Maybe 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…
If I was to review (CS-related) math after a break of many years, what would be a good starting point? High school math? Discrete math? What are the foundations and the path like for the stuff more closely related to algorithms and core CS stuff?
Math education: It’s not about numbers, it’s about learning how to think
211–220 of 343 posts
Re: Math education: It’s not about numbers, it’s about learning how to think
#212Maybe 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…
> By the time people go to high school it is to late
Just anecdotal evidence, I know, but for me it was the exact contrary that happened. As a 5-year old kid I had learn to read all by myself (my dad had taught me to recognize the letters and how to pronounce them), but I had huge conceptual difficulties in teaching myself things like subtraction (what do you mean you can take/subtract a thing from another thing? what happens when the thing which you're taking/subtracting is bigger than the thing from which you're subtracting? you cannot do that, it would result in a non-thing, a thing that is below 0, and all things below 0 technically do not exist).
In elementary and middle-school I was passable in maths, while very good in humanities, but all this changed in high-school, mostly because of my teacher, who taught me (and my colleagues) how to think about math. He's one of the reasons why I decided to pursue a STEM education and why I'm currently a programmer.
Re: Math education: It’s not about numbers, it’s about learning how to think
#213I'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.
Many undergrad programs (in the U.S.) now have a course in math proof, which is intended to introduce students to proof techniques. It seems helpful to learn how to write proofs without the pressure of learning the content of a subject (like linear algebra, abstract algebra, and so on) at the same time.
Re: Math education: It’s not about numbers, it’s about learning how to think
#214Earlier quoted context omitted.
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 m…
> I think the idea of focusing on concepts over computation 100% in the right spot Concepts of what? Without intuitive understanding of basic computation procedures, what concepts can anyone build out of nothing? Math is concrete. It's from observation of actual quantitative phenomenon.
And yet it's completely abstract as sets and numbers don't need a universe like ours to exists, they only need axioms.
Modeling the world is a job that math the tool was created for, but it certainly isn't concrete just because the world that needed it was or is.
Re: Math education: It’s not about numbers, it’s about learning how to think
#215Wow ... this blows me away ... in a few short hours, so many people chimed in sharing thoughts ... It is great ... Would like to share mine as well.
Fundamentally, math to me is like a language. It's meant to help us to describe things a bit more quantitatively and to reason a bit more abstractly and consistently ... if it can be made mechanical and reduce the burden on one's brain, it would be ideal. Since it's like a language, as long as one knows the basics, such as some basic things of set theory, function, etc., one should be ready to explore the world with it. Math is often perceived as a set of concepts, theorems, rules, etc. But if one gets behind the scene to get to know some of the original stories of the things, it would become very nature. At some point, one would have one's mind liberated and start to use math or create math like we usually do with day to day languages such as English.
Re: Math education: It’s not about numbers, it’s about learning how to think
#216Earlier quoted context omitted.
> Maybe 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 indeed believe you are wrong. I started getting interested in STEM when I was 16 and now have a masters in mat…
Anecdote time: I was a late bloomer in math too. From about 6th to 10th grade, I was terrible at math. I think it was because of inapplication problems. Yes, you get a word problem about Jack and Kwanzeeah trying to fill up a pail or something, it was nonsense to me. Why are they filling up a pail? Why are they using a bucket that is 173/9th's the size of the hose? Why do they have to not let it overflow? Why do I ha…
I'd agree with this completely. It's less about teaching what, how and when than it is teaching WHY you need to know this.
In high school, if you can convince a single person that they need to know calculus, more power to you. 99% of people don't and it's literally a waste of time. Teach people finance though...suddenly we have math that people will know is important.
Re: Math education: It’s not about numbers, it’s about learning how to think
#217My 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…
In my view, speed plays a role, though I can't say that it should play such a central role.
I think that in a lot of disciplines, you have to become fluent at manipulations, and at seeing and thinking in higher level patterns. Being able to think your way through a more complex problem would seem to benefit from, if not require, seeing multiple steps ahead in a progression. At least this is my perception.
My experience in school math was that it wasn't enough to satisfy myself that I knew how to solve a problem. I then had to work my way through a whole bunch of similar problems until I could perform the manipulations quickly. This is also how I managed to commit the definitions, axioms, and theorems to memory. If I didn't do that stuff, then I got my arse handed to me on the exam. I gave my kids pretty much the same advice.
Re: Math education: It’s not about numbers, it’s about learning how to think
#218I 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…
In lower-level courses, Greek letters aren't used that much (at least, I don't use them that much), so I wouldn't describe it as a fixation (or a major problem). Examples of use:
College algebra: pi.
Calc 1/2: pi, theta, capital sigma for sums. epsilon and delta to state the definition of a limit (but epsilon-delta proofs have largely disappeared from mainstream calc courses, so those two letters actually don't appear much).
Linear algebra: pi, capital sigma. Some people like to use stuff like phi or psi for linear maps. I don't.
On the one hand, in writing my course notes I have moved away from Greek letters somewhat in courses like abstract algebra. On the other hand, there is at least one good reason to use them: It helps you distinguish different kinds of objects by using different kinds of symbols. So, for example, I can write "f(x)" for a group map f applied to a group element x. If you follow the rule of picking letters from disparate parts of the alphabet for different kinds of things, that's okay. But I think some people prefer "phi(x)" exactly because the function phi "stands out" from the argument x.
Also, if I need 4 functions "in parallel", I could use alpha-beta-gamma-delta. If I try to do this using Roman letters, I use f, g, h, ... now what? ("i" is sort of reserved for indexing; "e" might mean the constant, but in any case is not often used to name a function.) You can subscript (f_1, f_2, f_3, f_4) but that introduces another kind of notational density and possibility of confusion. There are tradeoffs both ways.
There are places where I'm definitely in sympathy with the your general point. I've sometimes read something and wondered why the writer didn't just use a Roman letter (particularly when I don't recognize the Greek letter - is that a capital Greek whatsis or an emoji?). Sometimes, people use calligraphic script caps (e.g. for ideals), not realizing that they will be a pain for handwriting or TeXing notes. And in some subfields of math, people seem to like Fraktur or Hebrew characters which are a real nuisance (it took me a few tries to learn how to handwrite a passable aleph), particuarly when they're handwritten and the person writing has shitty penmanship so you can't tell what letter they wrote.
I agree with the general point, though: Greek letters should be used like seasoning.
Re: Math education: It’s not about numbers, it’s about learning how to think
#219Earlier quoted context omitted.
> Maybe 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 indeed believe you are wrong. I started getting interested in STEM when I was 16 and now have a masters in mat…
So basically you're saying that it is a preferred solution to teach kids math poorly, let them develop insecurity and aversion towards it and then spent a lot of money to let them overcome it than just do it right in the first place? Yeah, I know that's not what you mean, but you're missing a point. The premise of the article is that US has a STEM problem. I merely suggested that they're allocating resources wrong to…
Went to college with a guy. Freshman year calculus. He was really slow with it. I helped him, and we did homework together. Now he's a math major. Totally surprised me, but he's now majoring in a subject that he wasn't super good at.
People need good teachers at all ages. I'm glad his college professors helped him out enough to lead him to major in math.
Re: Math education: It’s not about numbers, it’s about learning how to think
#220My theory is that math anxiety is really anxiety about a cold assessment. In other subjects you can rationalize to yourself in various ways: the teacher doesn't like me, or I got unlucky and they only asked the history questions I didn't know. But with math, no rationalization is possible. There's no hope the teacher will go easy on you, or be happy that you got the gist of the solution. Failure in math is often (but…
Marking a math test is only objective when an answer is entirely correct. What mark do you give a student who makes a minor arithmetic error on a single step of a multi-step problem?
0 pts - space shuttle exploded.