# A good way to retain mathematical understanding?

What is a good way to remember math concepts/definitions and commit them to long term memory?

Background:

In my current situation, I'm at an undergraduate institution where I have to take a lot of core classes and easy math classes because there are no higher level courses available. Therefore, I have to study math on my own. It worked out fairly well, but I have a big problem when there's a lapse in my studying. For example, in February, I was working on transfer college applications, so I was unable to study math. Right now, I have less than a month left in school, so I need to work on all my term papers. It seems that when I take these extended breaks from studying math, like 2+ weeks, I have trouble going back, and occasionally I have to re learn a lot of the material. Previously, I was able to understand the definitions, theorems, etc and solve some of the exercises. Now, I need to look back on the definitions to understand what I wrote down! It's frustrating that I'm regressing in my mathematical education when I'm not actually studying math.

• Try to solve difficult problems; the harder, the better. You'll be forced to think of theorems and definitions in many different, perhaps new ways. This strengthens understanding. – Shaun Apr 14 '14 at 11:02

Teach.

Seriously it is absolutely the number one way to learn.

• +$\infty$. If your problem is that you don't have anyone to teach, write up the material that you want to teach and put it on a blog. – eykanal Apr 10 '14 at 17:19
• I just babble into my friends ears for a few minutes until they get it. – Cole Johnson Apr 11 '14 at 0:46
• @eykanal or write a textbook on WikiBooks.org or individual lectures on WikiVersity.org, or other well-known platforms for sharing teaching material under a free license. – ignis Apr 11 '14 at 11:44
• Make up homework exercises and their solutions; work other exercises; pretend you're explaining them to a student. For me, reading is not enough; I have to write. – Reb.Cabin Apr 11 '14 at 12:47
• Except when everyone hates your notes and your effort, especially if you write on a topic which is already full of documentation. – Hakim Apr 11 '14 at 14:07

I find that this site is very helpful for retaining understanding. A short visit to this site a couple of times per week is helpful, I think.

Committing things to the long term memory is not easy. There is not generally accepted magic trick that will make you remember something forever. From what I understand the things that we do end up storing in our long term memory are what we have made a routine and have done for a long time. (There are exceptions such as traumatic events that you can't remove from your long term memory)

If you have memories in your long term memory you can try in some way to associate the concept that you memorize to that memory. For example, if you remember the way your childhood room looked like, then you can associate each of the things to parts of a definition. Doing this, will make it a lot easier to remember definitions and is what many people do when they compete in competitions about remembering.

I know that some teachers recommend doing something similar to the above in that they recommend trying to visualize the concept. Can you make up a mental image of the definition? Can you find an image of what a normal subgroup is? (kernel of homomorphism maybe ...)

Many researchers don't actually try to remember the exact definition of something, but they remember the concepts by knowing the deep meaning. That means that you should ask yourself why is the definition as it is? Definitions in mathematics are not arbitrary, they have a motivation. Figuring out the motivation behind a definition can, however, be difficult. And so you might want to ask your professors about some of all of this.

I think, though, that best way is to make it routine. You have to do it a lot. Reading something one time will not put it in the long term memory. I also want to echo what others have already said: Teach. If you can find a way to teach the concepts some someone else, then that will definitely help both with the understanding and the memorization.

I went through something similar and I think I can contribute by sharing my experiences. I began a course in Topology 5 months ago. During the first week, I was really pumped up, ready to embark on a new journey. After a month or so, I stopped attending classes regularly. I kept taking a break of 1-2 weeks, because I had some important work that I had to get sorted out. After 2 months or so,it was hard keeping up with the course. They started doing Algebraic Topology and that just did my head in. I even forgot what an open set was. I had trouble understanding "openness" in different contexts and settings. The course kept moving forward. They were doing simplical complexes which at the time seemed like martian concepts to me. All the motivation I had in me drained out. I stopped studying and failed my exam. It was not that the course was hard, it was just the simple fact that I wasn't regular. What I have realized after this experience, well three things -

(1) Write down notes! Don't just copy everything the teacher writes on the board. Try and understand it first and then write it down in your own words. Counterexamples, different approaches to the same problem or whatever that pops up in your head at that time, note them all down. That way you won't forget how you managed to come to an understanding the first time.

(2) Don't let the work load pile up. Even if you are busy, you can revise what you have learned for about 30 minutes or so everyday. That way, things will be fresh in your memory.

(3) Discuss with your friends or classmates! Challenge each other, work in groups and discuss topics you find difficult. Try to have a good time. It will be like, " Oh I remember how Larry proved that thing about fundamental groups".

Just my two cents, don't know if it'll help you. Oh and one more thing, if you have less than one month left, the best thing you can do is just try and work out old sample papers, questions and homework problems. That is the best strategy, I know of, to pass the exams with a good grade, if that's what you are looking for. Extract out the important stuff, you can learn the details later

I use the Richard Feynman technique and I find it helps me retain information much better and for longer compared to when I just blindly did wrote-memorisation.

The way it works is that if you are trying to understand a particular topic, you take a blank piece of paper and write down what you know about that topic (as if you are trying to explain it to somebody). Where your understanding breaks down or you have forgotten a crucial step, you go back to the books and reinforce your understanding or refresh your memory. Then go back to the piece of paper you were working on and complete the explanation of that topic. I just find that I remember information much better this way as I am forced to understand every bit of it and to connect all the dots.

Hope it helps.

• For a second there, I thought you were going to say the Feynman Problem Solving Algorithm: Step 1: Write down the problem. Step 2: Think really hard. Step 3: Write down the answer. – Joseph DiNatale Apr 24 '14 at 2:06
• This is probably a good system, but what makes this specific to Feynman? I think it could equally well be called the Gell-Mann technique or something, because I don't recall Feynman discussing this technique specifically. And in particular I don't think Feynman used this technique when he was challenging mathematicians to stump him with one of their theorems -- I think he was answering on the fly without writing things down. – littleO Jun 9 '14 at 7:52

Forgetting is natural phenomenon of human brain. If you want to remember something you have to refresh your knowledge from time to time. It is also good to really understand some theorem/definition this way it stays for lot longer than just learning it by heart.

Agree with Jp McCarthy: Teach. I will never forget a Trig identity. Forever.

If that's not an option, you'll have to fall back on the oldest trick in the book: repetition. For abstract algebra, which was an ocean of meticulous definitions during my first year of Grad School, I made index cards with one definition or one theorem/lemma/proposition each and spent 15 minutes each day reciting the top several cards. (There were more cards than could be handled in 15 minutes.) This caused these definitions and results to become ingrained, like paths in a forest, due to repetition. In fact repetition is one of the best ways to acquire information that does not come naturally to you. Repetition. Yup. Repetition.

Figure out as much as possible and memorize as little as possible. When you get to a theorem, instead of memorizing the proof, try to prove it yourself. If you get stuck, go for a walk while thinking about the problem, and/or briefly look over the proof of the theorem just to get clues as to how to solve it yourself. Repeat until you understand the proof.

You are dealing with the forgetting curve of human memory decay.

Two solutions:

1) Progressively spaced repetition. Either using SRS programs or manually. eg Review 30 mins later, 1 day, 5 days, 3 weeks, 6 months.

This keeps moving you back to the left of the curve.

2) Make the initial memory more vivid using mnemonics, emotions or drawings. eg to remember the definition of Hausdorff I drew a picture of two houses around each point to represent the neighborhoods that separated them.

This changes the curve shape so that it decays more slowly. Moving from the red curve to the blue or green curves.

Teaching the material can use both methods because it will up your emotions as you want to do a good job and will force you to repeat the material too.

Adding a layer of understanding helps you shift memory decay curves too because it provides a meta structure in your memory for the definitions, theorems etc. This is similar to advanced mnemonic techniques where you put the items to be remembered in an imaginary walk through your house.

I wrote more on math note taking methods and how they help studying and memory in my answer to that question. Rewriting mindmaps from memory of key concepts, definitions and theorems from memory and comparing to the original helped me see what I recalled well and what to improve plus provided repetition and mnemonic structure of the map.

I learned a lot about memory and speed learning techniques in my last year at High School before I did my math degree from the book "Use your Head" by Tony Buzan. This is an article on advanced memory methods that may also give you ideas. If you are interested in MindMaps then I got some good ideas from Mapping Inner Space by Nancy Margulies , Nusa Maal

Learn concepts and only strive to retain abstract ideas. The details can be figured out when the circumstance calls for it. Learn the general ideas, leave the details for the reference books that you'll keep handy.

One thing I have done with some decent success is using "spaced repetition" flash card software. It works like standard flash card software, except you subjectively rate how hard the flashcard was. Based on that, the software schedules the next review. Basically, it tries to push you just slightly beyond your limit each time. If you managed to get it right, your brain will work a little harder to remember it next time, and the next cycle can get longer again.

It is time consuming at first, but I have about 5000 definitions and theorems, and it takes me less than 5 minutes a day to review the 20 or so that are due on an average day.

Like Jp McCarthy said...I would suggest you to teach. If that's not possible, you can discuss it with your friends. Try to learn it in a bigger group circle.

The more the number of people, the more the doubts. And it doesn't matter how silly the doubt is. Whats important is how you solve it and satisfy the asking person.

According to my experience, I don't think human mind can store all this kind of information for a long time, especially if those are complex or something similar. When I had to study, I used to pretend to be an actor who had a show about math. Audience knew everything about I was saying and I forced myself to be carefull and well prepared before acting. This improved a lot my memory and my focus when studying formulas, theorems and so on.
I would like to suggest you to not remember everything you are going to study. Try to break your problem in simplier parts and try to memorize those. I think it is easier now to remember a more difficult concept because you can derive it from the smaller parts you have stored in your mind.
For instance, suppose you have to study how to evaluate the volume of a sphere but you don't remember the formula. It happens. But if you know the basis of integration and the circumference you can derive it very easily. Thanks to "low-level" concepts ( perimeter + integral ) you found out a more complex formula ( volume ).

• "I used to pretend to be an actor who had a show about math. Audience knew everything about I was saying and I forced myself to be carefull and well prepared before acting. This improved a lot my memory and my focus when studying formulas, theorems and so on."... sounds like teaching!! – JP McCarthy Apr 11 '14 at 8:46
• Yep, I would have voted your suggestion up but I still can't! :( It quite similar, I keep on losing focus if "my class" - I taught math teens in summer and it was frustrating when nobody hears you. I went losing my inspiration. Acting, instead kept me focused and it was effective in my own experience! By the way, you were absolutely on point with teaching! – Emi987 Apr 11 '14 at 9:07

Take a look at http://www.supermemo.com/

This system (and the method it describes) is based on repetition over a changing interval of time - short time intervals first, then each interval longer than the preceding interval. The author is rigorous in his approach to analyzing how memory is reinforced. What is repeated is putting items INTO memory, and and TAKING THEM OUT (recalling them). Both actions are components of memory, that is, being able to "memorize" and being able to "recall" from memory. The continual use of this technique also results in a second-order effect: an improved ability to memorize things in a shorter time, and with fewer repetitions.

DISCLAIMER: I have no relationship to the author of this method, nor do I know the author personally. For me this was just a useful discovery.

There is a difference between remembering and understanding, and remembering is only a step on the way towards understanding. You really understand a subject only when you know it so well that you can reconstruct it yourself after having forgotten most of it. In order to do this, you will have to develop key stepping stones, which will contain the essense of the subject.

Mathematical concepts can be approached from different angles. Try to understand them all! The exponential function can be defined by using rational powers and limits, or as the inverse of the natural logarithm (which is then defined by an integral), or by power series, or by the differential equation $y'=y$, $y(0)=1$. How do you derive the fundamental properties of the exponential function in each case?

Write out complete solutions to all (nontrivial) problems you solve. Make them as neat and clear as possible. Revise them until you are not able to improve upon them.

And draw pictures! Lots of pictures!

I think it was Einstein who said, "You really only understand a concept when you can explain it to your grandmother." If your grandmother is not available for duty, any friend who can think abstractly will do. Try to find a way to explain the mathematical concept in plain English, and in a way so that almost anyone can grasp the essential idea.

The method I use and that has been really helpful but time-consuming is to process the knowledge I want to acquire and then produce beautiful colorful book-like notes in such a way that they are written for my future self, explaining things from the bottom up, and making insights I found important easy to understand.

Afterward, I store them in folders tagged with a theme, such as "real analysis" or "abstract algebra"

Usually, it's a long and iterative process where I sometimes re-write the same page 3 or 4 times until it's aesthetically pleasing, short easy to understand.

Later during my studies when I randomly forget about some detail I know I have studied, I know exactly where to look for and i have notes specially tailored for me from my past self.