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Learning algebra in my 60s

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Re: Learning algebra in my 60s

#101

Post by WalterBright »

BeetleB wrote: post #93
gavinray wrote: post #23

Earlier quoted context omitted.

Lol, I've been kicked out of class for asking the teacher to explain why something works in math here in the US I think it was long division and lattice multiplication in elementary school Doing math by drawing numbers in predefined shapes so that it magically worked out was the most ludicrous thing I'd ever seen "Because that's how it works." Wasn't really a satisfactory answer lol.

It's hard to explain to young kids what "out of scope for this class" means. And that the number of people who need these skills vastly the number of people who need to understand the derivation. Lattice multiplication will probably take algebra to explain. Long division definitely will require algebra. In my school, there's a gap of 5 years between teaching the two. A lot more people in the world need arithmetic tha…

Much of the math part was taught in math classes. Engineering classes worked with the math classes to ensure the students were ready for the math there.

Chemistry, physics, thermo, dynamics, electronics, fluid mechanics, electronics, etc., were all math classes, in addition to a solid slate of required math classes.

If one didn't care for math, Caltech was a very very wrong place to attend :-) You either got good at it, or you left. I definitely felt that 4 years of that rewired my brain.

Re: Learning algebra in my 60s

#102

Post by scythe »

jonathanlydall wrote: post #69
jjice wrote: post #54

Earlier quoted context omitted.

> ...still ended up going down an execution rather than concept focused route. This is something that made me really disappointed when I realized it in school. It was always a step by step process taught to me, and I loved math. I took my AP Calculus class in my senior year of high school and my teacher (fantastic man) showed us how the formula for a derivative is derived. I was blown away. We had just learned and it…

When learning trigonometry in high school (in South Africa, ‘98), we were taught how to use the correct operation for a particular situation and then use a calculator to get the correct answer. I asked the teacher where the numbers which were spat out by the calculator came from and was only told “in my day we didn’t have calculators and had to lookup the answers from a table on a book!”. Which was such a thoroughly…

>I asked the teacher where the numbers which were spat out by the calculator came from

This is a bit of simple knowledge which is sadly unbeknownst to even most math majors and educators. Often, students are taught in calculus that cosines (and hence other trig functions) are computed with Taylor series, which is not really correct. In fact they use CORDIC, a highly optimized algorithm.

But CORDIC is based on repeated use of the sum formula, and a simple version of this can be taught without calculus. Just notice that:

cos(2x) = 2 cos(x)^2 - 1

Then for small enough x, you have (can be shown by drawing, but kind of annoying):

cos(x) ≈ 1 - x^2 / 2

Or rigorously: 1 - x^2 So divide x by 2 until you get a small number (x I'm not entirely sure this would work (i.e. improve understanding), though — I have not taught high schoolers before. And there is a decent bit of work required.

Re: Learning algebra in my 60s

#104

Post by BeetleB »

WalterBright wrote: post #99
BeetleB wrote: post #95

Earlier quoted context omitted.

Can't speak for the others, but you may come across as too one sided. I had your mindset, and one of the reasons I started getting left behind in grad school whereas others didn't was my stubborn refusal to memorize anything and insisting only on understanding and deriving as needed. Once you go deep enough, you'll often find yourself relying on N random theorems you learned some courses ago to solve a problem, and t…

1. none of the exams at Caltech required memorization. They were open book and open note. Memorizing simply wouldn't have helped. For example, one physics exam question was: "Assume magnetic monopoles exist. Derive how Maxwell's Equations would then look." If you didn't understand the ME derivation, you'd be completely lost. The same for FFTs, where the exam question was derive the hyberbolic transforms. 2. one winds…

Much of your experience was Caltech specific, and will simply not translate to other schools. Caltech is famous for this - more so than schools like MIT, etc.

1. I learned the hard way, as did others, that one should still memorize with open book exams (or at the least make a 1-2 page cheat sheet). Why? Because there was a time limit and most professors would not alot enough time for people to even look up everything they needed in the textbook. Sure, if they increased the time by 50-100%, you'd do just fine with memorizing.

I'm not even hypothesizing there. After one open book exam I went and asked everyone who got a good score - most had incorporated some level of memorizing. And clearly most who did not memorize at all got a poor score (which is less surprising than it should be, because most students will do poorly regardless ;-) Still, I was the clear exemplar of one who improved from "below average" to "one of the top students" within the duration of one semester when I finally embraced that some level of memorization would be needed.

2. True, and I can relate to trig identities - most of which I remember 25 years later - and even after over a decade of not needing them. However, when I got to more advanced topics in math, the frequency with which I would need to use them dropped significantly, and the approach of "Just do lots of problem sets and you'll passively memorize" failed me in grad school.

> we were not expected to use the Laplace transform until its derivation was demonstrated. This applied to all the formulas used. FFTs too.

Did they teach you measure theory before those transforms? Did they teach you measure theory before probability? Did they teach you the theory of distributions before the Dirac-Delta function?

> Which is why some engineering schools are better than others. I applied to the USN&WR list of "top 10 engineering schools in the US". Caltech was #2.

I went to a top 5 engineering school. Can assure you Caltech's approach is not the norm.

Re: Learning algebra in my 60s

#105

Post by bslorence »

BeetleB wrote: post #97
bslorence wrote: post #77

Earlier quoted context omitted.

> there ARE points in a student's academic path where they HAVE TO memorize stuff and do rote operations like the multiplication tables. Sure, but imagine learning multiplication tables without having any idea what it means to "multiply"; literally just memorizing sequences of symbols, without ever looking at piles of coins or whatever. Multiplication is so basic that this is hard to imagine, but I'm sure that, say,…

> I first encountered difference-of-squares in late middle school, memorized the procedure and used it handily through twelfth grade, and had no idea that there was a visualizable geometric basis to it until I read Book 2 of Euclid's Elements in college. Many mathematicians would disagree with your characterization. For them, the difference of squares is an abstract concept in algebra, and the geometric interpretatio…

Fair enough. Maybe I should have just said that it's possible to give concrete illustrations of basic algebraic concepts, and that doing this would probably help some students learn algebra, and might help others retain it.

But for whatever reason this is generally skipped in middle/high-school algebra.

Re: Learning algebra in my 60s

#106

Post by BeetleB »

bslorence wrote: post #105
BeetleB wrote: post #97

Earlier quoted context omitted.

> I first encountered difference-of-squares in late middle school, memorized the procedure and used it handily through twelfth grade, and had no idea that there was a visualizable geometric basis to it until I read Book 2 of Euclid's Elements in college. Many mathematicians would disagree with your characterization. For them, the difference of squares is an abstract concept in algebra, and the geometric interpretatio…

Fair enough. Maybe I should have just said that it's possible to give concrete illustrations of basic algebraic concepts, and that doing this would probably help some students learn algebra, and might help others retain it. But for whatever reason this is generally skipped in middle/high-school algebra.

I'm torn. I think it's always good to show these - it certainly makes the subject more interesting!

At the same time, if one is to use algebra for future studies/work, one really needs to be able to manipulate those symbols in the abstract, without feeling a need for some deeper understanding. I can see teachers not wanting to deal with "But what does that really mean?" for every detail in algebra.

Re: Learning algebra in my 60s

#107

Post by synu »

This quote reminded me of learning chess as an adult as well:

> In the paper Acquiring Skill at Mental Calculation in Adulthood, Neil Charness and Jamie Campbell say that middle-aged people perform as older ones do, but if they practise, they perform more as younger people do. If speed is valued more than accuracy, the decline in ability is obvious. If accuracy is valued more than speed, the decline is less obvious and maybe not even very pronounced.

Re: Learning algebra in my 60s

#108

Post by bslorence »

BeetleB wrote: post #106
bslorence wrote: post #105

Earlier quoted context omitted.

Fair enough. Maybe I should have just said that it's possible to give concrete illustrations of basic algebraic concepts, and that doing this would probably help some students learn algebra, and might help others retain it. But for whatever reason this is generally skipped in middle/high-school algebra.

I'm torn. I think it's always good to show these - it certainly makes the subject more interesting! At the same time, if one is to use algebra for future studies/work, one really needs to be able to manipulate those symbols in the abstract, without feeling a need for some deeper understanding. I can see teachers not wanting to deal with "But what does that really mean?" for every detail in algebra.

This goes to the question of why anything other than basic arithmetic is compulsory. The famous 10th-grader's whine "what are we going to use this for", which infuriated my own 10th-grade Algebra 2 teacher, and even made me roll my eyes at the time, is actually a fair question when algebra is taught as abstractly as it typically is.

I think you've explained exactly why here -- because the emphasis on abstract manipulation presupposes that this is useful for something that we need to get on to. But that's just false for almost all students. And yet they're required to take the class to get a diploma.

My vote would be to treat any math beyond basic arithmetic as a liberal art, and do a lot less of it in compulsory curricula, but spend a lot more time on deep understanding. This would benefit everyone. The current approach pretends that everyone in the class is going to be a certain kind of engineer or scientist some day.

Re: Learning algebra in my 60s

#109

Post by hgomersall »

mkl95 wrote: post #75
jonathanlydall wrote: post #69

Earlier quoted context omitted.

When learning trigonometry in high school (in South Africa, ‘98), we were taught how to use the correct operation for a particular situation and then use a calculator to get the correct answer. I asked the teacher where the numbers which were spat out by the calculator came from and was only told “in my day we didn’t have calculators and had to lookup the answers from a table on a book!”. Which was such a thoroughly…

> but I was also sad and a bit upset the unit circle was not taught in high school I'm not a mathematician, but that sounds like poor teaching to me. Even if it's not part of the programme, learning the unit circle takes literally a few minutes and is an invaluable tool afterward. I don't know if it's possible to develop an intuition for trigonometry without learning the unit circle.

The problem is, these tools always feel like they should be "the way" it's taught. I feel complex numbers should be taught as a special case of geometric algebra, though I suspect that would make things much more difficult. My understanding is that lots of university level maths _is_ taught like this: start from the most general case.
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