Comment by cs702 | Hacker News Reader

Comment by cs702 | Hacker News Reader

Parent Full thread

As always, this is written by someone who admits to never having learned to reason abstractly about algebraic objects, because it is hard.

Quoting from the OP:

> The traditional arithmetic I learned in the early grades made sense to me because I could relate it back to real-world things. Later, when I was learning how to design, finding the area of a circle proved essential to practical tasks such as sizing hydraulic and pneumatic cylinders. I could do trigonometry by visualizing, for example, the cables on a suspension bridge. But algebra: no.

For a different perspective, I recommend reading instead Susan Rigetti (nee Fowler)'s posts about overcoming challenges to learn Math and Physics. She says she had to solve what felt like "millions of exercises" to become familiar with many abstract concepts, until they finally clicked for her and she was able to reason about them. In her words: "Solving problems is the only way to understand mathematics. There's no way around it."

https://www.susanrigetti.com/math

https://www.susanrigetti.com/physics

https://www.susanjfowler.com/blog/2016/8/26/from-the-fledgli...

It really does feel like all of this hand-wringing around the state micro-managing syllabi (esp in Math), is sort of skirting around the core issue of having better math teachers.

This involves paying teachers paying more and treating them with respect? Nah, lets have endless conversations about how whether we should not learn math at all!

This involves paying teachers paying more and treating them with respect? Nah, lets have endless conversations about how whether we should not learn math at all!

That issue is highly political though. Even when setting aside the issue of selling to voters a pay raise for teachers. Good math teachers are specialists with a math education. To attract them to teaching you’re going to have to pay them more than other teachers. How do you sell that, politically, to teachers’ unions?

There’s a lot of cognitive dissonance going on with non-specialist teachers teaching math. They’re teaching bad habits and passing on their own math anxiety to students but if you confront them about it they get very defensive and start talking about school funding and textbooks and materials and computers.

The truth is: that’s all bunk. Third world countries are better at teaching math. They don’t get distracted with computers and they don’t waste time and money redesigning the curriculum and printing new textbooks all the time. They teach math in the old, straightforward style and emphasize practice.

No point comparing ourselves to the third world. The American middle-class has different expectations (of pay, of lifestyle, of cultural occupational conditions) than the third world, we just have to accept that. And FWIW, having had gone to a "third world" school, those teachers are great at teaching people to do well on tests, not to think critically.

Paying english teachers less also means you get bad english teachers that are bad at teaching critical reading (which..yikes!), so I don't really see the point of that pay discrimination.

I would absolutely advocate for removing administrative bloat, but we should be recruiting teachers like tech workers.

Paying english teachers less also means you get bad english teachers that are bad at teaching critical reading (which..yikes!), so I don't really see the point of that pay discrimination.

I haven't seen any evidence that any particular teaching method delivers tangible improvements in critical thinking skills over the school term. Do you know of any? It's important to measure skills at the beginning and end of a term (and a followup years later) in order to show improvements at the individual level. Otherwise you risk selecting for students' prior ability.

It maybe worth it to talk about universally superior teaching methods but as far as I can tell no teaching method can survive a teacher not caring. I understand this is not true for all professions. In manufacturing you can write a handbook (Just follow steps 1 thru n) and any min-wage worker will give good results as long as they follow those steps.

As far as I know, professions like teaching, nursing etc. don't work that way. I personally know about 3 teachers who genuinely cared about their students but could not survive on teacher salaries (2 are now real estate agents and miss their kids :/).

There are ~23,500 high-schools in the country.[a] If the US had allocated, say, 10% of the $8T cost of the "war on terror"[b] for math education, each public high-school would have had an additional $34M in its budget -- more than enough to hire multiple amazing math teachers at each high school, drawing only from the top 1% of college graduates, as in Finland.

[a] https://nces.ed.gov/FastFacts/display.asp?id=84

[b] https://www.brown.edu/news/2021-09-01/costsofwar

We pay our teachers well in the US compared to our better-performing peers. And who doesn't respect teachers?

More money isn't the solution.

Teachers get a very superficial respect. They get lip service, but are not treated in accordance with the nominal respect they get.

Do you believe that the current cohort of teachers would better serve our students if they received more respect?

Or do you believe that if the profession were more respected, it would attract a better, more qualified class of teacher, who could then better educate the students?

I speak for myself, not the original comment, but more the latter than the former. I think of it this way: "What needs to change to make teaching children and young adults appealing compared to going into industry?"

And, to be clear, I think teachers are generally qualified. Math just represents a very acute failure mode of our education system.

Honestly, they don't even get lip service anymore. Parents micro-managing and mistreating teachers made teaching the leading edge profession of the Great resignation.

>We pay our teachers well in the US compared to our better-performing peers. And who doesn't respect teachers?

Have you been to the US? Because most people here don't respect teachers. And we pay teachers almost nothing.

> And we pay teachers almost nothing.

That is largely not true, though certainly true in some cases. Teacher salaries are set by the local school district and range from "reasonable" to "not great".

Overall, teachers make about $51,000/year while bachelor's degree holders as a whole make about $59,600/year. Against that, you have to recognize that the teacher pay is for nine months, while the bachelor's degree number is for twelve months. If the typical bachelor's degree holder worked only nine months, one would expect their salary to be about $44,700, considerably less than the typical teacher.

Sources:

https://mint.intuit.com/salary/teacher

https://nces.ed.gov/programs/coe/indicator/cba

Many teachers have a masters, although places like Florida are looking to change that.

> This involves paying teachers paying more and treating them with respect?

It seems like paying the same people more money to do the same job won't change outcomes in any meaningful way. Why would anyone expect it to? Paying teachers more in a competitive market, on the other hand, might has a dramatic impact as we might likely end up with very different teachers.

> "Solving problems is the only way to understand mathematics. There's no way around it."

I'll read the linked documents at some point, but taking it per se, it should be noted that doing problems as a method of understanding may be necessary, but is not sufficient. I had a professor who taught how to solve problems and then assigned way way too many problems. I could manipulate the symbols, but I didn't grok what I was doing until I picked up a different text and started learning on my own. How many problems are needed definitely varies from person to person, and I'm not entirely convinced there aren't other ways to understand mathematics; but I suspect problems are the fastest and easiest.

Yes - building intuition around a subject is key (the aha moment perhaps), which is exactly what Grandin struggles with in the abstract disciplines.

I believe most people who underperform in math had bored and uninspired teachers in their education.

I found a joy with math only by accidentally reading biography of Newton, Godel Escher Bach, a book about Cantor, and a compendium of math papers for educators.

Then I could see all my reasons for avoiding math were excuses for poor insight to the beauty and interrelatedness of mathematics.

You do the work for intimate understanding for each move. If I could learn mathematics like learning sports (simple ideas worked into muscle memory) I would appreciate the result, and you can keep the abstract problems. I’d rather solve real problems.

I’d argue that people need to be taught basic group theory and abstract algebra first, preferably in the form of puzzles to get familiar with more abstract thinking, and to see things in terms of composition of objects instead a of mixing numeracy in the mix.

I think that numbers create all sorts of problems because people focus on them as they are familiar and miss the bigger picture.

I don't think it really works, at least not for most people.

Abstract algebra (i.e. actual algebra, as opposed to doing useless stuff to poor polynomials) requires one to have developed the mathematical maturity to iron out informal thinking.

Good luck explaining "prime" and "irreducible" are not the same.

I think a path that would work better is through problem solving, like olympiad problems. But even that definitely isn't for everyone.

This reminds me a bit of the New Math pedagogy, popular in the US during the 1950s-1970s.

Without the proper context, group theory and abstract algebra are deeply uninteresting subjects. It would be like trying to read Bertrand Russell before you learned your ABCs, ridiculous. It's important to start with numbers because they are familiar.

This is something I’ve always wondered about. It seems individual differences in innate capacity explains a lot when it comes to math ability, but don’t young math prodigies also need to do the brute force time consuming stuff too? Does it take them as many practice problems to learn to differentiate polynomials or are they able to skip over much of the drudgery somehow? If so, how? There are many concepts which require exposure and practice in order to master, so how can one speed up that part as some children seem able to do.

In music I’ve heard the answer is that no, you can’t speed it up, you have to do the practice. But then how to explain the Mozarts of the world who somehow seem to have, innately, the knowledge that for others requires training and time?

> But then how to explain the Mozarts of the world who somehow seem to have, innately, the knowledge that for others requires training and time?

Those kids didn't really start out with "innate" knowledge, or anything like that. They get intensive training starting at a very early age, that's how they do better than others. Also, practice is not necessarily boring. When you're actively learning/mastering a skill at close to optimum effectiveness, practice is quite enjoyable and it's easy to rack up a lot of it.

I suspect that’s true, it sort of has to be, but then it seems to contradict various anecdotes. There’s a notion of the prodigy as a different type of person, a qualitative difference in kind rather than a difference in degree (of effort, iq, focus etc).

> Does it take them as many practice problems to learn to differentiate polynomials or are they able to skip over much of the drudgery somehow? If so, how?

Why would it take everyone the same amount? It isn't like there is a natural law that you need to drill the same problem 20 times to understand it, why wouldn't someone's brain be able to process it and understand it after doing it the first time or even just looking at someone else doing it?

It is also written by an autistic person. Cognitive barriers are easier to overcome for some people than others.

Reply on news.ycombinator.com