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Let's be clear from the start - I'm NOT a maths educator.

I have a reasonable aptitude for maths: I've passed my high-school level exams in the subject, taken a life science degree and gone on to work in computing. In my experience I can understand moderately complex mathematical techniques (such as calculus, or probability) when they're explained to me by a patient teacher.

However, my ability to recall mathematical concepts once learned is appalling. I can literally spend an hour learning something and have forgotten how to answer questions on the subject within another hour.

I don't have this experience with any other subject. Most things I read about, learn and can recall a decent amount without significant effort.

As I've gotten older my struggle with remembering how to do things in maths has become more and more burdensome: I need it for aspects of my job, and now, as my daughter approaches high-school age, I need it to help her with her learning.

So: I find mathematical concepts interesting, and I must possesses a reasonable level of capability. Is there anything I can do to help my ability to remember and apply mathematical concepts?

EDIT: Examples were asked for.

The worst example was several years ago when I joined a work study group. We were tasked with sitting through a lecture a week of Standford's Machine Learning course (https://www.coursera.org/course/ml). Most of my colleagues were in a similar position to me having come from a numerate but not entirely mathematical academic background and didn't practice maths on a regular basis. Yet they were able to comprehend the course much better than me. Worse, they'd spend the study session explaining to me what I'd missed, and I'd feel like I understood, but then when I watched the lecture again that evening, it was all gibberish again.

A more recent - and less taxing - example was binomial distribution. I wanted to calculated the chance of a particular set of outcomes after some repeated trials so I looked up how to do it, and went through the maths, digging a little deeper into the algebra behind it before performing the calculation. A week later I wanted to do it again, and couldn't remember the first thing about how.

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Will you provide an example of something you've learned, forgotten, and would like to remember? –  Matt F. May 21 at 12:17
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I think somebody told me some techniques... but I don't remember them now. *rimshot* –  David Richerby May 21 at 17:22

3 Answers 3

You may have been able to remember a lot of math for a short time, but you ma have lacked understanding.

When you talk to people who excelled in school mathematics, they might be perplexed by your plight, or they may think you just didn't stick with it long enough. They might mistakenly tell you that you're just not good at math. However, traditional approaches that many of us experienced in school often especially privileged people with good memories or an ability to see patterns quickly, which might leave many of them to wonder why everyone does not come out of math education with the same insights they did.

The phenomenon your talking about is addressed in the Learning Principle of the NCTM Principles and Standards for School Mathematics. That is to say, the phenomenon you are describing is a familiar thing to mathematics education researchers, and is something often seen when an approach to mathematics focuses on rote learning and memorization. Stylianides and Stylianides (2007) wrote:

An important point set forth by the LP [Learning Principle] is that memorization of facts or procedures without understanding often results in fragile learning. This remark corresponds to research which has shown that mastery of facts and rote performance of procedures are not sufficient in thinking mathematically (Schoenfeld, 1988), getting the right answers does not necessarily imply mathematical proficiency (Erlwanger, 1973), and learning computational formulas is a poor substitute for developing understanding of the underlying concepts (Pollatsek et al., 1981).

The learning principle, stated by the NCTM, can be found here. In short:

Research has solidly established the importance of conceptual understanding in becoming proficient in a subject. When students understand mathematics, they are able to use their knowledge flexibly. They combine factual knowledge, procedural facility, and conceptual understanding in powerful ways.

There are a few models for understanding this phenomenon. One is that your view of math becomes more simple over time as you understand concepts that draw different aspects of mathematics together. It's a process of abstraction in which you have to remember fewer things, because abstraction compresses knowledge (that's one way to look at it). Another way to look at it is that greater understanding is not just about knowing more things, but it's about the connections among those things that you know. Without these connections, the things you have learned would remain in isolation, unuseful, neglected, and eventually (possibly) forgotten. A representation of this idea is in diSessa's (1988) Knowledge in Pieces, in which he talks about how people can have intuitive physics knowledge, but not a way of bringing that knowledge together into what we would see as understanding. Imagining these things in a sort of network, expert knowledge might look like a very efficient set of connections among ideas, tailored to the types of situations most useful to someone in a particular field (whether it be math or science).

OK, but how do you become more expert? The Learning Principle is meant to address that by promoting the notion that teaching approaches should be mindful of how student understanding is not a given (or a goal, even) in some approaches to teaching mathematics. It tells us that avoiding the fragility you describe requires a conscious effort to make understanding an educational outcome, not just a hopeful result of the application of memory or a practiced facility with procedures. So, seek to learn from people who take this view and the results will be a less fragile knowledge of mathematics.

One notable example, is Jo Boaler, who just happens to have an online course in which she addresses some of the issue of learning mathematics.

How to Learn Math is her attempt to educate teachers and parents about the ongoing process of learning mathematics. It's a good start if you want to understand what you or your daughter need to do to learn for understanding, and what teachers can and should be doing (based on the last couple of decades of research). You should also get some idea of next steps to address areas of mathematics you're interested in.

Hopefully, this has given you a way of understanding that your experience is not all that unusual, and can be addressed. But how you learn is important in how you will be able to retain and use your knowledge.


Works Cited

diSessa, A. (1988). Knowledge in pieces. In G. Forman & P. B. Purfall (Eds.), Constructivism in the computer age (pp. 49–70). Hillsdale, NJ: Erlbaum.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA.

Stylianides, A. J., & Stylianides, G. J. (2007). Learning Mathematics with Understanding: A Critical Consideration of the Learning Principle in the Principles and Standards for School Mathematics. Montana Mathematics Enthusiast, 4(1).

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While this would be a very good answer to a question which specifically asked about the difference between rote memorization of procedures/formulas versus deep understanding of mathematical concepts, this is a bad answer to the question above, which is specifically about the ability to retain deep understanding of mathematical concepts, rather than the ability to retain procedures and formulas. The OP's problem is, as best I can tell, not a problem in understanding mathematics, but rather is a problem in remembering that understanding. –  DumpsterDoofus May 21 at 16:11
    
I personally have dealt with this sort of problem for years: it's a bizarre conjunction of being able to quickly understand proofs, being able to quickly create proofs of theorems, and yet repeatedly forgetting virtually everything which has been done. –  DumpsterDoofus May 21 at 16:11
    
For example, while working on a mathematics REU in network theory for graph rank-numbers, I discovered and proved the correctness of a closed-form expression for the rank-number of a certain class of loop-like graphs; two weeks later I was presenting some results along with my coworkers, and I had completely forgotten how I had even proved the result, or what the formula was; fortunately my coworkers remembered the formula and proof. –  DumpsterDoofus May 21 at 16:12
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@dumpsterdoofus I appreciate your comments, but they're comprehensive enough to stand as am answer in their own right. I think it would be valuable if you copied to together into a single answer, then deleted them. –  Matt Thrower May 21 at 18:23
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@DumpsterDoofus - I hope you take Matt's advice, repost this as an answer, and remove them as comments here. I disagree fundamentally with the idea that the answer is not applicable to the question. Matt's reference to having things explained patiently, and use of techniques implies to me a focus on procedure rather than concept. And I point out that you associate Matt's problem directly with proof, though he never mentioned proof and there is no similar basis in his question to ground your answer. In any case, I still think it's a worthy contribution, and worth reformulating as an answer. –  JPBurke May 22 at 0:15

Following points have always worked for me:

  • Learn the concepts, not the formulas.

  • Practice. If you really want something to stick in your brain, you have to convince the brain, you need it. The clue is not how many times you repeat something, but how many times you try to recall it.

  • Only do one manageable theoretical step at a time and let yourself time to exercise it.

  • Write down notes in a consistent manner to help your memory and do not hesitate to use the notebook, if your brain doesn't want to remember. The worst paper is still better than the best memory.

  • Do not let non-understood or misunderstood topics aside. Take your time to disassemble them alone, or with help of someone who knows the topic better. Unless you try something really crazy, you can always find someone in the neighborhood, or online to help you further.

  • Read your notes at loud, when you need to reread them. Acoustic feedback will provide additional association layer to the information network in your head.

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The last point came in via a suggested edit of some other user, I approved it as it seemed in the spirit of the post. Of course feel free to revert. –  quid May 22 at 8:15
    
The answer and the edit come both from me. Something went wrong with my login to stack exchange. –  Thinkeye Jun 3 at 12:12
    
Thanks for letting me know. It seems your problem is fixed, but if you need assistance do not hesitate to contact me. –  quid Jun 3 at 12:16

This is an opinion. Based on JPBurke's answer, I'd say that (as with everything memory-related) the problem is one of associative memory (to give one reference that comes up quickly in Google). Their answer suggests understanding as a possible association, but this is a vague and insufficient concept if the comments are anything to go by. My experience is that the relevant quantity is interest, which helps the understanding remain active in my mind.

Interest is not just appeal, but also a form of obsession or fascination. It implies an urge to constantly revisit the subject outside of scheduled study sessions, turn it over and question it, rebuilding its foundations as you forget the received ones. Being interested is more than having an interest, as in the sense of career preparation, and more than just a pleasant reaction to the facts as received. It's also more than understanding in the sense that teachers often test in their students: the ability to reproduce known responses to standard challenges (even conceptual challenges: the interaction "Why is $x = 1$ not a solution of $\sqrt x = x - 2$?", "Because you have to check both solutions after squaring an equation" reflects some competence with algebra, but misses the big picture).

Anything you learn you have to teach yourself. You may have heard it from a book first but that's only factual memorization, and unless you have an eidetic memory those facts won't be accessible after a short while. To teach yourself, you need to think of questions and answer them. I define interest as "having questions"; this facilitates memory because the questions come from a place that existed before the material you're trying to absorb, and the answers connect that material with that place. That's associative memory, and it's how the brain works. It also works through repetition, but as you've noticed, learning based only on repetition is fragile. Instead, I mean the kind of repetition I mentioned above: compulsively revisiting your understanding. Keep it fresh in your mind, remind yourself of what you have already worked out, and the brain will (possibly literally) carve out a niche for it.

Again, this is my opinion, my experience, based on a superficial understanding of brain science that may as well be mysticism. But there are a lot of facts I remember from classes I took more than ten years ago because I was inspired to dig into them, and also several classes that are complete blanks because nothing in them made me care. You should try to learn not because you need to know, but because you want to be the kind of person who knows.

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The description of interest in your second paragraph can be found in all sorts of literature under intrinsic motivation. To this end, I recommend the writing of TM Amabile. (But I also like your mention of problem posing...) –  Benjamin Dickman May 21 at 22:21

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