White Elephant Students and Charters: A Proposal

I was re-reading a barely started essay (you don’t want to know how many I have) on reform’s bait and switch, in which I quoted Jersey Jazzman on reformers finally admitting they cream the easy to educate. This reminded me of white elephants.

Our faculty holiday party had a white elephant gift exchange . Everyone brought an item of questionable value, nicely wrapped, and turned it in for a ticket number. The person who got ticket #1 opened a present of his choice. Oh, look, it’s a mug gift with some hot cocoa mix! Oooh, ahh. Then the person with ticket #2 could either “steal” the mug gift with hot cocoa mix, or select a new present, open it, and oh, look, it’s coal in the stocking! (a joke gift, it’s candy.) Then person with ticket #3 could “steal” one of the previous gifts, and so on.

Each person could steal a previous gift or take a new present. But once a gift has been stolen, it’s off limits.

I very much enjoyed this game because my proffered white elephant, a 9 year old digital photo frame that sat in my trunk for six years before I finally needed the room and stuck it in a closet through three moves until I happened to be cleaning out the closet 3 days before the party, was stolen! Someone wanted it! I felt very high status, I can tell you. Plus, I stole a gift when my turn came. All this and lumpia, too. A great party.

And so the white elephant metaphor stood fresh in my mind, ready to hand when I reviewed that draft essay. I’ve been trying to write about this topic forever, specifically about the restraints public schools face with disruptive students. (Charters aren’t public schools. They just use public money. ) But like many issues I feel strongly about, the essay began life as a cranky rant. I do better with humorous rants, so I abandoned delayed the effort.

But thanks to the faculty party, I’m ready to take this on.

Charter advocates’ constraint: caps. They want more schools.

Public school constraint: laws. They are bound by laws that charters can ignore or game, and bound by law to hand their district kids and associated monies over to charters, who aren’t bound by those law when they kick some students back, with no feds chasing after them for racially imbalanced rejects.

So publics can’t reduce their unmotivated misbehaving population; charters want more room to grow because, after all, they provide a superior education.

And it came to me: let public schools create white elephant students, by making a “gift” of a disruptive, unmotivated student, something the public school has and doesn’t really want.

Give public schools the right to involuntarily transfer up to 1-3% of their students to charter schools in their geography, with the limit set by the number of available charters. “Involuntary” to both the students and the charters, neither of whom are given any say in the matter.

In exchange, charter caps are significantly increased.

Involuntary transfer, not an expulsion. Students have rights in an expulsion hearing. White elephant students have no say in an involuntary transfer. Parents couldn’t appeal. They can accept the assigned school or try to convince another public school or charter to take their student, now identified as difficult.

But remember the other condition of white elephants gifts: they can’t be handed about indefinitely. Parents “gifted” the public schools, public schools “gift” charters. Game ends. The receiving charter has no involuntary transfer rights for that student. The transfer occurs without regard to the charter population limits or backfilling preferences.

Moreover, the transferred students maintain their public school protections. The charters can’t refuse admission in subsequent years. Unless the students can be expelled, the charters are stuck until the transfers age out or graduate. This restriction means that some kids at charter schools would have more rights than others. Welcome to public education, folks. Public schools have been dealing with this tension for decades.

So public schools would continue to have no choice on incoming students within their districts, but would win a (limited) choice to send students away. Charters would continue to have considerable selection benefits on incoming and outgoing students, but would lose those benefits with a few students.

Logistical issues would need ironing out. Transportation comes immediately to mind, as do actual numbers on transfer limits, but I’m sure others would show up.

Ironically, given the name, the white elephant students would be almost entirely black and Hispanic. Literally and figuratively, that’s where the money is. White and Asian districts aren’t facing heavy competition for their students. Billionaire philanthropists don’t give a damn about poor white kids, which is one big reason why West Virginia’s charter ban doesn’t attract a lot of interest. We could speculate why (perhaps they aren’t really interested in educating kids, just killing teacher unions), but never mind that.

Parents of white elephant kids would lose any real sense of school choice. Sorry about that. But at least the kids will be at a charter, with far fewer peers to help them get in trouble.

On the other hand, the white elephant kids would have a real incentive to behave better in public school. They’d see charters as a real threat. “Behave or I’ll send you to a school that makes you SLANT!”

Public schools would see this purely as win-win. They’d still lose money on the transferred students. This incentive, coupled with the involuntary transfer cap, will limit their desire to cavalierly toss out kids for minor offenses. But even if publics did act capriciously, what would the feds say? “I’m sorry, but you are dooming these children by sending them to a charter school, trapped with well-behaved children in smaller classes!”

Never mind whether or not it could be enacted as policy; consider the white elephant proposal purely as a thought experiment, because everyone knows this is true: Charter operators, the highly regarded “lottery” schools, would reject this proposal out of hand.

Why? Because KIPP failed miserably the one time it tried to turn around an existing school. Because to get the results that reformers brag about, charter schools have to control their student population: selection bias at the start, sculpting as needed, uniform learning schedule.

But this proposal on the surface makes perfect sense, based solely on the reform and choice rhetoric over the past decades. Charters have absolutely no grounds for bitching. They want the caps lifted, they want to end charter bans. They’ve been bragging about their superior schools for twenty years. They swear they aren’t creaming, aren’t selecting, aren’t cherrypicking. Great. This policy gives charters everything they want, in exchange for educating students they claim they could educate in the first place. What do they have to lose?

As Jersey Jazzman and countless others have pointed out, this makes a lie out of their boasts. They aren’t getting better results than public schools; they just have better kids and fewer laws to follow.

Now, just for fun, pretend that charter operators took the deal: the occasional mandated student in exchange for additional growth.

Motivated students are desirable, but without the guarantee of high scores, they aren’t in and of themselves a competitive strategy. White elephant students, in contrast, are ideal for horsetrading.

Public schools can designate white elephants only to the extent that charters exist to receive them, and based on the number of public schools affected. So, imagine a district with three elementary schools: one high poverty, two low poverty. When a new elementary charter opens, the state declares that three white elephants per grade per school are allocated for dumping transferring to the charter. The charter primarily skims from the high poverty school. But the other two elementary schools don’t want charters popping up, and see an advantage in a hostile environment, so they “gift” their allocations to the high poverty school, which can now move nine white elephants per grade.

The “lottery” charters will naturally want to opt out of this involuntary transfer program. Sure! For a small fee, of course. How about shaving off 50% of per-student fees charters get for their willing transfers? In that case, the charter would be doing less damage to the public schools by creaming. Moreover, any charter that publicly opted out of the involuntary transfer program has revealed its Achilles heel. Choice advocates couldn’t maunder on endlessly about the superior education charters offered if all the best ones paid to cherrypick.

To recap:

1. Public schools restricted from selecting their students can use an involuntary transfer mechanism to move troublesome students creating disruptive learning environments to charters.
2. The maximum number of students subject to involuntary transfer depends on school and charter populations.
3. Public schools can trade or gift their transfer vouchers to other district schools.
4. Charter growth caps are significantly increased.
5. Charters required to give full weight of education law to white elephant students.
6. Charters can opt out of involuntary transfer program by accepting substantially reduced per-student fee for voluntary charter attendees.

How would this play out, given some time?

Long term, the white elephant program could ironically limit charter growth. The fewer the charters, the fewer involuntary transfers possible. One charter could probably handle 3-4 white elephants per grade without sacrificing too much control and wouldn’t take too many motivated students to damage the public schools in the area. Additional charters, each taking 5-6 troublemakers? Suddenly the charters are struggling with difficult students while the public schools have considerably improved environments, potentially enabling them to lure many prospective charter students back. The fewer charters, the less likely the public schools can dump all their white elephants.

But then, many charters aren’t choosy and don’t have lotteries. They need butts in seats, and could use the white elephant students as a growth strategy. Hire teachers who specialize in handling tough kids, advertise for desperate parents, take the public school white elephants and expulsions. Win win for everyone. Collaboration, not competition. In fact, districts would probably set up their own white elephant charter school, in absence of an outside enterprise for their own schools to use as an outlet. Alternative high schools, you ask?Best avoided.

In an environment where white elephant charters work synergistically (oooh! Big word) with district public schools, any other charters would have to compete with public schools on merits, without the added appeal of “no knuckleheads”. That, too, is going to limit growth.

And of course, it’s entirely possible that typical charters–no excuses, discipline oriented, progressive, whatever–accept white elephants and the disruptive kids thrive. In many cases, disruptive, unmotivated kids with no other options improve in a stricter environment, or perhaps one with a higher percentage of motivated students.

However, this outcome is only likely in a district not drowning with white elephants—that is, a suburban district. Suburban charters operate under entirely different premises, geared towards a progressive curriculum and a “diverse” student population. Suburban districts consider charters an annoyance and an aggravation, not a threat. So if they can dump some white elephants on the earnest do-gooders, it’s all good.

I could go on, but the New Year approaches and this piece is long enough. One final point, for any new reader who comes across this piece: I am kind of the go-to math teacher for low ability and/or poorly motivated kids. This isn’t personal; I don’t have a gift list of white elephants.

But I’ve said before now that I stick with the suburban poor, because when Ta Nahesi Coates casually describes the disruption he routinely inflicted on his high school classes, threatening substitutes, disrespecting teachers while getting violent at any hint of disrespect (and remember, none of his friends or family considered him a “thug”), I get slightly ill at the utter chaos that must have reigned in his school. So I work in Title I suburbs, where my daily tales shock my friends with the disrespect and disruption my students dole out daily, while I know full well it ain’t all that.

Meanwhile, all the signals are pointing in the opposite direction, what with federal discipline “guidelines” and that god awful spare me restorative justice nonsense.

So let’s try gifting. After all, it’s the thought that counts.

Assessing Math Understanding: Max, Homer, and Wesley

This is only tangentially a “math zombies” post, but I did come up with the idea because of the conversation.

I agree with Garelick and Beals that asking kids to “explain math” is most often a waste of time. Templates and diagrams and “flow maps” aren’t going to cut it, either. Assessing understanding is a complicated process that requires several different solutions methods and an interpretive dance. Plus a poster or three. No, not really.

As I mentioned earlier, I don’t usually ask kids to “explain their answer” because too many kids confuse “I wrote some words” with “I explained”. I grade their responses in the spirit given, a few points for effort. “Explain your answer” test questions are sometimes handy to see if top students are just going through the motions, or how much of my efforts have sunk through to the students. But I don’t rely on them much and apart from top students, don’t care much if the kids can’t articulate their thinking.

It’s still important to determine whether kids actually understand the math, and not just because some kids know the algorithm only. Other kids struggle with the algorithm but understand the concepts, Still others don’t understand the algorithm because they don’t grok the concepts. Finally, many kids get overwhelmed or can’t be bothered to work out the problem but will indicate their understanding if they can just read and answer true/false points.

If you are thinking “Good lord, you fail the kids who can’t be bothered or get overwhelmed by the algorithms!” then you do not understand the vast range of abilities many high school teachers face, and you don’t normally read this blog. These are easily remediable shortcomings. I’m not going to cover that ground again.

So how to ascertain understanding without the deadening “explain your answer” or the often insufficient “show your work”?

My task became much easier once I turned to multiple answer assessments. I can design questions that test algorithm knowledge, including interim steps, while also ascertaining conceptual knowledge.

I captured some student test results to illustrate, choosing two students for direct comparison, and one student for additional range. None of these students are my strongest. One of the comparison students, Max, would be doing much better if he were taught by Mr. Singh, a pure lecture & set teacher; the other, Homer, would be struggling to pass. The third, Wesley, would have quit attending class long ago with most other teachers.

To start: a pure factoring problem. The first is Max, the second Homer.

Both students got full credit for the factoring and for identifying all the correct responses. Max at first appears to be the superior math student; his work is neat, precise, efficient. He doesn’t need any factoring aids, doing it all in his head. Homer’s work is sloppier; he makes full use of my trinomial factoring technique. He factored out the 3 much lower on the page (out of sight), and only after I pointed out he’d have an easier time doing that first.

Now two questions that test conceptual knowledge:

Max guessed on the “product of two lines” question entirely, and has no idea how to convert a quadratic in vertex form to standard or factored. Yet he could expand the square in his head, which is why he knew that c=-8. He was unable to relate the questions to the needed algorithms.

Homer aced it. In that same big, slightly childish handwriting, he used the (h,k) parameters to determine the vertex. Then he carefully expanded the vertex form to standard form, which he factored. This after he correctly identified the fact that two lines always multiply to form a quadratic, no matter the orientation.

Here’s more of Homer’s work, although I can’t find (or didn’t take a picture of) Max’s test.

This question tests students’ understanding of the parameters of three forms of the quadratic: standard, vertex, factored. I graded this generously. Students got full credit if they correctly identified just one quadratic by parameter, even if they missed or misidentified another. Kids don’t intuitively think of shapes by their parameter attributes, so I wanted to reward any right answers. Full credit for this question was 18 points. A few kids scored 22 points; another ten scored between 15 and 18. A third got ten or fewer points.

Homer did pretty well. He was clearly guessing at times, but he was logical and consistent in his approach. Max got six points. He got a wrong, got b, c, & d correct, then left the rest blank. It wasn’t time; I pointed out the empty responses during the test, pointing out some common elements as a hint. He still left it blank.

On the same test, I returned to an earlier topic, linear inequalities. I give them a graph with several “true” points. Their task: identify the inequalities that would include all of these solutions.

(Ack: I just realized I flipped the order when building this image. Homer’s is the first.)

Note the typo that you can see both kids have corrected (My test typos are fewer each year, but they still happen.) I just told them to fix it; the kids had to figure out if the “fix” made the boundary true or false. (This question was designed to test their understanding of linear concepts–that is, I didn’t want them plugging in points but rather visualizing or drawing the boundary lines.)

Both Max and Homer aced the question, applying previous knowledge to an unfamiliar question. Max converted the standard form equation to linear form, while Homer just graphed the lines he wasn’t sure of. Homer also went through the effort of testing regions as “true”, as I teach them, while Max just visualized them (and probably would have been made a mistake had I been more aggressive on testing regions).

Here I threw something they should have learned in a previous year, but hadn’t covered in class:

Most students were confused or uncertain; I told them that when in doubt, given a point….and they all chorused “PLUG IT IN.”

This was all Max needed to work the problem correctly. Homer, who had been trying to solve for y, then started plugging it in, but not as fluently as Max. He has a health problem forcing him to leave slightly early for lunch, so didn’t finish. For the next four days, I reminded students in class that they could come in after school or during lunch to finish their tests, if they needed time. Homer didn’t bother.

So despite the fact that Homer had much stronger conceptual understanding of quadratics than Max, and roughly equal fluency in both lines and quadratics, he only got a C+ to Max’s C because Homer doesn’t really care about his grade so long as he’s passing.

Arrgghhh.

I called in both boys for a brief chat.

For Max, I reiterated my concern that he’s not doing as well as he could be. He constantly stares off into space, not paying attention to class discussions. Then he finishes work, often very early, often not using the method discussed in class. It’s fine; he’s not required to use my method, but the fact that he has another method means he has an outside tutor, that he’s tuning me out because “he knows this already”. He rips through practice sheets if he’s familiar with the method, otherwise he zones out, trying to fake it when I stop by. I told him he’s absolutely got the ability to get an A in class, but at this point, he’s at a B and dropping.

Max asked for extra credit. He knew the answer, because he asks me almost weekly. I told him that if he wanted to spend more time improving his grade, he should pay attention in class and ask questions, particularly on tests.

We’ve had this conversation before. He hasn’t changed his behavior. I suspect he’s just going to take his B and hope he gets a different teacher next year who’ll make the tutor worth the trouble. At least he’s not trying to force a failing grade to get to summer school for an easy A.

I was extremely direct with Homer,  expressing (snarling) my disappointment that he wouldn’t make the effort to be excellent, when he was so clearly capable of more. What was he doing that was so important he couldn’t take 20 minutes or so away to finish a test, given the gift of extra time? Homer stood looking a bit abashed. Next test, he came in during lunch to complete his work. And got an A.

Max got a B- on the same test, with no change in behavior.

I haven’t included any of the top students’ work because it’s rather boring; revelations only come with error patterns. But here, in a later test, is an actual “weak student”, who I shall dub Wesley.

Wesley had been forced into Algebra 2, against his wishes, since it took him five attempts to pass algebra I and geometry. He was furious and determined to fail. I told him all he had to do was work and I’d pass him. Didn’t help. I insisted he work. He’d often demand to get a referral instead. Finally, his mother emailed about his grade and I passed on our conversations. I don’t know how, but she convinced him to at least pick up a pencil. And, to Wesley’s astonishment, he actually did start to understand the material. Not all of it, not always.

This systems of equations question (on which many students did poorly) was also previous material. But look at Wesley! He creates a table! Just like I told him to do! It’s almost as if he listened to me!

He originally got the first equation as 20x + 2y = 210 (using table values); when I stopped by and saw his table, I reminded him to use it to find the slope–or, he could remember the tacos and burritos problem, which spurred his memory. You can’t really see the rest of the questions, but he did not get all the selections correct. He circled two correctly, but missed two, including one asking about the slope, which he could have found using his table. He also graphed a parabola almost correctly, above (you can see he’s marked the vertex point but then ignored it for the y-intercept).

He got a 69, a stupendous grade and effort, and actually grinned with amazement when I handed it back.

Clearly, I’m much better at motivating underachieving boys than I am “math zombies”. Unsurprising, since motivating the former is my peculiar expertise going back to my earliest days in test prep, and I’ve only recently had to contend with the latter. However, I’ve successfully reached out and intervened with similar students using this approach, so it’s not a complete failure. I will continue to work on my approach.

None of the boys have anything approaching a coherent, unified understanding of the math involved. In order to give them all credit for what they know and can do, while still challenging my strongest students, I have to test the subject from every angle. Assessing all students, scoring the range of abilities accurately, is difficult work.

As you can see, the challenges I face have little to do with Asperger’s kids who can’t explain what they think or frustrated parents dealing with number lines or boxes of 10. Nor is it anything solved by lectures or complex instruction. My task is complicated. But hell, it’s fun.

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).

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.

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:

“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.)

“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(³⁄₅). “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 ^-33⁄₆₅“.

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.”

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(^-33⁄₆₅))), which is just using the inverse to find the degree measure and taking the tangent in one step. Shazam.”

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.

“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.