# How can I retain the mathematics that I've supposedly learnt?

First off sorry if this question isn't suited to matheducators.stackexchange I just thought here would be a good place to ask seeing as you would probably have experience with students. Anyway here is my question and other related information:

So my question simply is "What is the best method to make sure you retain what you have learnt?"

Okay so I've tried learning mathematics up to where I should be at in the past. Every time though I end up stopping for some reason, whether that be health reasons, family reasons etc. sometimes not even for a long time, such as for a couple days/week. Once this happens the information seems to just drop out of my brain no matter and I have to start again.

I am only trying to currently get up to year 12 level for here in South Australia so I will be caught back up and ready to continue my university study and move onto Discrete Mathematics among other things.

Anyway I've tried the practice, practice, practice approach a while back where I spent probably 6+ months like 5 hours a day on Khan Academy.

To add to the problem I am currently an external student so teaching other people for example maybe difficult as I hear teaching is the best way to learn. Finding other people to teach that will listen would be difficult even as an internal student. I probably wouldn't even feel confident enough to teach due to a lack of understanding myself.

Anyway I'm not giving up yet, still looking for a method that might work. Sure you might argue why waste time asking this question, go do problems and practice more but as I said that doesn't seem to work exactly. I don't have this problem with programming for example so I don't know what is wrong. Plus once I'm back studying at university I doubt I will have enough time to dedicate to just doing problems everyday to make sure the information stays fresh in my mind.

So yeah any help is appreciated (hopefully I'm posting in the correct stackexchange site...), also hopefully not too specific to my case or something too, which I just realised might be a problem...

Thanks.

• There is a question that is somewhat related on MathOverflow mathoverflow.net/questions/3951/memorizing-theorems It is likely not really what you are looking for as it is more concerned with more advanced things, but perhaps some pieces of advice are still helpful. – quid Oct 6 '14 at 16:43
• One simple way to try to teach is to answer new low level questions on the math stack exchange. Or, even to add answers to existing questions. That gives you an exercise which is not too far removed from teaching. – James S. Cook Oct 7 '14 at 1:39
• Since your question title explicitly refers to this as the mathematics that you have "supposedly learnt," I think one place to start is to consider how deeply you learned the math in the first place. One common framework for classifying learning is using APOS theory (Piaget-inspired Action-Process-Object-Schema approach to discussing e.g. mathematical learning). One possibility (hard to know given your description) is that your learning is only getting to the A or P stage; I'd think this would make it easier to forget than if you reached O or S pre-hiatus. – Benjamin Dickman Oct 9 '14 at 4:27
• math.stackexchange.com/questions/748197/… – JP McCarthy Oct 9 '14 at 12:14

Keep doing math and thinking about problems, even if you can only devote a little bit of time each day! I'm no expert in neuroscience of the brain, but my understanding is that a lot of learning constitutes forming connections (synapses) between different neurons in our brain; the more we think about and learn about a particular topic, the stronger (more interconnected) the network of related neurons becomes. Conversely, as we stop thinking and learning about a given topic, the more likely it is that the connections between the associated neurons will weaken and our brain may eventually use that neural space for other knowledge.

Even if some of that attempt at neuroscience is incorrect or overly simplified, it is an observable fact that people retain knowledge, especially mathematical knowledge, much more if they are constantly practicing it. So one thing to do is to try to find some time each day, even if it is only a small amount of time, to think about math and try problems. Even if it is only 30 minutes out of your busy schedule, it will make a difference. The other solution, as a complement or replacement for finding available time to work on math at a desk, is to think about math problems in your head when you are walking somewhere, in the shower, or eating a meal. Even if you cannot fully engage your thoughts because of lack of immediate access to paper and pencil, you'll be consolidating what you do know and keep your brain's neural network developing its mathematical connections.

There are many ways to increase the chances that you will retain what you learn. I list here a few that I know:

Review regularly

The more often you review what you learn, the more likely you will retain it even longer. By review, I mean stopping to look back over what you have learned to remind yourself of the major ideas and how they are connected. You may want to skim over what you've done, or think about it, or say it to yourself, or write a short summary, or draw a picture.

People recommended to me when I was a student that you should review what you learned at the end of every study session, at the end of each day, at the end of each week, and at the end of each month.

So, when you have been studying for an hour and you want to stop, before you leave, try to summarise what you learned. At the end of each day, plan a short study session where you look back over the whole day and summarise what you learned. At the end of each week, plan a study session where you look back over the whole week and summarise what you learned. Similarly at the end of each month.

Personally, when I was a student I used to use my daily train journey home to quickly jot down in a diary what I learned today. On the weekend, I would have a session where I looked back through the week's diary and thought about the week as a whole.

In your particular situation this has an advantage that if your study is interrupted at any moment, then you have been reviewing as you go anyway, so you are more likely to retain across a gap in time.

Explicitly make connections

You are more likely to be able to recall things you learned if they are connected to other things in your mind already. Imagine your mind as a map with towns on it an roads between them. If you want to get to town A, the more roads there are between any town and any other town, the more chance you have of hitting the right road to town A eventually. But if town A is only connected to one other town, you have much less chance.

Also, the sensation of understanding happens when new ideas are connected to existing ideas in your mind (literally the brain cells connecting to each other).

So you should explicitly try to make connections. Constantly ask yourself "How is this similar to other things I have learned?", "How is this different to other things that look similar?", "Where does this idea come up?". Another strategy is to draw the connections on paper, in a diagram like a mind-map. Don't just try to connect the ideas within one topic, but try to connect to other topics too. A memorable example of this for myself was when I explicitly made connections between algorithms in polynomial arithmetic to algorithms in whole-number arithmetic.

The above paragraph would seem to suggest that the only connections are to other ideas in maths, but the more connections there are to anything the better chance you will have of understanding and retaining. So ask yourself if the things you are learning remind you of other tasks you know how to perform, or life events you've been part of, or movies you once watched, or people you know, or foods you ate etc. Personally for me the movies I have watched often give me great connections to draw upon.

Use multiple senses

The connections mentioned earlier are not just to ideas, but also to sensory input. Each of your senses is stored in different places in your brain and is associated in your memory with different experiences and ideas. So a way of increasing learning retention is to use more than one sense.

When you are summarising the day, write a list, and also say your list aloud. Construct models out of play-dough or paper. Associate a sound-effect to a particular numerical operation. Write different types of ideas in different colours. Draw pictures with your finger on the back of your hand so you can feel them. When you explain an idea to yourself, use hand-gestures as well as words.

Hand gestures are particularly important -- there is research to suggest that the movement of your body and in particular your hands plays an important role in all learning.

Practice using the ideas and choosing which one to apply

As others already mentioned, using the ideas to do something will help you retain them. So seek out problems to do. The best problems are the ones that involve more than one idea, or involve a bit of playing around to get somewhere. These ones will force you to make more connections and really use what you have learned to do something new. Puzzles are another good thing to do because they often use ideas in unexpected ways. Don't forget to review the things you learned while doing problems at the end of a problem-solving session!

While it is obviously good to practice each separate skill or type of problem separately, it is also important to have practice at choosing what skill to use or what type of problem the problem is. Many excercises are in blocks where all the problems are about the same thing or require the same skill (such as those on Khan academy). You also need to seek out exercises which mix together different types of problems and skills. Choosing one from each block in a textbook is a good start.

• this is a really superb answer! i would just to like to add one more thing to this great list, try to find a buddy doing math can be tough and having someone to bounce ideas off of can really help both people learn :) – celeriko Mar 17 '15 at 2:32

If you try to memorize things, you will lose them. If you try to understand and make connections, you can retain what you learn much longer. Perhaps finding puzzles related to what you're trying to learn. If you give some examples of topics you've learned and lost, I can give examples of different ways to approach those topics.

Here is a somewhat off-the-wall approach (not really, though) that may do well for you.

You can worry about retaining, so that you have instant (or near-instant) recall of what you have learned. You might consider a slightly different goal of recovering what you have learned, so that you can recover what you have learned when you need it. How exactly that goes depends on how you want to use it in the future, but I present a scenario for establishing a system for recovery, and then using that system. Incidentally, this has the benefit of learning by teaching, primarily by teaching yourself.

I will take as an example learning about computing with permutations in group theory. The system I describe looks a lot like annotating an existing text. However, what you actually do is create a couple of documents (on computer, or in a paper journal, or scrawled in beet juice on the wall as you prefer), one of which is your main set of notes to remind you of particulars of the subject you are learning, and the other document is a glossary or list of references. For example, you may want to use a.o.l as an abbreviation for "acting on the left", a notion used when thinking of a permutation as a function on the group of permutations induced by left-multiplication; record the definition in the glossary document, and then use the abbreviation as needed in the main document. Or you may want to note errata in a text that you like, but in the reference list include the edition of the volume, date of printing, and other things that focus on the text you are using.

So suppose you have your documents after a few months of study, and you print them out and put them on the shelf. 14 months later, a project pops up where computing with permutations would be useful. You take your binder off the shelf, look at the document of references and glossary (and other high-level information you record, perhaps an index to the larger document), and locate the information you might need in the document of notes about the subject. Or you find the pointer to the page in the textbook you need, as well as to your page of notes that contains the explanation you needed or insight required for you to understand the page. This way you recover the important information you wish to do the computation. You might even use this to explain to another person how to do the computation.

Of course, some variation on this is what you would do in researching a problem. I recommend two documents to separate the kinds of info to be recorded. If you use a computer format, you can have the machine perform certain jobs such as indexing and cross-referencing, but doing it by hand as you go is not much harder when you have a system in place for recording the information. If you approach your study as preparing for the future, without the burden of testing your knowledge in exam format, you may find that doing recovery (with testing of the recovery system occasionally to make sure it works for you) is as good if not better a goal than recall.

There are already several long answers here, I'll try a shot at a short one; and while your question is not strictly a good fit to this SE site, it is easy to interpret as a good fit: I'll try to answer briefly "what to say to student to help them recall what maths they are supposed to learn?" (Sorry I don't have the time to edit the question in this direction).

The most important thing to learn maths is to try to deepen your understanding. This goes with practicing, but more importantly with practicing in various directions (note that below I am assuming quite a commitment to the task of learning maths):

• get your hands dirty, and do the messy calculation that are needed in the standard exercises of your topic. Then drop your sheet, and the next time you'll have to do them again from the beginning,

• make connections, as was said in another answer: try to gather results by type, by similarity, by differences; connect multiplication to additions, taking powers to multiplications;

• understand why each rule you want to learn must be how it is: to understand a theorem, construct explicit examples, and seek counter-examples that show that the hypotheses are mandatory; try to construct a counter-example of a true statement you want to learn, and see why it has to fail and how the particular example does fail,

• to remember a formula, apply it to silly base cases and look how easy it is (e.g. for $a^{x}a^y=a^{x+y}$, expand $a^3\times a^2$ and $a^5$),

• each time you apply a rule, ask yourself again why is that rule correct, etc.

The bottom line is: don't do the same thing over again, but turn your math upside down until you are familiar with all its aspects. You are a baker, and math is your bread dough. Bread is learned math, it needs kneading.

• @BenjaminDickman: Hum. Thanks for letting me an honorable way out. – Benoît Kloeckner Mar 17 '15 at 12:24