It would probably be helpful to start at a level above where you left off and look up specific gaps in your knowledge as you find them. It’s always good to come to a topic with the understanding of every previous topic, but often those deeper understandings come from moving into more advanced subjects anyway. It might be more slow and frustrating at times, but it will give you a better understanding of where your specific gaps in knowledge actually lie and you’ll be able to address them as you go.

I guess first consider that you might have math trauma -- read about it, own it (if it applies, of course), and recognize what it feels like when it rears its head. Lots of people -- mainly in the US, but it's a thing in other countries too -- were taught math in a pretty shitty way, often by people who don't want to be teaching it in the first place because they themselves have math trauma.

Then build up fluency in the basic manipulations: do lots and lots and lots of exercises until those manipulations become second nature. You might need to start with fractions[2], and that's fine. One of the nice things about math at the elementary level is that you nearly always are guaranteed to get better with practice. This absolutely isn't the case for proof-based math, where you really need to be intentional about truly digesting the material and thinking careful about the ideas. But if you're shaky on absolute fundamentals, you can get incredibly far with grinding.

At some point you'll need to engage with the ideas, but I think that's easier once you've built up some pattern recognition. But others will (surely) disagree

[1] https://www.edweek.org/teaching-learning/opinion-how-to-help...

[2] https://www.edweek.org/teaching-learning/fraction-phobia-the...

The book that made things click for me was Maths: A Student's Survival Guide by Jenny Olive. Give it a shot, use a calculator for everything. IMO in the age of computers let computers do the calculations and just learn to have intuitions.

I'd have to be convinced that you can develop those intuitions without basic arithmetic. Mind you, I'm not saying spend a lot of time mastering long division or whatever. (I'd probably have to look up the technique myself.) But I think some level of numeric fluency is probably necessary.

“I'd probably have to look up the technique myself.”

No offense but I am pretty sure that was the point. If it’s math you forget, it’s not math worth learning. Learn what you want or need to know. The rest is dross.

I have the same problem and have still not found a workable solution.

Only one there seems to be is to basically rerun 12 years of math, which is really unpleasant because I know 80% of the work making it unrewarding and slow. I don’t know what I don’t know that is super demotivating, indeed.

Same for me, let me know if you find the solution! I think it's extremely hard to get into this kind of stuff when you basically have to start from 0 meanwhile everyone else has a better foundation than you

Too many false starts too lead to doing the same fundamentals all over again, making one feels like they are not progressing at all.

I remember reading somewhere, do the implementation of it along side fundamentals. The reason you are studying fundamentals to progress ahead, do that course along side too. This helps one grasp the fundamentals quickly and more importantly to know which fundamentals you really need than to try to do everything and forget aspects of it later.

That being said, I am yet to implement that concept and get over false starts

Maybe 5 years ago, I was in a similar place. I had a particularly embarrassing moment at work when it clicked that I just... didn't know the basics. I was, to use an overused term, "mathematically immature".

So I made a commitment: I decided I would work through Khan Academy math for 1-hour a day for 1 year. I started with pre-K [1] (specifically counting) and watched every video and did every single exercise in order. I focused on mastery. I didn't rush myself, and I did not continue until I felt completely confident in the material. I just did this for a year. I think I go through roughly algebra 2. In my mind, it is critical to combine explicit knowledge (watch videos) with tactic knowledge [2] (do exercises). For example, you need to understand what a logarithm is conceptually but you also just need to do problems to get a feel for it. So this is fundamentally different than learning-by-grazing or just reading a book.

I could go on and on, but let me just say that it changed my relationship to math in a deep way.

[1] https://www.khanacademy.org/math/k-8-grades [2] https://commoncog.com/tacit-knowledge-is-a-real-thing/

if you would have to do it again, would you change something on your methodology? I assume there were some unnecessary time sinks here & there

I don't think so. My problem was that I had a weak grasp of many basics concepts, and more critically I did not know in which areas I was weak. So while it's easy ex post to say "I could skip such and such section", it would have been impossible to make this judgment ex ante.

And in fact, I think a failure mode many people make is trying to predict which things they already know and then skipping those. This allows for blind spots to persist.

I suppose the one way to skip things correctly would be to have a coach. But that comes at a new cost ($), but maybe that works for some people.

I have been seeing more and more usage of 'ex post' and 'ex ante' lately. What do they convey that isn't conveyed by 'after' and 'before'?

it conveys that you know what ex post and ex ante mean

Or, quite often, conveys that you don’t know what they mean.

"Before" and "after" are generic terms. A car might stop before the crosswalk (space). You might eat dinner after work (time). But "ex ante" and "ex post" specify a relationship to an (random) event or to specific information. For example, a data scientist might compute a quantity "ex ante". This means that the quantity was estimated using only forecast data. No historical data was used. It would not make sense, however, to say that a car stops ex ante the crosswalk.

I could have easily said "afterwards" and "beforehand", but I like "ex post" and "ex ante" when referring to before/after having access to specific information.

Got it. I will try to leverage the synergies between Latin and English ex post.

If you, or someone else is seriously considering learning math from the basic at a high level, I’d recommend picking up “art of problem solving, pre-algebra” book, and walking up from there.

These sets of books are universally considered to be among the best math education resources by mathematicians and others, and they start from the very basic (such as the number line and basic operations), but without the need of practicing elementary school material like counting.

I think the TAOPS curriculum would be incredibly challenging for somebody who has actively incorrect intuitions about math.

I think if you've literally never seen the material before, you might be right.

But for anyone who's graduated high school, a lot of it would be at least a second encounter so you'll be reviving forgotten knowledge which is much easier while also diving deeper in your second pass.

What did you do after you completed the K-8?

I worked from the top of this page [1] downwards and ended after algebra 2.

[1] https://www.khanacademy.org/math

This is really inspiring, thanks for sharing. I should do likewise. I wound up dropping out of school (had kids too early, don't ask), which was... not a good decision. I do work as a developer but I need more domain knowledge in mathematics. Never too late to start, I guess.

This is inline with a Zen practice called "beginner's mind", and is so useful in many areas, thank you for this inspiration.

I loved reading your comment. Your story is very similar to my own! A little over 8 years ago, I also started at the beginning of Khan Academy. Due to reasons related to my childhood, I had essentially no education growing up. When I was in my early twenties, I had only an elementary education. The highest level of math I knew was basic fraction arithmetic. I had never written an essay and I did not have a scientific understanding.

Having no education, I only did menial work for money. Yet in my early twenties, I was contemplating my lack of scholarship and realized I wanted to fill the holes in my education. I went to Khan Academy and, as you did, started with pre-K and worked my way linearly through up to pre-college math. Thankfully, I was soon laid off from my job, which was an opportunity to start attending community college.

I then transferred to a state school and double majored in applied math and computer science. Now I’m doing theoretical research as a PhD student in computer science.

The 8-year path from pre-K math to graduate-level math classes and now being published has been a journey. And I’m deeply grateful for resources like Khan Academy.

Deciding to commit to a daily study of math transformed my life.

Have you seen "An Infinitely Large Napkin"? I saw it posted here a couple years ago. It is Evan Chen's project "aimed at making higher math accessible to high school students."

https://web.evanchen.cc/napkin.html

For what it's worth, if you want to go bottom-up and give yourself a super-solid foundation, go through the videos I recorded several years ago specifically for this purpose. They are for children as well as adults: https://www.youtube.com/@thinkingmathematically

Please let me know if they were helpful!

Paradoxically, self-teaching is most difficult at the "101" and "pre-101" level. Once you have the basics, self-teaching becomes much easier.

Teaching yourself Calculus I and Calculus II after obtaining High School mathematical literacy is not too bad. Teaching yourself Differential Equations or even Functional Analysis after obtaining the pre-reqs is actually quite easy. But teaching yourself high school or especially middle school mathematics requires a TON of dedication.

It's not impossible, of course, but "The Basics" are where you really benefit from the help of a professional educator.

For this reason, I recommend eschewing self-directed resources and enrolling in an "Applied Math" or "College Algebra" course at your local community college. These courses are basically "high school mathematics for people who never learned or forgot high school mathematics". Depending on where you live, the "College Algebra" course at your local Community College is probably very cheap or free and available as an evening and/or online course. You usually do not need to enroll in a degree program to take the course.

Once you make it through "College Algebra" you can return to self-directed learning.

Community Colleges are a vastly under-utilized resource, particularly for these "very fundamental knowledge gaps" where self-directed learning is much more difficult.

It might not be the same for you. But this is what I enjoy when I study anything. Filling up the gaps in my knowledge and learning about new ways to think or understand about a concept. Maybe you shouldn't dive so deep into the concepts. Just write all the topics/concepts you didn't understand while learning something and watch one video about each of them.

Why not start from the beginning, ie pre-K on up, as the other commenter mentioned? I've done something similar for teaching myself web development (since it's not taught often in CS degree programs), starting from bare HTML, adding CSS, adding Javascript, adding TypeScript, adding React, and so on. Now I have a good foundation from which to build anything I want. So too with mathematics or really any subject.

My 13 year old is a very good student, currently taking high school math classes while in middle school... one of the things that the public school system hasn't figured out is how to make any of it relevant. So sure, she's brilliant and flies through the material, but she'd be the first to tell you it's just puzzles, it has no actual meaning to her. I was the opposite - because it had no meaning to me, I couldn't understand it and was a terrible math student. But, once I could view it through the lens of computer science, suddenly it had meaning and I did fine.

My oldest daughter was a terrible student. But she would say to me "If they taught history class the way you explain historical events around the dinner table, I would have been a lot more interested in studying."

So point being - it's probably not you, a lot of it really is the way we approach k-12 education. In hindsight, I'm not sure college is any better and may be worse what with the approach of hazing and weeding out.

The Khanmigo chatbot that KA is rolling out supposedly helps with this problem of relevance.

Weirdly enough, mathematics is a very emotional subject for me. I never understood how it was taught in school, and had serious difficulty understanding it. As a result I just ignored it, believing that I didn't have the right brain for it.

Even though I taught myself how to code, I never went back and learned math, so my level of knowledge is at a basic high school level. It can be embarrassing at times, and every now and then I think about trying to learn it.

I honestly think this is one of the bigger problems with people who "can't do math", probably most people can learn quite a bit of (at least basic) mathematics. But oftentimes when you did not understand the subject well previously, like in high school because of different interests at the time or a bad teacher or something. And then because of this whenever you see an equation or hear the word 'mathematics' you just shut off your brain. I know some people who can understand me when I explain some physics or mathematics concept without explicit mathematics, but they would be scared off at the first equation.

> Whenever I try to catch up with maths (in Khan Academy and elsewhere), I always end up in an awkward state where I keep recursively researching less and less advanced subjects because of gaps of various sizes in my fundamental knowledge. It's incredibly demotivating.

Try this course: https://www.edx.org/course/college-algebra-and-problem-solvi...

It uses the ALEKS system which identifies your weak points and brings you up to speed. Take notes during the process so you have something to reference in the future and won’t forget what you learned.

After you knock out the algebra course you’ll be ready for the precalculus course: https://www.edx.org/course/precalculus

The ALEKS system in the Precalculus course will also remediate anything you forgot from the Algebra course.

Hopefully this will help give you the confidence to go after more advanced maths once you finish both courses. Be kind to yourself, math, like anything, is a skill, it takes time and practice.

At Math Academy, we've created a sequence of math courses called Mathematical Foundations I, II, & III that cover everything from 5th Grade Math up through Calculus I & II and will allow anyone to get up to speed on the skills required for university-level mathematics in the most efficient way possible. Our adaptive diagnostic exams will create a custom fit course no matter where you are mathematically and our algorithms will continually adapt to your individual strengths, weaknesses and learning curve.

https://mathacademy.com/courses/mathematical-foundations-i https://mathacademy.com/courses/mathematical-foundations-ii https://mathacademy.com/courses/mathematical-foundations-iii

In addition, we have courses on Linear Algebra and Mathematics for Machine Learning, with more coming soon:

https://mathacademy.com/courses/linear-algebra https://mathacademy.com/courses/mathematics-for-machine-lear...

The system is mastery based, lightly gamified, and completely automated. Our algorithms intelligently apply spaced-repetition to a hierarchical knowledge graph of over 3,000 mathematical concepts to make it substantially more efficient than a traditional course (typically on the order of 4X or more).

I'm a founder and would be happy to answer any questions.

Another resource I wanted to mention here is OpenStax. Specifically, their Prealgebra book: https://openstax.org/details/books/prealgebra-2e it's a fully online-native book format with exercises, etc.

That happens to me all the time the solution is I need a push or some deadline. For high school it was a strict (we thought) teacher who brought me from an average of 70% to 95%. For computer programming it was going back to school as an adult and timed in class tests. For music is was paying for lessons at a certain time.

If I don't get motivated from some external push I never get past a certain level. Every single thing I do in my life is like that even laundry. I must be chronic or maybe permanent procrastinator.