Relearning math as an adult • Hacker News Board

Re: Relearning math as an adult

#231

Post by apwell23 » Sat, Jan 20, 2024, 12:24 AM UTC

tuukkah wrote: post #226

apwell23 wrote: post #222

Earlier quoted context omitted.

> I cannot imagine passing the math tests based on rote learning. why not. Its not exactly rote learning like a parrot. You just learn tricks and patterns in problems. Tests usually have a limited amount of patterns.

These are not tests generated from patterns. You may be asked e.g. to solve an applied problem you've never heard of or to sketch a proof to a given (simple) lemma you've never proved before. OTOH, you are not supposed to solve every problem in the exam, so perhaps you can get the best grade even if you skip all the problems where there's no pattern to apply. In that case, that's a loop hole which the exam creators s…

> solve an applied problem you've never heard of or to sketch a proof to a given (simple) lemma you've never proved before

Can you link me to finnish high school question that is asking for 'sketch a proof to a given (simple) lemma you've never proved before '

I agree that this cannot be rote learnt .

Re: Relearning math as an adult

#232

Post by sn9 » Sat, Jan 20, 2024, 12:40 AM UTC

melagonster wrote: post #146

baq wrote: post #128

Earlier quoted context omitted.

> set her nose to the grindstone Worth calling out the key observation: it takes practice, and lots of it (for most people, anyway) to get good at math. Just reading about it does zilch.

please tell me that how can I get question for practice in math of university level? most of textbook even don't contain answers of practice in the book...

You just find books with the solutions.

Some like Hubbard and Hubbard's Vector Calculus come with a solutions manual.

Some like Knuth's Concrete Mathematics contain full solutions.

There's also whole genre of books like Schaum's Problem books and their Outlines which contain thousands of solved problems. And they're quite cheap.

For any given math subject X, you can probably search for an "X problem book".

Re: Relearning math as an adult

#233

Post by danielmarkbruce » Sat, Jan 20, 2024, 1:51 AM UTC

mjburgess wrote: post #117

The "math" used in ML/AI papers is usually just a sort of 'whiteboard math' which is a domain-specific mishmash of linear algebra, calculus, set theory and statistics. If you could find a book just going through the relevant bits you wouldnt really have to "learn math again", it can be translated into english straightforwardly -- very very few ML papers relevant to industry have extended proofs, etc. that require eg.…

This is definitely true. It's largely just notation for communication.

I just paste sections of dense math from AI/ML papers into chatgpt and it explains it. Almost none of it is complicated. It's just really awful notation.

Re: Relearning math as an adult

#234

Post by aleph_minus_one » Sat, Jan 20, 2024, 1:51 AM UTC

gmadsen wrote: post #227

aleph_minus_one wrote: post #183

Earlier quoted context omitted.

> No school in 5th or 6th grade is giving formal definitions of relations and functions. I remember that my math teacher pretty surely did. > There is a reason nearly all undergrad math books in analysis, topology, algebra, all devote an entire first chapter to it. Indeed there exist multiple good reasons: - recapitulation - setting up the notation - clarifying how the textbook defines the relevant mathematical objec…

how do you motivate formal functions and relations to 5th graders, and more importantly why? Relations are important because of the natural partitions of a set they create and the development of group theory. Functions are useful in calculus, but not really the algebraic properties, those are glossed over, i.e kids learning calculus are usually not learning the formalization of functions. That isn't important until a…

> how do you motivate formal functions and relations to 5th graders, and more importantly why?

The word "you" in English has two meanings:

1. how would I motive this to 5th graders ("you" as "tu/vous" in French or "du/Sie" in German)?

2. how is this topic motivated in school to 5th graders ("you" as "on" in French or "man" in German)?

For 2: Well, it isn't. The pupils have to accept that in future, they will hopefully get why it is useful. Until then, better learn the material so that you won't fail on the tests.

For 1: If the prophet does not come to the mountain, the mountain must come to the prophet. If you need group theory to motivate functions (as you implicate in your answer), then teach group theory to 5th graders, so that the pupils get the motivation that they desire. If you additionally need to motivate group theory: well, I do know some quite interesting applications of group theory. :-)

Just to make it clear: I do have quite some experience in teaching mathematics, but to highly gifted students.

> Functions are useful in calculus, but not really the algebraic properties, those are glossed over, i.e kids learning calculus are usually not learning the formalization of functions. That isn't important until analysis or abstract algebra, hence why its included in the textbooks

In Germany, there is no distinction made between calculus and analysis. At the university, this subject is taught from beginning on in the abstract way. In school, what you call "calculus" is often taught in a more "hand-waving" way by bad math teachers. Good teachers rather attempt to teach calculus/analysis in the abstract way in school.

Re: Relearning math as an adult

#235

Post by kirso » Sat, Jan 20, 2024, 2:20 AM UTC

apwell23 wrote: post #195

baq wrote: post #128

Earlier quoted context omitted.

> set her nose to the grindstone Worth calling out the key observation: it takes practice, and lots of it (for most people, anyway) to get good at math. Just reading about it does zilch.

OP described a specific type of practice to get good test grades using 'Anki cards' . Standardized tests test for grinding and dedication not expertise or 'getting good'.

Thinking about how to learn is fascinating. A few short things that helped me:

Essentially we need to get fundamentals first.

Then apply the knowledge and get challenged (feedback loop) -> build a project, play in front of others, speak the language.

Improve - not by force, but by understanding (remembering something and understanding something are two different things).

Synthesise - learn about a topic in a different manner or try to find similar concepts in completely different context /

Use mentors to amplify the knowledge and get feedback quicker.

Immerse yourself in practicality (work in a field, live in a country etc.)

Re: Relearning math as an adult

#236

Post by mr_mitm » Sat, Jan 20, 2024, 2:24 AM UTC

tayo42 wrote: post #90

jorgesborges wrote: post #28

I’m a thirty-something with an arts degree who decided to learn math. Basically I was tired of reading popular science and being fed metaphors to understand concepts. I wanted to “see” it for myself. I spent time online at Khan Academy and friends for a year or so on and off. It was fine but meandering? So I enrolled in community college! It’s great. I have a clearer path, immediate feedback, teachers, and an obligat…

> working toward a bachelor of science in physics. Thats cool, I kind of want to do that. But also Im stuck with wondering, I put all this work into that, what do I do at the end?

Then you will have gained insight into how the world works on a fundamental level. Isn't that something in its own right?

I can only speak for myself, and did go on to get a PhD, but even on a bachelor level, studying physics changed how I see the world and how I think.

Re: Relearning math as an adult

#237

Post by bsdpufferfish » Sat, Jan 20, 2024, 2:38 AM UTC

woolion wrote: post #203

bsdpufferfish wrote: post #24

Earlier quoted context omitted.

> leading to overcomplication and obscurantism Mathematicians attempt to express ideas in the most readable and clear way possible. It's actually code that must be obfuscated by the constraints of the language and computer. Mathematicians have no constraints preventing them from presenting something in the way that makes the most sense. The part that can be called "Obscurantism" is when they use a high-level abstract…

No, "there exists a group" is not constructive, in general. You just need to prove that non-existence leads to a contradiction. The whole deal about "constructive mathematics" is to have that if you can prove something exists, you can also construct it. I think the success of the constructive mathematics program is really debatable, but in any case I don't think it leads to more 'natural' mathematics. (The terms used…

> You just need to prove that non-existence leads to a contradiction.

Indeed however this is the exception, not the rule. The general way to do an existence proof is to construct it.

Re: Relearning math as an adult

#238

Post by mr_mitm » Sat, Jan 20, 2024, 2:40 AM UTC

boxfire wrote: post #3

I may be biased as I am a trained Mathematician, but I always feel when someone says "Math is Hard", that is because they had bad teachers. Math is easy if you build up from fundamentals, not like physics education where you say "but lets delete everything before because it had an oversimplifying assumption", rather if you build your knowledge entirely sequentially from things you know or assume, you build up a toolb…

"Bad teacher" often strikes me as a face saving excuse. Not that having a bad teacher will not make it harder to learn, but there seem to be a lot of bad maths teachers out there, if I go by how many times I heard that.

I mean it's okay to be bad at something. I sucked in history class, and I'm not blaming the teachers. I simply had zero interest in it as a teenager, unlike maths and physics.

Re: Relearning math as an adult

#239

Post by libso » Sat, Jan 20, 2024, 3:11 AM UTC

boxfire wrote: post #3

I may be biased as I am a trained Mathematician, but I always feel when someone says "Math is Hard", that is because they had bad teachers. Math is easy if you build up from fundamentals, not like physics education where you say "but lets delete everything before because it had an oversimplifying assumption", rather if you build your knowledge entirely sequentially from things you know or assume, you build up a toolb…

Now tell us where to find the ladder with all the steps in order.

Re: Relearning math as an adult

#240

Post by gmadsen » Sat, Jan 20, 2024, 3:23 AM UTC

aleph_minus_one wrote: post #234

gmadsen wrote: post #227

Earlier quoted context omitted.

how do you motivate formal functions and relations to 5th graders, and more importantly why? Relations are important because of the natural partitions of a set they create and the development of group theory. Functions are useful in calculus, but not really the algebraic properties, those are glossed over, i.e kids learning calculus are usually not learning the formalization of functions. That isn't important until a…

> how do you motivate formal functions and relations to 5th graders, and more importantly why? The word "you" in English has two meanings: 1. how would I motive this to 5th graders ("you" as "tu/vous" in French or "du/Sie" in German)? 2. how is this topic motivated in school to 5th graders ("you" as "on" in French or "man" in German)? For 2: Well, it isn't. The pupils have to accept that in future, they will hopefull…

fair enough, I was only speaking for the US, which has a slower pace and lack of motivation throughout k-12 + calc series. I still have no idea why they don't teach basic algebra with matrices, so it doesn't seem like a giant bad of tricks, or why linear algebra is so separated from multi-dim calculus, it really makes it more difficult with busy work and you never really comprehend anything.