I often worry that mathematics education is strongly supported on the grounds that it is about "learning how to think", yet the way it is executed rarely prioritizes this goal. What would it look like if math curriculum were redesigned to be super focused on "learning how to think"? Different, for sure.
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
#62I wonder if a large part of our math problem is our legacy fixation on Greek letters. Would math be more approachable to English speakers if we just used English? I like to think about math as language, rather than thought or logic or formulas or numbers. The Greek letters are part of that language, and part of why learning math is learning a completely foreign language, even though so many people who say they can't…
If the use of greek letters is a major hurdle to learning maths, I'm guessing that the learner has misunderstood something fundamental about symbols in maths. It shouldn't make any significant difference if the symbols are Latin, Greek or even made up by the lecturer (as long as they are sane).
Re: Math education: It’s not about numbers, it’s about learning how to think
#63Maybe 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 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 that year's standardized test don't land the teacher in hot water (though the kids won't get it at all). The result is that the kids fall farther behind every year but keep failing up to the next grade. They'll enter e.g. 6th grade with ~Dec. of 3rd grade math skills (more often than not their other subjects aren't much farther along, though not always). If they get an LD diagnosis they'll get some extra help but by then it's too late.
It sucks for all concerned.
Re: Math education: It’s not about numbers, it’s about learning how to think
#64Disclaimer: I'm CTO of https://www.amy.ac an online math tutor.
From our experience most people struggle with math since they forgot/missed a curtain math skill they might have learned a year or two before. But most teaching methods only tell the students to practise more of the same. When looking at good tutors, we could see that a tutor observes a student and then teaches them the missing skill before they actually go to the problem the student wanted help with. That seems to be a usefull/working approach.
Re: Math education: It’s not about numbers, it’s about learning how to think
#65Earlier quoted context omitted.
Symbols are symbols, are symbols. Instead of trying to compress everything into a very limited set of "daily language" symbols we should be trying to expand our ideas, including our language(s).
Let's add Chinese, Amharic and Arabic characters. Why not, expanding is expanding.
In some branches of engineering (traditional engineering, not software), there are sometimes too few symbols in the Latin and Greek alphabets, so they have to rely heavily on subscripts. Expanding the set of symbols would be useful here, but I guess it's hard to get people to accept change unless they absolutely have to.
Re: Math education: It’s not about numbers, it’s about learning how to think
#66I wonder if a large part of our math problem is our legacy fixation on Greek letters. Would math be more approachable to English speakers if we just used English? I like to think about math as language, rather than thought or logic or formulas or numbers. The Greek letters are part of that language, and part of why learning math is learning a completely foreign language, even though so many people who say they can't…
Symbols are symbols, are symbols. Instead of trying to compress everything into a very limited set of "daily language" symbols we should be trying to expand our ideas, including our language(s).
I'm not sure what that means. Are you saying the use of Greek letters is a good thing, and that we need more of them, not less? Or do you mean that all symbols are equally easy to learn for beginners, whether they're English or Greek or Chinese or abstract? Or do you mean that we should do away with symbols that don't mean anything and use real language to describe math instead?
The symbols we use in math aren't to be confused with the ideas, they are shorthand. Ideas and symbols are independent, and we still have to explain what Greek letters do using English anyway. We use Greek letters for convenience, not for expressive power. Writing "∫ x ∂x" is only faster and more compact than writing "The integral of the equation x over x", it is not an improvement of the concept, right?
Use of symbols is the act of trying to compress everything into a smaller set, not the other way around.
For the modern audience, the use of Greek letters may hamper learning because the symbols are unfamiliar and foreign looking. It takes cognitive effort to learn the Greek letters in addition to the cognitive effort to learn the mathematical ideas. What if we could remove the cognitive effort on letters, and focus on the pure math ideas? What if the Greek letters are a speed bump to learning? That we use ∑ for sum and ∆ for change are arbitrary - canonical, but completely arbitrary. History could have decided that a circle was the integral symbol, or the capital letter I. Maybe math would be easier to learn if we switch to capital letter I.
Re: Math education: It’s not about numbers, it’s about learning how to think
#67My problem with Math education was always that speed was an enormous factor in testing. You can methodically go through each question aiming for 100% accuracy and not finish the test paper, while other students can comfortably breeze through all the questions and get 80% accuracy but ultimately score higher on the test. This kind of penalizing for a lack of speed can lead to younger kids who are maximizing for grades to move away from Math for the wrong reasons.
Source: I'm slow but good at Math and ended up dropping it as soon as I could because it would not get me the grades I needed to enter a top tier university.
Re: Math education: It’s not about numbers, it’s about learning how to think
#68Maybe 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…
Absolutely. See Toom (2010) Word Problems in Russia and America, http://www.de.ufpe.br/~toom/travel/sweden05/WP-SWEDEN-NEW.pd...
Re: Math education: It’s not about numbers, it’s about learning how to think
#69Earlier 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…
Tying students' progress through the curriculum to their age is silly. If you think about it for a single second, the whole thing is totally absurd. Vague pre-puberty/tween/teenager distinctions make sense, but within each grouping/building? totally unmotivated.
Yearly age-to-competency distinctions continue by sheer force of tradition, and are harmful to all but the totally mythological "average" student.
Re: Math education: It’s not about numbers, it’s about learning how to think
#70> 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 skill somewhere, just not in a math class.
Abstraction and reductionism are unusual capabilities to acquire on your own. As I recall, they're almost nonexistent in pre-linguistic societies like hunter-gatherers. If you have no words for abstract concepts, your thinking will be strictly concrete.