Relearning math as an adult

Yeah, it sounds like you're not far from the point where you can start jumping ahead and working backwards to fill in the bits that you're missing - that's what many people naturally and instinctively try and it can work but can also be frustrating if one misjudges one's degree of proficiency. You don't often see 'I'm just going to give myself a full secondary school maths refresher' which is more demanding on time and self-discipline but at least we know it's pretty reliable given those things.

Ah, okay. I actually took calculus in 8th grade. I studied another two years past that, dropped out, and then later did a complete 180 and graduated with a literature degree.

I'm now over 40 and interested in relearning the math I learned long ago and pushing a bit further than I had before.

I bought the whole set for my kid. He's also doing Brilliant.

It starts at somewhere that the kids are at the end of primary school (at least in the UK) and ends somewhere in high school. My kid could already do all the pre-algebra stuff, so that book went fast. The way I see it, the kids waste a lot of time in the middle years when they already know the arithmetic and pre-algebra, but might as well be doing a bunch of more interesting things.

I think it's a weird way to learn math, and I learned it this way in school. Most of these courses just teach information memorization and recall. sin(x)^2 + cos(x)^2 = 1, etc.

I would start with something like Elementary Analysis: The Theory of Calculus, and work from there. You'll eventually arrive at the same place -- Calculus but from a much stronger mathematical foundation.

Are you doing the online classes or only the books? I wanted to register for the online classes but they seem to be heavily oriented towards interactive learning.

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

Heh knowledge graphs sometimes don’t feel acyclic, at least not to me anyway. Sometimes I’m stuck bouncing back and forth :)

Right, it must be acyclic. Which means it can be presented as a tree with some duplicate nodes. The important thing is the student must understand in which order they can try to understand the topics.

100% agreed

> Math Academy is much more dense and on-point that Khan's. You don't have to sit through 15 minutes of video when 2 minutes worth of text explanation does it.

This is precisely what bothers me about KA. I guess they're trying to ease into the topic, but I find that kind of repetition annoying and distracting.

Hi, I'm Alex, curriculum director at Math Academy.

Thanks for your comments. In response to the things you'd like us to have:

"ability to skip lessons" - we plan on introducing "mini-diagnostics" sometime soon, hopefully within the next few months. This will allow students to "place out" of certain content they know. The primary diagnostic assessment will have done most of the grunt work here, but mini-diagnostics can be used for fine-tuning the knowledge frontier.

"a reference page to track unlocked material" - This is an interesting idea that we can discuss.

"fewer multiple choice questions" - We're actively introducing "Free Response" across the entire curriculum. Complete coverage across all courses will likely take several months, maybe over one year. Many of our lower-grade students should be seeing lots of free-response questions already.

"more in-depth problems" - we have multipart problems in most courses. We plan to add many more. Introducing "challenge problems" into the curriculum is also something we have planned for the near future.

"proof-based math" - We plan on launching our "Methods of Proof" course within the next 6-8 weeks. This course is designed to introduce students to all fundamental concepts related to proof building: sets, logic, functions, relations, cardinality, proof by induction, direct proofs, counterexample, contrapositive, contradiction, and trivial and vacuous proofs, to name a few. Most of the content is already ready. We have a few technical challenges to overcome before it can be launched due to our new "proof" question format, but we have a clear idea of how these challenges are to be resolved, so 6-8 weeks is certainly realistic.

FYI this is the guy that wrote the comment you replied to: https://www.justinmath.com

He's true math nerd, and definitely not a marketer. In fact, I don't think a single marketer works at the company. It's a bootstrapped labor of love and I've been following their journey for almost a decade.

I get the skepticism, but some things are legit.

Don't be nasty.

Especially ridiculing a tech person giving detailed and interesting answers as a "marketeer".

Every state sets its own curriculum, with varying degrees of freedom for the districts and schools within the states. The federal level has standards, which students are generally tested against in various standardized tests, that are tied to federal funding.

Multiple curricula can satisfy the same standards, at least on paper if not in practice. So states are, more or less, free to teach things how they want. However, they're also strongly driven by the textbook industry, which turns on the two biggest textbook purchasers: Texas and California. So a lot of the textbooks (and associated curriculum material) available for purchase in the rest of the states are driven by whatever those two states are pushing.

I would have still appreciated that refresher in a college class on content I learned 7-8 years prior.

> I guess education in the US has local standards?

Very much so! The US didn’t have a Department of Education until the 90s.

10th grade is the “on level” time to take geometry.

It is, which is why I was interested in the first place. But physics the way it is taught is pretty rigorous and study is intense for someone with a passing curiosity. And idk if there is a way around the math for some of the stuff.

> 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.

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 seems not to be entirely obvious to the students in my classroom. Today second day of the semester, job one is to explain that they need to have a notebook. They need to write in the notebook. And they need to do the homework in the notebook. They seem not to know that, many of them.

It depends entirely on the cards and what's done when a card is missed. If the cards are straight definitions then probably not the best way to "get good at math". But if the cards (you can do this with plugins) are things like generated problems using specific mathematical techniques (solve a system of linear equations, for instance) and after failing a card you can go back and study the material, then it's effective. You can also throw textbook problems into cards if you don't want to use a generator, but that's a lot more effort than most people are willing to put in. You don't even need to put in the solutions if you're willing to keep the book around. It does mean cards take longer than a few seconds, but it still has the effect of taking advantage of spaced repetition to develop mastery of a technique.

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.)

I had great success using Anki cards during my undergrad to fully internalize important definitions (i.e. trig identities, etc.) for utilization within practice problems.

Sometimes, you just know a thing or don't, and for me, spaced repetition cards helped a ton with the static things you just have to memorize (or derive). Knowing definitions is important—you're absolutely right that it's just one piece of the puzzle though.

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-choice tests.

It's also a bit different from learning to play music in that almost everyone appreciates music even without being able to create it themselves. There's a much more direct connection between the craft and its outcomes than with math.

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".

That's mostly because proofs are too long to write out easily in the back of a textbook, imo. Your best bet here is to learn the proofs of the theorems proved in the book first - they're usually much harder to come up with at first glance than the problems one bases on them. Or, if you're using a commonly used textbook, you can usually find people's own proofs online. Sourcing an easy to grok and easy to check proof is a very good skill to build in any case, it minimizes wheel reinvention at the neural level.

If you have a textbook you're interested in studying from, you can usually find courses from searching on Google that use that text. If you're lucky, they will make the homework and answers available online.

If you search long enough and look for older courses (before all of these online learning platforms like Canvas became popular), you can usually find lots of worked examples. There is no central location to find what you're looking for, but you should be able to find supporting resources.

Read the book and whenever you come to a theorem, stop and try to prove it on your own. If you can't, read some of the proof, then try to continue on your own.

Hop on the mathematics discord and ask in their book channel. It's the big server with a picture of a torus as the icon. There are some very knowledgeable people there

Hi, I'm Alex, curriculum director at Math Academy.

As Justin mentioned, there are several criteria that we must meet in our high-school pathway that aren't needed for studying higher-level (e.g., undergraduate) math, or they can be postponed. We decided to remove some of these in the Foundations series.

The idea behind the foundations series is to provide adult learners with the most efficient path possible to get onto the higher-level material.

Examples of topics that were removed from the high-school series to create the foundations series include some of the following:

* Various Geometry topics: All of the _essential_ geometry is covered. However, we removed topics on inscribed angles, Thales' Theorem, Triangle congruence, and similarity criteria (apart from the AA, which is the only one that seems to come up in practice), midpoint and triangle proportionality theorems, a fair amount of solid geometry, except what's fairly standard for calculus (volumes and surface areas of spheres, volumes of cones), lots of stuff on different types of quadrilaterals.

* Conic sections: The essentials are covered in both pathways. But in the high-school path, we go into a little more detail about foci, directrices, eccentricity, and utilizing their geometric definitions (e.g., focus-directrix properties).

* Trig identities and Equations: Covered in both pathways, but the high-school versions go into more detail and consider more cases.

* Some word problem/modeling topics.

* Other arbitrary Prealgebra topics: Divisibility rules, going into more detail about ratios in contextual settings, scientific notation, and some basic data representation topics that one would normally meet in Prealgebra.

* Slope fields. This will be covered in our upcoming differential equations course.

* Some analytical applications of differentiation that are quite specific to the BC Calculus exam: Identifying and removing point, jump, and infinite discontinuities and analyzing graphs of first and second derivatives.

* There are also fewer topics on related rates and optimization, though these topics are still covered.

* Some contextual applications of integration, like volumes of revolution and volumes of known cross-sections.

* Convergence tests for infinite series. When we get to that, these will be covered in real analysis, but other than infinite geometric series (which _is_ covered in Foundations), these tests don't show up too often anywhere else.

* Some ODE models, such as exponential and logistic growth and decay. We cover ODE basics in the foundations course, but particular models will be covered in the differential equations course.

* Taylor series. Again, this can be covered in the differential equations course for anyone wishing to take that course when it's ready.

Happy to answer any further questions you may have.

Hi, I'm Alex, Curriculum Director at Math Academy.

It is on our radar to allow students to "place out" of certain topics and modules if they feel ready. We'll call them "mini-diagnostics" or something similar when they're ready. I believe it will be worked on within the next few months.

Unfortunately, these things do take time to implement, but we are listening. FWIW, we're a tiny, bootstrapped company with literally two programmers (Justin, our ML/backend dev, and Jason, our founder and solo UX/UI) working on the entire codebase.

The content team is a little larger. We have around 12, mostly PhD mathematicians, working on the content, which is why the context is probably a little further ahead in some respects.

> sure, but you dont need to understand actual set theory

How would you know? You sound like you understand set theory. Ironically, a body needs a rather extensive grounding in the theory and practice of set theory to look at something and think "I don't need to use the tools of set theory here" with deserved confidence.

A lot of maths is about having a huge repertoire of tricks that you know won't help solve the problem at hand. Saves weeks of fruitless attempts. But I'll bet you didn't notice the things you know not to do. That does matter for interpreting these papers.

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!

Oh, I read that paper about a year ago. I feel like it could have been explained better, but the concepts aren't that hard if you've taken numerical methods.

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.

Adding this to my arsenal of memorization tricks.

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, I needed the above simple rule in addition to the fact that n x 10 is always what you get when you take the digit 'n' and add a 0 after it.

So learning multiplication tables was hard, until I learned the rule of looking for an easier-to-remember result and then adding or subtracting something to it. Of course I also had to understand that multiplication is really just repeated addition.

My teacher never taught me this trick, just told us to recite the multiplication tables in out heads again and again. But after doing that for some time I figured out the above trick myself.

Learning math beyond multiplication is hard if you cannot multiply numbers in your head, because lots of math presentations assume that of course you know that 3 x 9 = 27. Or something similar. It is not just about understanding the concepts, it's about being able to perform calculations, in your head. Else you cannot understand the explanations of new concepts. Even though we have pocket-calculators, we still need to be able to do calculations in our heads to understand new topics. in math.

So, learning what is 3 x 9 is not hard AFTER you have learned n * 10, and this trick. I assume something like that happens in the minds of mathematicians. They know a lot of math already which makes it easier to understand new results when they already know a lot. To learn what is n * 10, you had to learn 1 x 10, 2 x 10, 3 x 10 etc. and then understand the pattern in there.

Learning something is easy if you already know lots of related stuff. So it's not about learning more and more difficult things, it is about just learning more and more, related things. It is about having more and more (learned) data in your head.

I assume that is also why LLMs work so well: They have lots of data.

In summary: Learning math is not "difficult", it is tedious.

> The difference is that I find this process enjoyable, so I don't mind spending the time.

This is definitely the difference for at least some of the people out there, however...

Imagine however that you do enjoy it at the start so you move on from topic Y to topic Y+1, then to to Y+2. However you find that you no longer understand Y and you need Y when you are doing trying to learn Y+3 so you study Y+3 and Y, now your progress in Y+3 has been slowed down.

Really your goal was to get o Y+7 though that is where you can start breaking new ground and contributing but as you try Y+4 and Y+5 the gains stop and maybe even reverse. You are now on a learning treadmill(perhaps sometimes falling off and having to restart too) redoing Y-1,2,3,4,5 not moving forward. Often it is possible to find a trick/skill/simplification/etc to continue moving forward to get to Y+6,7.

How long would you find the process fun on that treadmill though? I think it is common to not find covering the same ground over and over fun or never being able to make it to the point where you are part of peer group where you can contribute. An understandable result is when those people invest elsewhere, where they see better returns.

I wrote "I had excellent teachers", not that they were super good at math.