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Learning algebra in my 60s

theguardian.com

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Re: Learning algebra in my 60s

#131

Post by Viliam1234 »

lake_vincent wrote: post #17
As_You_Wish wrote: post #12

Earlier quoted context omitted.

Right. >Fundamental understanding gives exponential results yet most courses try and get the axioms and basics out of the way as fast as possible, working through basic proofs or derivations is cute sideshow, if anything. This is so true. I was exactly like the author. I could not understand any math beyond algebra. Trigonometry was way beyond my ability. But one day, I found an old trig book lying around when I went…

Very well said! It is true that once you grok this technique, you are basically unstoppable. It does make for slow progress at first, but it's worth it. Edit: I went to grad school for math, and I got there by pretty much doing what you described!

The problem with using the right techniques is that there are essentially three kinds of techniques:

a) fast at the beginning, then gradually slow down, e.g. rote memorization;

b) slow at the beginning, then gradually speed up, e.g. deep understanding;

c) slow at the beginning, then remain slow forever, e.g. doing something chaotic and stupid.

The problem is that sometimes you have incompetent people who use the third kind of method. Then, when things blow up, people start paying close attention to the speed, and reject any method that is slow at the beginning, because they suspect it would be the same story.

(In other words: premature optimization, technical debt.)

Re: Learning algebra in my 60s

#132

Post by Viliam1234 »

bowsamic wrote: post #117
mattwilsonn888 wrote: post #8

It's a shame he went to all the trouble of avoiding conventional education (for the wrong reasons), but then still ended up going down an execution rather than concept focused route. As soon as you start memorizing operations and treating math more like a narrow grind only approached through arbitrary problems solved primarily through computation or use of rote application of poorly understood technique, you lose the…

I disagree. My favourite lecturer for general relativity said the important thing is fluency, which comes from repetition. It’s great to know everything, but you should be able to do the basics, such as calculus or algebra, very quickly and without thinking. You will need that brain power for the new stuff. The best way to gain mathematical fluency is by lots of practise.

You can understand something first, and practice for the speed later. Understanding does not prevent practice.

I know the result of 8×8 immediately, and I don't think I would ever forget it. But, hypothetically speaking, if I ever made a mistake here, I have the option to slow down and verify the result. If I wouldn't have the option, there is a risk that I would make a mistake and then keep making the same mistake forever, because once the lessons are over, there would be nothing to correct me.

Re: Learning algebra in my 60s

#133

Post by shard »

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…

SOHCAHTOA (https://mathworld.wolfram.com/SOHCAHTOA.html) is how I learned about the trig functions back in school. It's stuck with me all these years, along with the quadratic formula (https://en.wikipedia.org/wiki/Quadratic_formula) from algebra class.

Re: Learning algebra in my 60s

#134

Post by jsmith45 »

dhosek wrote: post #116
deelowe wrote: post #111

Earlier quoted context omitted.

Common core. It's become a taboo term in certain circles.

Arrrgggghhh! I was getting my math teaching credential when common core was getting its start and what I want to shout from the rooftops is: COMMON CORE DOES NOT SPECIFY PEDAGOGICAL METHODS! It is, essentially. a list of topics that should be mastered at each grade level.

Seeing what people complaining about "common core" mathematics are complaining about, and it is understandable that they are complaining.

At some point the idea came about that for things like addition, subtraction or multiplication, we should teach multiple different ways of viewing the same concept (like addition) in the hopes that even if students don't really understand the classic approach one of the other approaches makes sense. So perhaps the classic explanation of borrowing in subtraction does not make sense to some students, but one of a few other equivalent ways of handling it does make sense.

But then it got turned into a system where all students need to learn all these different different methods, and apply them in both homework and on tests. Which totally defeats the point. The who idea was that some students may find some methods useful and intuitive, and others find different methods useful and intuitive, and as long as every student finds some method can can work with we are better off than only teaching the classic method of the concept.

Re: Learning algebra in my 60s

#135

Post by ikrenji »

mattwilsonn888 wrote: post #8

It's a shame he went to all the trouble of avoiding conventional education (for the wrong reasons), but then still ended up going down an execution rather than concept focused route. As soon as you start memorizing operations and treating math more like a narrow grind only approached through arbitrary problems solved primarily through computation or use of rote application of poorly understood technique, you lose the…

it's way easier to understand the concept if you see it repeated many times in action though...

Re: Learning algebra in my 60s

#136

Post by WalterBright »

BeetleB wrote: post #104
WalterBright wrote: post #99

Earlier quoted context omitted.

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…

> Much of your experience was Caltech specific

I am sadly well aware of that. Caltech was known as unique at the time, I wonder what it is like 40 years later.

1. It was rarely necessary to look anything up. What was on the exam was reliably in the notes or in the assigned textbook. I don't recall ever memorizing things, and managed an A- average. My innate abilities were completely average there, and we all knew who the really smart ones were, like Hal Finney. What I did do to prepare, however, was ensure I attended every lecture and took comprehensive notes, make sure I could solve every homework problem, and every midterm problem (in trolling for the final). I did not look at prior year's stuff.

2. I didn't attend grad school, so can't comment there. But I did match wits with Masters engineers at Boeing, and would wind up fixing their work, too, though far more rarely. That group eventually offered me a position, though they had a Masters as a requirement.

Measure theory wasn't taught, at least in the undergrad courses I took. Neither was the theory of distributions.

> Can assure you Caltech's approach is not the norm.

So I found out later :-(

Re: Learning algebra in my 60s

#137

Post by mlyle »

ravi-delia wrote: post #94
mlyle wrote: post #89

Earlier quoted context omitted.

A lot of what you describe has become the standard way to teach elementary math-- from the "new math" onwards to Common Core pushing aspects of looking at problems the same way Our school does the much-lauded Singapore Math in elementary, which definitely tries to build intuition and looks at many approaches, and supplements with drills. And I teach a competitive math class which definitely is all about finding diffe…

I'm thrilled to hear it's popular! In my hometown it was killed by doubters, but perhaps with expanding evidence they'll reconsider. I think perhaps it was lumped in with disastrous testing efforts, but the math at least was pretty great.

:D I'm not sure it's popular: it's controversial with parents, in general. But I think it's a good thing, overall.

I do think most programs implementing it end up taking out a little too much rote and algorithms.

Re: Learning algebra in my 60s

#138

Post by auggierose »

Viliam1234 wrote: post #127
auggierose wrote: post #16

Earlier quoted context omitted.

Ever heard of new math? https://www.americanheritage.com/whatever-happened-new-math-... I agree with you, especially if you are in your 60s and have a math professor as your teacher, one would think a new math approach works best!

If you know the history of education, the "new math" thing is a sad story of a mind-blowing stupidity that keeps hurting math students for decades, because it divided most people into two camps that keep promoting two different wrong ideas. Theories of education are often based on some psychological theory -- you have a theory how people think in general, and you use it to design a process to teach people. If we skip…

I don't really know much about the history of education, I only read the article I linked. From what I understand from that, it is not so much that constructivism has been debunked, but rather that in order to explain something properly, you need to understand it yourself deeply. Now, many teachers don't actually understand math on that level. So it will be difficult for them to teach it based on constructivism. That seems to me to be the main difficulty. Also, everybody understands things differently, so even if the teacher is capable, how do you scale that for an entire classroom?

Re: Learning algebra in my 60s

#139

Post by mattwilsonn888 »

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…

This is a really important point on this topic.

I don't believe (nor do I think you were claiming) that this means math cannot be taught more intuitively. An even broader example to your point is that Calculus itself was invented without foundation, at least without foundation modern mathematicians would find satisfactory; intuition patched some holes around 'infinity.' But Calculus could none the less be developed by blackboxing its inner workings and justifying its existence by just how spectacularly useful and predictive it was.

Likewise with arithmetic and algebra itself: ancient civilizations who factored numbers or solved equations did not require Peano's work to justify inventing some math. Calculus and all the math before it would indeed rest on foundations absent any Set Theory. The wider perspective is that discovery starts in the middle of a concept, and works out towards its implications and inwards towards its axioms.

The justification taught to those learning math (or anything) need not start from its axioms, but it should start from its history in context to how humans found use for the concept - that is justifying enough and likely more satisfying than learning axioms developed after the fact when the question is 'why?'

Re: Learning algebra in my 60s

#140

Post by y0ungarmanii »

Math is so much easier when your using it to model something that is real. I remember in Physics 1 in college when the professor was going over how a baseball hits a bat and how it is modeled with all of its nth order derivatives. Everything clicked! Not to say theoretical math isn't fun though!
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