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Relearning math as an adult

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Re: Relearning math as an adult

#221

Post by Jtsummers »

galaxyLogic wrote: post #208
ginnungagap wrote: post #105

Earlier quoted context omitted.

If to understand D you need to know both B and C, each of which requires familiarity with A, the graph is not a tree

Right, but if it is not acyclic, in which order should I try to understand them all?

If it's not acyclic then you haven't broken down the knowledge graph enough. But that's probably a waste of time, trying to come up with a perfectly ordered plan of study for all of mathematics. When you find an apparent cycle, it means the two domains are strongly interrelated and you'll be studying part of one, then the other, then the first again, repeat until you're done (whatever that means to you). No need to try and break every subject down into one-week or one-day chunks and finding a perfect ordering, just figure out the roughly course-lengthed chunks of study and start working through them, concurrently if needed as described.

Re: Relearning math as an adult

#222

Post by apwell23 »

tuukkah wrote: post #204
apwell23 wrote: post #195

Earlier quoted context omitted.

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'.

There is no (multiple choice) standardized testing in the Finnish school system and I cannot imagine passing the math tests based on rote learning. I have to assume she took the test somewhere else. EDIT: The exams at the end of high school can be considered standardized tests, but they are taken at your own school, graded by your own teacher and only verified by the national test organisation. They are not multiple-…

> 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.

Re: Relearning math as an adult

#223

Post by mlyle »

galaxyLogic wrote: post #97
ChainOfFools wrote: post #66

Earlier quoted context omitted.

> abstraction stops being relevant I found this to be the points where abstractions being learned today are only precursors for abstractions that will be learned tomorrow. Another way to put it is at the stage where you're learning to make tools that are themselves only used to make other tools, not used to get results outside of the domain of tool making. These stages have no apparent relevance outside of math, and…

I remember memorizing multiplication tables in school. I learned that 3 x 9 = 27. You just had to memorize that, right? Well then I realized that if 3 x 10 = 30, then 3 x 9 must be one fewer '3' added together by the multiplication, which means take out one '3' from the set of 3s you are adding together by multiplication when doing 3 x 10, which comes to 30 - 3 = 27. That means I didn't really need to memorize 3 x 9,…

> So, learning what is 3 x 9 is not hard AFTER you have learned n * 10, and this trick.

The tricky thing here is that you have a limited amount of working memory, energy, and focus.

To do well at math you need:

- practice at being focused and confronting things that are hard

- an understanding of the problem space you are facing and how your tools work

- enough stuff memorized so that you don't have to context switch too much

You can have some missing pieces in the third area and do okay. But for a lot of students, needing to context switch to do simple arithmetic throws them off. I encounter students who can do any step of a problem, and can even describe the steps of what to do, but when I observe them thunk down to arithmetic and struggle, they aren't able to find their place again and make mistakes.

Most students are better served by getting their multiplication tables firmly committed to memory; perhaps a mnemonic or a simple algorithm of multiplying by 9 helps them get there. But you still don't want to be leaning on that when you're trying to factor a quadratic or cancel things in fractions or whatever.

(Seeing patterns, and learning why the pattern works is perhaps more valuable than multiplication tables... but that doesn't mean you don't need the multiplication tables.)

Re: Relearning math as an adult

#224

Post by ndriscoll »

gmadsen wrote: post #174
tux3 wrote: post #139

Earlier quoted context omitted.

It does a little more than zilch. Exercise is important. Reading is also important. If you want to learn fast, you need a balance of both that works for you. Some people will spend months grinding hard problems on their own in a single chapter to make sure they really understand. Some people will read too fast and have to go back because they don't have solid foundations, and only thought they understood. Reading is…

This is just not true. Reading is necessary but hardly about as important. If you are lucky and get a good text, the exercises guide you to "invent" the important parts of the theory. Otherwise its just definitions and theorems, and its up to you to make your own examples to gain an intuition, which to be fair is closer to life outside of a classroom.

It depends a bit on what you call "reading" (i.e. whether you are reading passively or actively). If you stop and think through each proof yourself before reading the one in the book, that more-or-less turns the main content into a series of exercises.

Re: Relearning math as an adult

#225

Post by AlchemistCamp »

KallDrexx wrote: post #23
AlchemistCamp wrote: post #18

Earlier quoted context omitted.

Compared to what? They have a track-record of students not even old enough to be in high school yet passing AP Calculus BC exams. Here in Taiwan, where I live, it's not that uncommon for people to pay 5x that price per month on supplementary math courses for their kids. https://twitter.com/_MathAcademy_/status/1708542077695574292

Compared to Kahn Academy, which also has a proven track record, is free, and has been around for a long while.

If your time is worth even $1/hour, the "free" Kahn Academy option will be far more expensive than this program.

I respect what Sal Kahn built, especially in the early days, but it's just not anywhere near as time-efficient.

Re: Relearning math as an adult

#226

Post by tuukkah »

apwell23 wrote: post #222
tuukkah wrote: post #204

Earlier quoted context omitted.

There is no (multiple choice) standardized testing in the Finnish school system and I cannot imagine passing the math tests based on rote learning. I have to assume she took the test somewhere else. EDIT: The exams at the end of high school can be considered standardized tests, but they are taken at your own school, graded by your own teacher and only verified by the national test organisation. They are not multiple-…

> 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 should plug.

Re: Relearning math as an adult

#227

Post by gmadsen »

aleph_minus_one wrote: post #183
gmadsen wrote: post #175

Earlier quoted context omitted.

that is a little bit of hyperbole. No school in 5th or 6th grade is giving formal definitions of relations and functions. There is a reason nearly all undergrad math books in analysis, topology, algebra, all devote an entire first chapter to it.

> 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 analysis or abstract algebra, hence why its included in the textbooks

Re: Relearning math as an adult

#228

Post by maximus-decimus »

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…

Sure, but it's like saying "not dying of dehydration in a desert is easy. Just drink water!". Where do you find the water?

The clarify my terrible analogy, where do you find a curriculum that tells you exactly what to learn in what order? When you don't know math, you can't even tell if you ladder is missing steps.

Re: Relearning math as an adult

#229

Post by maximus-decimus »

MrDresden wrote: post #116
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…

> ... it is because they had bad teachers. > Math is easy if... The one constant I observed in most parts of my mathamatics journey (math major in college, software engineering & computer science at university) was the lack of understanding by the person doing the math teaching that not everyone will be able to follow along if steps in the ladder are missing. Words and sentences like 'it is obvious', 'clearly', 'as c…

The recurring phrase in physics was "deriving this is left as an exercise to the reader."

Re: Relearning math as an adult

#230

Post by dimal »

mjburgess wrote: post #140
sjducb wrote: post #134

Earlier quoted context omitted.

I think you don’t realise how much math you know. I’ve heard of set theory, but never studied it.

sure, but you dont need to understand actual set theory -- it's just notation for mostly obvious programming stuff, R means float, Z means int R^2 is actually notation from linear algebra, but here it means a point is 2 floats with a measure of distance between points etc. A lot of this could just be given in a "crib sheet" for tech people, and you'd get 80% of it straightaway. It's years of work to understand this n…

What they’re saying is that you may be falling for the curse of knowledge. From Wikipedia:

> The curse of knowledge is a cognitive bias that occurs when an individual, who is communicating with others, assumes that others have information that is only available to themselves, assuming they all share a background and understanding. This bias is also called by some authors the curse of expertise.

You know which parts of the AI math notation is shallow and which not. Someone else might not have that same knowledge, so they might not know exactly what to put on the crib sheet and what they need to go deeper on.

By the way, if you wanted to make such a crib sheet and publish it online, I think a lot of people would be very grateful!

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