x) = x^2 is one way; 'y' is a label for a function of one variable. But in math there is no fixed typing, or distinctions. The symbolic equation represents a relationship. Hence: y(x) = x^2 x(y) = (+/-) sqrt(y) the domain and domain can be switched and the entire equation rearrange so that a 'variable' becomes a 'function'.
Here is an interesting observation on what you wrote:
x(y) = (+/-) sqrt(y)
is not valid notation. On the left hand side you are stating that x is a function of y. But on the right hand side your usage of (+/-) indicates that y can go to two different values. Hence x is not a function of y. To be a function one must have each input going to a distinct output.
Re: Math education: It’s not about numbers, it’s about learning how to think
#104Earlier quoted context omitted.
I think math education needs to be reworked from top to bottom to focus more on building blocks. E.g. I'm not sure there's enough focus on what multiplication is before you're expected to memorize the times tables. Think of stuff like the quadratic equation, or formula for the surface area of a cone. How many of us were just taught these magic formulae without first deriving them? Why do we start talking about pi wit…
> I think math education needs to be reworked from top to bottom to focus more on building blocks. E.g. I'm not sure there's enough focus on what multiplication is before you're expected to memorize the times tables. This was tried. It was called "New Math". Spectacular failure. Do you want to know what worked? Memorizing times tables. There's just no way around the fact that drilling is key to early mathematical lea…
It's way too early to judge "New Math" to be a failure. I think the idea of focusing on concepts over computation 100% in the right spot. The biggest problem to its acceptance is cultural. Parents don't feel comfortable with math concepts, and in my opinion, most of the negativity is coming from that insecurity. So they demand for things to be taught the "old way", even though that's produced a generation or two of mathphobic Americans.
EDIT: As pointed out, I mistook OPs invocation of New Math to be talking about the much maligned Common Core rethinking of math education. I was briefly a high school math teacher, but before the roll out of these changes, so I can't comment first-hand on what the new curriculum looks like in the actual teaching. But I do know how poorly prepared my students were for math beyond arithmetic. They were trained with similar curriculum that I had experienced growing up in the 90s, which I think is poorly thought out.
Apologies for the confusion caused by me not recognizing the term New Math.
Re: Math education: It’s not about numbers, it’s about learning how to think
#105> it's about learning how to think It's about learning a set of thinking skills, not how to think . Many people who know no math can think and function very well in their domains and many people who know lots of math function and think poorly outside of math.
Math skills (esp proofs) shape the way you approach problems, giving the problem structure and providing you the tools to reduce it into components, relations, and dependencies. To make mysteries unmysterious, such tools are indispensable. I'd go so far to disagree with you that people do well without learning math skill (those tools of thinking). If you can deconstruct a problem, then you certainly learned that skil…
I feel like there's at least three ways of thinking about problems that are practical:
- systematic (math, rule-based)
- social (history, emotions, people-based)
- creative (art, invention)
These all overlap and feed into each other because we're a complex species, but we really only ever hear about the necessity of math to "learn to think".
Re: Math education: It’s not about numbers, it’s about learning how to think
#106Programming 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.
Scheme doesn't really look like normal algebra notation. Though, arguably, it resulted from McCarthy's attempt to invent an algebraic notation for general expression evaluation (the original Lisp).
Re: Math education: It’s not about numbers, it’s about learning how to think
#107Earlier quoted context omitted.
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 freshm…
Schools (at least around here) do not hold kids back anymore. No matter what. Some of them get behind in one or more subjects (math, say) as early as 1st or 2nd grade and all their later teachers are stuck trying to give them the remedial math help the desperately need, while teaching their other students on-grade-level material, and also trying to teach the remedial kids enough of the new stuff that their scores on…
Yeah. To my mind, the problem is that learning is very individualized (not even taking into account learning disabilities!), and yet mainstream U.S. pedagogical theories really only target the middle of the bell curve. What's needed -- at least for a huge chunk of students -- is something more like Montessori, where a learner who's struggling can receive individualized attention from a teacher or a peer, and where the learner has free access to tactile/visual/whatever aids which may be overkill for most other learners, AND -- perhaps most importantly -- where the natural human tendency to sit and focus for hours at a time on a single task is fostered and encouraged.
Re: Math education: It’s not about numbers, it’s about learning how to think
#108Learning "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…
I think the most important part, especially for people who do not intend to be mathematicians, is how to apply math to solve problems. The math courses I took in high school and college were all about theory and nothing about application. These courses were taught as if it were still the 19th century and computers don't exist.
I would rather see teachers accept that computers exist, and teach students how to solve problems, even very complicated ones, using software (e.g. Mathematica, R, or whatever). Keep the theory to an absolute minimum (leave the theory courses to people who are interested.)Really, so what if you don't learn how to do calculus manually?
Re: Math education: It’s not about numbers, it’s about learning how to think
#109Maybe 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 I was merely a passive observer of everyday struggle for some people.
This is the opposite of my experience, I did horribly in math and hated it until I took calculus (which, I guess I should have taken in highschool but I tried to avoid it since the rest of math sucked.)
EDIT: just want to clarify: I did have a reasonably intuitive understanding the whole time, the earlier stuff just felt like busy work.
Re: Math education: It’s not about numbers, it’s about learning how to think
#110I studied Mechanical Engineering, and it was my experience that several professors are only interested in having the students learn how to solve problems (which in the end boil down to math and applying equations), instead of actually learning the interesting and important concepts behind them. My wife went to school for Architecture, where she learned "basic" structural mechanics, and some Calculus, but still cannot…
Nailed it.
I remember "hitting the wall" in Honors Algebra II, when I just couldn't keep up with all the new things I was expected to compute: radicals, synthetic division, logarithms, etc. It was a major dent in my math confidence, which up until that point had be reinforced by being told I was "good at math".
I didn't start regaining that confidence until Chemistry, when I learned what pH actually meant. That sparked the epiphany that scale was deeply important in nature, and with that motivation, logarithms seemed much simpler.
We so quickly forget that most math wasn't discovered as some abstract exercise. But we still teach it that way, with the concepts before the applications.