"How to learn mathematics" (general study method)

What do you think about this study method in general? It’s designed to focus on the information that you don’t know instead of the information that you already know.

  1. Find somewhere quiet. Turn off the radio (and your portable digital music player).
    A library reading room is best if it is a quiet library reading room.
    Preferably have a large table or desk to work on.
  2. Open the book or lecture notes which you wish to study.
  3. Start copying the relevant part of the book or the lecture notes to a (paper) note-pad by hand.
    If you are studying a book, you should be writing a summary or paraphrasing what you are reading.
    If you are studying lecture notes, you should copy everything and add explanations where required.
  4. Whenever you copy something, ask yourself if you really understand it completely.
    In other words, you must understand every word in every sentence.
    As long as you are completely comfortable with what you are copying, keep going.
  5. If you read something which is difficult to understand, stop and think about it until you understand it clearly.
    If a mathematical expression is unclear, determine which set or class each term in the expression belongs to.
    All sub-expressions in a complex expression must belong to some set or class.
    Every operator and function in an expression must act only on variables in the domain of definition of the operator or function.
    Draw diagrams of the relations between concepts.
    Draw diagrams of everything!
  6. If you find something that you really can’t understand after a long time, copy it to your notebook, but put an asterisk in the margin.
    This means that you have copied something that you did not understand.
  7. While you continue copying, keep going back to the lines which are marked with an asterisk to see if you can understand them.
    If you find an explanation later, you can erase the asterisk.
  8. When you have finished copying enough material for one sitting, look over your notes to see if you can understand the lines which still have an asterisk.
    If you have no asterisks, that means that you have understood everything.
    So you can progress to the next section or chapter of the text.
  9. If you still have one or more asterisks left in your notes after a day or more, you should keep trying to understand the lines with the asterisks.
    Whenever you get some spare time and energy, just look at the lines with asterisks on them.
    These are the lines that need your attention most.
  10. If you discuss your work with other people, especially with teachers or tutors, show them your notes and the lines with the asterisks.
    Try to get them to explain these lines to you.
4 Likes

Well, I’ll talk about high school mathematics in Turkey. I think upper advice is not very time efficient. A brief description of the topic would be enough in the first place. Then, the one who wants to study should watch already solved problems videos and if one thinks he is fine to do it by himself, one can try to solve the problem before the video. The key is practicing. Starting from easy is the best. And I should add self-confidence, helping your classmates with their math problems will boost your confidence. Hope It’s helping ^^

2 Likes

Focusing on things that you don’t know is essential to understanding math. In other topics like history and science you can miss a few topics and still have a pretty good idea about whats going on in the class but with math everything builds off of each other so completely that you will struggle to understand the next topic if you didn’t completely understand the preceding ones but if you do understand the preceding topics then the future topics will seem obvious.

Most people struggle with math because school never taught us how to truly understand math it just gave us a list of formulas and told us they worked without ever really explaining why.

This video goes into a bit more depth about this idea.

Now if you are learning math I would heavily suggest that you go back to the beginning and really try to deeply understand why everything works the way it does also if you haven’t read this book yet it’'s extremely useful for building some deep intuition for a lot of different math topics:

3 Likes

As a mathematics graduate, one of the things that did help me a lot at that level was to rewrite all my lecture notes in my own words. This helped me remember the material later (much more effective for this sort of knowledge than any memory technique that we would usually consider on this forum) and showed me the parts I didn’t understand (because I couldn’t rewrite the notes confidently unless I understood them).

My grade jumped up in my 3rd year (compared to my 2nd) when I started doing this and I’d recommend it to others—thanks for sharing!

5 Likes

That’s a pretty bad video actually, let’s stick with his history argument that you also picked up…

It would really help to understand WWII if you know about the reparation payments post WWI ( Article 231 of the Treaty of Versailles). In turn, to understand where those came from, you’d need to understand about the war of 1870/71 and where the German Empire was proclaimed (hint: there was no Eiffel Tower yet). To make sense of that you’d then need to understand the Congress of Vienna after the end of the Napoleonic Wars. Ultimately, the Cold War was Napoleon’s fault. :wink:

Now, do I need high school geometry for differential equations in college? Absolutely not. The same is true for quite a few areas of math and please note that you’re also just talking about algebra in the pdf you’ve lined below and math and algebra is simply not the same thing.

Decent but backwards…

So, get your syllabus at the beginning of the semester and go over the outline of every class. Skim through the appropriate chapters of the book and then do the reading and look at the homework before the actual lecture rather than after. That way you can use the lecture for clarification because you know what you should listen to and/or ask about during the lecture.

If you still need to go see the prof or the TA afterwards if using this approach, then you should maybe consider changing fields.

Of course to understand some history topic you need to understand the information within that topic but even if you completely understood all of the events that happened in world war two that would do you very little good in understanding a completely different historical event while understanding a geometric concept can easily deepen your understanding of math as a whole which can ripple outward to completely different topics.

To put it a different way: If I locked you in a room with nothing but a complex historical event and a couple of different websites that go over that event you could probably understand it in a few months but if I instead gave you a complex math topic and a few different websites that go over that math topic you would be completely unable to understand it even if I gave you centuries because there would be so much background logic that you would need to understand that it would be almost impossible for you to figure it out on your own.

The pdf goes over more then just algebra and of course there are many different math topics.

So, that topic is supposed to be “European History” then? I mean you can take that particular thing back to 800 AD and Charlemagne and the fact that his Empire got split into three after his death when his sons took over. If you get to see math your way (i.e., similar to the guy in the video), then all is “World History.” Napoleon because of the Revolution, Revolution because of money that went to the US (I’ll just call it US)… etc, etc.

Trust me, completely useless as far as diff eq.

Can I ask… are you in high school? What is this “complex math topic” you speak of that “even if I gave you centuries” would be “almost impossible for you to figure”… Fermat’s Last Theorem? :wink: Similarly, I could ask a linguist to reconstruct the Proto-Indo-European language or a biologist to create life… for that matter, doesn’t foreign language (classes) build on top and on top and on top?

I could take some high school kid, show them a QR code and give them enough background in group theory and tell them about Galois, etc. so they’d be able to “write” a QR code by hand by the end of the day. Plus, they’d know better than to fool around with other people’s wives, but that’s a Galois joke for which you’d need to know some history rather than just the particular branch of math that this guy started.

The pdf (just read the TOC and skimmed a few pages) would be considered high school math where I went to school. Looks more like a week long primer for the week before college starts if you’re weak in math. Also, second sentence in the preface:

If you were like me, by the time you got to high school or college, you would have learned a considerable amount of math, mostly algebra.

Plenty of examples of this happening throughout history. There’s a reason why people we view as “extremely intelligent” (euler, pythagoros, euclid) were struggling to figure out problems we view as trivial by today’s standards. I guarantee that if you got dropped into the middle of a high level history class you could still get some idea about whats going on but doing the same with math would be nearly impossible.

Anyway I would ask that you try to find a comparable example were large groups of highly intelligent people spent decades struggling (and continuously failing) to understand some past historical concept because it was “too complex.” If you can do that then I will change my mind.

  1. When did I claim that the pdf was the all inclusive bible of mathematics?
  2. Also yes the pdf was supposed to be a primer. My point this entire time has been that you need to understand math from a fundamental level in order to understand higher level concepts.

I also do something like that and agree that it’s one of the best memory techniques. My brain is a little weird, and if I don’t write things down in my own words I sometimes can’t understand them at all. If I’m stuck on something, I try to explain it to an imaginary student like I’m giving a class or tutoring someone, or I might write a blog post about the topic, which cements the knowledge.

I haven’t studied math, but one thing that has been especially useful to me when studying programming is to encourage people to ask me a lot of questions about programming, even if I don’t know the answers off the top of my head. It results in a similar kind of effect where explaining the answer in my own words forces me to understand it, especially because computers are generally unforgiving. :slight_smile:

Insights are often found by simply describing the problem aloud.

— Duck, Quack Overflow

(Related: Feynman technique, rubber ducking.)

2 Likes

Side note: I removed the link to the PDF in one of the posts above, because filesharing of copyrighted material isn’t allowed in the forum, but if anyone wants to look it up, I think the title was Re-Introduction to Algebra with Intuition: Algebraic Foundation for Calculus, Computer Science, Physics, and Engineering by Ruchira Sasanka.

1 Like

Interesting that you bring this up. I have recently been trying to push this idea further where instead of simply explaining the idea to yourself you imagine the idea as a physical character and then you see yourself become that character and then explain that idea to yourself. I would recommend trying it out because it forces you to see the idea at a much deeper level then simply explaining it to yourself.

I made a post about this a few days ago: New System Idea

2 Likes

Instead of rewriting text I write questions to each paragraph. It prevents me from falling into state of blindly rewriting.
Then I doddle answer on erasable board and check if it mach. If I do not understand I write more questions until I find answer or decide to bring it to someone else.

4 Likes

Math is an abstract science. So you should use the methods that make the best fit for this sort of information.

At first, these are metaphors. If you have trouble with understanding something you should try to find more concrete examples of how it works in real life. Graphics also make a great tool for it. You also need to deeply explore new ideas and understand how this relates to the principles already that you have already learned.

And remember that mathematics is something that you can do. You need to adjust your understanding by solving mathematical problems.