Quadratics have become my new nadir, which is cheerier news than it sounds since it means I’ve kicked linear equations into obedient submission. For the first two and a half years of my teaching career, I felt good about quadratics because if nothing else, most kids remembered how to factor, and remembered that factors had something to do with zeros on the graph. Which was a big step up compared to what they retained of linear equations. But then, last year, I cracked linear equations in a big way, which is great except now I just feel bad about quadratics, because as I develop as a teacher I realize the suckers are absurdly complicated and don’t model very easily. The kids learn a lot, but at their level of ability I’d need to do two months to have them internalize quadratics the way most of them internalize linear equations. And I don’t have two months. I just tell myself they still learn a lot. Consequently, I am relieved to see quadratics in the rear view as I move them onto the third of the models that define second year algebra (at least, as I teach it).
Exponential functions are awesome. First, they’re absurdly simple compared to both lines and quadratics. Second, they model actual, honest to god, real life situations. I’m not a big teacher for “Hey, this is something you’ll use again” but automobile depreciation or interest payments are, in fact, something they’ll use again. Third, they provide a memorable and again, useful, reason to review (or learn for the first time) percentage increase and decrease. Finally, they present a situation in which any kid who has even somewhat grasped the course essentials can see hey: Given y, I can’t solve for x. This leads beautifully and meaningfully into logarithms.
So like linear equations, I can kick off the unit with a modeling activity and get the kids moving easily into the math.
I begin with a brief lecture reminding them of the two previous models.
No. Quadratics aren’t repeated multiplication. Exponential functions involve repeated multiplication, as they’ll see in the lesson.
Then I review percentage increase and decrease. I am of two minds about this review. On the plus side, it’s immediately relevant, easy to apply, and gives them a good reason to remember it long term. The downside: the kids never remember what I taught them when they get to the percentage problems. So I explain it up front, knowing that 90% of the kids will forget everything I said just 20 minutes later, when they get to the first percentage exponential increase. 
So I explain it, go round the room asking “So, if I want to increase a number by 8%, what do I multiply it by, Jose?” “1 point…..8?” “Watch that leading zero!” “Oh, 1.08.” “Right.” Do that with five or six times, think everyone gets it, and set them to working on models. This is one side of the worksheet, crunched for space so I could “snip” it.
And sure enough, the kids work through the models, making great progress, and stop cold at the third one.
“I can’t do this. How do you increase by a percentage?”
“Excuse me while I beat myself on the head with this whiteboard.”
“What?”
“Nothing. Do you remember me just talking about percentages?”
“Yeah.”
“Do you see it on the board there? All the stuff about turning it from two steps into one step, and why you need to do that?”
“Yeah.”
“DO YOU SEE ANY POSSIBLE CONNECTION BETWEEN THAT CONVERSATION AND THIS PROBLEM?”
“Man, I don’t see why you’re so mean.”
“Read what it says on the board. Right there. In red.”
“Increase x by a%.”
“Yes. Can you read problem 3 and tell me what you think might possibly qualify as x?”
“The population?”
“Yes. And do you see the value that might possibly qualify as a%?”
“Um.” Long pause as the student stares at the problem, and finds the ONLY OTHER VALUE MENTIONED. “Twenty percent?”
“Indeed.”
“Okay.”
I repeat that four or five times to four or five groups and then, miracle of miracles, find a student with a full table of five values for the population problem. There is a god.
“Great.”
“But I don’t know how to find the equation for this one like I did the first two. This one isn’t repeated multiplication. I had to take 20% of 250 and then add it….why are you hitting yourself on the head?”
“We need a function. We need an operation in which we can plug in x—do you have any thoughts on what x might be?”
“How many months?”
“How is it you know that, you smart child, and yet make me go through this torture? Yes. We need an operation that we can plug in the number of months (x) and get the population (y).”
“Right. But this is like three steps.”
“And we need only one.”
“Right.”
“Wouldn’t it be cool if there were a way to increase a number by a given percentage in just one step?”
“How do you do that?”
“LOOK AT THE BOARD!”
“Oh, is that what you were talking about? I was already doing the worksheet.”
And still, the lesson is largely a success. Kids are absolutely freaked out at the cell growth caused just by doubling and yes, I bring up the million dollar mission example, but at the end of the lesson, not as part of it. Most of the kids correctly graph the models, although a few end up with lines that I correct. The flip side of the handout is a blank graph, which they use to take notes on the basic exponential growth model.
Total Amount = Initial Amount * Ratetime
Initial Amount > 0
Rate > 1
One thing I mull over—the book, and the state test, go through the exponential equation (basically, Initial Amount = 1), along with the transformation model (f(x) = ax-c +- k. I haven’t focused on this in previous classes, because in my experience the kids don’t even get tranformations of lines and quadratics. But I’m going to give it a try on Monday.
Anyway. Day 2 is exponential decay, but I start by going over percentage decrease. I am nothing if not optimistic.
“So if I take away a third of something, how much is left?”
Pause. Pause some more. Pause still more. I grab three whiteboard pens.
“Rhea, decrease these pens by a third.” Rhea obediently takes one pen.
“Class, how much is left after she decreased the pens by 33%, or a third?”
“TWO!!!”
“Two……?” I wait. No. I sigh, and grab three more pens, getting the one back from Rhea as well.
“Paul, take away a third of these six pens.” Paul takes two pens.
“Class, he’s taken away 33% of the pens. How much is left?”
“FOUR!”
“AUUGGGGHHH!”
It all works out. Seriously. By the end of the exercise, most of the class is shouting back the correct answers as I ask “I take away 30%, how much is left?” 35%? 23%?” and the only mistakes they make are place errors—that is, 100-23 does not, in fact, equal 87.
The second day is always better, because it has slowly permeated their skulls that I’m serious about this percentage nonsense, that it has some relationship to the worksheet. So when they ask questions, it’s more of the “could you run this whole percentage decrease by me again? If they take away a third, I have two thirds left? But what’s two thirds as a decimal?” and trust me, this is a big step up for my blood pressure. Well, a step down. And they do the decay modeling and notes with no small degree of interest:
They have the model graph on the back, too, for exponential decay:
Total Amount = Initial Amount * Ratetime
Yes, it’s the same equation, so what’s different?
Initial Amount > 0
0 < Rate < 1
By day’s end, they have registered the import of the realization that Estefania has 95 cents left after ten days, and they’ve figured out that Jose is right, that his car is worth more than Stan’s after five years, which they managed by using an equation they built themselves, by golly, rather than decrease 25,000 by 5% 5 times.
You notice, of course, that I’ve spent most of this post talking about the percentage issue, something the kids learned were first taught back in middle school, than the exponential growth/decay functions, the actual new material. This should not come as a shock to regular readers.
Back in March, there was much fuss about a study revealing that algebra and geometry classes aren’t rigorous enough.
Of course the classes aren’t rigorous enough. They can’t be. I refer you again to the false god of elementary school test scores and the Wise Words of Barbie.
This twitter debate between reformers Mike Petrilli and Rishawn Biddle is typical of reform debates about “rigor”. Petrilli wants end of course exams to stop us teachers from pretending to teach a subject. Biddle wants more of the same, just shout louder and MANDATE instruction, particularly to those disenfranchised black and Hispanic youth who are being let down by lousy teachers with low expectations.
Both of them assume that the problem is ineffective teaching, that all us math teachers could actually teach percentages and fractions to all seventh graders if we were just smarter and better. Or maybe they just think we take the easy way out, that it’d be really really hard to teach the kids properly, and what the hell, we get paid no matter what and behind close doors it’s easier to just go through the motions. Well, sure.
Petrilli’s proposal, end-of-course exams, would trigger a bloodbath. People really don’t seem to understand how I’d be all in favor of that, if the result were a rethinking of expectations. But of course, what would actually happen is that we’d end the end-of-course exams. That’s what always happens whenever a state or district tries to enforce higher standards (cf Oklahoma and now Texas). And of course, that’s what’s going to happen with Common Core standards, assuming that anyone actually takes them seriously after the testing bloodbath this year. But I’d be all for end-of-course testing if reformers would accept responsibility for the 80% decrease in graduation rates among blacks and Hispanics who would never get past algebra I and understand, finally, that they believe in a myth.
But I digress. And I’m still going to like exponential functions, at least until I crack quadratics. Because you know what? The kids do make progress in understanding percentages, and they learn for the first time not only about exponential functions, but about asymptotes, as I explain Zeno’s Paradox. I don’t use Achilles and the tortoise as an example, but instead talk about how I could throw a stapler right at BTS’s head and know that the stapler would never draw blood because it wouldn’t reach his noggin, so I couldn’t get fired. Or that I could walk to the door and never get there. I do get to the door, of course, and alas, the stapler would eventually crack BTS’s skull. But even though we know that this is true, the tools for proving the paradox false, as opposed to demonstrating it, don’t come around until calculus. They get a kick out of that.
If all that’s not fun enough, I see genuine, honest-to-god intellectual curiosity among most students as they realize that they don’t have the tools to isolate x in the equation 8 = 3x. That for all these years they’ve been getting along fine with addition/subtraction, multiplication/division, nth power/nth root, but none of those will work here. Which sets us up beautifully for both logs and a proper discussion of inverses, leading into inverse functions. Yes, their skills are still basic, but I can see the glimmering of understanding of the underlying concepts. If the damn state tests would just ask questions about those underlying concepts instead of demanding underlying concepts and advanced operations, I might even be able to get the kids to show that understanding.
And in writing up this essay, I am struck by the obvious solution to the percentage problem on day one: I need a worksheet. They fill it out, and not until they are done with that do I give them the worksheets on growth and decay. Naturally, this solution is again a lowering of expectations, a realization that a clear explanation on a blackboard that they can refer to isn’t enough, that I need to give fifteen to seventeen year olds an activity so the information will sink in and they use the method right away without asking me to explain it all again group by group. But to hell with expectations. It will be much better for my bloodpressure.
educationrealist 
May 6th, 2013 at 1:47 pm
Here is a joke about Zeno paradox.
Conditions of the paradox were changed. A nice naked lady is placed in one corner of the room. A contestant starts at the opposite corner [i.e. at initial distance L from the other corner] and is allowed to make one approaching move per 5 minutes (this resolving Aristotle’s objections about convergence of the time sequence). Each next step covers only half of the distance remaining to the lady. Here is what happens to a mathematician and to a physicist.
Mathematician has heard about Zeno. He calculates that (total time) = (5 min)*SUM {1}= (5 min)*{N}=approx= (5 min)*log_(base_2) of [L/L_new] to reach the lady completely, i.e. to get L_new= L*2^(-N). Since L_new has zero limit, this sum _diverges_. So he quits.
Physicist realizes that he does not need to reach _zero_ distance to the lady; the distance about 2 inches will be sufficient. So in a time (5 min)* log_(base_2) of [L / 2 inches] = about (5 min)*7 = 35 min he is happy.
My best to Educatinrealist. Your F.r.
May 6th, 2013 at 8:04 pm
Following the adage, “If you want to show off, disagree publicly; if you want to convince someone, disagree privately,” I had hoped to email you. But I can’t figure out how, so this will have to do.
I always look forward to reading you. I wish more people did and took what you say seriously. The five paragraphs beginning, “Back in March” should be carefully pondered by anyone who cares about “education reform.”
Or at least the ideas should. I fear most people will get to the links at the end of the second paragraph, think fleetingly, “I wonder what they say,” and just read on. The next three paragraphs will then be a lot less convincing than if the linked posts had been read. To have more of an impact, I think it would be better to summarize the argument of the two links in a few sentences each.
Otherwise, those three paragraphs come across too much like blogger snarkiness. Insult rather than analysis.
I think the problem is bigger than snarkiness. Someone who hasn’t been reading you carefully over the last year or two won’t know what you mean by “they believe in a myth” or how seriously they should take the expression “testing bloodbath.”
Certainly the linked posts are more detailed–and in many ways better–than a summary. It’s easier to link to them than to summarize and repeat yourself. And no doubt, repeating yourself can be annoying for a writer. But I think it would increase your influence. And you can always include a link for those who want the details.
May 9th, 2013 at 3:56 am
I’m not really clear on what you were saying, unless it’s that I shouldn’t have put the rant in there. But in reading it again, I think I was needlessly harsh to both so I edited that out. I’ll see if I can decipher the rest of the edits, because I think fundamentally you’re right that if I’m going to go off on a rant, I should watch the language.
May 9th, 2013 at 3:24 pm
My problem wasn’t intemperate language. It was that you threw out a number of important ideas, and rather than saying what you meant by them, you required the reader to click on a link to discover what you were saying and why.
So, for example, you underline “the false god of elementary test scores.” A reader who didn’t click through would have no idea what you mean by that. I think I do but that’s because I’ve read that post several times.
Now, maybe this is an inappropriate criticism of a short “Rant” that is not the main point of the post. I was just trying to put myself into the position of an interested reader who has a somewhat open mind but finds you hard to understand because you are so far out of the mainstream. I want it to be as easy as possible for that person to understand. Yeah, easy for me to say; I’m not doing the writing, and like math, writing is hard.
(BTW, I liked the edit. It tightened up the “Rant.”)
May 13th, 2013 at 3:42 am
So, I came upon your blog after a “hmm I wonder what I will find out if I click here” click and was immediately sucked in. Your view points are well different than anything that I have read, pretty much ever in my 15 years of teaching high school math. Arguments aside (I need time to digest and revisit more posts – have only read 5 so far), I really enjoy are your discussions with your classes. You are good at asking questions, very good. I think (opinion based on personal observations) that asking good questions is an important key to being an effective teacher. I am sure there is data from studies somewhere, I might have even read some of them before, that shows a connection. My opinion is, having been through ed school, is that teachers are not taught how to ask good questions. I mean we are told that we have to (Bloom’s, Essential Questions, etc) but I don’t remember practicing in any way except to write up lesson plans with a smattering of questions in them. The great teachers are naturals, good teachers realize it is important and work at it, the rest may do it but are not flexible enough to ask the next perfect question to lead students in the right direction or don’t do it at all.
It’s late and tomorrow’s full of state testing. I will be back to read more. Thanks
May 13th, 2013 at 5:42 am
Hey, thanks for the good word. If you’re interested in my teaching posts, they are mostly under the pedagogy category.
I think ed school only really gets “lecture” (which they say is bad) and “group work” (which they say is good). In my view, few teachers know how to have a class discussion, where the teacher’s doing most of the talking but there’s lots of back and forth. Most teachers say they can’t have class discussions, and I do think that’s something ed schools could teach–except I’m not sure they know how to do it, either.
December 1st, 2013 at 12:00 am
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May 30th, 2016 at 3:03 pm
I wonder if you need to spend so much time “internalizing” at the beginning of introducing a topic. For myself, I find doing manipulation and mindless plug and chug is easier at the beginning, then come back and deepen the intuition.
Maybe a small intro on the rationale for the concept (repeated multiplication) would be enough. It’s just that having to internalize something and have a deep understanding before you start to use it, makes a big hurdle at the beginning. Also, some just familiarity/practice with rote algorithms makes it easier for me to afterwards learn the deeper intuition. I’m not a teacher, so just an idea.
I like the worksheet exercise idea, if the kids are not following the blackboard instructions. May as well break it up into little drills. Like a gymnast coach teaching a complicated skill with some little drills to emphasize different parts of the body position to start.