# Got Any Mathematical Mnemonics?

August 28, 2008 12:12 PM Subscribe

Do you have some mnemonic you came up with to help memorize formulas or concepts in mathematics (particularly for university math)? Or was there a different way of looking at a concept that really helped it sink in?

I'm looking for things that you might have come up with yourself that aren't so common or widely used. For example, when I was trying to memorize the formula for integration by parts, I always thought about the fact that uv was like "uv rays" and vdu sounded like "video." Or, when I learned about span, the professor compared vectors to colours and said that the span of "red" and "yellow" would include all the shades of orange.

I'm looking for things that you might have come up with yourself that aren't so common or widely used. For example, when I was trying to memorize the formula for integration by parts, I always thought about the fact that uv was like "uv rays" and vdu sounded like "video." Or, when I learned about span, the professor compared vectors to colours and said that the span of "red" and "yellow" would include all the shades of orange.

*Or was there a different way of looking at a concept that really helped it sink in?*

I stopped using mnemonics after high school, I found trying to remember an unrelated odd phrase ended up being more distracting than useful after a while.

Here are the methods I used (YMMV):

- A lot of my classes gave out printed formula sheets or allowed students to make their own one page cheat sheets. In those cases I purposely didn't memorize the formulas themselves, just the meaning behind them.

- If I knew that no formula sheet was going to be given/allowed, I would purposely do all of the homework without a formula sheet, to try to force myself to remember them.

- I did practice tests the night before a test. In my experience that's the best way to keep everything "fresh".

- For straight memorization, I would write everything in one big column on a single sheet of paper, and go through the list over and over. To test myself, I would try to remember the items on the list without looking on the sheet.

- In the actual test itself, if I had trouble remembering something that I knew I had written down in my notes or used in studying, I would visualize the page in my mind to try remember it.

- Also in the test, if I couldn't remember something important for a question after trying for a minute of so, I would completely skip the question and go on with the test without thinking about it. Many times, one of the later questions would jog my memory and I could go back and finish the question.

posted by burnmp3s at 12:39 PM on August 28, 2008

*Or was there a different way of looking at a concept that really helped it sink in?*

I noticed that a significant difference between mediocre math students and outstanding math students, was that the mediocre students would try to use mnemonics and other memory aides to learn the formulas, while those who excelled were instead learning how those formulas had been derived in the first place and so how they worked - doing so not only made the formula easier to memorize (because you can see its structure and know why it is like it is, rather than trying to memorize a meaningless random sequence), but even if you did forget the formulas, you could just rapidly re-derive them from whatever point you could remember - or even get them from nothing if you completely blanked.

posted by -harlequin- at 1:41 PM on August 28, 2008

I second the notion of trying to make the math "meaningful." In your example of integration by parts, for example, the formula follows from the product rule for derivatives if you keep in mind the inverse relationship between derivatives and integrals. I would think of the "span" of a set of vectors as being all the vectors you can "build" from that set using linear combinations.

I also second the notion of doing as much practice as you possibly can, especially if you have access to a practice test. I usually try to work my way through the odd problems in a section of the textbook, checking my answers after each problem. If you're short on time, do this with the hardest ones.

If you have access to a sheet of notes on an exam, I'd include the following:

- Notes on definitions and theorems

- Formulas, with variables and parameters explained (for example, the formula for the equation of a parabola should include how to find the parabola's focus and directrix from a, b, and c)

- Example problems from a practice test or your textbook, worked out, with notes added where necessary to explain each step

- Reminders to yourself on the mistakes you might make (for example, ALWAYS ADD A "+ C" ON INDEFINITE INTEGRALS)

I would also consider brushing up on symbolic logic. Testing series for convergence and divergence requires mastery of a lot of rules, and understanding which rules have converses that also hold, which rules are logically equivalent, and which rules are contrapositives of other rules is a lot easier with some background in logic. Predicate logic with quantifiers is also really useful for manipulating mathematical statements (for example, knowing how to negate a statement will help you produce a proof by contradiction or prove its contrapositive).

Also, look for multiple ways of understanding a given concept. A good calculus textbook will include verbal, algebraic, and visual explanations of important topics, and may have several verbal explanations, for example.

And of course, the quotient rule for derivatives:

d/dx f(x)/g(x) = d/dx hi/ho = (ho-d-hi - hi-d-ho)/(ho ho)

posted by alphanerd at 2:06 PM on August 28, 2008

I also second the notion of doing as much practice as you possibly can, especially if you have access to a practice test. I usually try to work my way through the odd problems in a section of the textbook, checking my answers after each problem. If you're short on time, do this with the hardest ones.

If you have access to a sheet of notes on an exam, I'd include the following:

- Notes on definitions and theorems

- Formulas, with variables and parameters explained (for example, the formula for the equation of a parabola should include how to find the parabola's focus and directrix from a, b, and c)

- Example problems from a practice test or your textbook, worked out, with notes added where necessary to explain each step

- Reminders to yourself on the mistakes you might make (for example, ALWAYS ADD A "+ C" ON INDEFINITE INTEGRALS)

I would also consider brushing up on symbolic logic. Testing series for convergence and divergence requires mastery of a lot of rules, and understanding which rules have converses that also hold, which rules are logically equivalent, and which rules are contrapositives of other rules is a lot easier with some background in logic. Predicate logic with quantifiers is also really useful for manipulating mathematical statements (for example, knowing how to negate a statement will help you produce a proof by contradiction or prove its contrapositive).

Also, look for multiple ways of understanding a given concept. A good calculus textbook will include verbal, algebraic, and visual explanations of important topics, and may have several verbal explanations, for example.

And of course, the quotient rule for derivatives:

d/dx f(x)/g(x) = d/dx hi/ho = (ho-d-hi - hi-d-ho)/(ho ho)

posted by alphanerd at 2:06 PM on August 28, 2008

I've always liked the theory that math is a language. So, you can use it by memorizing rules about how to respond, or by learning what it means. If you do the former you have to come up with a lot of memory tricks and mnemonics, and you'll be thrown for a loop whenever some unspoken assumption is violated. WAY better to work on learning "what are these symbols supposed to represent?"

For example, derivatives represent change. dy/dx is just "how much is y changing when x changes?", which is the same as the slope at one infinitesimal point. How easy this sort of thing is to grasp will depend on how your mind works, such as being willing/able to try and wrap your head around things like limits/infinitesimal amounts.

For formulas that are too complex to reach like that, I find it helps to think about them from multiple different angles. Knowing the shape of the rule for integration by parts, for example, or being willing to figure out the quotient rule for derivatives (given above) as just the product rule using f(x) * (1/g(x)). The more ways something is stored in your brain, the harder it is to lose it entirely. This is also why learning derivations can be useful.

Of course, I

posted by Lady Li at 4:08 PM on August 29, 2008

For example, derivatives represent change. dy/dx is just "how much is y changing when x changes?", which is the same as the slope at one infinitesimal point. How easy this sort of thing is to grasp will depend on how your mind works, such as being willing/able to try and wrap your head around things like limits/infinitesimal amounts.

For formulas that are too complex to reach like that, I find it helps to think about them from multiple different angles. Knowing the shape of the rule for integration by parts, for example, or being willing to figure out the quotient rule for derivatives (given above) as just the product rule using f(x) * (1/g(x)). The more ways something is stored in your brain, the harder it is to lose it entirely. This is also why learning derivations can be useful.

Of course, I

*had*to learn to do that because mnemonics per se have never really worked for me. They all end up like "stalagmites MIGHT grow from the ceiling, but they don't!" - I can switch things around and it still makes sense, so I'm not sure whether it's ho-d-hi that comes first or hi-d-ho.posted by Lady Li at 4:08 PM on August 29, 2008

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posted by cmiller at 12:33 PM on August 28, 2008