A while ago, I posted some of the quirks/concrete things that I’ve developed for my class that seem to work. I think those sorts of things are SO useful. And I love getting them from other teachers. In fact, someone posted about teaching distribution as THE CLAW and I was helping a middle schooler with that today and it was so helpful. So yeah, if you have some concrete things that you use to teach specific topics, blog about ’em or if you don’t have a blog, throw ’em in the comments below.
Currently I’m teaching Riemann Sums in Calculus, and I don’t teach it rigorously. My kids don’t need to use summation notation or anything. I’m focusing on the concept. So I have them do a few problems like this (with the picture) by hand:
Many of them struggle with three things: the left handed vs. right handed thing, finding the endpoints of each rectangle, and being able to calculate the Riemann Sum without drawing a picture of the function.
So I created a way for them to represent the Riemann Sum so that they (a) don’t mix up the Left Handed and the Right Handed sums, and (b) they can still sort of “see” the picture. It also helps them if the interval isn’t totally nice.
Before I show it, I want to say that it isn’t innovative or ground breaking. I almost expect people to say how stupid and obvious it is in the comments — or that everyone does something similar. But heck, I don’t care. It does help my kids who have trouble organize all the information.
Note, when you’re watching the video, the difference in how I set up the left and right handed sums…
Vodpod videos no longer available.So that’s that… If you didn’t catch it, I put little tabbie things on either the left hand side or the right hand side of each rectangle base, to show which one we’re doing. That tabbie thing is there to remind students (a) that’s the side we’re looking at and (b) we’re concerned with the height of the rectangle. That’s also why I write it vertically, instead of horizontally. To show we’re talking height.
I also will probably use this setup when showing them functions that go below the x-axis. (And I will probably write the height of the rectangle under the rectangle base to highlight that the function itself is going below the x-axis.) And use that to parlay into a discussion of “signed areas.”
I can easily see this being extended in a more rigorous course to dividing the interval into 
Continuous Everywhere but Differentiable Nowhere 
Well I think it’s genius, and if anyone ever lets me teach calculus someday, reading your archives will be the first thing on my list.
Have you noticed everyone’s smartboard writing is kind of the same?
I must know about this CLAW trick–could you provide a link (I didn’t see it in the other post’s comments)?
Did you know that every time a student says that 2^-3 = -8, a jet ski hits a dolphin? (Yes, I totally stole that from http://theoatmeal.com/comics/misspelling ). There have also been rumors of attacks by ninjas if you try to take part of a binomial when you’re reducing rational expressions.
When do you use law of cosines? Write the “c” as < and the "o" as a triangle–so you use it when you have two sides & the included angle (the <) or all three sides (um, the triangle). Then I put a smiley face in the S, which half the students also added to their paper when they wrote it down on their quiz.
I always have the hardest time when we reduce the quadratic formula. So, yes, we circle all three parts to see if we can reduce. We also talk about how it's like a seesaw. They want to change (2 + 2i)/2 into i. But then you've gotten your seesaw off balance! Not cool! You've just let your friend hit the ground! But it is ok to write it as 1 + i, you don't have to write the denominator, as long as the entire seesaw is on the ground.
One last one (because these mnemonics are my favorite thing to collect), I use the "woosh" method for converting logs. Start at the base number, then draw an arrow around the equation, back to what you're taking the log of. That's the order you'll write your equation (of course they have to know that it equals an exponent). It also works from exponential form to log form. So when I ask, "how can we get this into a form we can solve," they reply, "woosh!"
I hope these made sense–I wish I had a whiteboard online where I could draw these things! :)
Thanks for such a fun and insightful blog!
ok, one really last one from my friend: when finding slope, think of "right ON" and "NO way" with on/no spelled vertically. O in the numerator, right ON! 0 in the denominator, NO way!
Thanks for all your mnemonics! I put a link to THE CLAW up in the post, so you can see where it came from.
I LOVE the WOOSH method. I didn’t totally understand it until I actually drew it in the air and saw that you’re giving them the ORDER of how to write the equation. I think this is my new favorite thing and I’m going to use it when teaching logs.
Sam
Thanks so much for the link! Do you get to say it like the aliens in Toy Story? “It’s the claaaaawww!”
I’ve seen a lot of integral calculus students at my university use similar pictures, but without the tabs, which are quite a nice addition.
I’m wondering, though, why you would want them to be able to calculate a Riemann sum without drawing a picture (possibly a somewhat inaccurate picture) of the function. I guess I can see it if they get too upset about not being able to draw the picture precisely enough, or if they frequently draw pictures so misleading as to cause them to make errors in setting up their calculations– but I would see these reasons as the problems, and not their need to draw pictures of the functions. I draw at least a generic function every time I do a Riemann sum calculation, and I don’t think it speaks to my lack of understanding of the concepts involved.
Allison, you ask a fantastic question. (“why you would want them to be able to calculate a Riemann sum without drawing a picture (possibly a somewhat inaccurate picture) of the function.”)
One I don’t have a really good answer to.
I don’t have ANY problem with my kids drawing/sketching/graphing-calculatoring the functions. But when it comes to somehow marking up that graph to get the Riemann sum, two years ago I saw some major epic fails. Some students in that class couldn’t/wouldn’t draw the graph in an organized and neat enough way to make it useful. Their graphs were small, wrong, and/or inaccurate. They couldn’t figure out left/right. It They kept messing up the distance between the x-values, and they just estimated the correct height based on their poor graphs. It was their *graph* (and their inability to graph it neatly, accurately, and large enough) that was getting in the way. So when I didn’t do it with thee little __ __ __ __ things, some of my kids just couldn’t get the right answer because it was just too much for them. Not all my kids had this problem, but a good enough number that I wanted to come up with a different approach.
Yeah, I have seen all these mistakes in marking up the graphs, except possibly the mistake of estimating the height of the boxes based on an inaccurate graph. Which is the mistake I’d be happiest to see, so maybe I just don’t recall it. But every time I see a student do a Riemann sum problem with no graph (or worse, no graph and no _ _ _ _) I worry that that student is blindly applying a formula and is not, in fact, aware that the Riemann sum is estimating the area under a curve. And this is the only thing I really want them to know!
My students, at least, tend to be drawing more the more toward the middle of the pack they fall. Some of the ones who are doing really well draw a very neat graph and show lots of work, others write the bare minimum required to get an answer. (I tend to assume a lots of these students, of both types, saw the material in high school.) The ones who don’t seem to understand the concept don’t know what kind of picture to draw. The ones who seem to be learning the concept tend to draw pictures, and make some mistakes in that process, but overall come out better than I think they would without the picture they’ve drawn. I’d hate to have them encouraged to keep from drawing it.
Disclaimer: I’m the TA. I see them once a week, unless they are so motivated as to come to my office hours. I don’t usually have the chance to compare the same student’s work with and without the picture.
I’ve had a really tough time this past unit with getting students to see the difference between right/left and midpoint sums…the idea that you stop at the function when drawing. They can often find the right side, but if the function isn’t increasing, they’ll actually be drawing left hand sums. Thanks for the tips!
Being able to understand a Riemann sum without drawing the picture is actually important, and if you can’t do that, you are indeed missing an important concept. As an analogy, consider a student who could only remember that (a+b)^2 = a^2 + 2ab + b^2 by drawing a square of side length a+b and dissecting it into 2 squares and 2 rectangles, but could not see the relation algebraically.
Understanding the formula for a Riemann sum by itself, without thinking of areas or volumes is important to understand. Think of the many applications of the integral. For example, if you have to consider finding the mass of a 3-dimensional solid, you would break up the solid into many little pieces, then given a continuous density function for the solid, you would pick a point in each piece to evaluate the density function, then multiply the density obtained by the volume of the piece, and then sum this all over the many pieces.
Now, you could imagine in your mind that you are finding the volume beneath the graph of the density function in 4D coordinate space, and sometimes it is useful to do this kind of thing. But you would be missing the essential idea of breaking something up into many pieces and approximating the quantity for each piece by using something of the form function times volume element.
To use my analogy above and relate it to your experience, Allison, I would say that many students have trouble understanding the geometric interpretation of the algebraic relation above. It is easier for them to do the algebra. But that doesn’t mean it’s not important for them to be able to do the algebra! I would say the difficulty your students have is due to their weakness with visualization in general. Yes, it’d be good, even essential, for them to understand the “area under the graph” perspective, but in many situations, that’s NOT the way you should be thinking about the problem.
I’m teaching non-AP calculus for the first time this year, and have just stumbled upon your blog! It’s chock full of awesome ideas! I’m wondering what text you use for your calc class. Do you use Foerester? Do you know of him? What do you think of his stuff?
We just switched texts this year from Anton (NOT my choice – awful for the HS level in my opinion) to Rogawski. I like Rogawski a lot, but I tend to shy away from textbooks and use them as supplements. For no other reason that that’s a consequence of switching texts.
I have a copy of Foerester and I liked it. I really liked the book of activities/tests that came with the sample copy. I flip through it everysooften when I remember I have it, and wonder WHY DON’T I USE THIS MORE? There are some nice things in it.
This is stupid and obvious and everybody does something similar…I think I will from now on, too. Thanks.
Okay, Sam…I wish you would’ve posted this 3 weeks ago. j/k I like it. I will pull it out as my class starts review time as I’m sure any additional strategies will help!
As a student, the thing that bugs me about Riemann sums is how imprecise they are. We’re always given a precise function and then we’re asked to come up with some approximation(s) from it. If we have the function, what is the point? I bet it would make more sense if it were taught alongside function fitting, and using data points as input.
We’ve burned weeks of classtime on Riemann sums (from Calculus 1 through 3) but only in 3 did we even touch on the concept of fitting, and then only to show that it can be done.
I might be partially biased against Riemann sums because I’m also uninterested in drawing precise graphs. I’m in a math class because I’m interested in math. If I were interested in drawing I would take an art class. Ya know? If I were to vote I’d toss out any requirements that students draw graphs (from start to finish, anyway), replacing it with multiple-choice “which graph is the graph of function f” and “fill in the area of the graph that has not been drawn” problems. The latter would be set up with a coordinate system and the scale already in place, just requiring the student to calculate some interesting points in the middle and fill in the blanks.
I’m by no means suggesting dumbing down math curriculum, I’m suggesting refocusing efforts on the math itself, rather than the ancillary art aspects. (While we’re at it, no more writing assignments, too. I’m taking a math class because I’m more interested in math than I am in English or writing. As evidenced by my tangential rant here.)
Hi @dpk, thanks for your comment. If you don’t see the point of Riemann Sums, and don’t quite like the fact that they’re approximate, and the fact that they seem just like precise drawings that take up time… then as a total outside observer who may be completely off the mark, I suspect that the true connection was never made between Riemann Sums and the integral — or at least not well. (Of course, I may well be totally be off.)
However, I agree completely that they are tedious. (Hence the point of this post, and allowing my kids to write a program to calculate ’em.)
But the *concept* of Riemann Sums is so important! I return to it again and again, when we’re calculating arc length or volumes of rotated curves, or in multivariable calculus for any of a million different reasons.
Not knowing about Riemann Sums would be like knowing about the derivative operation, without knowing what it means or where it came from (limit of secant lines). It would be MAGIC, not math.
Sam
But how would you numerically evaluate the integral of a function like (sin x)/ x ? This is just one of many easy-to-state functions that do not admit an antiderivative formula utilizing a finite combination of the elementary functions. In other words, you can only evaluate the definite integral of (sin x)/ x (except for certain exceptional intervals) by using some kind of approximation scheme, and a Riemann sum is the most obvious way to go about it.
I just wanted to thank you for this and all of the random tidbits that I’ve taken from you – I’m a 2nd year American math teacher teaching non-AP Calc and physics at a boarding school in Jordan. the thing that my students struggle most with is NOTATION, and any tricks like this I love. thank you for your filing cabinet too – it’s often the first place I look when trying to get random ideas.
one thing that worked with my students (which is definitely not unique) is using the idea of a pregnant lady for the chain rule – derivative of the mom with the baby still inside then you take the baby out and differentiate it. now whenever chain rule comes up we pretty much always say “derivative of the baby” which has actually been pretty helpful. for a double chain, i made a whale eat the pregnant lady. i had fun drawing that one. i’m not sure if i will continue the metaphor with antiderivatives and do something like stick the baby back in…
haha, stick the baby back in… ewwwwww. (and hilarious.)
you know there’s this comic that i put on my differentiation test that does this baby thing:
http://brownsharpie.courtneygibbons.org/?p=90
Hey, if it works… and they remember it!
Once again, you open your mouth and out tumble great pearls of mathematical wisdom. I’ll be stealing this, fo sure.
And my students all LOVE The Claw too! They turn into a whole roomful of those little green cyclops-beings from Toy Story, all chanting, “The Claw is our master! We worship The Claw! You have been chosen!”
It’s kind of sad, really, when it doesn’t totally crack me up.
– Elizabeth (aka @cheesemonkeysf on Twitter)
I’m late to the party here, but I just wanted to say this is simple and great, and I’ll be doing this next Friday when I introduce Riemann sums in my college calculus courses.
(Minus the smartboard)