Teachers Say (and do) the Darndest Things

We all have catch phrases. Things we say, purposefully or accidentally, enough times that the kids have taken note. You know, these are things kids probably mimic when doing impressions of us. Which I know they do. I mean, don’t they?

I bet someone doing an impression of me teaching any of my classes would say “ooooh, CRUST!” as an expletive a lot. That’s my curseword in class, whenever I lose track of time or make a mistake. I also often deny mistakes, jokingly. A brave student will note “you forgot the negative sign there.” I’ll carefully add it in and say “Um, hell-O, no, I DIDN’T. I have no idea what you’re talking about.”

We also have catch phrases related to math. In calculus, you all know my motto, which gets said at least once a week if not two or three or four times a week:

turn what you don’t know into what you do know

And in Algebra 2, I have one class rule for safety. Which I pull out of my pocket a lot:

don’t divide by zero! if you do, the world BURSTS INTO FLAMES!

[And then I take the red smartboard marker and draw flames around the thing that would have a zero in the denominator.]

Kate Nowak’s recent post talked about the changing of traditional teaching phrases in her class[1]:

Today in Algebra 2 we reviewed negative exponents and the children acted like they had never seen it before. I told them about the phrase “move it, lose it” for dealing with a negative exponent, as in, move the term to the other side of the fraction, and lose the negative sign. A student who moved here from another state (where, you know, they get to spend enough time on things to actually learn them) told us about the phrase she learned “cross the line, change the sign.” Which the kids liked better. “You know, because it actually rhymes, Miss Nowak. Unlike yours.” Um, last I checked “it” rhymes with “it.” I’m not an English teacher! You can tell because I’m not wearing cool shoes and I don’t give hugs.

Okay, a big giant *grin* for the best two lines I’ve ever read on a blog (the last tines, obvi). But it got me thinking more about these techniques we use to teach kids to remember things. Yes, I think kids should know the reason why particular algebraic manipulations / formulas work. But once they show me that they “get” it I have no problem with them using phrases and shortcuts to help them remember things.

I mean, how many of you always rederive the quotient rule when taking a derivative of a rational function in calculus? Or do you sing a little sea shanty like:

low de-high less high de-low
and down below
denominator squared goes

Or for the quadratic formula? In your mind you’re definitely saying the formula in a very specific way each time. Think about it.

THINK, I said.

Yup, I thought so. So inspired by Kate, I thought it would be a neat exercise to chronicle three of the ways I get kids to remember things or do hard things.

(1)

The “pop it out” rule for logarithms. When we are learning how to expand \log(x^3) to get 3\log(x), I always say “POP IT OUT!” and do a raise the roof hand gesture. I don’t know why I do that. I don’t know how it started. Maybe the raising of the roof is popping out of the exponent. Then when students are working and get stuck, I sometimes help ’em out by quickly doing a mini raise the roof. Then they always exclaim “Oh! Pop!”

(2)

[Note: I am pretty sure I stole this from someone from a couple of years ago.] When teaching how to visually find the domain and range of a function, I tell students: “Guess what? You don’t know this but besides your calculators you have another a highly sensitive and powerful mathematical instrument. It’s kind of awesome. It’s called a domain meter.” I throw a relation on the board:

I tell them to hold out their index finger way to the left of the graph. Then slowly move it rightward across the graph. As they do it, I do it with them, and as soon as my finger hits x=-2 I start annoyingly sounding “BEEEEEEEEEEEP” while continuing to move my finger. Finally when my finger reaches x=2, I stop beeping and I silently continue to move my finger. I tell them: “my domain meter only beeps when it hits the graph. What’s the domain?” (They get it.)

Then I say “Believe it or not, you have another amazing mathematical instrument. You guessed it, a range meter.” I then hold up my vertical index finger (“domain meter!” I exclaim) and turn it horizontal (“range meter!” I exclaim). Then I take my horizontal finger and start at the bottom of the graph and move it upwards. I only start beeping at y=0 and continue until y=2. They all can state the range at this point.

I’ll end up finally throwing something more complicated on the board:

and they’ll get it, first try.

(3)

In calculus, I want my kids to be able to see derivatives quickly. My first year teaching it, I focused a lot on u-substitution to take the derivative of \cos^4(x). Why? Because my kids just couldn’t get the hang of “seeing” the answer. So I came up with the “box method” of teaching the chain rule, which works great. (And yes, of course, I always teach u-substitution first and we talk about why this “box method” works.)

I have my kids first rewrite the function so that they can see “inner” and “outer” functions. So for example, they have to rewrite \cos^4(x) as (\cos(x))^4. That way they can see the “inner function” easily. Similarly, they need to rewrite \sqrt{\cos(x)} for the same reason. I then ask them to put a box around the inner and outer functions respectively. If there are more than two (functions within functions), they should make all the boxes.

So let me show you with a simple example from class today:

I have students write the functions in terms of “outer,” “inner,” “more inner,” etc. until they get to the gooey center of a composition of functions. Then I tell them to look at the outermost function, ignoring everything from the boxes inside (in our example above, they’d say “sine of blah”). I asked them “what’s the derivative of sine of blah?” and they all say “COSINE!” So I write

\cos(stuff)

and ask “what do I put in here?” They say “don’t touch the innards!” So I fill it in:

\cos(\cos(x^{1/2})).

Then I put a check next to the outermost function and say “we’ve dealt with you, so we’re done with you.” I then go to the middle function and say: “What’s the derivative of cosine of blah?” and they all say “NEGATIVE SINE!” So I go to the board and add

\cos(\cos(x^{1/2}))*-\sin(stuff)

and ask “what do I put in here?” They say “don’t touch the innards!” So I fill it in:

\cos(\cos(x^{1/2}))*-\sin(x^{1/2})

Then I put a check next to the cosine function and say “we’ve dealt with you, so we’re done with you.” I then go to the middle function and say: “What’s the derivative of x^{1/2}?” and they all say “\frac{1}{2}x^{-1/2}. So I stick that on at the end.

\cos(\cos(x^{1/2}))*-\sin(x^{1/2})*\frac{1}{2}x^{-1/2}.

And fin, we’re done. It goes pretty fast once they get the hang of it. And they actually secretly love having equations that are scrawled across a whole page.

[1] Kate, forgive me for cribbing so much wholesale. But I needed to have the last sentence in there!

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26 comments

  1. When I write an incorrect sign (or some other minor error) on the board, and am corrected, I make the correction and slowly say, “Yeah, I was just… uh… making… sure you were… paying… attention…”

    I’ll be coming up on the chain rule in January. When we reviewed functions in September, we did function composition as well as function de-composition. They can look at a Russian-doll of a function and identify its “layers” pretty well. I’m hoping that will aid in their eventual chain rule understanding. I will definitely keep your boxes in my back pocket…

  2. Re: Chain rule method

    I call it the “Shrek Method.” I begin by asking my students, “Why are ogres like onions?” They all know it’s “layers!” Then after some laughing and reminiscing and someone maybe even spouting some lines about parfaits, I remind them that ogres are also like onions because they’re smelly when you cut into them (but peeling the layers is not quite so bad). So, we do your box method, but I call them layers–and you don’t want to touch the innards of an ogre (or an onion).

    I’m all for the mnemonics and whatnot, but I am starting to see a few students who forget where it comes from and why it works. Some “back to basics” training may be in order for both my alg2 and calc students.

    1. Also re: equations scrawled across the page, I like to stop once in a while after doing a problem like that and say, “Take a step back. Now imagine I’m wearing a white lab coat and asking you to buy the new Super-Deluxe-Megenta Pill! As ‘simple’ as this calculation is to us now and we can make sense of what this is, probably about 95% of the school (students AND teachers) would walk into this room and freak out about the crazy stuff on the board right now. You can each now officially play a genius on TV whether you think you are or not.”

      1. I sometimes do something similar. And in fact, today, I said “okay, you just solved this problem. Trace back to me all that you needed to know how to solve this.”

        I didn’t diagram it, but I should have, because we got back to the formal definition of the derivative, and we ran out of time when we were going even further back to limits. We were beginning to outline the course with the “wow, look what you needed to know to solve this. WOW you know a lot.”

  3. Hee hee, my mistakes get them donut points. When they’ve racked up 30, I bring in the donuts. Calc classes have been known to get ’em more than once in a term.

    I love #2. I am so stealing your range and domain meters! Thank you, Sam!

    I may also steal your universe (uhh, world) bursting into flames.

    My students like my “I think I can…” when we’re factoring quadratics (beginning algebra, ya know). I say it while I’m putting the two sets of parens on the board.

    I also talk a lot about temptation leading us astray when we’re simplifying rational expressions. “So, shall we cancel these x squareds that we see on top and bottom?” I try real hard to trick them. Eventually they refuse to be tricked in class. But of course, some of them still do it on their tests…

  4. Great anecdotes, Sam.

    My students always struggle with domain and range…your meters just might help a bit.

    I did the u-substitution idea for chain rule this year…it went well, but was not as intuitive as I thought it would be. I like the box method (and Dave’s ogre method). It doesn’t all translate well, though. Thinking…”sine of blah”…how do I sign that?…

    I use “impossible” and “against the law” for things like dividing by zero. Example: Algebra 1 learning about slope: calculating slope from 2 points and they end up with something like 7/0. First instinct is to tell me the answer is 7, or 0. I generally make up some funny story about dividing 7 into 0 groups to get a “you’ve got to be kidding me, this woman is nuts” reaction or I pull out the “Impossible!” (One of their favorite signs). *smile*

  5. haha. As you can tell, I’m catching up on all my readings today.

    I love your domain and range meter! I’m gonna do that next time it comes up.

    For the chain rule, I bring in one of those Russian dolls, where there’s a doll inside a doll inside a doll inside a doll. You can work it inside-out or outside-in. Outside-in would be like yours, where you take the derivative of the outer doll, keeping it’s “innards,” then take the derivatve of the next layer, keeping it’s “innards,” etc. It’s my way of visually representing composite functions.

    As for dividing by zero, I showed them the Chuck Norris forward (http://sites.google.com/site/hwangclass/random-1/chucknorris) and tell them “only Chuck Norris can divide by zero!” and for my Calc kids and with limits and such, I tack on, “not even ZERO can divide by ZERO.” I really don’t know why, but they think that’s funny. *shrug*

  6. For dividing by zero, I usually start with something innocent like “your homework will burst into flames. Then the math police will show up and fine you if the math gods don’t get here first and strike us all dead.” Something like that.

    For the quotient rule, I have no memory when the word low got replaced with ho and the derivative order switched around, but now I say “hi-d ho minus ho-d hi over ho-ho.” The kids love it for the “hi-d ho” (like Ned Flanders) and the really deep musical “ho-ho” which around Christmas time becomes more Santa-like. The majority of my students are international so to explain the “hi-d ho” we get to experience Great American Culture Time (as I call it in my classroom) where we watch a few clips of The Simpsons and Ned Flanders.

    I love the domain and range meter as well. I do something similar with a meter stick but no beeps. Let the beeping commence!

  7. You’re killing me with the domain meter. That was always a hangup for my kids when I taught alg 2. Why couldn’t you have posted this about 10 years ago? ;-)

  8. I’m teaching all remedial Algebra 1 this year in an inner-city school (did I mention it’s my first year teaching, too? Wow.). I’ve been desperate for helpful stuff and my kids love it when I use music to help them rememeber things… especially old school stuff that makes me look waaaay older than them (this is a 2nd career for me… I AM way older!).

    Best use so far… When multiple variables are present in equations they seem to really struggle with solving for a specific variable. I started out the “solving for y so we can graph the equation” lesson by telling them I would hum a familiar song and they would have to complete the lyrics. Enter: “dum dum dum dum… dum dum… dum dum… CAN’T TOUCH THIS!” (thank you, M.C. Hammer). They think I’m a huge loser but they loved it and totally remember it every time. When they get stuck on a problem I can just start humming and they’ll jump in “oh! Is this ‘can’t touch this’?” It’s gone really well and it’s kinda fun. :)

  9. Erin, This sounds great, but I don’t know the song, and I don’t get what ‘can’t touch this’ refers to. Could you explain in more detail? This would make a great blog post, btw.

  10. I’m excited to try out your domain/range meter. My Algebra 2 students have lots of trouble with this concept. They have a lot of trouble with “seeing” the graph past the end of where the line was drawn, among other things like confusing the x- and y-axes, but maybe this is a place to start.

    By the way, I’ve been a fan of your blog for a few months now, and I’m currently a first year Algebra 2 teacher in Philadelphia through the Teach for America program. I only recently figured out, however, that you teach at Packer, where I graduated from in 2005! (Those SMARTboards and SFJC stuff did sound awfully familiar.) If you get a chance, do say hi to Ms. Danforth (my 7th grade math teacher!) and some of the administration (Feibelman, Genaro, etc.) who’ve been around for a while for me, and maybe I’ll get to meet you at my 5 year reunion (yikes…) next May.

  11. Thanks for such an awesome post! Reminds me of a recent study covered on the Newsweek NurtureShock blog that I thought you might find interesting (http://tinyurl.com/ya4xzkl)

    Basically, Dr. Susan Wagner Cook divided 3rd and 4th graders into three groups and taught them how to solve equivalence problems. One group was taught a phrase to say out loud, another group was taught a gesture, and the third group was taught a phrase and a gesture.

    4 weeks later, the kids got a pop quiz on equivalence problems. Only 10% of the students who learned the phrase remembered how to do the problems. But 90% of those who learned the gesture remembered how to do the problems– even though it had been a whole month!

    I wonder if that’s part of why “pop it out” is so effective — because of your “raise the roof” gesture.

    I think now that we math educators know how powerful gestures are in helping students remember things, we can come up with all kinds of awesome new ways of doing that!

  12. One more thing, about catch-phrases: I usually tell kids if they divide by zero they’ll be “breaking the laws of the universe.” I think it’s cool that so many of us seem to have independently come to similar conclusions on this one.

    Also, sometimes it helps them to visualize the zero as a big ball and the fraction line as a tabletop. It’s possible to balance a ball on a tabletop, so it’s possible to put zero above the fraction line. But it’s not possible to balance a tabletop on top of a ball, so you can’t put a zero below the fraction line. It’s kind of weird, but it works!

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