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Learning algebra in my 60s

theguardian.com

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Re: Learning algebra in my 60s

#71

Post by greymalik »

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…

What’s the alternative? How does a neophyte go about cultivating the mindset and understanding you describe?

Re: Learning algebra in my 60s

#72

Post by crispyambulance »

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…

> ...still ended up going down an execution rather than concept focused route.

Everyone will say, of course, that "concepts matter". But the reality is that there ARE points in a student's academic path where they HAVE TO memorize stuff and do rote operations like the multiplication tables.

One can't move on to new concepts in math until the previous dependent concepts have been mastered. Mastery means practice, practice means drills, and all of that is boring but necessary. The best we can do is to make the mathematics more meaningful by tying it to problem solving. At some point many students will experience an "ah ha" realization and be further stimulated, while for some others math will always be meaningless drudgery. For most it will be something in-between.

Re: Learning algebra in my 60s

#73

Post by Wistar »

Despite that — or, maybe, because — my father was a math academic, I did poorly in school with math beyond basic arithmetic. When I was about 25 years old, a friend gave me three algebra textbooks written by an ex-military pilot, John Saxon. In a marathon session lasting about two weeks, I went through all three books from end-to-end and really learned algebra. For whatever reasons, the Saxon books worked really well for me — better than any other learning I have ever gotten from a textbook. Yes, I was really motivated but I attribute a lot of my success learning algebra to those books. Learning that opened a lot of doors since then, mostly because my confidence in taking on matters technical was so much improved.

Re: Learning algebra in my 60s

#74

Post by stakkur »

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 agree, but the fundamental problem is how most western math education is structured top-down to be a goals-driven curriculum. In other words, teaching to the test.

In the US, it’s a sprint towards meeting goals tied to government funding (“core curriculum “, etc.) And so many get left behind in that sprint, often feeling they’re ‘not smart enough’ or ‘not a math person’.

Re: Learning algebra in my 60s

#75

Post by mkl95 »

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…

> 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.

Re: Learning algebra in my 60s

#76

Post by trebbble »

wawjgreen wrote: post #63

My litmus test of someone being intelligent or not--among other such tests--is what they say about Shakespeare. This might anger the Brits, but Shakespeare is garbage--and the guy says: "I read .... most of Shakespeare." One famous guy lambasting Shakespeare (Tolstoy) was only marginally smarter, because according to his "Confessions", he read it many times over (incl. in English) and STILL could not find any artisti…

> EDIT: I see I have touched quite a few nerves.

It reads like a post you made for yourself, not for the benefit of any reader.

Re: Learning algebra in my 60s

#77

Post by bslorence »

crispyambulance wrote: post #72
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…

> ...still ended up going down an execution rather than concept focused route. Everyone will say, of course, that "concepts matter". But the reality is that there ARE points in a student's academic path where they HAVE TO memorize stuff and do rote operations like the multiplication tables. One can't move on to new concepts in math until the previous dependent concepts have been mastered. Mastery means practice, prac…

> 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, difference-of-squares rules feel like this to most beginning algebra students -- and for most people that probably never changes.

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.

Re: Learning algebra in my 60s

#78

Post by photochemsyn »

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 think an introductory course would ideally hop back and forth between the explicit execution of specific examples and the understanding of abstract concepts. As you say the concepts are very important - for example, the author never mentioned the associative, distributive, or commutative rules which apply to all cases of addition and multiplication (but not necessarily to subtraction and division). Those are universally applicable, and indeed understanding them is important for moving on to higher maths (group theory etc.)

However, without doing a fair amount of ditch-digging, abstract general theories of ditches might be hard to grasp. For a programming example, consider data structures - lists, trees, graphs, and so on. The abstract concept is clearly what allows larger constructs to be made, i.e. entire programs, but if you've never actually implemented a linked list in any particular programming language, and try to, the result is almost certainly going to be a buggy disaster.

The optimal math education might be one where for every hour the instructor spends on the conceptual approach, the student then spends about three hours on the execution approach as applied to specific examples of those concepts (ideally with a teaching assistant around to help if the student gets stuck).

Re: Learning algebra in my 60s

#79

Post by racl101 »

My grandpa used to enjoy doing Euclid's Elements compass and straightedge constructions and understanding proofs.

He also enjoyed reproducing geometric proofs of certain equations.

What's funny is he hated math as a student but for some reason or other got turned onto Euclid in old age.

He then tried to get my dad to read Euclid but that didn't take.

Then I came across the book as teen and took an interest so Grandpa and I bonded over that.

Re: Learning algebra in my 60s

#80

Post by wrycoder »

rongenre wrote: post #35

One of my retirement goals is to redo my math degree - turns out it's really easy to get hooked up with online tutors, and I figure it's totally worth paying $20-$40 an hour to work through a textbook.

If you have a math degree, you already know how to learn math. That hasn't changed. You're not going to try passing tests, so you don't really need a tutor. What you might want is a counselor to guide your "reading", as is done in places like Oxford. That can be worth the $40.

All you need is updated texts, a plan of action, and maybe Wikipedia for an alternate take on sticking points. (Just using Wikipedia doesn't work well.)

You could also look at the MIT or Harvard online curricula/syllabuses to get a feel for what the current subjects and texts are. Maybe take a couple of the free courses.

Also, scan Kahn Academy first - that site is good for identifying holes in your background.

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