Category Archives: math

Great Moments in Teaching: The Third Dimension (part I)

“How many other dimensions are there?”

“Well, four, according to Einstein, and five according to Madeline L’Engle, if you’ve read A Wrinkle in Time.”

“I have!” Priya’s hand shot up. “It’s a tesseract!”

I was impressed. Not many girls read that classic anymore. “But we’re going to stick to three dimensions.”

“Isn’t real life three dimensions?” asked Tess.

“Yes. But if you think of it, up to now, we’ve only been working in two. We’ve spent a lot of time in the coordinate plane thinking about lines. In two dimensions, a line can be formed by any two points on the coordinate plane. We’ve been working with systems of equations, which you think of as algebraic representations of the intersections of two lines. We can also define distance in the coordinate plane, using the Pythagorean theorem. All in two dimensions.”

Now, no mocking my terrible art skills” and I put up this sketch, the drawing of which occurred to me the night before, and was the impetus for the lesson.

3dclassroom

Everyone gasped, as they had in the previous two classes. My instincts about that clunky little sketch proved out, beautifully. No clue why.

“Holy sh**,” groaned Dwayne, the good ol’ country boy who offered to paint my ancient Honda if I gave him a passing grade. He doesn’t like math. He’s loud and foul and annoying and never shuts up. That last sentence is a pretty good description of me, so I’m very fond of him. “What the hell is that? Get it off the screen, it hurts my eyes.”

“That is awesome,” offered Talika, a senior I had last year for history. “How long did it take you to draw that?”

“What is that white stuff spread everywhere? Did someone get all excited?” asked Dylan, a sophomore whose mother once emailed me about his grade, giving me the pleasure of embarrassing him greatly by describing his behavior.

“Ask your mother,” I replied, to a gratifying “ooooo, BURN!” from the class, who knew very well what had happened.

“How did you draw that?” asked Teddy, curious. “It’s not ordinary graph paper, right?”

“No, it’s isometric paper, which allows you to draw three dimensional images. So…”

“This is really stupid,” said Dwayne. “I’ve taken algebra 2 three times and no one’s ever taught me this.”

“Best I can tell, no one’s ever taught you anything , and not just not in algebra 2,” I replied, earning another “Oooooo” from the class and an appreciative chuckle from Dwayne.

“It’s weird, though, because in two dimensions, you start in the middle,” offered Manual, who was consulting with Prabh, another bright kid who rarely speaks.

“That’s a good point! For example, if we were going to plot seating positions in this room in two dimensions, we’d start with Tanya,” I said, moving to the class center and indicating Tanya, who looked a bit confused. “So Tanya would be the origin, and Wendell would be (1,0), while Dylan would be (1,-1).”

“I’m not negative!” Dylan said instantly, talking over my attempt to continue. “You’re saying I’m negative. You don’t like me.”

“Hard to blame anyone for that,” said Wendell who is considerably more, er, urban than Teddy, with pants down to his knees and a pick that spends some time in his hair. Despite his occasional class naps, he maintains a solid C+, and could effortlessly manage a B if I could just keep him awake. “S’easy, dude. It’s like one of those x y things, like we’re all dots on the graph.”

“You’re one down and one to the right of me,” pointed out Tanya.

Dylan was interested in spite of himself. “So Talika’s, like, (0, 4)?”

“Yes,” several students chorused.

“Then I’m negative 8.” said Dwayne, unhappy with any conversation that doesn’t have him at the center.

“More like….(-2,2), yeah,” says Cal.

Ben speaks up, “But how come Tanya’s at the center for mapping the room’s people, but your sketch is, like, from the left?”

“Or right?” Sophie, from the back.

“Or is it….outside?” asks Manuel.

“Yes, it’s kind of like you’re standing on a desk in Ms. Chan’s room and the walls are transparent,” says Ben, more certainly. Ben is repeating Algebra 2 after having taken it with me last semester. Very bright kid who clowned incessantly, confident in his ability to learn without really trying, only to learn that Algebra 2 was different from other nights, and he wasn’t finding the afikoman. I advised him to repeat. The big sophomore not only agreed, but specifically asked to repeat with me. His attitude and behavior is much improved. I ran into him while walking across the courtyard a few weeks earlier, and he said “I just realized I was Dwayne and Dylan combined last semester, and it’s so embarrassing. I’m really sorry.”

“I’m not enough of an artist to know if I could have drawn this any other way. It just seemed intuitive to me last night, when I came up with the idea.”

“See, I knew it,” trumpeted Dwayne. “You’re making this up!”

“Yeah, I know this isn’t in any algebra book” said Wendy, a sophomore whose excellence in math is often hard to discern beneath her complaints. “This is just some weird thing you’re doing to make us think about math.”

I picked up at random one of the four algebra 2 books sitting on my desk (I’m on the textbook committee) and walked over to Wendy’s desk, opening it to the “Three Dimensional Systems” chapter. She looked, and said “Ok, maybe not.”

“So just as we can plot points in two dimensions, we can plot points in three. Take Aditya here,” my TA, who was watching the circus in amusement. “How could we represent him as a point on my graph?”

Teddy said instantly, “Yeah, I’ve been working that out. I can’t figure out which the new one is, and what do we call it? Where’s x, where’s y?”

Sanjaya said, “I think the part along the ground is x. Like if you go along the bookshelf?”

“Like this?”

3dstartoutline

“Yes,” Sanjaya said, confidently. “That has to be x. So you could count to Aditya, right?”

“Count which way?”

“The bottom!” “The bottom line!” “the bottom..axis, thing. The X!” comes a chorus of voices.

I start counting, and while I do, Sophie objected. “But hang on. I still don’t see what the new thing, direction, is. What’s the third?”

“Up! said Calvin, who rarely participates and often tunes out so far he can’t keep up. But he was watching this with interest. “You know how the class map with Tanya was going north and south and east and west. But it’s all flat, like. This picture has an up.”

“Yeah!” Ben got it. “Cal’s right.”

Dwayne has begun to grasp this. “So you can’t just draw a line? You have to follow along the…things?”

“The axes.” I finish counting along the “bottom” axis and go over to my bookshelf in the furthest corner of my room. “So the sketch starts here…QUIET! One conversation at a time, and I’m the STAR here. The origin starts at this bookshelf. I am walking along the wall, hugging it, on my way to Aditya. Does everyone see how they could track my progress on the axis?”

“YES!” from various points of the room.

“What the hell are you doing?” Dwayne is watching me carefully hug the wall.

“Everyone except Dwayne?”

“YES!” much louder.

I walked along the wall to the table where Aditya sat (fourth along the wall) and stop.

“So now what?”

“You’re there.”

“No, not yet.” countered Nadine. You have to go out….” she waved me towards her. “this way.”

“Yeah, towards Aditya,” this from Talika.

I stepped out 2 steps or so. “That all?”

Josh frowned. “Yeah. You’re there. Except…”

“UP!” Sophie shouted from the back. “That’s the third axis!” General approval reigns loudly, until I wave them all quiet, or try to.

“You go up 4!” Teddy shouted.

“OK. So Aditya is about 40 units out along the wall, 2 units out towards…the door, and 4 units up. Yes?”

“Yes!”

“So let’s draw that.”
3dfinalprism

Of course, while I’m drew this, general mayhem is ongoing with my back turned. I shouted “QUIET! or “Could someone stick a sock in Dwayne?” a few times.

“Wow, so it’s a…cube?”

“A prism, yes. So here’s what we’ve done. We’ve taken the two dimensional x-y coordinate plane and extended it.”

“We extended it up,” from Sophie.

“Yes. And now, instead of a rectangle, we have a three-dimensional rectangular prism. And we can describe things now in three dimensions. But we can do more than that. So let’s step away from my classroom sketch….”

3dcoord

“Whoa. What’s that?” Dylan.

“Man, that’s f***ed up. I just started to get this, and now you’re….” Dwayne, of course.

“No, it’s fine,” Manuel said. “It’s just like the whole thing moved to the center.”

“Oh, I see. It’s like there’s four rooms, all cornered.” Wendell.

“Yes, exactly. Except now, you want to stop thinking about it as a room and think of it as a coordinate plane. As Sophie says, the new plane is the up/down one. So the old x is now here. The old y is now here. The z is the straight up and down one. I think of it as taking the 2 dimensional plane and kind of stepping back and looking down on it.”

“That’s just….”

“DWAYNE BE QUIET. One thing to remember: when you see a 3-dimensional plane, they may be ordered differently. There’s a whole bunch of rules about it that make potentially obscene finger orientations, but I promise I won’t test you on that.”

“So let’s say we’re plotting the point (8,4,5). I’m going to show you how to do it first. Then I’ll go through why. Start by plotting the intercept along each of the planes.”

“Man, does anyone else get this?”

“YES. Shut up, Dylan,” says Natasha.

3dplot845beg

“The trick to remember when you’re graphing in 3-d is to stay parallel to the axis you’re drawing along. So never cross over the lines when plotting points. Now let’s add the yz and xz planes.

“What? This is weird. Why are you drawing so many rectangles?” Patty, frowning.

3dplot845mid

“What you have to visualize is that it’s like we’re drawing sides. So far, I’ve drawn,” I look around and grab three of my small whiteboards, “the bottom and two of the sides. Hold this, Natasha, Talika.” and I build the walls. The kids in the back stand up and look over.

“Oh, I see,” Teddy again. “You’re drawing the prism again.”

“Right. It’s just looking different because the axis is in the center.”

“You do all this just to plot one point?” Sophie, ever the skeptic.

“Yes, but remember this is more just to illustrate, to see how you can extend the dimensions. So after you draw the three sides, joining the intercepts for xy, yz, and xz intersections, you extend those out–again, along the lines.”

3dplot845

“So the point we’re graphing is going to be at the vertex, the intersection of the three planes, the furthest point from the origin–just like in two dimensions, the point is at the intersection of the two lines.”

“That’s really complicated.” Wendy sighed.

“No, it’s not” “Don’t you see the…” Ben, Manuel, Teddy, Wendell and others jump in at the same time, while Dwayne bellowed, Wendy and Tess were asking questions of the room, and, as the writer says, pandemonium ensued. It was a shouting match, yes, but they were shouting about math. The Naysayers, the Doubters, and the Apostles were all marking their territory and this was no genteel, elegant, “turn and discuss this with your partner”, no think-pair-share nonsense. This was a scrum, a brawl, a melee conducted across the room with the volume up at 11—but just like any good fight, there was order beneath the chaos, a give and a take at the group level.

And for you gentle souls wondering about the quiet kids, the introverts, the shy ones who need time to think, they were enthralled, watching the game and making up their mind. It may not look like everyone gets time to talk, but pretty much every time you read me call on a kid, it’s a quiet one. And I shush the room. Then the quiet kid sits there in shock as he or she realizes oh god, I’ve got the mike and I can’t be a spectator anymore.

Anyway, the story goes on with a second great moment, but I’m getting better at chunking and this half had too many details I didn’t want to give up. I’ll stop here for dramatic effect. Because oh, lord, I was high as a kite in this moment, watching the room, realizing I was riding a tremendous wave of energy and excitement. Yeah. ME. On Stage. Making Drama.1

Now I just had to come up with a good ending.

*********************
1I’m not congratulating myself, saying I’m proving kids with the great moment. No, the great moment is mine. I’m standing there going oh, my god, this is a great moment in teaching, in my life. For me! The kids, hey, if they liked it, that’s good.


Making Short Math Tests

A trig student told me he was hanging out with a group of friends, some who’d had me, some who hadn’t. One was bitching about his four page test.

My students snorted. “Ed’s tests are double sided single pages. Once we had a three page test, but only for the space.”

A debate ensued, and those with the widest range of math teacher experience agreed: My tests are shortest, and hardest.

I’m not sure what this means. I don’t try to make my tests difficult. But periodically I’ve perused other teachers’ tests off the copier, and…wow. They are four or five pages. The questions are straightforward. They are typically of what I would call rote difficulty–they could have peeled off a few pages of one of these tests. When math teachers snort about regurgitating algorithms, these are the tests they have in mind.

I used to have more traditional looking tests, but even back then I wasn’t an exact match for typical. Once I started down the multiple answer path, it became even easier to wander miles off the reservation. But without question, multiple answer tests make it easier to assess understanding on multiple topics—thus shorter tests.

This semester, I finally decided to start my class with a functions unit. Regular readers know that I’ve been beefing up my functions curriculum, after initially (as a new teacher) giving it a perfunctory treatment. But I still began the year with linear equations. This last semester start, though, I went back to the textbooks. Why do they always start with functions? I finally started to grasp the logic: beginning with functions allows the teacher to work with transformations, parent functions, mapping, as well as challenging algebra (solving for x in a square root or quadratic function, etc).

So I mapped out a basic plan:

  • Function definition: domain, range, criteria
  • Function notation
  • Transforming functions
  • Four parent functions (line, quadratic, square root, absolute value). I told them we’d be introducing lines to ignore them until the next unit.
  • Transforming parent functions.
  • Solving for input and output

I originally planned to introduce inverses, but the kids were maxed out. This was a much tougher first unit than linear equations, and a good chunk of the lower ability students were struggling with the abstractions. Generally, I was pleased.

Some new questions from my first functions unit test–which was a single page, double-sided.

ftgraphquest

Notice I slipped in a couple function notation questions? That’s how I save space.

Here’s a mapping question:

ftmapquest

Again with the function notation! Am I the only math teacher whose kids simply can’t compute the difference between “f(3) = ” and “If f(a) = 3, a=”???? I do my best to beat it into their heads.

Here’s another way I use space effectively, I think:

ftsqfunct

So a graph, some free-response algebra, and conceptual understanding. (Most of them DO NOT understand how to read graphs, and missed d.) Time and again, I had to show the students how to write the equation, but they are learning how to isolate. Relatively few order of operations errors.

I didn’t ask them to graph this next one, but again, practice at setting up an equation to find the input given the output. Another plus of doing functions early is an introduction to quadratics, which is a tremendously tough Algebra 2 unit.

ftquadquest

Hands down, this next question had the weakest response. The strongest students understood it, but many of the same students who were able to graph the square root were flummoxed by this one. Go figure. But again, notice that I assess several different knowledge areas with the same question.

ftabvalfunct

A New Quiz

I don’t usually discuss my quizzes, which are often relatively straightforward compared to my multiple answer assessments. But I created a quiz on Thursday that I’m really pleased with. It’s my second quiz for linear functions. The students have learned the three different linear forms. The first quiz covers slope intercept and standard form, which are the forms for modeling situations. This one focuses on point slope and creating equations from points, as well as parallel and perpendicular points. We actually did much more modeling of real-life situations than this quiz shows. Usually, my quizzes are a very reliable guide to what the students have done in the previous week, but this was an attempt to see how well they could transfer knowledge and work several concepts in combination.

The quiz itself, I think was cool. I stole the nuggets of two ideas from textbooks, but the presentation and questions are mine own.

lmcricketsquest

I’ve seen this crickets question in both Pearson and Holt Harcourt books. I built the graph on Desmos, and was dismayed that a number of kids counted the barely visible lines, rather than use the points. But most of them didn’t.

Notice that this is a relatively easy question. I didn’t want to focus on the algebra needed to find the y-intercept. I wanted them instead to look at the patterns (the 120 chirps is exactly halfway between 0 and 240), and think about what graphs say vs. what they mean. Most kids confused question c and d, explaining that the temperature was too cold for chirping, or that the crickets died. But after a few pushes, they go…”negative chirps?” which is fun.

Here, I’m just testing their fluency:

lmfluencyquest

Lots of room for self-correction. One student asked me why all her solutions were “Neither”, and I suggested that perhaps she should check her algebra, where she’d handled a negative value incorrectly. Other students plotted the points incorrectly and, because they were only able to find slopes from the graphs, couldn’t catch their mistake–thus giving me an opportunity to reiterate the importance of using different methods to validate and self-correct.

As part of the work leading up to this quiz, they’d derived the Celsius to Fahrenheit conversion algorithm, given two points. I decided to give them the formula to see if they could recognize the errors in verbal description and work a solution using fractions.

lmcelsiusquest

And then my favorite:

lmhelicopterquest

I got the basic idea from my new favorite textbook series, Big Ideas Math, then played with the goals a bit. Big Ideas has wonderful scenarios.

As always, if you spot any errors or ambiguities, let me know.


Jake’s Guest Lecture

Our well-regarded local junior college is the top destination for my high school’s graduates, a number of whom are more than bright enough to go to a four-year university but lack the money or the immediate desire to do so. Case in point: Jake, my best case for the hope that subsequent generations of Asian immigrants will adopt properly American values towards education, now at the local community college with a 4.0 GPA. He earned it entirely in math classes, having taken every course in the catalog–and nothing else. This from a kid who failed honors Algebra/Trig for not doing homework, and didn’t bother with any honors courses after that.

Jake visits four or five times a year, usually coming during class to see what’s up, working with other students as needed, then staying afterwards to chat. This last week he showed up to my first block trig class, with the surly kids who mouth off. We were in the process of proving the cosine addition formula.

The day before, I started with the question: “cos(a+b) = cos(a) + cos(b)?” and let them chew on this for a bit before I introduce remind them of proof by counterexample. A few test cases leads to the conclusion that no, they are not equal for all cases.

Then we went through this sketch that sets up the premise. I like the unit circle proof, because the right triangle proofs just hurt my head. So here we can see the original angle A, the original angle B, and the angle of the sum. Moreover, the unit circle proof includes a reminder of even and odd functions, a quick refresher as to why we know that cos(-B) = cos(B), but sin(-B) = -sin(b).

cosineadditiondiag

Math teachers often forget to point out and explain the seemingly random nature of some common proof steps. For example, proving that a triangle’s degrees sum up to 180 involves adding a parallel line to the top of the triangle and using transversal relationships and the straight angle.

Didn’t I make that sound obvious? You have this triangle, see, and you wonder geewhiz, how many degrees does it have? Hmm. Hey, I know! I’ll draw a parallel line through one vertex point! Who thinks like that? The illustration of a triangle’s 180 degrees is much more compelling than any proof.

So when introducing a proof, I try to make the transition from question to equation….observable. Answering the question requires that we define the question in known terms. What is the objective? How does the diagram and the lines drawn get us further to an answer?

Point 1 in the diagram defines the objective. Points 2 and 4 allow us to represent the same value in known terms–that is, cos(A) and cos(b). And thanks to some geometry that is intuitively obvious even if they’ve forgotten the theorem, we know that the distance between Point 1 and Point 3 [(1,0)] is equal to the distance between Point 2 and Point 4.

So I’d done this all the day before in first block, setting up the equation and doing the proof algebra myself, and the kids were lost. In my second block class, I turned the problem over to the kids at this point.

cosineadditionmath

The solution involves coordinate geometry, algebra, and one Pythagorean identity. No new process, nothing to “discover”. Familiar math, unfamiliar objective. Perfect.

I grouped the second block kids by 5 or 6 instead of the usual 3 or 4 (always roughly by ability), giving each team one distance to simplify (P1P3 or P2P4). Once they were done, they joined up with kids who’d found the other distance, set the two expressions equal and solve for cos(A+B). The group with the strongest kids were tasked with solving the entire equation, no double teaming.

Block Two kids worked enthusiastically and quickly. I decided to retrace steps and do the same activity with block 1 the next day. Which is when Jake—remember Jake? This is a story about Jake—showed up.

“Hey, Jake! You here for the duration? Good. I’m giving you a group.”

Jake got those who had either been absent or were too weak at the math to be comfortable doing the work. I kept a watchful eye on the rest, who tussled with the algebra. I tried not to yell at them for thinking (cos(A) + cos(B))2 = cos(A)2 + cos(B)2, even though they all passed algebra 2 (often in my class), even though I’ve stressed binomial multiplication constantly throughout the year but no, I’m not bitter. Meanwhile, Jake carefully broke down the concept and made sure the other six understood, while they paid much more attention to him than they ever did to me but no, I’m not bitter.

Result: much better understanding of how and why cos(A+B) = cos(A)cos(B) – sin(A)sin(B). One of my most hostile students even thanked me for “making us do the math ourselves” because now, to her great surprise, she grasped how we had proved and thus derived the formula.

And then she went on to ask “But we have calculators now. Do we need to know this?” She looked at me warily, as I’m prone to snarl at this. But I decided to use my helper elf.

“Jake?”

Jake, mind you, gave exactly the same answer I would have, but he’s just twenty years old, so they listened as he ran through the process for cosine 75 (degrees. 75 degrees. Jake’s a stickler for niceties.)

“But why is this better?” persisted my skeptic.

“It’s exact,” Jake explained. “Precise. When we use a calculator, it rounds numbers. Besides, who programs computers to make the calculations? You have to know the most accurate method to better understand the math.”

“Class, one thing I’d add to Jake’s answer is that depending on circumstances, you might want to factor the numerator, particularly if you are in the middle of a process.” and I added that in:

cosinefactor

“Yeah, that’s right,” Jake confirmed. “like if you were multiplying this, I can think of all sorts of reasons a square root of two might be in the denominator. But other times you need to expand.”

I suddenly had another idea. “Hey. How about if we use right triangles?”

“Like how?”

I sketched out two triangles.

“Oh, good idea. Except you forgot the right triangle mark.”

I sighed. “Class, you see how Jake is insanely nitpicky? Like he’s always making me write in degrees? He’s right. I’m wrong. I’ve told you that before; I’m not a real mathematician and they have conniptions at my sloppiness. But…” I’m struck by an idea. “I don’t need to mark it here! These have to be right triangles. Neener.” (I nonetheless added them in, although I left them off here out of defiance.)

cosinepythagexamp

“This is good. So suppose you want to add the two angles here. These right triangles have integer sides, but their angle measures are approximations. Let’s find those values using the inverse.”

Ahmed has his calculator out already. “Angle A is…53 degrees, rounded down. Angle B is 67.38 degrees.”

Me: “Just checking–does everyone understand what Ahhmed did?” I wrote out cos-1(35). “He used the inverse function on the calculator; it’s just a reverse lookup.”

” Let’s keep them rounded to integers. So 53 + 67 is 120 degrees, which has a cosine of ….what?” Jake paused, waiting for a response. Born teacher, he is.

By golly, my efforts on memorization have paid off. Several kids chimed in with “negative one half.”

“Meanwhile, if we multiply all these values using the cosine addition formula…” he worked through the math with the students, “we get -3365“.

Dewayne punched some numbers and snorted. “-0.507692307692. That’s practically the same thing!” .

I had another idea. “You know how I said you should look at things graphically? Let’s graph this out on the unit circle.”

cosdesmoscircle

Jake was pleased. “This is excellent. So where would cosine(A+B) show up? We need to find the sine of each to plot it on the circle.” We worked through that and I entered the points.

Isaac: “Yeah, Dewayne is right. The two points are the same on the graph!”

“But this is a unit circle,” Jake said. “Just a single unit. As the values get bigger….I wish we could show it on this graph. Could we make a bigger circle? Or that probably wouldn’t scale.”

“How about if we just show all the values for every x? We could plot the line through that point? From the origin?”

“What would the slope be?” Gianna asked.

“Yeah, what would the slope be? Rise over run. And in the unit circle, the rise is sine, the run is cosine, so…”

“Tangent!” everyone chorused.

Jake was impressed. “See, this is why I should have taken trigonometry. I never thought about that.”

“OK, so I’m going to graph two lines. One’s slope is the tangent of 120, the other’s is the tan(cos-1(-3365))), which is just using the inverse to find the degree measure and taking the tangent in one step. Shazam.”

cosdemotangentline

We then looked more closely at different points on the graph and agreed that yes, this piddling difference became visible over time.

“So the lines show how far apart the points would be for 120 and the addition formula number if you made the circle to that radius?” Katie asked.

“Yep. And that’s just what we can see,” Jake added. “The difference matters long before that point.”

When second block started, after brunch, Abdul rushed in, “Ahmed said we had a genius guest lecturer? Where is he?”

I faced a cranky crowd when I told them the genius had to go to class, so Jake will have to come back sometime soon.

*************************************************************************

Two months ago, Jake stopped by for a chat and I asked him about his transfer plans.

“Oh, I don’t know. Four year universities, I’ll have to take other classes, instead of what interests me.”

“You can’t be serious.”

“Well, maybe in a few years. But I have to wait a while for the computer programming classes I need to take, and the math classes are more fun.”

“Computer programming?”

“Yeah. That’s what I want to….what. Why are you laughing.”

“Do you know anything about computers?”

“No, but it’s a good field, right?”

“I think you’re one of the most gifted math students I’ve bumped into, and you’ve never shown the slightest interest in technology or programming.”

Jake sat up. “My professor told me that, too. He said I should think about applied math. Is that what you mean?”

“Eventually, probably, but let’s go back to why the hell you don’t have a transfer plan.”

“Well, should I go to [name of a local decent state university]?”

I brought up his school website, keyed in “transfer to [name of elite state university system]”.

Jake looked on. “Wait. There’s a procedure to apply to [schools much better than local decent state university]?”

“You will go to your counselor, tell her or him you want to put together a transfer plan. Report back to me with the results in no less than 2 weeks. Is that clear?”

“OK,” meekly.

Just five days later, Jake’s cousin, Joey, my best algebra 2 student, reported that Jake had a transfer plan started and was getting the paperwork ready.

So after this class, I asked him about transfer plans.

“Oh, yeah. I’m scheduled to transfer to [extremely elite public university] in fall of 2017. I’ve been taking all math classes, so I have a bunch of GE to take. But it’s all in place.” He grinned wryly. “I didn’t think I’d be eligible for a school that good.”

“And that’s just the guarantee, right?”

“Yes, I want to look at [another very highly regarded public]. Do you think that’s a good idea?”

“I do. You should also apply to a few private universities, just for the experience. It’s worth learning if they give transfer students money.” I named a few possibilities. “And ask your professors, too.”

“Okay. And you don’t think I should major in computer programming?”

“Do you know anything about programming right now? If not, why commit?”

“I don’t know. I never knew about applied math possibilities. It sounds interesting.”

“Or pure math, even. So you’ve got some research to do, right? And keep your GPA excellent with all that GE.”

“Right.”

“And at some point, you’re going to think wow, I never would have done any of this without my teacher’s fabulous support and advice.”

“I already think that. Really. Thanks.”

Just in case you think his visits pay dividends in only one direction.


Tales from Zombieland, Calculus Edition, Part 2

The comments on part I have been fascinating. I want to reiterate that my math zombie’s teacher is not encouraging this behavior; I have no idea if she lectures or teaches using a more “progressive” style, but she certainly doesn’t believe that “procedural fluency leads to conceptual understanding”. A commenter also argues that “We Are All Math Zombies”. No. “Zombie” doesn’t mean “ran into the math ability wall”, nor does it mean someone who struggles with a topic and decides to forge through an obstacle, putting a black box around the difficulty to be returned to later, with more experience. I refer readers to the Brett Gilland definition of “math zombies” who “who can reproduce all the steps of a problem while failing to evidence any understanding of why or how their procedures work”.

Back to it–we are now into the “rules” questions, 3 through 8. She did question 3 easily. Please remember that my knowledge of calculus is being pushed to the limit in this entire sequence. I found this nifty derivative calculator so non-calculus folks can see how much rote algebra my zombie was doing, mostly correctly, again with no understanding.

Problem up: question 4: g(x) = (x2 + 1)(x2 – 2x)

She began by just taking the derivative of both terms and multiplying them.

“Um, no.”

“You don’t just multiply them?”

“Didn’t you do a bunch of rules? Product, Power, Chain, Quotie….”

She looked vague, but I was pretty firm on this point. “Look, you have to stop being so helpless. This math hasn’t been imposed on you by some fascist regime. Turn back a page or two in the book again.”

And then, a page or two back, when she spotted the product rule, “Oh, yeah.”

And she instantly started into the procedure.

“Stop. STOP!!! What the heck are you doing?” She looked at me in confusion.

“You’ve done this before. You have no memory of doing this before. Now you’re all oh, yeah, mindlessly working a routine you didn’t even recognize 30 seconds ago. Your next two years are going to be a case of lather rinse and repeat if you don’t start forging some memories, some connections.”

“I’ll just forget it again.”

“Then stop making yourself crazy and go take actual pre-calc.”

“I don’t even think that exists in my school.”

“Then listen up. What you know how to do is find derivatives of individual terms added together. First step is to realize that multiplying, dividing, or exponentially changing functions is more complicated. So there are separate rules that build on the easier, basic task of finding derivatives of individual terms.”

I wish I could say I broke into her drive for “just do something”, but at least she slowed down a bit. “But I wrote it down.”

“You did that the first time. So let’s try something different. Repeat this. The Product Rule: multiply the derivative of the first term by the second. Add it to the derivative of the second term times the first.”

“Yeah, I wrote it down.”

“No, you wrote down an abstraction. Say it.”

“What, like in words?” I looked at her sternly.

“Okay, I take the derivative of the first term. Then I…multiply it…”

“Stop. You’re into memorization, so memorize. But words, not symbols. The Product Rule: multiply the derivative of the first term by the second term. Add it to the derivative of the second term times the first.”

She repeated it patiently; I made her do it two more times.

“Okay, now you can work the problem.”

(I have no evidence for the notion that auditory/oral repetition helps, but intuitively, it seemed to me that the many rules are easier to remember by focusing on what the actions are, rather than what they look like. I lunched a few days later with my friend the real mathematician and department head, who told me that he requires his students to write–yea, write, Barry and Katherine!–a description of the product, quotient, and chain rules in addition to the algorithms. “Whenever I had to recall them in college, I remembered them verbally first.”)

Did you know there were online derivative calculators? So for those who want some kind of idea what she did, I’ll link these in.

“I always wondered if you can just distribute the product and use the power rule,” I mused, scratching through the steps. “Looks like you can. (x2 + 1)(x2 – 2x) expands to x4-2x3+x2-2x which…has a derivative of 4x3-6x2+2x-2.”

“That’s what I got. But why would you multiply it out when you can use the Product Rule?”

“Oh, I dunno. Maybe some people forget the Product rule temporarily. But if they actually understood the math, they could just think hey, no problem. I’ll just expand the terms until I can look up the rule. Or until it occurs to me to look up the rule, since you were stuck on that step until I showed up.”

She allowed as that was true. “But you can’t do that with the quotient rule.”

“I’m not good enough at this to know for sure. But most of the time you’d have a remainder, which would be expressed as a quotient, so it’s kind of reiterative. Question 5 is a fraction that is, I think, always going to be less than 1, so I’ll take a crack at doing the division on question 6 while you work out the quotient rule on both problems.”

“But how can I find a derivative of a cube root?”

“Gosh, wouldn’t it be great if there were a way to express a root as an exponent?”

“Oh, that’s right.” And she set to work on some rather complicated algebra and then stopped. “How do you know that this will always be less than 1?”

“Well, look at it. I’m dividing the cube root of a number and dividing it by its square. So think about taking the cube root of, say, 8? which is 2. Then dividing it by 8 squared + 1, which is 65. Even if x is less than 1, I’m adding 1 to the square of the fraction, so that sum will always be greater than the cube root of a positive fraction less than 1. I think, anyway.” Her eyes had long since glazed over, but I confess–I graphed it just to brag.

cuberootdividedbysquare

“I finished question 5, but it doesn’t match the book.”

I looked. “No, you didn’t drop the power on the cube root. It’s going to be negative two-thirds, which will move it to the denominator.”

She redid the problem while I did long division on problem 6, getting -1 with a remainder of -2x+2. Since the derivative of the constant was zero, I then had to take the derivative of the remainder (divided by x2-1).

“It just occurred to me I could use the Chain Rule here, too. Huh. I wonder if that means all quotient derivatives could be worked with the chain rule.”

Our answers to number 6 matched up, and my student was mildly interested. “So I can find derivatives with more than one method?”

“As is usually the case with demon math. But file this away with ‘repeat the processes verbally’ as a means of survival strategy.”

She worked her way through the next group, enduring my comments patiently but with little interest. I kept plugging away, trying to get her to think about the math–not because I wanted her to share my values, but I thought the conversations might create some memory niches.

So when she worked the derivative for problem 10: “hey, that’s interesting. That graph will always be negative, which means the slope at any point on the original graph will be negative.”

“What? How can you tell?”

“No, you can figure this out. Look at it closer.”

“It’s negative 8 divided by…oh, I see. Squares are always positive. So it’s a negative divided by a positive.”

“So that means that no matter what point we put in…” I prompted.

“Wait. Every slope is negative? No matter what?”

“I wonder if it’s always true for reciprocal functions. Huh.”

“Is that a reciprocal or a hyperbola.”

“Huh. I….think… they’re the same thing? Or a reciprocal is a type of hyperbola? Not sure. Good question. A hyperbola is a conic, I know, and I’m more familiar with transformations than conics.” (Answer is yes, a reciprocal function is a rectangular hyperbola.)

Then, when we got to problems 11 and 12: “Look, you need to remember that a square root function will in all cases turn into some sort of reciprocal function. You keep on messing up the algebra and aren’t catching it because you aren’t thinking big picture.”

“I don’t see why it’s a negative exponent.”

“What do you always do with exponents in derivatives?”

“You subtract….oh! I’m always subtracting 1 from a fraction.”

“Bingo. And negative exponents are..”

“they’re reciprocals, you’re dividing. Okay.”

“But look at the bright side. You actually understood this question.”

“I do! You really have helped.” I beamed. And she was able to work problem 13, finding a derivative given a graph, without help when an hour earlier she couldn’t. Progress, at least in the short term.

Problem 14 was interesting. “Determine the points at which the graph of f(x) = 1/3x3 – x has a horizontal tangent line.”

“Should I use implicit differentiation?”

“What? No. Well. I don’t really grok implicit differentiation, but that’s not what this one is asking. What does a horizontal line have to do with slopes?”

“Horizontal lines have a slope of zero. So the rate of change is zero? It’s asking where the rate of change is zero? The derivative is….x2 – 1.”

“Which factors to (x-1)(x+1). Hmmm.”

“So it is implicit differentiation?”

“No. Look, I don’t know what implicit differentiation is specifically, but it always involves y. This is….I’m just confused, because the point at which this parabola has a slope of 0 is the vertex, which is x=0.”

“Yeah, the slope of the parabola isn’t what I’m looking for, right? That means the slope of the other graph is 0 and I should plug in 1 and -1.”

I looked at her, impressed. “My work here is done.”

“What, I’m wrong?” She quickly worked the problem. “It’s positive and negative 2/3. That’s what the book says, too.”

“You’re not wrong at all. I was the one who was confused and you spotted the problem. Very good!”

“But why couldn’t I have used implicit differentiation?”

“Look, you need to talk to your teacher about that because it’s at the edge of my knowledge. I know that working the math of implicit differentiation is easier than understanding it. But at 90,000 feet, what you need to remember is that you use implicit differentiation when you can’t isolate y, so your equation has two variables. Circles and ellipses, for example. Or some of those other weird circular graphs. Look at problems 16-19, for example. Anyway, the derivative on this one was simple. The crux of the question was the link between the zeros of the parabola and the rate of change on the original graph.”

And with that, our ninety minutes were up. I tried to talk the mom out of paying me, since I’d learned a lot and wasn’t an expert, but she insisted.

Some observations:

She was capable of some pretty brutal algebra without any real understanding of what she was doing, time and again. That’s the zombie part–that and the fact that she really didn’t much care about anything other than plowing through. She wasn’t ever really interested but hey, all this stuff the tutor was saying seemed to help, so play along.

I learned a great deal, in ways that will further inform my pre-calculus class curriculum. Can’t wait to try it out. I also wrote out a lot of equations and may have made typos, so bear with me. And yeah, that’s how I remember implicit differentiation–it’s the one with “y”. I get the basics–normally it’s just x changing, this is saying they both change with respect to each other, or something. Implicit differentiation is the point at which I start to realize that the algebra of the differentiation language (dy/dx) has its own logic and wow, a chasm of interesting things of which I know nothing about opens and threatens to swallow me up so I look away.

I’ve really increased my understanding in advanced (high school) math over the past few years, and going back into calculus armed with that additional knowledge has led me to think—really, for the first time—about the lunacy involved in high school calculus instruction. I am starting to understand how math professors could be dismayed at the total ignorance demonstrated by students who scored 5 on the BC Calc test.

Finally, consider that this student is taking pre-calculus. Her transcript reflects pre-calculus. Yet the content is clearly calculus. Meanwhile, I teach a lot of second year algebra with an analytic geometry spin in my pre-calc class. Most schools fall somewhere in between. This is why I laugh when people propose doing away with tests and using grades and transcripts. I still believe in good tests, despite my increased awareness of cheating and gaming.

This enormous range of difficulty and subject matter reflects the bind faced by high schools kneecapped by our education policy. We must offer all students “college level” material, and our graduation and class enrollments are scrutinized closely by the feds and civil rights attorneys ever in search of a class action suit. So we have to move kids along, since we can’t fail them and can’t offer them easier courses. So we have to try and teach good, solid math that isn’t too much of a lie. That’s what I do, anyway.

Maybe things will change with the new law. I’m not counting on it.


Tales from Zombieland, Calculus Edition, Part I

A couple weeks ago, I met with a charming math zombie who I coach for the SAT. “Could you help me study for a pre-calc test instead?”

She brought out her book, a hefty volume, and turned to chapter 4, page 320

I took one look and skidded to a stop.

“What the hell…heck. This is calculus.”

The mother sighed. “Yes, they cover calculus in pre-calculus so that everyone is ready for AP Calc next year.”

Huh. Remember that, folks, the next time you hear of a school with a 100% AP pass rate. They are teaching the kids some of the calculus the year before.

“OK, I can maybe help you with this but before we start: I don’t usually work in calculus. I’m pretty good conceptually, and my algebra is awesome, but at a certain point I’m going to have to send you back to the teacher.”

“That’s fine; I really need any help I can get.”

First up. “Use the limit process to find the derivative of f(x) = x2 – x + 4.”

“What on earth is the limit process?” I turn back in the book, leafing through the pages.

“I have no idea.”

“Well, you must have worked the problem before.”

“I don’t know how.”

“Maybe they mean the definition of a limit, the slope thingy.” I look at the next problem, which also focuses on slope, and decide that must be it.

“So you know the definition of a limit, right?”

“No, not really. I know the derivative of this is 2x-1.”

“Yes, but what is the derivative?”

“I don’t know. I don’t understand this at all.”

“Um, okay. The derivative of any function is another function, that returns the slope of the tangent line for any given point on the original function. The tangent line represents…um, .not just the average rate of change between two points, but the instantaneous rate of change at that point.” (I am not using math terms; whenever mathies get together and talk about the “intuitive” definition of a derivative I want to slap them. I checked a few places later, like this one, and I think I’m on solid ground.)

“Yeah, but why do we care about the rate of change?”

I should mention here that her teacher and I went to ed school together, and I’m certain she (the teacher) explained this multiple times from various perspectives.

“You say you know the derivative is 2x-1, yes?”

“Right. You’re saying that’s the slope of the line?”

“Almost. The derivative is the means of finding the slope of a tangent line to any point on the function, with various caveats I’m going to skip right now. Remember, most functions do not change at constant rates. You can find the average rate by finding the distance between any two points, and finetune that average by picking two points closer and closer together. The slope of the tangent line, which means the line is intersecting only at one point, is the….” I can see she doesn’t care, and her understanding is definitely ahead of where it was just five minutes earlier, so I stopped for the moment.

She sighed hopelessly. “Look, can’t I just find the derivative?”

I scrawled something like this:

“Oh, I remember that. Okay.” And she plugged it all in and calculated rapidly. “How come I have an h left over?”

I was a tad flummoxed, but then remember. “Oh, h approaches 0, so it’s basically negligible. I think that’s right, but check with your teacher. Now, what does this represent?”

“I have no idea.”

“Suppose I ask you to find the derivative when x=1, or at the point, um, (1,4).”

“I plug 1 in for x in 2x-1, which is 1. Then I write the equation y-4=1(x-1).”

“So graph that.”

“I don’t know how. It’s a line, right?” She thinks a bit, then converts the equation to slope intercept. “Okay, so it’s y=x+3.”

“Now, graph the parabola.”

“Um…” I sketched it for her, and marked (1,4). “Now sketch the line.”

calcex1graph

“See how it just intersects at the point, perfectly tangent? That’s what a derivative does–it returns the slope of the line through that point that will intersect at just one point.”

“Yeah, I saw this before.”

“And it made quite an impression. Stop waving this off. You want to feel less hopeless about math? This is why you have no idea what’s going on. So gut it up and focus.” She nodded, somewhat chagrined.

“The slope of the line at that point indicates the slope of the original function at that point, which is the instantaneous rate of change. Remember: most functions don’t change at a constant rate. Finding the rate of change at a single point is an essential purpose of calculus. So pick another point and try it.”

“OK, I’ll try -1. What do I do first?”

“What do you need to know?”

She looked at the graph. “I need to know the slope of the line….which I get from plugging in -1 to the derivative 2x-1, which is….-3. And then I—”

“Stop for a minute. Say it. What did you just find out?”

“The derivative for x=-1 is -3, which means…the slope of the line where it meets the graph is -3?”

“Slope of the tangent line. And what does that represent?”

She frowned in concentration and looked at the sketch I’d drawn. “That’s the rate of change at that point. But where is that tangent line intersecting? Oh, I need the plug that in…” She did some work. “So the point is (-1,6), and the slope is -3, and that’s why I use point slope, because I have a point and a slope.”

“And remember, you don’t have to convert from point slope to slope intercept. I just do it because I find it easier to sketch roughly in y-intercept form.”

calcex1graph2

“But how does this work in problem 2? They don’t give me an equation but they want me to find a derivative.”

calcexfig2

“You can find the equation from the graph.”

“Oh, that’s right. But I checked the answer on this, and it’s just -1, which makes no sense.”

“Sure it does. Graph the line y=-1.”

She thinks for a minute. “It’s just a horizontal line.”

“And the slope of a horizontal line is…”

Pause. “Zero. But does that mean the derivative is 0?”

“Which would mean what?”

“The rate of change is zero?”

“How much does a line’s slope change?”

“It doesn’t.” I wait. “You mean a line has a zero change in its rate of change?”

“There you go. And doesn’t that make sense?”

“So….because a line has a slope, which is the same between every point, its derivative is zero. So the derivative is….oh, that’s what you mean when you say other functions don’t change at a constant rate. OK. So lines are the only functions whose derivative is zero?”

“Um, yes, I think. But a derivative can return zero even if the function isn’t a line. ”

She sighed. “It’s much easier to just do the problem.”

I’m going to stop here, because I want to go through several of the conversations in detail so I’ll do a Part 2.

In my last post, I pointed out that Garelick and Beals and other traditionalists are, flatly, wrong in their assertions that procedural competence can’t advance well in front of conceptual understanding.

At the risk of stating the obvious, here is a nice, charming, perfectly “normal” calculus student who understands how to find a derivative, how to work the algebra to find a derivative, and yet has absolutely no idea or caring about what a derivative is—and complains in almost identical words to the middle school girl in G&B’s article. She just wants to “do the problem.”

Our entire math sequencing and timing policy is based on the belief that kids who can do the math understand the math. Yet increasingly, what I see in certain high-achieving populations is procedural fluency without any understanding.

In case anyone wonders, I’m not engaging in pointed hints about East Asians (I tend to come right out and say these things), although they are a big chunk of the zombie population. The other major zombie source I’ve noticed is upper income white girls. I have never met a white boy zombie, or a black or Hispanic zombie of any gender, although perhaps they are found in large numbers elsewhere. But the demographics of my experience leads me to wonder if culture and expectations play a big part in whether a student is willing to put the time and energy into faking it. Or maybe it’s easier for people with certain intellectual attributes (a really good memory, for example) to fake it.

Anyway, I’ll do a part 2, and not solely to reveal zombie thinking. I was planning on writing about this session before the G&B piece appeared. Not only did I enjoy the chance to work with calculus, but I also have really started to understand how unrealistic it is to teach calculus in high school. I’m moving towards the opinion that most kids in AP Calc don’t understand what the hell’s going on, thanks to the unrealistic but required pacing.

Oh and yes, I don’t know much calculus. Forgive me if my wording isn’t correct, and feel free to offer better in the comments.


The Test that Made Them Go Hmmmm

So school has begun and despite my palpitations about the boredom of only two familiar preps, I’m pleasantly busy. Last year was a hell of a lot of work, and given the nosedive that my writing time took, I should maybe not be so eager for a less…familiar schedule. So instead of demanding new classes, I accepted the first semester, threw a minor temper tantrum when no one listened about second semester and all is well. Algebra 2 in particular is proving a delightful challenge, given my new emphasis on functions.

In no small part because of this planning breathing room (is anyone noticing I’m saying my panic was a total overreaction?), the senior Water Park Day registered in my awareness ahead of time. In prior years, I didn’t heed the warnings that half my class would disappear, and so would be forced to dump my lesson plan on the Day itself, when the smaller classes would just have a day to practice. But thanks to this old, familiar schedule that gives me more time, I anticipated the impact.

So for the first time, I was able to give serious thought to having a day to pursue math without regard to subject matter or schedule. I could have a “math day”! Then I remembered Grant Wiggins’ challenge to math teachers everywhere in the form of a conceptual knowledge quiz.

hmmquiz

Grant proposed this as an actual test: I will make a friendly wager: I predict that no student will get all the questions correct. Prove me wrong and I’ll give the teacher and student(s) a big shout-out.

What math teachers think their kids would know the answers? I certainly didn’t. In some cases, they probably were taught, but in others, I doubt an elementary school teacher would ever think to bring them up. But even if all the concepts were taught by fifth grade, how many kids of that age could really appreciate the questions?

Most of the questions tease at the paradox….wrong word? tension? between the functional day-to-day applications of arithmetic, and the amazing truths that underlie them. John Derbyshire wrote, in Prime Obsession, that “arithmetic has the peculiar characteristic that it easy to state problems in it that are ferociously difficult to solve.” (I was rereading Prime Obsession last night; there’s tons of useful thought material for math teachers. I need to go get his book on algebra.)

Arithmetic looks easy. (And certainly in the last twenty years, the rush to shove everyone into calculus has led to a certain contempt for “basic arithmetic” classes.) But even if elementary school age children are capable of understanding its ideas fully (and most of them aren’t), they haven’t experienced several years’ utility of arithmetic. They haven’t had time to get bored of the routine rules that they are expected to remember (mind you, many don’t, but leave that for another day.) Yeah, yeah, invert and multiply. Yeah, yeah, you can’t divide by zero. Wait, what the hell do you mean multiplication isn’t repeated addition?

To really enjoy this test, to be fascinated by the underlying truths–or misconceptions–behind certain everyday math tools, requires familiarity with “the rules”. Time spent in the trenches of doing math just because.

That’s when a teacher can spend an enjoyable hour taking the kids back through a re-examination of the basics and what they really know. I’d much rather discuss these concepts with adolescents who have survived two or three years of high school math than try to force sixth graders to “demonstrate conceptual understanding” of dividing by zero.

I had no real expectations—no, that’s wrong. I had hopes. My sense was the students would be interested in the exploration, if I didn’t take on too much or dive in to the wrong end of the pool. But which end was the wrong end?

So for each of my four classes–two Algebra 2, two Trigonometry–I gave them the test and 20 plus minutes to write down their thoughts. I was alert to the possibility that kids would use five minutes to doodle and fifteen to giggle, but in each class the bulk of students asked for and got an additional five minutes to finish up. I collected their answers and will share some of them in later posts; they were often detailed and thoughtful.

After the writing time, the students had a few minutes to “share out” in their groups, so they could learn what questions puzzled their classmates—and also as reassurance that they weren’t alone in their befuddlement. Again, this seems different from Grant’s intent; he considered it a real test that the students would either answer correctly or leave blank in confusion. I listened in on many conversations; they were rich with exchange as the students realized they weren’t alone in their uncertainty.

But certain questions also sparked genuine debate and interest. More than a few students offered up multiplying negatives as an example of multiplication being something other than repeated addition. In every case I witnessed, their group members, who had written something to the effect of “isn’t it always repeated addition?” instantly recognized the roadblock that negative numbers posed to their definition. I came across more than one group arguing whether multiplying by zero counted as repeated addition (“yes, it does. If I have zero groups of five, I have zero!”). Interestingly, no one came up with the roadblock I was interested in, and I’d never once considered negative numbers until my students brought it up.

Their discussion time was about ten minutes. My goal wasn’t to have them determine the answers; rather, I wanted them all to have a shared experience before we discussed them as a class, and I gave them the “answers” (to the extent I knew them). That way, there’d be more of a sense of “we”–yeah, we thought of zero, too! yeah, we all have 3F=Y–that’s not the answer? yeah, we think dividing by zero gives you zero–it doesn’t?

So then we went through the answers as a group.

I had taken a subset of Grant’s list, ignoring the last three items. Doing it again, I would have swapped out question 2 for question 11 “appropriately precise”), because while question #2 is good, it really requires its own day. The rest of them are easily covered and discussed in at most 15-20 minutes each.

The questions I really wanted to spend time on, to explain in at least introductory depth, were 1, 3, and 5. From a practical standpoint, I wanted to be sure everyone understood why they got questions 4, 6, and 8 wrong, assuming most missed at least one of them. I was genuinely interested to see what they had to say about 7 and 9 but was going to take most of my lead from them. Question 10, I wanted to know if the trig students knew it; obviously, my algebra 2 students learn about imaginary numbers for the first time.

My trig classes are quite different in nature. Both are small, just 25 in each. Both are doing quite well; I have no kids who simply shouldn’t be there, as I did last year. My first block class is stronger, on average, but has more surly kids who mouth off. It’s very irritating, frankly, since the five or six kids giving me quite nasty sass are seniors who are doing relatively well (Bs and Cs), and who openly acknowledge that they think I’m a hell of a teacher. Two of the surlies had me last year for algebra 2, when they were much less trouble, and had been switched into my class because they were failing with another teacher. But these other teachers, who they didn’t like (and often failed, forcing them to retake a fake summer school course if they couldn’t switch to my class), didn’t get nearly the lip. I’m a tad flummoxed. My second block class has more kids who are amiable and interested but not taking the class as seriously as they should, so several more low scores on the first test. First block has a stupendous top tier, but it’s just three or four kids. Second block has a top tier of close to eight, but they aren’t quite as strong.

Anyway, I was expecting more interesting conversation from second block, and I had it backwards. First block was on point, even the cranky ones. They loved the test, wrote detailed responses, discussed it thoroughly in group, and were wildly participatory in the open discussion. Easily 90% of them came up with the correct response to imaginary numbers (and the ones from my algebra 2 class identified multiplying by i as 90 degree rotations in the complex plane, which was quite gratifying, thanks so much). Second block, the amiable, mildly uninterested ones pulled things down slightly, goofing around and making jokes while the stronger kids would have preferred more time to explore things. The conversation was still great, the students learned a lot and enjoyed the discussion, but I had the enthusiasm levels backwards.

My algebra 2 classes, I nailed in terms of expectations. Block three is a fairly typical profile, except I have a lot more sophomores than usual (which is due to our school successfully pushing more kids through geometry as freshmen). But still a good number of seniors who barely understood algebra I, a lot of whom are just hoping to mark time til graduation without ending up in summer school. (One of my specialty demographics.) And in between, juniors and seniors who are often thrilled to find themselves actually understanding math and succeeding beyond anything they’d ever hoped (another specialty of mine). Typically, many of the seniors were in class, as they lacked the the behavior or grade profile (and sadly, in some cases, the money) to go to the water park. So I expected conversation here to be a bit lower level, with less interest. Happily, everyone engaged to the best of their ability and many told me later how much they loved just “talking about math”. I spent much more time on questions 4, 6, and 8, and could see them all really registering why they’d made the mistakes they did. But they still were enthralled by questions 1, 3, and 5, which is great because it’s going to give them some memories when we review percentages in preparation for exponential functions.

Last up was block 4 algebra 2, a ridiculously strong class; only five students are of the usual caliber I expect. The seniors are all well above average ability level. Two of the kids are so skilled that I’ve already introduced three dimensional planes and the matrix, while still forcing them and the other really strong kids to deal with complex linear word problems (mixture questions! I usually skip them, so it’s a trip). They stomped all over the test, writing at great length, discussing it with their teams and then shouting out to other groups to see what they’d answered for multiplication. The class discussion took so long that I actually allowed it to continue for 20 minutes into the next day, when I invited one of my mentees to watch. He came away determined to try the test in his honors geometry class.

Look, the whole day was teacher crack. Take a day. Try the test. I’ll be discussing individual questions and my explanations in future posts, but this introduction is offered up as invitation. High school teachers working in algebra 2 or higher would be a good starting point. Honors classes in algebra and geometry would also benefit. Every math teacher can find links from this test to their math class—but then, that’s not the point.

As for me, I started out the day with hope, but also a determination to see it through as part of a way to honor Grant Wiggins, who felt very strongly that students needed to do more than just march through curriculum. I promised myself I wouldn’t abandon the effort even if it went wrong. It didn’t go wrong. Quite the contrary, the test sparked delighted interest and intellectual curiosity among students who are often hard to push into exploring mathematics in depth. So hey, Grant, thanks for the idea–and the inspiration.


Functions vs. Equations: f(x) is y and more

I wanted to talk about function algebra, which naturally would include a reference to function notation.

So here’s the frustrating thing about writing this blog. I try to include links to other sites that explain a concept, so that I don’t have to reinvent the wheel for my reading audience. But a google gives me these results: useless links that do little more than say “f(x) is the same as y”. That’s not math. That’s test prep. And there’s nothing wrong with test prep, but every one of these sites purport to be math teaching sites, and hey, I’m not a mathematician, but shouldn’t we be explaining what f(x) means?

Someone somewhere is saying “See, this is why we need teachers to be math majors, instead of English majors who get 800 on the GRE quant section. You can’t substitute math understanding that comes with the study of these important principles.” That someone somewhere is wrong. I used to think that in my early days, until I had too many conversations like this:

Me, to AP Calculus teacher WHO MAJORED IN MATH: Hey, what do you tell your kids about function notation?

AP Calculus teacher WHO MAJORED IN MATH: f(x) is the same as y.

Me, nonplused: Well. Yeah. But I mean about why we developed function notation, what it serves that can’t be served by….

AP Calculus teacher WHO MAJORED IN MATH: It’s just notation. Don’t be confused.

Me: I’m not confused. But they serve different purposes, and I’m just trying to be sure I accurately capture…

AP Calculus teacher WHO MAJORED IN MATH: They don’t serve different purposes. It’s just notation f(x) is the same as y.

Me: Ok.

In my experience, very few math teachers WHO ACTUALLY MAJORED IN MATH care about these things either. My beer drinking buddy is an exception (and he’s now department head), and he’s the only math teacher I’ve found so far who was interested in my work on this subject.

Textbooks? McDougall Litell, CPM has a lot of those function machines. But no explanation. Holt does a little better but I didn’t understand that until I understood what I was looking for.

So I spend more time looking for a good link. Otherwise, I have to spend a lot of time figuring out how to explain function notation accurately, or at least inoffensively, so that people reading this blog don’t make me remind them that, for chrissakes, I’m an English major not a mathematician! That takes time. It’s not time I wanted to spend. I don’t want to tell you what function notation is, in a way that will pass expert muster. I want to tell how I build on function notation to teach function algebra. But I can’t do that well without explaining function notation, which I didn’t set out to do. This leads to many blog entries taking much more time than they should. The original intent for my function algebra post was to be just a quick little throwaway.

I began writing this post nearly a month ago, and got stalled looking for a way to characterize the explanation. You may be wondering why I would explain something I don’t understand—but that’s not it, really. I just don’t know what to call it. And that’s fine for teaching, not so much for writing, and so I spend hours trying to figure out the correct query. Which took me, literally, up until today.

Just fifteen minutes ago (as I write this sentence) I finally found the kernel in this discussion on function notation before Euler, in which someone writes:

but [Newton] refers to these as equations, not functions, and admittedly (written the way they are) that is exactly what they are. It seems anything that we would today write as a function, Newton described in words, such as:

HA. I learned something I hadn’t quite understood completely before–a function and an equation are not the same thing. Googling “what is the difference between an equation and a function” led me to the right websites. I realize now that I wasn’t just looking for an explanation of function notation, but rather why and when we use functions vs. equations.

Here’s an explanation that covers what I was trying to say.

So my research paid off. In practice, what I’ve been doing in this lesson is introducing function operations and function notation as a way to overcome a constraint in using equations.

******************************************************************

Sami needs $15 more to buy the new hoodie that he wants. But if Sami skips the hoodie, he needs just three more dollars to buy a ticket to the pizza feed on Friday. If Sami has x dollars, how much money, in terms of x, does Sami need if he wants both the hoodie and the ticket to the pizza feed?

The first thing the kids think is that Sami needs $18 more.

I say okay, Sami has $20. How much does the hoodie cost? $35. How much does the pizza feed cost? $23. How much ….oh. Huh, say the kids. He needs a lot more than $18.

Depending on how goofy I feel, I might get out some fake money. I count out $20, give it to a quiet student. How much more for the hoodie? Count out another $15. Now how about the…Right about then, a student gets it: you need the $20 twice.

So then we go to the board and model the two different equations for each purchase.

y=x+15
y=x+3

So if we are getting both things, what are we doing? Adding, the class choruses.

Ah, now there’s a new wrinkle. The kids have been adding equations for a while now, in systems. So I say, let’s try to add these equations.

2y=2x+18.

Is that right? We test it with $20 and the kids realize that the right side “works” (that is, we get $68) but the left side says we still need to divide by 2, which would be…wrong.

“So what’s happening is that we are running into the limits of an equation. An equation tells us that two expressions occupy the same point on a number line–that is, after all, what “equal” means.”

“But when we use multiple variables in equations, then the equation becomes a relationship between two variables, an if-then. If y=x + 15, then the point (3, 18) is a solution because setting x=3 and y=18 creates an equation that has both sides occupying the same point on the number line. If 3x + 2y=12, then (2,3) is a solution because setting x=2 and y=3, etc.”

But in an equation, the variables are values. So in the Sami case, we can’t treat y as a collection point. We can’t keep track of the dependent variable because it varies, obviously. The y in the first equation has a different value from the y in the second equation. If we wanted to keep them separate, we could use two different variables, like z = x + 15 and y = x + 3. Or we could number the ys: y1 = x+15, y2 = x+3.

“Using the language of functions makes a lot of these constraints disappear.”

“First, logically. Functions are different in a key way from equations: a function is an output. An equation is a relationship between variables. Yes, y=x+3 and f(x)= x+3 yield the same results, which is why we teachers always tell you to remember that ‘y and f(x) are the same thing’. However f(x) isn’t a variable, but an output. So when we add two functions, we’re adding outputs. Remember, too, that a function doesn’t even have to be an equation, like in the cell phone code example.

Then there’s function notation, invented by Euler. Function notation enables unique names, usually a single letter. But it doesn’t have to be. You can get creative with the letter names and the input values.”

“Function notation is just more elegant and efficient, too. Instead of saying ‘if x=7’ you can just say f(3). Once you define the function named ‘f’, anything can be input, even another expression, like f(a+7). And then, instead of saying ‘y=’ and solving for x, write f(x)= 3.”

“So let’s call Sammy’s cash on hand c, and then create a function h for hoodie, and p for pizza feed.

h(c) = c+15
p(c) = c+3

In both cases, c represents the money Sami has, so the input value is the same. But the output value varies based on the function used.”

“Now, this is a small difference. But how many have you been told that f(x) is the same as y?” Bunch of hands raised.

“Yep. And in a lot of ways, it is. But you have to be wondering why, if they’re the same thing, we bother teaching you about function notation.” Lots of nods.

“So as you move on into advanced math, you’ll start to learn other reasons why we sometimes use functions and other times use equations. For now, it’s enough to know that function notation allows us to keep track of our different outcomes.

“Once we can do this, we can actually create an entire math with functions. They can be added, subtracted, multiplied. They have inverse operations.”

“But then why do we use equations?”

“Well, for one thing, functions don’t do systems well. Remember, when we solve systems, we are expecting both the x and the y (and any other variables) to be equal. Functions don’t handle that well. So you’ll see that we switch back and forth between equations and functions as needed.”

When you need to add expressions, functions are great. So now we can add h(c) and g(c).

h(c) + p(c) = (x + 15) + (x + 3) = 2x + 18

“Because we are adding outcomes, and have a unique way of tracking each outcome, we can add them properly. Remember, too, that since a function doesn’t need to be an equation, I can add or subtract outcomes without even having an equation. If a(x) = 9 and b(y) = 17, then b(y) – a(x) is 8, and I don’t have to care if a(x) and b(y) are generated by an expression or a rule or a code or a random happenstance—provided, of course, that random happenstance is only one per input.”

******************************************************************************

I know. You’re wondering why I don’t just follow the AP Calculus teacher’s “f(x) is the same as y”. Well, it turns out that function operations are a big part of pre-calc, so they’ll use this later.

In the meantime, I give them some practice with function notation (I stole this at random). Not enough. Kids don’t really know it later. But at least they’re exposed to it.

Then I go on to linear function addition and subtraction. I usually just put problems on the board.

Sample quiz:

functionquiz

Here’s a test question:

functionma

And from here I go on to linear function multiplication (aka quadratics) and, eventually, rational expressions (linear function division).

Like teaching congruence with isometries, I can’t argue that using functions to further our work in linear and quadratic equations is better. I find it more…elegant, maybe?

But the execution isn’t quite there. This is the first year I’ve really taught this whole sequence: introducing functions, function addition/subtraction/notation, function multiplication, inverse functions, rational expressions. Writing it up has revealed an obvious improvement. Up to now, my function illustration has been a quick standalone lesson. Then later I introduce the notion of function addition and in doing so, bring up function notation.

This is goofy, now that I look at it. In the future, I’ll introduce functions and then go into function notation. I can spend a day or two on that, quiz that early. Then I can go back into linear equations or inequalities (the placement is flexible) and then bring up function addition and subtraction, with function notation already covered.

You know what’s irritating? The huge effort described at the beginning of this post to figure out how to describe what I was teaching led me to this. The huge effort underwent solely in order to write this post. Which I was griping about. In learning how to describe function notation for my readers, I learned that the proper way to characterize my work is as a difference between functions and equations, and that led to an idea for better sequencing.

This is kind of a placeholder post. Obviously, I’m in flux about this right now. My linear equations unit has been in good shape for a while. This gives me plenty of room to add flourishes, introduce more complicated topics onto a subject the students know well. Meanwhile, linear function multiplication has proven to be a great introduction to quadratics. So now I’m involved in putting it all together.

Next up in this sequence: the post that I really wanted to write, on my quadratics introduction.

Sorry for the slow rate of posts lately. I did five in April, then got lazy.


Illustrating Functions

Function definitions aren’t usually tested on either the SAT or the ACT and since I never worked professionally with math, functions were something I’d barely considered in algebra a billion years ago. So for the first few years of teaching, I kind of went through the motions on functions: unique output for each input, vertical line test, blah blah. I didn’t ignore them or rush through them. But I taught them in straight lecture mode.

Once I got out of the algebra I ghetto (which really does warp your brain if that’s all you do), I accepted that a lot of the concepts I originally thought boring or unimportant show up later. So it’s worth my time to come up with the same third way activities and lessons for things like functions or absolute value or inverses that I do for binomial multiplication and modeling linear equations and inequalities.

So every year I pick concepts to transfer from pure lecture/explanation to illustration. Sometimes it’s spur of the moment, other times I plan a formal curriculum change. In the case of functions, the former.

Last year I was teaching algebra II/trig and–entirely in passing–noted a problem in the Holt book that looked something like this:
functionoriginalexample

and had two simultaneous thoughts: what a boring question and hey, I could really do something with that.

So the next day, I tossed this up on the board without comment.

functionactivity

I’ve given these instructions three times now–a2/trig, trigonometry, algetbra 2–and the kids are always mystified, but what the heck, the activity seems simple enough. No student ever reads through the entire list of instructions first. They spend a lot of time picking the message, with many snickers, then have fun translating the code twice.

But then, as they all try to translate someone else’s message using the cell phone code, bam. They realize intuitively that translating the whole-alphabet code would be an easy task. And with a few moments of thought, they realize why the cell phone code doesn’t offer the same simple path. They don’t know what it means, exactly. But the students all realize that I’ve demonstrated a difference that they’d never considered.

From there, I graph the processes, which is usually a surprise as well. The translation process can be graphed?

alphabetgraph

cellphonegraph

At this point, I can usually convince kids to remember the Vertical Line Test, which they were taught in algebra I. At that point, I go through the definitions for relation, function, and one-to-one function, using a Venn diagram (something like this with an addition inner circle for one to ones). Then I go back through what the students vaguely remember about functions and link it to the correct code example.

Thus the students realize that translating the message into code is a function in either code key. I hammer this point home, because the most common misconception kids get from this is that all functions must be one to one. Both are functions. Each letter has one and only one number assigned, and the fact that one translation key puts several letters to the same number is irrelevant for its determination as a function. Reversing the process, going from numbers to letters, only one of them is a function.

Then I sketch parabolas and circles. Are they both functions? Are either of them one-to-one functions?

In Algebra 2, I do this long before the inverse unit. In Trig, I introduce it right before graphing the individual functions as part of an overview. In both classes, the early intro gives them time to recognize the significance of the difference between a function and the more limited case of the one-to-one function–particularly in trig, since the inverse functions are very limited graphs for exactly the reason. In algebra II, the graphs reinforce the meaning of the Horizontal Line Test.

I haven’t taught algebra I recently, but I’d change the lesson by giving them a coded message and ask them to translate with the cell phone code first.
functionalgebra1version

This leads right into function and not-function, which is all they need in algebra I.

I have periodically mentioned my mixed feelings about CPM. Here’s a classic example. The CPM book introduces functions with the following example.
cpmfunct

Okay. This is a terrible example. And really boring. Worst of all, as far as this non-mathie can tell, towards the end it’s flat out wrong. A relation can be predictable without being a function (isn’t that what a circle is?). But just looking at it, I got an idea for a great test question (click to enlarge):

functionvendingmachine

And I could certainly see some great Algebra I activities using the same concept. But CPM just sucks the joy and interest out of the great starting ideas it has.

Anyway. I wanted to finish up with a push for illustrations. What, exactly, do the students understand at the moment of discovery in this little activity? I’ve never seen anyone make the intuitive leap to functions. However, they do all grasp that two tasks that until that moment seemed identical…aren’t. They all realize that translating the message in the whole-alphabet code would be a simple task. They all understand why the cell phone code translation doesn’t lend itself to the same easy translation.

I look for illustrative tasks that convince kids to think about concepts. As I’ve said before, the tasks might kick off a unit, or they might show up in the middle. They may demonstrate a phenomenon in math, or they might be problems designed to lead the students to the next step.

The most common pushback I get from math teachers when I talk about this method is “I love the idea, but I don’t have enough time.” To which I respond that pushing on through just means they’ll forget. Well, they’ll probably forget my lessons, too, but–maybe not so much. Maybe they’ll have more of a memory of the experience, a recollection of the “aha” that got them there. That’s my theory, anyway.

There’s no question that telling is quicker than illustrating or letting them figure it out for themselves. Certainly, the illustration should be followed by a clear explanation with much telling. I love explaining. But I’ve stopped kidding myself that a clear explanation is sufficient for most kids.

That said, let me restate what I said in my retrospective: The tasks must either be quick or achievable. They must illustrate something important. And they must be designed to lead the student directly to the observations or principles you want them to learn. It’s not a do it yourself walk in the park. Compare my lesson on exploring triangles with this more typical reform math example. I resist structure in many aspects of my life, but not curriculum.

In researching this piece, I stumbled across this really excellent essay Why Illustrations Aid Understanding by David Kirsh. I strongly recommend giving it a read. He is only discussing the importance of visual illustrations, whereas I’m using the word more broadly. Kirsh comes up with so many wonderful examples (math and otherwise) that categorize many different purposes of these illustrations. Truly great mind food. In the appendix, he discusses the limitations of visually representing uncertainty.

kirshappendix

On reading this, I felt like my students did when they realized the cell phone message was much harder to translate: I have observed something important, something that I realize immediately is true and relevant to my work–even if I don’t yet know why or how.


Assessments with Multiple Answers

Multiple Answer Math tests are my new new thing, and I’m very pleased with how it’s going so far. I thought I’d talk about some of the problems in depth, see if anyone has suggestions.

Most of these questions come from an A2/Trig test I wrote this weekend, focusing on systems of equations, but my tests are always cumulative.

MAexamp1

One of the things I really like about this format: I can combine free response and selected responses very easily. So here they had to graph the plane, then answer questions which may or may not have to do with the graph. So I could both test their ability to graph a plane see if they understand how distance works in three dimensions, check out their attention to detail, and see if they remember what a trace is. Query: is “slope of a trace line” acceptable? I’ve never taught 3-dimensional systems before, and the book only said “trace”. But when I was teaching it, I kept forgetting and say “trace line”. I wanted them to demonstrate they could visualize the plane in three dimensions and see the slopes of the lines forming the plane, and I couldn’t find any sample questions. Probably an oddball question.

“a” and “e” contain typos. I originally had a different line, until I realized it’d be too hard to graph on the coordinates I provided. So I changed everything, or tried to. Missed two things. First, I intended “a” to be correct, but forgot to change the constant. That’s okay, it will allow for attention to detail. But “e” is just a kluge question, since I changed the points but forgot to change the distance. Before, it was a test of evaluation; now it’s a more obvious wrong answer.

MAexamp4

This question makes me very happy. Transformations, function operations, evaluation, and then a transfer of knowledge test! We’d never done any problems like “e” before. No one squawked, and I even saw some kids solving it graphically.

MAexamp3

(I stole this graph from online, but can’t find it any more. If it’s yours, let me know and I’ll provide a link.)

I tiptoed conceptually into linear programming, but we did a lot with feasibility regions and of course, systems of inequalities. I describe my approach for Algebra II, but I step it up a bunch of paces for A2/Trig. I expect them to be able to graph lines and inequalities. They get review during the modeling section, but that’s all.

MAexamp2

Another one I just think is elegant because it approaches the absolute value from so many different angles: algebraically, graphically, and then a function conceptual question for good measure.

MAPCexamp3

I use this on both Algebra II and Algebra II/Trig. We math teachers try to beat into the kids’ heads the idea that a function can be defined or expressed in four ways: verbal, algebraic, graphic, and tabular. If this were a multiple choice question, students would just test one value and see what happens. But it’s multiple answer, and plugging in numbers takes a long time. Plotting the points and sketching the lines, on the other hand, works very nicely and very quickly—if you know how to graph those lines.

Every so often you can really mess with the kids’ minds, like this:

MAexamp5

None of the “obvious” answers are right. The kids really have to trust their abilities.

MAexamp6

This is almost pure concept. I introduced the algebra of rational expressions; we’ll do the graphs later. Well over half the kids correctly selected e, but a lot missed b. Ack.

Here’s a couple that work for either pre-calc or algebra II. The quadratic runs the gamut from conceptual to technical. The circle question is more purely technical, but that’s because there’s a lot to test.

MApcexamp1

MApcexamp2

I’m having a much easier time grading these now, once I realized I was actually creating True/False tests.

Still to be resolved, however: I have to distinguish between “left the problem blank because I didn’t know” and “not true”. Right now, I evaluate the test to determine what the student is doing, but in the future I think I’m going to have a field they can mark “T” or “F”. If it’s blank, it’s wrong.

So, for example, take a look at this question again:

MApcexamp1

Answers A, D, and E are true. The others are false. I give this question 14 points, 2 for each letter.

Almost all my students correctly select A as true, because they’ve built the equation themselves as an exercise and understand the parameters. They likewise know that B is false. Some of them read “maximum” as “initial” and wrongly select C, but many otherwise weak students with good attention to detail get it correct. So even my weak students are likely to get 6 points on these three letters.

Then we get to the tougher ones (they aren’t always in order of difficulty). Students have to understand what elements of the parabola equate to max height, time to max height, and zero height. Obviously, I cover these extensively, but kids have a harder time with this. I don’t just teach them a method. I expect them to know that max height is the parabola’s vertex, so that the x value is time to max height, and the y value is the height.

I had at least 12 students who correctly factored the problem, thus correctly NOT selecting E, but also NOT selecting D. Strong technically, weak on the concept of a “zero”. I gave them partial credit (a point) and yelled at them on the paper: things like “Noooooooooo!” and “Arggggghhhh.” and arrows and question marks and “Yo! What do you think (2t-3) means, exactly!?!”

The vertex questions E and G give students the most trouble, but that seems to be less about concept and more about a reluctance to work with fractions. My algebra II students actually do better than my precalc students because we spend a whole unit on this, as opposed to a few days in precalc.

So an average weak student will get 8-10 points out of 14. Very few students get all 14 points, maybe 8 out of 60. Most get 10-12. If they show their work and I can see they were on the right track with just an algebra error, I give partial credit. Other times, I can clearly see their math was terrible, even if they got the right answer. In those cases, I mark the question correct and then dock them 2 points for bad math.

While I don’t normally review tests, I always go through these and give the correct answers and discuss grading decisions.

I strongly recommend giving these a try. They’re lots of fun to make and again, typos are a lot easier to hide.


Assessing “Upper Level” Math Students on Algebra I

A2/Trig

I am teaching Algebra II/TRIG! Not Algebra II. First time ever. Last December, I gave the kids a packet with the following letter:

Hi! I’m looking forward to our course.

Attached is a packet of Algebra I review work to prepare you for our class. If you are comfortable with linear and quadratic equations, then you’re in good shape. If you’re not, it’s time to study up!
Our course will be challenging and fast-paced, and I will not be teaching linear equations and quadratics in their entirety—that is, I expect you to know and demonstrate mastery of Algebra I concepts. We will be modeling equations and working with applied knowledge (the dreaded word problems) almost constantly. I don’t just expect you to regurgitate solutions. You’ll need to know what they mean.
I’m not trying to scare you off—just put you on your toes! But you should put in some time on this, because we will be having a test when you come to class the first full day. That test will go in the gradebook, but more importantly, it will serve as notice. You’ll know if you’re prepared for the class.

Have a great holiday.

Reminder: My school is on a full-block schedule, which means we teach a year’s content in a semester, then repeat the whole cycle with another group of students. A usual teacher schedule is three daily 90-minute classes, with a fourth period prep. I taught algebra II, pre-calc, and a state-test prep course (kids killed) last semester, and this semester I have A2/Trig and two precalcs.

(Notice that I am getting more advanced math classes? Me, too. It’s not a seniority thing. It’s not at my request. It’s possible, and tempting, to think they noticed the kids are doing well. I know the first decision to put me in pre-calc last year was deliberate, a decision to give me more advanced classes because they wanted a higher pass rate. But I honestly don’t know why it’s happening. Maybe they cycle round at this school, moving teachers from high to low and back again.)

So I said the first full day, and today was a half day, but the kids had a whole packet to work on and I wanted to understand I wasn’t screwing around. If they’d done the work, they’d do fine on the test. If they were planning on cramming, too bad so sad.

I was originally going to do a formal test, but decided to just throw a progression of problems on the board. Then I typed it up for next time, if I teach the class again.

A2PrelimAssess

How’d they do? About a third of them did well, given the oddball nature of the test. A couple got everything right. Most of them stumbled with graphing the parabola, which is fine. Some of them knew the forms (standard, point slope), but weren’t sure how to convert them.

Another three passed–that is, answered questions, showed they’d worked some of the packet. The rest failed.

Of the ones who failed, easily half of them had just blown off the packet but have the chops. The other half of that third I’m not sure of.

If you are thinking that kids in Algebra II/Trig should know more, well, they were demonstrably a step ahead of my usual algebra 2 classes. And I think some of them just didn’t know I was serious. Wait until that F gets entered, puppies. Like I told them today: “There’s a lower level option here. Take it if you can’t keep up.” Whoo and hoo.

Pre-calc

I’ve now taught pre-calc twice. The first time, last spring, I was stunned at the low abilities of the bottom third, which I didn’t really understand fully for two or three weeks, leaving some of them hopelessly behind. I slowed it down and caught the bulk of the class, with only four to five students losing out on the slower pace (that is, they could have done more, but not all that much more). So when I taught it again in the fall, I gave them this assessment to see how many students could graph a line, identify a parabola from its graph, factor, and use function notation. If you’re thinking that’s pretty much the same thing I do with the A2/Trig classes, well, yeah. Generally, non-honors version of course is equivalent of honors version of previous year.

I don’t formally grade this; the assessment happens while they’re working. I can see who stumbles on lines, who stumbles on parabolas, who needs noodging, who works confidently, and so on. I was able to keep more kids moving forward in the semester/year just ended using this assessment and a slightly slower pace. One of the two classes is noticeably stronger; half the kids made it through to the function operations before asking for assistance.

This assessment also serves as a confidence booster for the weaker kids. Convinced they don’t understand a single bit of it, they slowly realize that by golly, they do know how to graph a line and multiply binomials. They can even figure out where the vertex should be, and they might have forgotten about the relationship between factors and zeros, but the memory wasn’t that far away.

precalpreassess

While I just threw together the A2/Trig course, I put a huge amount of thought into this precalc assessment last fall. I think it’s elegant, and introduces them to a lot of the ideas I’ll be covering in class, while using familiar models.

Part II is just a way of seeing how many of them remember trig and right triangle basics:

PrecalcAssess2b

PrecalcAssess2a

If you’re interested in assessing kids entering Algebra (I or II) or Geometry, check out this one–multiple choice, easy to grade, and easy to evaluate progress.


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