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Relearning math as an adult

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Re: Relearning math as an adult

#241

Post by galaxyLogic »

mlyle wrote: post #223
galaxyLogic wrote: post #97

Earlier quoted context omitted.

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…

Good point about working memory. And you are right if it is in memory you can read stuff that assumes you know it and just glide through without stopping.

For me the tricks like above were like a backup solution, using it a few times it became obvious that 9 x 3 == 27. Indelible. It is. For some cases it was like "It can only be 27 OR 26" and then I would use the trick figure out which.

But whether you use a simple trick and a trivial calculation or don't have to do that at all the point is the same it should not take much thinking which would cause you to lose your focus and train-of-thought, as you say.

Re: Relearning math as an adult

#242

Post by galaxyLogic »

Jtsummers wrote: post #221
galaxyLogic wrote: post #208

Earlier quoted context omitted.

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 t…

Right when you read something you don't need to understand it all to understand something which may be needed to understand something else elsewhere.

But still I think it would motivate me to keep on learning if somebody could show me an accurate acyclic pre-requisites graph and tell me: "These are the thing you need to understand before you should go to the next topic. If someone could come up with the time to come up with an accurate acyclic "knowledge-graph" it would help millions of students of mathematics.

If you try hard and long enough you will understand what you're trying to understand, you will. The question is what would make that more fun and less tedious. It is about precision and not needing to learn something you don't need to learn, to understand something that you need to learn. Spend your time on learning stuff you need to learn to understand what you want to learn.

Re: Relearning math as an adult

#243

Post by ajmurmann »

wharvle wrote: post #102
ajmurmann wrote: post #82

Earlier quoted context omitted.

So US school kids don't encounter any geometry in school till they are ~16?

Shapes, areas of simple polygons and circles, Pythagorean theorem, volume of solids, graphing, translation, all that stuff and more that one might call “geometry”, is scattered between roughly ages 6 and 12. I think the high school level “geometry” class—which may be the only one named such, but primary school math is full of geometry—is an atrophied organ left over from when it was still common to teach directly fro…

Thanks for the explanation! That is a lot less surprising.

Re: Relearning math as an adult

#244

Post by tuukkah »

apwell23 wrote: post #231
tuukkah wrote: post #226

Earlier quoted context omitted.

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 .

I hear these problem types are not as common as they used to be, but here's some recent examples in this direction.

Fall 2023: "Prove that 2^12345678910 - 1 is divisible by 1023."

Spring 2022: "Using induction, show that the sum of the numbers on the line n of Pascal's triangle equals 2^n."

Here's the full exam from fall 2023: https://yle.fi/plus/abitreenit/2023/syksy/matematiikka_pitka...

Re: Relearning math as an adult

#245

Post by Tainnor »

aleph_minus_one wrote: post #159
Tainnor wrote: post #158

Earlier quoted context omitted.

Not every mathematician needs to understand axiomatic set theory, but probably every working mathematician needs to understand at least basic set operations, simple identites like De Morgan's laws (I don't care if you know the name or not), as well as what cartesian products, relations and functions are.

> every working mathematician needs to understand at least basic set operations, simple identites like De Morgan's laws (I don't care if you know the name or not), as well as what cartesian products, relations and functions are. This is stuff that you learn in the 5th or 6th grade in school, and is about as far removed from what mathematicians call "set theory" as basic arithmetic operations are from college math cou…

You may have learned this in 5th or 6th grade, but I certainly didn't.

The point is that any working mathematician is comfortable manipulating sets in algebraic expressions and that's not something you expect from your average high schooler.

I could have added that mathematicians need to know at least about different cardinalities, but I guess strictly speaking you could be working in discrete maths and not care about any of this.

Re: Relearning math as an adult

#246

Post by ykonstant »

Since we're doing some ad posting anyway, I am available for online undergraduate and graduate mathematics tutoring one-on-one. So we can solve problems together, I can give you roadmaps for specific math topics, or we can analyze mathematical aspects of software development. You are not getting any certifications, though! (´• ω •`)

Re: Relearning math as an adult

#247

Post by pprotas »

nicholasbraker wrote: post #114

I work in IT as a network architect and when I tell people I graduated high school with one F on my final list being math they frown a bit. In my high- schooldays (90'ies era) they made a distinction between "math-A" which was more statistics and "math-B" which was more algebra and calculus. I really sucked at math-A which was considered by some as not even real math. I will definitely check this one out and hope to…

Sounds like The Netherlands, no? At the very least, the same system is used in The Netherlands to this day.

I graduated high school with a barely-passing grade in Math A, and years later was able to finish exams in Math B with a passing grade. All it took was dedicated practice, and I suck at math! You can do it as well.

Re: Relearning math as an adult

#248

Post by apwell23 »

tuukkah wrote: post #244
apwell23 wrote: post #231

Earlier quoted context omitted.

> 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 .

I hear these problem types are not as common as they used to be, but here's some recent examples in this direction. Fall 2023: "Prove that 2^12345678910 - 1 is divisible by 1023." Spring 2022: "Using induction, show that the sum of the numbers on the line n of Pascal's triangle equals 2^n." Here's the full exam from fall 2023: https://yle.fi/plus/abitreenit/2023/syksy/matematiikka_pitka...

I cannot seem to access that link outside finland.

That is indeed a tough proof to solve sight unseen. Any reason you say that students are seeing that question for the first time in the test. Seems like a famous questions, even chatgpt got the proof correctly .

Re: Relearning math as an adult

#250

Post by apwell23 »

kirso wrote: post #235
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'.

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 differen…

This is a great summary. Thank you!
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