Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Learning algebra in my 60s

theguardian.com

Post reply on HN
Print view View original

51–60 of 158 posts

Re: Learning algebra in my 60s

#51

Post by jorvi »

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…

Yes. The clearest example of this for me was the fact that many things I was taught in high school were taught to me as an arbitrary concept first, without the accompanying visual relation.

Being taught a² + b² = c² with the triangle and accompanying squares drawn out much much later is the best example I can give.

Re: Learning algebra in my 60s

#52

Post by politician »

jerf wrote: post #38
sirsinsalot wrote: post #22

Earlier quoted context omitted.

I left school at 15, never to return to education, in part because of how maths was taught. There was never an explanation of the _why_ of things. It was very frustrating, and the teaching was very poor. I ended up a programmer, in part, because I figured if teachers weren't teaching, i'd teach myself. This has been a good strategy for me, however there's serious holes in my pure maths knowledge.

One of the thoughts I've been developing over the past 10 years or so is that you can't teach someone the solution to a problem they don't have. This encompasses the "why" question but even goes beyond it, because even the answer to "why" is often just another level of "why" and/or "who cares", quite reasonably. To learn something, you need a problem, you need to grapple with the problem for a bit, and then you can b…

I’d subscribe to your math app. Khan Academy Kids has greatly accelerated my kids’ language and reading skills to the point where they were reading chapter books to us before Kindergarten.

Brilliant and KA seem to be the leaders in self-directed learning, but I’m still waiting for the device described in Neil Stephenson’s Diamond Age.

Re: Learning algebra in my 60s

#53

Post by jerf »

agentultra wrote: post #40
jerf wrote: post #38

Earlier quoted context omitted.

One of the thoughts I've been developing over the past 10 years or so is that you can't teach someone the solution to a problem they don't have. This encompasses the "why" question but even goes beyond it, because even the answer to "why" is often just another level of "why" and/or "who cares", quite reasonably. To learn something, you need a problem, you need to grapple with the problem for a bit, and then you can b…

There is plenty of pure mathematics that is beautiful and worth learning on its own without any practical application. A great deal of joy I get from mathematics is the delight in seeing a novel structure I hadn’t before where the proofs fall effortlessly out of the definitions.

To be honest, even in pure math this approach ought to be taken. I love math too. But it's not really a very good pedagogical approach even in pure math to start out with a full week of unmotivated definitions.

I have no issue with the problems being posed being very abstract at a suitable level for the student. By the time you hit college, I have no issues with a professor introducing group theory with "Hey, look at this aspect of graph theory, and this aspect of topology, and this aspect of algebra... what commonalities do you think we could abstract from them?" But that's a way better introduction even at that level than "Let's spend 90 minutes giving unmotivated definitions and hoping you pick up the pieces later."

In a conventional school setting I expect the problems to be more concrete, by their nature. I can give another example myself: Taylor polynomials. In my opinion, they're one of the more important things to learn at that level. You can give the students a simple problem: "Having learned sin, cos, and tan, and by this point memorized some of the common values, please develop a procedure for taking an arbitrary sin/cos/tan of an angle." Give them some time to chew on it. They may even come up with some modestly clever things, maybe cover some more special cases or something. But then you can go into how we only "really" know how to add, subtract, multiply, and divide, and here's a tool that allows you to take a wide variety of functions that up to this point only existed in calculus and as magic buttons on your calculator, and turns them into problems we can do with real pencils on real paper using real human brains that do not come with a "sin" button. (And then, heh, be grateful you live in the 21st century and you don't actually have to.)

That's now how I learned them. I learned them as just "Here's some Taylor polynomials. Do these homework problems." And I did. I learned them, and could do the math. It wasn't until years later in my computer hardware class that I realized this is what was motivating them. (Not the literal hardware, because of course Taylor polynomials greatly predate that, but the need to be able to calculate these things prior to computers.) And I'm not saying "oh, that's what they are"; math very often has the characteristic that something is discovered for reason X but then has both mathematical and practical applications well beyond it. My point here is that my understanding of Taylor polynomials is now much richer than what I got in the class I learned them in... but there was no reason for that insight to be delayed and almost coincidentally obtained. It could easily have been conveyed via a different teaching method.

Re: Learning algebra in my 60s

#54

Post by jjice »

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.

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 were told to memorize it the previous year and I had just accepted it.

After that, math changed for me. Everything had a reason that was connected to everything else. If you understood where it came from, you could do even more than if you were just given a formula to follow. How beautiful.

I realized that I had 5 years (assuming we only cared to do so after algebra) of match education that could have been better taught if done conceptually. I don't blame teachers. You need to get everyone to pass your class for state requirements, so you try to streamline it to get this year done, and not prepare for the future. Also, most math teachers I had did it as a job, not because they loved it, and there's nothing wrong with that. When I had that calculus class during my senior year though, I realized that that man loved math. Hard but fair. I learned so much and that teacher taught me how to develop a work ethic.

I wish meta-learning and a more concept based approach could be applied to high school courses (at least looking back). I understand why they aren't, but man would I love to see how it would play out.

Re: Learning algebra in my 60s

#55

Post by erehweb »

tankenmate wrote: post #28
zozbot234 wrote: post #14

Earlier quoted context omitted.

There's nothing wrong with practicing formal manipulations: these are just "the rules of the game" and are 100% rigorous. Now, if you want to know "why these rules and not others?", they just happen to follow from basic properties like associativity, commutativity, distributivity etc. There's not much else to it. The author's "misintuition" of variables is not critically important; variables can literally just be arb…

The one thing that gave me deep insights into algebra was discovering Peano's axioms.[0] I had been taught the associative, transitive, and distributive laws in 6th grade at school, but discovering the axioms really opened my eyes in high school; especially under what circumstances they could and couldn't be applied. The last penny to drop was to realise that where you have numbers (e.g. x) you could also substitute…

People understood algebra just fine before Peano came up with this axioms.

Re: Learning algebra in my 60s

#56

Post by roenxi »

WalterBright wrote: post #42
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…

When I remarked yesterday that engineering curriculums that emphasized memorization of engineering formulas rather than understanding how to derive them were inferior, I was dismissed as arrogant and egotistical :-)

> I was dismissed as arrogant and egotistical :-)

Was that related or unrelated to the comment? I can manage arrogant and egotistical without even remarking. #engineering.

But more seriously (although that is normally how engineering formulas work) if an engineer is actually using the formula for anything they tend to pick up the intuition quickly. The memorisation in formal teaching is more limbering up the mental muscles so that it is easier to learn when & if the time comes.

Re: Learning algebra in my 60s

#57

Post by gamerDude »

WalterBright wrote: post #42
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…

When I remarked yesterday that engineering curriculums that emphasized memorization of engineering formulas rather than understanding how to derive them were inferior, I was dismissed as arrogant and egotistical :-)

I work with K12 schools in the U.S. and one of our really sad stories is how a calculus students failed to solve a "A t-shirt that costs $15 is 20% off today. How much does it cost after the 20% discount (don't include tax)?" They responded, "I don't remember the formula for a sale". Only knowing formula's is terrible, it makes knowledge super fragile. So, I for one support your idea of emphasizing understanding over formula memorization!

Re: Learning algebra in my 60s

#58

Post by scott_s »

jerf wrote: post #53
agentultra wrote: post #40

Earlier quoted context omitted.

There is plenty of pure mathematics that is beautiful and worth learning on its own without any practical application. A great deal of joy I get from mathematics is the delight in seeing a novel structure I hadn’t before where the proofs fall effortlessly out of the definitions.

To be honest, even in pure math this approach ought to be taken . I love math too. But it's not really a very good pedagogical approach even in pure math to start out with a full week of unmotivated definitions. I have no issue with the problems being posed being very abstract at a suitable level for the student. By the time you hit college, I have no issues with a professor introducing group theory with "Hey, look a…

Agreed with your overall point, and your specific example. Myself and one of my good friends I met in my physics classes in college both felt the importance of Taylor series had been massively undersold in our calculus courses, because it just kept coming up in our various physics courses. I learned it just as a thing that existed, but we kept relying on them when deriving things in courses like thermal dynamics or mechanics. We would joke that calculus professors should stop the class and just emphasize, "This is really important!" But of course, that wouldn't make the material land any better, for the reasons you've explained.

Re: Learning algebra in my 60s

#59

Post by gmays »

jerf wrote: post #38
sirsinsalot wrote: post #22

Earlier quoted context omitted.

I left school at 15, never to return to education, in part because of how maths was taught. There was never an explanation of the _why_ of things. It was very frustrating, and the teaching was very poor. I ended up a programmer, in part, because I figured if teachers weren't teaching, i'd teach myself. This has been a good strategy for me, however there's serious holes in my pure maths knowledge.

One of the thoughts I've been developing over the past 10 years or so is that you can't teach someone the solution to a problem they don't have. This encompasses the "why" question but even goes beyond it, because even the answer to "why" is often just another level of "why" and/or "who cares", quite reasonably. To learn something, you need a problem, you need to grapple with the problem for a bit, and then you can b…

Re: computer based math curriculum, this is what Jason Roberts is doing with Math Academy here: https://www.mathacademy.us It's mainly targeted to kids, but has adult users as well. It is by far the best self-paced math program I've come across with the widest breadth (from the basics up to graduate level).

He's been working on it for years and talks about it a lot on his podcast: https://techzinglive.com

Edit: The link to use the beta software is here with more details on how the system works: https://www.mathacademy.us/beta-test-information

Re: Learning algebra in my 60s

#60

Post by agentultra »

jerf wrote: post #53
agentultra wrote: post #40

Earlier quoted context omitted.

There is plenty of pure mathematics that is beautiful and worth learning on its own without any practical application. A great deal of joy I get from mathematics is the delight in seeing a novel structure I hadn’t before where the proofs fall effortlessly out of the definitions.

To be honest, even in pure math this approach ought to be taken . I love math too. But it's not really a very good pedagogical approach even in pure math to start out with a full week of unmotivated definitions. I have no issue with the problems being posed being very abstract at a suitable level for the student. By the time you hit college, I have no issues with a professor introducing group theory with "Hey, look a…

I think you have the right of it: it's hard to teach something like maths to someone who isn't curious or interested. And it is definitely difficult to hook someone's attention.

When my children were still babies and quite young I was reading Zvonkin's book, Math from Three to Seven. And when they reached that age I started playing games with them myself to try and introduce these ideas to them. Like Zvonkin I found that one of my kids was more keen than the other... but the only way to keep them hooked was to avoid the "M" word: maths.

What I think helped was to remind ourselves that we were playing games. Any time I went into an area that required calculation: determining some value -- they would catch on to that and shut down. However if we stuck to exploration and fitting things together and exploring games together I could keep them interested for an hour some days.

And as an adult that's what has kept me interested: Martin Gardners' articles in Scientific American and books; John Conway's playfulness (ONAG, the bloody game of life, etc) -- the stuff that wasn't simply rote calculation which I find many attempts at practical applications seem to focus on.

I can appreciate definitions and proofs now because I've learned the language well enough to piece things together. However it was the fun, the absurd, and the playfulness of the completely impractical that kept me going. Games, thought experiments, what-ifs. That sort of stuff.

Post reply on HN