Ask HN: Best way to relearn basic math? | Hacker News Reader

Ask HN: Best way to relearn basic math?

137 pointsnon-entity29 comments

I'm considering going back to school, but upon further thought, the first major roadblock is that I seem to have forgotten most high school math and probably even some easier stuff! I'm sort of concerned about my ability to relearn 4 years of mathematics in a year or two and was wondering if anyone had and books / tips / etc. to make it easier

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First of all, good on you for going back to school—that’s not always an easy thing to do. What subject are you considering studying?

Khan Academy is definitely the best one-stop-shop for this purpose, as someone mentioned. The key is to consistently do lots of practice problems over time. KA is adaptive+gamified, and helps select problems in your “zone of proximal development” (not too easy, not too hard).

Depending on what you want to get a degree in, you can focus on particular areas and gloss over others. If CS, then discrete mathematics and logic are the most important (plus stats/probability/linear algebra for machine learning and AI). If engineering, then trigonometry/calculus/physics is more important.

Learning also requires motivation, which for me personally requires seeing the big picture of why the math theory matters and how it was developed. Read the NYTimes series by Steven Strogatz. I also like “Mathematics for the Nonmathematician” for an overview of HS-level math that weaves in historical context (e.g. how Renaissance art or agriculture and mathematics are intertwined). Watch some YouTube videos on mathy subjects (like Numberphile or 3Blue1Brown or any of the zillions of channels along these lines). I try to do this as a “learning by osmosis” sort of activity that I fit in to my daily routine, e.g. when folding laundry or commuting.

Learning is also a social activity, so maybe enroll in a community college course or find a local study group. I find it’s especially important to have someone to discuss things with when learning math. I also recommend finding good public spaces to work in—libraries and coffee shops are timeless math spaces.

Along those lines—as you’re learning/reading/practicing new concepts, imagine explaining them to someone else (Feynman method).

Do lots of problems by hand on pen and paper, there is research and eons of practical experience that shows that doing math is a kinesthetic experience (that is, there’s literally “muscle memory” for math). Draw pictures and graphs on paper. Keep all this scratchwork and doodles and stuff in a notebook.

Learn to process the “I have no idea what’s going on” thoughts and feelings you get when you’re faced with something new and challenging, or seem to continually forget things you’ve just (re)learned. That’s par for the course, you just have to “feel the burn” and keep going. Cheesy but extremely true.

Oh yeah, and use tools like WolframAlpha to check your work and explore problems in more depth (like visualizing functions, seeing alternate forms). Get the app and use the solution explainer thing—essential for e.g. solving integrals.

It might be worthwhile to learn to solve and verify problems using some sort of (mathematical) programming language or a CAS. Could be anything, but something like SageMath comes to mind. Honestly, even Excel is pretty good for this. Being able to do simple things plot functions, verify work by plugging in values, simulate random numbers, etc. goes a long way. Developing this skills becomes even more useful (essential) when you’re at the college level and beyond.

For reasons that are mundane, I started college at 24 without any math foundation. I did a full year of remedial mathematics as 0 credit courses, and by the end of that, I was prepared for pre-calculus, the first course with credits for an undergraduate degree.

From there, a bevy of other math classes - Differential and Integral Calculus, Math Proof, Combinatorics and Graph theory, and many half-math, half-something classes like physics, regular expressions, and everything between.

But all of it was based on the foundation of that year of 0 credit courses. At the time it seemed like a risky, almost foolish gamble, but in hindsight it has paid off several times over - by that I mean I have been able to pay off the entire college tuition, several times - there is zero doubt that it was a good move, for me, at that time of my life.

I encourage anyone facing the uphill journey of (re)learning math to take a deep breath, make an itinerary, and sleep on it. And if you decide to start that journey, don't go alone: Resources exist along the way, like Interstate rest stops, to help you recharge and get ahead. Remember, that a tutor is also improving themselves as they teach you, for each time they learn a new way to explain a concept, they too gain from the experience. It's not a charitable action to receive extra help learning math.

As you continue, you will possibly feel inclined to pay back and tutor those following. You too will rediscover the path and improve your math foundation as you are generous with sharing your perspectives and techniques.

Good luck! It seems like it will take forever, but is over before you know it.

Along these lines, if you are in the US I recommend a local community college. You'll get a solid foundation, you be on the correct pace, and you'll be working with people whose primary mission is to teach you. You'll also find that you're not alone.

I don't recommend self-study, simply because if that was for you then you probably wouldn't have asked the question. It may turn out after a couple of classes jump-starting you that you then want to self-study: fine, you can make that decision then (we'll even give you advice lol). But don't start there.

Source: I used to teach in a community college, and saw a lot of students just like you.

suggestion for you. "Maths - A Student Survival Guide" by Jenny Olive: https://www.amazon.com/Maths-Students-Survival-Self-Help-Eng.... She basically starts out with the simplest algebra (fractions) and gradually works up to topics in 1st semester Calculus. And she starts each chapter with a short quiz to test yourself and skip ahead if already know the material. This book is great for what you are describing, if I'm understanding you. I picked it up when I was preparing to return to college after being away for many years. I supplemented it with another book I highly recommend: "Mastering Technical Mathematics" by Stan Gibilisco and Norman Crowhurst: https://www.amazon.com/Mastering-Technical-Mathematics-Third.... I found that Jenny Olive's book was well designed and preferred it's style to any math textbook I have ever used. Even so occasionally I would get bored while working thru it. That is when I flip thru the Stan Gibilisco's book, which was full of interesting looking problems and examples. When I would try to solve one of them, it would become apparent that I still need to work on the fundamental concepts that were prerequisites for solving the problem. Thus I would return to Jenny Olive's book right where I left off, re-energized by the desire to master those fundamentals that she covers so well

An intelligent comment. Good one.

Khan Academy (https://www.khanacademy.org/math) is excellent. I recommend working through the exercises as far as you can and then re-learning concepts grade by grade. I go through this process every few years just to stay fluent.

Before you do anything, you should take Coursera's Learning How to Learn to make sure you have efficient study habits.

Khan Academy is probably your best bet. Paul's Online Math Notes, PatrickJMT, and BetterExplained can be useful supplements.

If you want to "bring a nuke to a knife fight", you could work your way through all of the Art of Problem Solving books [0]. They go much deeper than a standard curriculum so your foundation would be extremely strong (especially if you use Anki to schedule your review of problems/definitions you've understood and solved). Completing it would mean there's unlikely to be any math book that's outside of your reach. This is complete overkill, though.

[0] https://artofproblemsolving.com/store

K.A. Stroud is good, lots of how to do maths, very practical:

https://www.amazon.co.uk/Foundation-Mathematics-K-Stroud/dp/...

https://www.amazon.co.uk/Engineering-Mathematics-K-Stroud/dp...

This is a good site: http://tutorial.math.lamar.edu/

Also, if you have interest in a specific subject, exploit your own interest and start with what you're interested in. Don't force yourself to relearn from the beginning just for the sake of starting from the beginning.

Also, your goal will be one or more of: pass a math class, learn to solve real world problems with math, and/or learn math for fun. All of these are possible even if you have some gaps in your knowledge.

I have read all the comments as of this typing. Some good, others okay. Ish. While there is no doubt that Khan Academy is really good, it depends on good for WHOM?

You don't sound like a typical KA learner, from what I see from your post, so a more accurate answer would be to ask you: how motivated are you to re-learn math?

If you are 110% motivated, I think you will have to first read Morris Kline's "why the professor cannot teach" which is an attack on the way math is taught starting from late 50's, especially in the U.S or the West.

After that, you will know what to do next. Depending on your future choices, you could start with building a strong algebra foundation before going to other branches.

To do that, you could read Polya's "how to solve it" as a general motivator, but as your main book, college algebra by Bittinger (or was it Bettinger) is the best that I know so far.

Make sure it is the pre 90's edition.

Find out the book your school is using for first year, it's likely Stewart's 'Early Transcendentals' Calculus or possibly Gilbert Strangs book (my online school used it for Math 101) https://ocw.mit.edu/resources/res-18-001-calculus-online-tex... with the bonus of Strang's book being Calculus is explained in the first chapter and the entire rest of the book is just exercises and applications.

Go through all the exercises, looking up the things you don't remember in the rest of the resources mentioned here already. Maybe your school offers some kind of advanced placement and by doing this you can also skip a semester of single variable calculus saving yourself tuition costs.

As others have said, Khan Academy is great. I used it to prep for the GRE after being of college for 5 years. I took my answering very seriously, so I would only move forward if I got 5 questions in a row correct. If I continued to answer wrong, review the session.

If you have the time, community colleges are also a worthwhile approach (if you have one). Their entire purpose is to help strengthen the community. Many are taught as distance courses or even night courses because they recognize that the people that need this education probably can't afford to attend during traditional school hours. Finally, it adds a small level of accountability to your education. You paid money for the class, there is a specified time and grading rubric for class which forces you to continue while something like KA or other MOOCs have very high attrition rates.

You don't say what you're going back to school for or what you plan on taking when you get there, so my answer might be a bit different with that information. But even if you didn't pay all that much attention the first time, it won't take you a year or two to brush up on high school math. You should be fine with a month of nights and weekends with a problem book. You'll probably want to get something like a Precalculus Problem Solver or Schaum's Outline and just do all of the exercises, or look for a textbook with full solutions. Give each problem a good try, don't beat yourself up if you don't come up with the answer, but make sure you understand the solution. You could probably ask on the forums on the Art of Problem Solving if you're really stumped. Good luck!

I would recommend Khan academy, but I don't think the resource is that important.

What is important is consistency, if you grow by 1% every day for a year will be 37X where you started(1.01^365 =37). So do a little every single day and you will be fine. I find this to be better than to do something every week. It is easier to get in rhythm if you do it every day, you can find fifteen minutes every day, on your way to and from work, during lunch, heck you can even do it while you're in the bathroom. I always tell myself the Bruce Lee quote “Long-term consistency trumps short-term intensity.” Take your time and be consistent.

I’m going through a similar process. Over the years, I had fits and starts trying to relearn what I had forgotten and clarify what I didn’t intuitively get. With the usual caveats about different people having different learning paths, I didn’t find the knowledge stuck with me if I tried picking a broad math topic (ex: “statistics”, which encompasses a lot) and trying to re-learn everything about it.

Over the last few months, I realized a different approach that had been most effective for me. If I picked a specific thing I found challenging to do or wanted to develop as a skill, I could work backward from there to probe through ever more fundamental concepts where I had a gap in knowledge until I hit something I didn’t need explained to me. Having gone through that backward journey, the path forward now was more clear and obvious. It helped as well that I was motivated to dig deeper because increasing my knowledge within this narrow context had an immediate practical utility for me.

If you would like to take this approach, you need to start somewhere specific. For example, I had a generic goal to learn how I could deploy machine learning in an actually valuable way. After many failed attempts to meet that goal satisfactorily, I decided to pop open MindNode, write TensorFlow 2.0 in the main node, pull up the TF 2.0 Alpha documentation, and start reading from the top. For each thing I didn’t immediately understand (ex: eager execution, input layer shape), I created a node. For each node, I began Googling and reading pretty much everything on the first page of results. If I encountered something within the explanations I didn’t get, that became a new child node. Among the many things I explored, I realized a grain that stuck in my mind was “but why do the rules for matrix multiplication feel so arbitrary”. Exploring that question led me to 3Blue1Brown’s linear algebra playlist on YouTube. I can’t begin to describe how it felt when those videos helped me “get” matrix multiplication. With that fundamental bit of math starting to make sense, the more complex concepts are becoming clearer too.

In a nutshell, the approach I’m taking is to start with something specific that’s personally meaningful, and then dig in to it to find out what exactly isn’t clear to me. I hope it helps to think about this approach as an option.

FWIW, when entering my CPGE [0] in France, our math teacher asked we forget everything about math except: natural numbers (0, 1, etc.), addition and multiplication. Everything needed to be scraped and "taught correctly", and it took only 2 years to get back up to speed (the entire program for the school year is available (in French): https://prepas.org/ups.php?entree=programmes).

Math is incredibly simple to build from scratch (as in: doesn't require a ton of knowledge) [1]. How long it takes for it to "click" though is another matter: I've had a very hard time with calculus and basic logic in first year, and thoroughly failed my second year.

I don't have a book to recommend though (everything was taught in class, no textbook); though I remember vaguely some books that others here do recommend.

[0] https://en.wikipedia.org/wiki/Classe_pr%C3%A9paratoire_aux_g...

[1] https://en.wikipedia.org/wiki/Axiom

I totally get it. My daughter took algebra last year (she is going into high school), and I had to relearn things I haven't had to do in a long time, and more so learn the way in which it was being taught. I use math quite a bit (and some is quite complex), but there are parts of Algebra that you just don't use commonly. Also, at least in the U.S. they teach Algebra (math in general) far different than how I learned it growing up and in college.

So to help her we went took Khan Academy lessons together and that way I could understand what it was they were going for and could help her understand if she had questions. I highly recommend them, really was super helpful and lets you move through stuff you know and slow down where you need to practice some. There are also a lot of places online you can download worksheets to do practice, which in the end is all you probably need to do to see where you are.

Another resource you can try is go to your local community college and see if they will let you audit a couple of the math courses. Or see if you can sit in on some of the adult learning math classes, usually those are geared to working people trying to get their GED or HS diploma but at least where I am they are usually super helpful to people who just need a refresher on math or english etc. I was helping a friends machine shop get more organized and improve their working situation and we sent a lot of the CNC operators to the adult learning classes for a very small fee ($50 or so) to get some help on math etc. So worth it, and what they all appreciated was there was no judgement of young 18-20 year old kids (that can be intimidating for some people) like there could have been in regular remedial math classes in college.

You'll do good, it comes back to you for the most part and you will crank through it faster then you initially think. Good luck!!

I went back to University in my late twenties after six years away. While I worked in the tech industry as a developer, I hadn’t done any actual math since University calculus nearly nine years before.

My first semester back, I had a calculus course. It was an absolute nightmare. I knew that if I had taken the class fresh out of high school, I would have dominated it. But, I didn’t. I fucked up and dropped out. Now, I was completely fucked...

But, I passed that class, went on and finished a degree. I got through that class with a few strategies:

1.) I told my professor the truth. When I went to speak to him, I was prepared for him to tell me that I just wasn’t qualified. To my surprise, when I told him how long I had been away from school, he wasn’t actually surprised. He told me that because of the work I did, I likely had the capacity to pass the class, I’d just need to work a little harder to “re-remember” that I knew it. He suggested two strategies. I had made some friends in the class, so he gave us permission to hand in one copy of the weekly assignments as a group. The second strategy was to go and investigate the tutoring opportunities my University made available to mature students.

2.) My university had several opportunities for mature students. They had regular math labs where people like me could go in, ask the most incredibly stupid questions and get really good answers. One particular graduate student helped me way more than I deserve and even took me to the bookstore to find a good introductory Calculus text.

3.) Working in the group was really amazing. First, I learned to trust myself. A large percentage of the time, I was actually doing things correctly. But, my time away convinced me that I couldn’t trust my instincts. Having two friends to say “you dumb fuck, you’re doing that right” helped so much. Second and most importantly, that first semester back was a huge shock and I wanted to drop out many times. Being part of a group where I had to meet up with two people twice a week kept me in school.

4.) “Re-remember” become a personal mantra. Writing that ‘word’ still hurts, but holy shit, was that ever an important concept for my ego.

Many unis offer a prereqs class just for people like you and for others who might have had inadequate secondary schooling. You might want to check out whether your prospective college/university does this.

For very basic math have a look at past-papers and marking schemes for the uk "Functional skills" maths.

https://qualifications.pearson.com/en/qualifications/edexcel...

I'm in a similar position, so I have blown the dust off my Nintendo DS and bought Brain training for 50p at a local shop. You can get the Math focused version "Math training" too.

The games track your progress well and adapt to your skill level. It is away from your connected devices so reduces your chance of getting distracted. Also it's a beautifully designed game.

Take a look at the math portion of the test prep material for the GMAT or other examinations. These materials include answers and solutions to questions. There are also online communities explaining concepts to each other in more detail. GMAT covers number properties, fractions, decimals and percents, basic geometry, and algebra. It's a very good refresher.

I concur with everyone who says Khan Academy - their UX will let you speed through the stuff you already remember well enough, so you won't waste too much time on the basics, and it will help you focus in on the gaps where you have forgotten the material.

Israel Gelfand, "Algebra", is a really good book on basic math.

I don't get why people mention this title so often, for a beginner it's confusing at best especially the first few chapters.

I highly recommend Khan Academy: https://www.khanacademy.org/

I suggest the following approach;

Start with some school textbooks for grades 8-12 i.e. Secondary Education. This is more for a refresher course in the absolute basics.

The above can be supplemented with the following books to develop intuition;

1) Who is Fourier - https://www.amazon.com/Who-Fourier-Mathematical-Adventure-2n...

2) Functions and Graphs - https://www.amazon.com/Functions-Graphs-Dover-Books-Mathemat...

After this is when you enter undergraduate studies and you have to fight the dragon of "Modern Maths" which is more abstract and conceptual. In addition to standard textbooks; i suggest the following;

1) Concepts of Modern Mathematics - https://www.amazon.com/Concepts-Modern-Mathematics-Dover-Boo...

2) Mathematics: Its Content, Methods and Meaning - https://www.amazon.com/Concepts-Modern-Mathematics-Dover-Boo...

3) Mathematical Techniques (i am linking this so you can see the reviews but get the latest edition) - https://www.amazon.com/Mathematical-Techniques-Dominic-Jorda...

Finally, if you would like to learn about all the new-fangled mathematics your best bets are;

a) The Princeton Companion to Mathematics - https://www.amazon.com/Princeton-Companion-Mathematics-Timot...

b) The Princeton Companion to Applied Mathematics - https://www.amazon.com/Princeton-Companion-Applied-Mathemati...

One important piece of advice that i have is to become comfortable with the Symbols, Notation and Formalism used in Mathematics. Most students are intimidated by the Formalism (which is nothing more than a precise form of shorthand to express abstract concepts) and give up on studying Mathematics altogether. This is a shame since it is merely the Form and not the Function of Mathematics.

A Mathematicians Delight by WW Sawyer

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