How do you remember mathematical rules?

Harry_the_cat

Harry_the_cat

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I just responded to another post with:

Here's the way I remember, and encourage my students to remember, the quotient rule:

If \(\displaystyle y=\frac{u}{v}\) then \(\displaystyle y′=\frac{vu′−uv′}{v^2}\).

Note the "superscripts" on the v's ... they increase as you write the RHS out (no superscript on first v, 1( actually ' ) superscript on second v, 2 superscript on last v). And you've got to remember a minus sign on the top.
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This got me thinking. How do you remember mathematical rules? What little tricks do you use?

SOHCAHTOA is another common example.
 
I wouldn't call them "tricks". What "tricks" do you use to remember your own name?

1. Use them a lot and they become "natural" to you! (That's one reason teachers give a lot of homework!)

2. Know how such rules are derived. The derivative of p(x) is, by definition, the limit of (p(x+h)- p(x))/h as h goes to 0. If p(x)= f(x)/g(x) then p(x+ h)= f(x+ h)/g(x+ h) so p(x+h)- p(x)= f(x+h)/g(x+h)- f(x)/g(x). To subtract fractions get the "common denominator", g(x)g(x+ h): f(x+h)g(x)/g(x+h)g(x)- f(x)g(x+h)/g(x)g(x+h)= [f(x+ h)g(x)- f(x)g(x+h)]/g(x)g(x+h). Divide by h, taking that h into the numerator:
[(f(x+h)/h)g(x+h)- f(x)(g(x+h)/h)]/g(x)g(x+h) and, finally, take the limit as h goes to 0. g(x+h)/h goes to g'(x), f(x+h)/h goes to f'(x), and, in the denominator, g(x)g(x+h) goes to g^2(x). The more you have invested in it, the easier it becomes to remember it. Make the students do the work! That's another reason teachers make students do a lot of homework.
 
Yes Halls, I agree with everything you said. Understanding where the rules come from and repeated use helps you recall them and get better at using them. No doubt about that.

I was thinking more of memory aids for quick recall. Like to name the planets (when Pluto was still considered a planet) in order from the sun, my kids were taught "My Very Easy Memory Jingle Seems Useful Naming Planets" and can rattle off the planets in order in no time flat.
 
This isn't a "rule", but illustrates something interesting about memory.

A high school teacher told us how to remember that [MATH]\sqrt(3) = 1.732[/MATH]: just remember that George Washington was born in 1732.

I didn't know either fact before then; but now that both are linked in my mind, I'll never forget either of them.
 
Acronyms work very well, like SOHCAHTOA. But another good way is the many finger and hand tricks that work by multiples of 90 degrees. Do a youtube search.
 
Dr.Peterson said:
This isn't a "rule" …
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I'm pretty sure the cat was referencing the mnemonic.

The brain is an association machine, so anything we do to increase cross-connections helps with recall (such as involving multiple senses, by linking words, sounds, images).

… just remember that George Washington was born in 1732 …
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That is weird, but it works!

Here's another goofy one. In college, I came up with: The war of 1828! The war of 1828! Because 45º makes 90º, with another 45º, to recall e to 15 decimal places. I didn't need to recall 15 decimal places; I just noticed the pattern:

2.718281828459045

Speaking of weird, I had trouble in middle school remembering that it's spelled 'weird' and not 'wierd'. Fixed that with WE are WEird.


@Harry_the_cat -- Mostly, I remember math facts from repeated exposure (usage). Yet, in some cases, I recall information based on deeper understanding. For example, I remember that -b/(2a) yields the x-coordinate of a parabola's vertex because that's what we get from the Quadratic Formula when the Discriminant is zero.

?
 
How do I remember the reciprocals of sin, cos and tan.
1st I have to remember the other trig functions, namely cot, sec and csc.

It is easy to remember that tan and cot are reciprocals.
Now I notice that sin and cos start with s and c AND that sec and csc also start with s and c.
How wonderful would it be if the two that start with s and the two that start with c were reciprocals of one another, but they are not.

So I remember that sin and csc are reciprocals because they start with different letters,same with cos and sec.
 
I remember the derivative of the trig functions just from using them so often.

Also if I haven't remembered some formula that I use regularly, rather than looking up the formula I derive the formula each time I need it and eventually I get TIRED of doing the proof on a daily basic and just remember the formula! It is like a punishment I impose on myself. It actually works!
 
Otis said:
For example, I remember that -b/(2a) yields the x-coordinate of a parabola's vertex because that's what we get from the Quadratic Formula when the Discriminant is zero.?
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I remember -b/(2a) but I see it as the average of the two roots. But then again, after reading your method I might go with that for now on!
 
Remembering things like sohcahtoa is great as it just helps you remember the definition of some of the trig function. There is nothing to understand in a definition. However there are things that some teachers say to their students to help them remember some important facts/rules/procedures. The problem is that in the end the student remembers the procedure but does not understand why the procedure is correct. For example, a teacher may drill into a students brain (maybe using a method like Dr P's teacher used in getting him to remember sqrt(3)) that to add fractions you need to get a common denominator. The problem is that yes the student will get the correct answer but will not know why this procedure works. I think that a good teacher of fractions will also every so often remind their class why they need common denominators.

Just to ramble on a bit. I think that too many teachers, especially k-12, only care about getting good grades out of students. I think that this is a terrible attitude for a teacher to have. As a math professor I never taught in way to get better grades from my students. NEVER. I always taught in a way that they could understand the material in a logical way. In the end my students grades were above my departments average. Even if it wasn't I would not care. I never seen the stats, but I will bet my last dollar that on average my students did better in their next class because of my teaching methods. Why would anyone want to take the thinking out of mathematics!?
 
Jomo said:
That works for me. Please send me 1 penny the 1st day, 2 pennies the 2nd day,...., 30 pennies on the 30th day. I'll be rich (I think).
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I'd rather have:
1 penny on the first day, 2 pennies on the 2nd day, 4 pennies on the 3rd day, 8 pennies on the 4th day, etc.
I'll be richer!
 
Harry_the_cat said:
I'd rather have:
1 penny on the first day, 2 pennies on the 2nd day, 4 pennies on the 3rd day, 8 pennies on the 4th day, etc.
I'll be richer!
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That's the one I use when my students start talking about bribing me. I tell them to fill a chess board like that. I take payment plans.

-Dan
 
topsquark said:
That's the one I use when my students start talking about bribing me. I tell them to fill a chess board like that. I take payment plans.

-Dan
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Sometimes on the first day of class I write the following on the board.

A= 500
B= 300
C= 150
D = Free

They look puzzled until they see that D is free!
 
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