Teaching Oddness #1: Teacher’s Aides, HS Version

Do outsiders know what TAs are? I went looking for research on this point, and could find none. These descriptions aren’t accurate, and most of the rest refer to employed teacher aides.

Teacher’s Aide is a student elective “class” in which the student provides the teacher with free labor as needed.

I get a bit stalled here, because the same practice can be used for neutral or ill. Arguably, there’s no “good”.

Neutral: Why would any student sign up to be a gofer? It’s not for the resume value, I assure you. But high school students are required to take a full slate of classes, and electives are in limited supply. So at a certain point, a mid-tier student with a good GPA but every intention of going to a junior college is left with no appealing electives. Every semester, students with schedule holes have to find somebody to work for, or they’ll get stuck in an actual class with responsibilities and grades, a class they have no interest in taking.

Some of them are assigned to run errands for the front office, taking notes out to the teacher rooms and back for counsellor call-outs, direct mail delivery to students, getting a teacher’s signature on a document, whatever. But schools only need three or four office TAs.

The rest of the students beg teachers to take them on as either doorstops or free labor, in exchange for an A. Because TA jobs get graded, and any grade less than an A raises eyebrows. (Colleges exclude TA from GPA calculation.)

Admins spend some serious cycles on TA assignment. First, the notes come out telling us that no teacher can have two TAs per class. (Yeah, what? OK.) Then out comes the notes begging teachers to take some TAs that still don’t have assignments. Then, occasionally, a TA shows up at the door with an administrator and a question, “Can you use a TA?” and while the answer would otherwise be “No”, the administrator keeps asking until the teacher says “Yes”. Then hours are spent entering these into the schedule for attendance and assignment and transcripts.

I’ve concluded tentatively that the student TA system is both a significant source of free labor to teachers and schools, and a non-trivial burden for teachers and administrators when willing users of that free labor can’t be found.

Many teachers, those teachers who come in each day with their task list set a week, or a month, a year, or three years ago—these people with a plan, they love TAs. Good, yes, I have a million little tasks to be done. Grade this quiz I created three five years ago, then enter the test scores. Create my bulletin board decorations, using this design. These are the ones who have to be told they can’t have more than two TAs per class.

I’m the teacher who was forced, year one, to take a TA. The AVP showed up at my door, just as described above. I was 4 days into my first job, and already knew I had no use for a TA, especially when they told me I couldn’t use him for the single most essential task eligible for delegation: copying.

That’s the insane part: WE CAN’T USE THEM TO MAKE COPIES. Moreover, no one seems to think that, so long as we have all this student labor going begging, a COPY CENTER MIGHT BE A GOOD IDEA. Nothing causes teachers more unforced stress than needing the copy machine when it’s broken or unavailable, thanks to a 20-teacher queue in the morning or lunch. A colleague and friend once took up lunch running 50 sets of 30 page documents. When a week later she announced her transfer, I told her she wouldn’t be missed. I wasn’t entirely joking.

What was my point? Oh, yes. So I had a TA year one who sat at my desk and surfed the web. He wasn’t a bad kid, although I can’t remember his name. His only responsibility was to sit in a nearby room during tests with my top kids, as my room wasn’t big enough to hold thirty kids and cheating was rampant. Pulling out the top kids ended that little game.

I don’t remember TAs being available at my next school, but I just texted a former colleague and will update this space.

My current school happily doesn’t pressure teachers to take TA, but who needs administrative pressure when students apply guilt? My first year at this school, one girl begged me to let her TA. She showed up late and texted each day. I vowed never to be suckered again. Except I did the next year, when a stoner begged me to let him TA my pre-calc class. He was worse than useless, but a better conversationalist than the girl, so there’s that.

But then, it all changed. Last year, Rufus, an exchange student and a top performer in my trig class, convinced me to let him TA, and then another favorite football player, Ronnie, begged me for a chance. I figured I may as well spend time with students whose company I actually did enjoy.

Rufus worked with my students, paying a little too much attention to cute girls, but with that exception, he was very good. Ronnie wasn’t as good in math and definitely liked distracting cute girls, but one day he volunteered to clean up my office space. Kid worked like a fiend, and no one recognized my room.

As I mentioned, Year 6 was busy and here, I have to break off a bit to explain something.

Full-metal, 4×4 block has killed my love of grading. We cover a year’s worth of instruction by the end of January. My assessments are difficult, and I’d rather give them less often, but I pretty much have to give a test or quiz every five or seven days to have grades for progress reports. Since I’m designing a new test system, I was spending much more time building and grading assessments already.

And that was before year 6, when I had two new subjects (trig and history) and three preps (subjects taught), four classes (no free prep period) and 110 students in the second semester.

My returned test lag time was now over a week, which really nagged at me. My work life was becoming something like create a test, created a key, grade a test, enter the grades, turn the test back, lather rinse repeat. Amd that’s without all the curriculum for the new classes. Mind you, this is a typical teacher complaint, but this is all work I typically enjoy. I was just running out of cycles.

Rufus was taking my history class as well as operating as my TA, and knew how slammed I was. He offered to grade. By this time, he’d proven himself reasonably trustworthy, so I decided to risk it. I’d create the key and the point system, have him grade a few samples, and then let him go.

Wow. Huge difference. I still reviewed the grading, adding or knocking off points, but time spent was cut from six to one hour. Rufus bragged to Ronnie (does this sound like Highlights?), who demanded he be trusted with grading as well.

Ronnie and Rufus provided the first really positive TA experience I ever had. I took them out for Starbucks at year-end, and am still in touch with both.

Last semester, Jacob, also from a previous trig class, asked if he could TA and I asked him if he minded grading. I worked Jake so hard I gave him cookies and Starbucks cards for Christmas, and told him I’d violate regulations to give him some service hours if he needed them. Jacob saved me dozens of hours. I couldn’t get over how I could use student labor to make my life easier.

This semester I have three, count ’em THREE, TAs: all previously successful students, all aware when they signed on that they’d be expected to grade or, occasionally, help students. I put one in each algebra 2 class.

I’m less conflicted about having three TAs cover the work than I was giving it all to Jake. While he didn’t seem to mind, I was bothered by the idea that one person was contributing so significantly to my workload reduction. Somehow three kids making life easier for me doesn’t seem as bad. When it was just Jake (heh) it took about 4 days to grade 50 tests, even with my working as well.

To illustrate how much they’ve decreased my workload, we’ve just done the cutover at mid-term, which is always tough. We have to both finish final grades while starting brand new classes with brand new kids. I’m teaching all four classes again, no prep, and 109 algebra 2 students, with another 20 in my geometry class.

I gave the first quiz on Tuesday. Each of the three TAs graded a class set, aand I had the grades in the book on Friday. Unreal. On my own, I probably wouldn’t have had that first quiz done for another week. Instead, I was able to review the tests, see who scored well on my pretest but tanked the function quiz, and vice versa. I’ve got time to redo seating, catch low scores early, call kids in to fix misconceptions. It’s great. My TAs also chide me on the state of my desk, and pressure me to collect all the papers and dump most of them, after review. I’m always worried I’ll toss something important.

I also enjoy talking to my TAs, who I chose because I liked and knew that I might be able to help in some way. I have at least once good talk with them a week, and advise them on college choices, course choices for the upcoming year, whatever.

But I can’t get over the fact that I’ve been freed from so much work without it costing me anything. I’m not alone, I know; many teachers brag about how much they turn over to TAs. I also remind myself that many teachers use scantrons and multiple choice tests. I spend substantial hours developing good tests, and I still review and evaluate all the tests before they are returned. I’m new to this; give me a few years and I’ll probably be one of those teachers griping I can only have two TAs a class.

So there’s the neutral use.

But the TA position can often be used to cover up scheduling shortfalls. As mentioned, schools are legally required to give students a full schedule of classes. In many struggling schools, the administration can’t keep enough teachers to offer all the classes needed, and so they use the TA slot as a stopgap. I very much doubt schools use this as a source of cheap labor for teachers, but many kids just can’t get the credits they need to graduate. I’ve mentioned before that the district controls the catalog; the catalog controls what can be assigned, so if a district offers “teacher’s aide” or “independent study” then they can use it to cover up a multitude of sins.

As I said, I can’t really make a “good” argument for TAing–at its best, it’s a way for kids to get out of taking a class and make some extra money by selling advance copies of tests. At its worst, schools use them to keep their doors open, rather than flatly refusing to fake it. We need more high schools refusing to take students, putting pressure on districts and states to address the problem.

Why is there so little data readily available about school’s “hidden work force”? Many tasks could undoubtedly be automated, particularly the office TAs. But then, strawberry farmers, schools will only automate when they lose cheap, free labor.


Note from a Trump Supporter: It’s the Immigration, Stupid!

(Or a la Dave Barry, “It’s the immigration, zitbrains!”)

Ann Althouse predicts a cascade of smart, educated Trump supporters in the coming months. I am kinda sorta in the ballpark of smart and educated–for a teacher, anyway—and came out early for Trump. So I thought I’d take a break from my usual education beat1 and add my voice to the many efforts to explain my people.

Why do I support Trump?

I want another forty year pause in immigration, putting a near-total block on every possible means of legal or illegal access. In part because I’m a teacher who sees no opportunities for far too many of my students thanks to immigration, network hiring, and the constant wage pressure of a never-ending unskilled labor supply. In part because the government is incapable of enforcing the laws so necessary to our national security and well-being, since even the best-intentioned state and federal employees see themselves as providing customer service, rather than ensuring taxpayer and citizen interests.

Finally, I want to turn the flood of immigrants to something less than a sprinkle because the influx is fraying America’s cultural fabric. Immigrants sensibly exploit our cultural and political mores to their advantage, usually without malice or intent to harm. They are supported by legal interpretation of laws that simply weren’t written with any consideration of non-Western cultures. Few of the countries sending us immigrants share American values.

I’m willing to negotiate. But in order to negotiate, shutting down access through visa restriction and border enforcement (land, sea, or visa overstay) has to be speakable. For the past twenty years, the cosmopolitan elite, as Sean Trende calls it, has deliberately shrunk and shifted the Overton window for immigration by punishing opinion violators with social and economic devastation. Ordinary people like me who come out for immigration restriction could lose their jobs. I don’t mind anyone opposing my immigration goals. I mind the attempts to shut down and ruin those who support them.

I don’t hate immigrants. Like all people, they range from fantastic to criminal to every possible characteristic in between. But their merit is not the issue.

Americans deserve a vote on every aspect of immigration. For thirty years or more, the public has opposed the generous federal immigration policy, rarely getting a chance to register their disapproval—and on the rare occasion when they were given a chance to express their opinion, the courts consistently overturned their effort.

The government and the media also conspire to present immigration as a shiny wonderful gift to the country, opposed only by a few nativists and xenophobes, withholding unpleasant facts and generally operating as cheerleaders and gatekeepers.

At present, 25% of the country support deportation and a wall with no immigration at all, with another 30% supporting a wall and very limited immigration, with deportation optional. Yet no major media outlet, no politician joins Trump in catering to that view. Why not? Doesn’t the media want eyeballs, the politician votes? I’ve concluded that the wall of silence is partly ideological, partly fear of repercussions from the powerful. But I don’t know.

What I do know is that Trump comes along, supports just the tiniest fraction of my agenda, and the media and political world goes wild trying to shut him down. They fail, and in that failure, everything changes.

Immigration wasn’t expected to be anywhere on the horizon this election. And certainly, the media has done everything to keep it out of the debates. The topic barely made it into the GOP debates, on weak-tea issues that barely scratch the surface. We saw Rubio and Cruz arguing not about reducing immigration, but which one had flipflopped on amnesty—which they both supported until quite recently, along with all the other GOP candidates, in the world Before Trump.

On the other hand, immigration hasn’t made the platform much at the Democrat debates, either. No rhetorical flourishes on Republican iniquity towards immigrants, no yammering about the Dream Act, no long tirades on the plight of Syrian refugees. The Democrats looked at Trump’s poll numbers and other recent events (Eric Cantor’s unemployment, for example), and got the hint. They’re worried enough that Trump’s immigration and trade talk might peel away their union vote. No one’s making big promises about immigration on the Democratic side.

I’m well aware that Trump’s actual beliefs on immigration, as reflected in his stump speech and, presumably, his private views, are considerably more welcoming than his satisfactory official policy position, but think it unlikely he’ll do a general election pivot. If he were to win the nomination and pivot against restriction, he’ll lose the general. Full stop. The Donald doesn’t need me to point that out.

He probably doesn’t feel this way, but from my standpoint, Trump has already won. From the moment his polls rose after NBC fired him, after Frank Luntz’s idiotic focus group said Trump crashed and burned, after many experts declared him a nuisance,a clown, a bad deal, a a false conservative and through the re-evaluations of his appeal (but not his chances), Trump has understood the strength of and reason for his appeal. He never worried about the media, didn’t give a damn about elite approval. Every additional day puts the hammer on the media and the political elite who have suppressed any discussion, much less a vote, on the issues so many Americans care about.

So Trump’s willingness to court social and economic punishment has already paid off by giving Americans a chance to show how utterly on board they are with limiting immigration. He has kicked the Overton window several notches back to center, and I’ll be forever grateful.

Excellent analyses of Trump’s success abound, but they all suggest Trump’s rise is due to a variety of factors. I believe this is wrong.

Without immigration, Trump is nowhere.

His call to “bring jobs back home” wouldn’t be nearly as appealing if voters were worried all those jobs would go to cheap immigrant labor. Yes, his ferocious assaults on political correctness and elite sensibilities are attractive, but more importantly, they are essential for withstanding the media and political assault that followed his proposal. Hit him, and he’ll hit right back, upping the ante and distracting attention from the original charge with increasingly outrageous insults. Had Trump stoically stayed on message, politely trying to explain his way through the outrage, he’d have been gone before Labor Day. I’m delighted that he’s rendered the media helpless in its self-appointed task of destroying people for the wrong opinions, but that’s not why he’s doing so well.

Without immigration, Trump is just a billionaire dilettante politician with good timing, a populist touch and big hair.

This election has been amazing.

For the past six or seven months, I’ve been watching, waiting for Trump to cavil or backtrack on the essentials, holding my breath. And instead of disappointment, I’ve had the ….really, the only word for it is elation…as I watched the frustration, the astonishment, the fury at Trump’s success. Watching George Will’s head explode is—forgive me—exhilarating. Watching the Republicans–some I count among my favorite writers and thinkers–who called me stupid and desperate eat crow time and again after their earlier assurances of the desperate idiocy of Trump supporters and his imminent decline has brought me so much joy.

But my personal satisfaction aside, these Republicans’ shock and dismay at the depth of Trump’s support is a necessary first step if the country’s going to change its immigration ways, because change has to come via the GOP.

I don’t know what will take Trump down, if anything does. He’s created a seismic impact just getting this far, and I’m not going to count the effort wasted if it all ends in Iowa, or at some future state primary. I sense it will not. I think those who, like me, have longed for the chance to be a single-issue voter, are going to come out in droves.  I hope enough Americans will vote on this issue to put him over the top.

But if he wins the primary to lose the election, then my side doesn’t have enough votes yet. So be it. Sing me no sad songs about the Supreme Court. I worry about Democrat nominees, yes, but conservative or liberal, the Court doesn’t seem interested in protecting the nation’s borders. Maybe this last executive fiat pushed them too far. If Clinton gets elected, the GOP Congress can just get serious about the “consent” part of its job.

Recently, Ramesh Ponnuru declared that immigration issues are the new conservative litmus test.

Wrong. I’m not conservative. I’ve supported Republicans for a decade not with any particular enthusiasm, but because the GOP politicians have on most issues reliably opposed Democrats in their brand of crazy. It’s not Ronald Reagan or William F. Buckley that has me voting GOP; it’s Nancy Pelosi, Al Gore, and Barack Obama, along with the causes they espouse.

The GOP has been pandering its electorate on immigration for long enough. What I guess the Republican elite didn’t understand until now is just how many GOP voters were, like me, pandering right back. We don’t really support the GOP’s goals intellectually or emotionally, but what the hell, if we vote for them, maybe our turn will come.

Trump is our turn.

******************************************************
1To my regular readers: I understand you range from liberal teachers to alt-right HBDers and everything in between; I’m not assuming a friendly audience. Feel free to fulminate.


The Myth of the Teacher Leader, Redux

To understand why outsiders don’t grasp teacher quality in meaningful terms, consider this list.

Which of the teachers described here are leaders, working with their colleagues to improve school quality? Which ones are speaking out in support of improving the professional community? Which ones forge the way to a new professional concept? Which ones have a clear vision of their teaching identities? Which ones are committed to student achievement? Which ones, in Rick Hess’s phrasing, are cagebusters?

Ignore trifling matters like whether or not the teachers agree with your own values and priorities. Focus on leadership, caring, professional commitment. Yes, this makes it a more difficult task.

  • Teachers work late into the evening developing curriculum and planning instruction, but violate their contractual obligations by occasionally or consistent tardiness to staff and department meetings. Their timely colleagues see the tardy attendance as unprofessional. But the gripers often leave campus three minutes after last bell.

  • An attendance clerk wonders why a student is skipping first period each day for two weeks. The clerk contacts the other teachers and learns that he’s actually been absent in all classes, but that unlike the first period teacher, they’ve been letting it slide, since the student told them he was joining the Marines. Further investigation reveals that the student was on a cruise. Shortly after this incident, the principal announced that failure to take attendance and submit completed attendance verification reports would be made an evaluation point, if needed.

  • A new teacher is confused as to what responsibilities are held by the “department head” and a “math coach”, since neither approached to offer assistance when he began his job. The teacher who did approach him with help (and has no official role) told him not to worry about it, as department “leadership” roles are meaningless.

  • A few senior math teachers informally agreed to improve advanced math instruction by holding students to higher standards and a demanding pace. A new teacher was brought on, who taught at a slower pace and had a much wider “passing window”. The senior teachers requested that the new teacher be fired. The principal refused. One of the senior teachers left. The newer teacher continued with the same priorities.

  • A team of teachers and counselors are enthusiastically discussing methods to convince colleagues to comply with a new district-wide initiative. One team member cautions against mandated compliance, suggesting they accept cynicism and caution as logical responses. The team decides to go much more slowly, realizing that they can’t really enforce compliance anyway. They introduce a smaller initiative that builds on existing interest, hoping to win more compliance through results.

  • A second-career teacher works unceasingly to help at-risk students get to college, achieving a decade or more of success getting first-generation kids to college. He is a valued and highly respected leader in the teaching staff—right up until he confesses to inappropriate contact with a student. He is arrested and fired.

These examples all reveal why Rick Hess’s 90-10 split makes no sense:

…[W]hat’s happened is to a large extent…there are these teachers out there who are doing amazing things and speaking up, there are lot of teachers who are just doing their thing in the middle, and then you have teachers who are disgruntled and frustrated. These teachers in the backend, the 10 percent, they’re the teachers the reformers and policymakers envision when they think about the profession. They’re the ones who are rallying and screaming and writing nasty notes at the bottom of New York Times stories.

Hess never says so, but presumably we are to assume that the “amazing teachers” are moving test scores, while the disgruntled, frustrated teachers demanding more money are out there on the picket lines, demonstrating against Eva or taking time off to bitch in Madison, while their students sit in a dull stupor.

Would that the dichotomy were that simple. Dots can’t be connected between teaching ability and political activism. The street corner screamers protesting merit pay and standardized testing might just as easily be the ones working until 9 at night, building memorable lessons. The slugs who check out each day at 3 using the same tests year after year might have worshipful students. The former teacher who cries on cue as a paid hack for Students First might actually be less admired than the much loved teacher identified as incompetent based on a single student’s opinion. (I am always flummoxed that reformers think anyone other than the already converted would find Bhavina Baktra compelling.) Political activism is one of the utterly useless proxies for teacher quality.

Teacher Quality–what is it, exactly?

What makes a good teacher? Let us count the many ways that broad circles can’t safely capture and identify teaching populations.

  • An engaging, creative teacher can be a terrible or indifferent employee, showing up to meetings late, missing supervisories, forgetting to submit grades on time.
  • An uninspiring or incompetent teacher can be a fabulous employee, impeccably on time with contract obligations: grades, attendance, and assigned tasks.
  • Teachers of any instructional or employee quality can be activists fighting against reforms they see as damaging to either their jobs or children—or on the reform payroll (yes, it does seem that way to us) pushing for merit pay or an end to tenure.
  • An ordinary, somewhat tedious teacher can have an outstanding attendance record, while a creative curriculum genius misses ten or more days a year;
  • Unlawful teachers–from the extremes of unthinkable sexual behavior to the seemingly innocuous falsification of state records—are, often, “good teachers” in the sense that reformers intend the word. (Just do a google on teacher of the year with any particular criminal activity.

No objective measure or criterion exists for teaching excellence. At best, most might agree on its display. Were a thousand people to watch a classroom video, they might agree on the teacher’s displayed merits. People might agree that certain opinions are unacceptable for teachers to have, or that certain actions are unacceptable. But those merits, actions, or opinions have next to no demonstrated relationship to test scores or other student outcomes.

So What Makes a Teacher Leader?

And if we can’t even know who or what defines a good teacher for any objective metric, then naturally the whole idea of finding “teacher leaders” is a lost cause.

Who’s a leader? The officially designated department heads or coaches, or the de facto mentors who offer advice and curriculum to the nervous newbies? The teachers who follow the contract obligations like clockwork, or the ones who work late and give hours to the kids but are weaker at the contractual obligations? The teachers who want to plow down resisters, or the teachers who suggest accommodating to the reality that the plowdowns will never happen? The teachers who want everyone to follow proven procedures, or the teachers who follow their own vision? The teacher who successfully manages a a site-wide program for at-risk kids, helping hundreds over the years while occasionally making sexual advances, or the teacher who just shows up every day to teach without ever molesting his students? The teachers who want to embrace reforms to improve schools, or the teachers who fight the reforms as the efforts of ignorant ideologues?

These aren’t rhetorical questions. There are people who can brief for either side–yes, even the molester. Just ask Mrs. Miller or, just to ratchet up the difficulty level, the hundreds of kids who weren’t abused by this predator, but found focus and purpose to achieve based on his advice and support.

So who wants teacher leaders, anyway? Reformers. Ed schools. Politicians. Administrators. Teachers who want to be teacher leaders—a handy group that serves as mouthpieces for the other organizations. The same people, in short, who believe the delusion that “good teachers” is an axiom, an easily defined, obvious trait.

Who doesn’t want teacher leaders? Teachers who don’t want to be teacher leaders. Which is most of them.

I repeat, for the umpteenth time: what the outside world sees as a bug, most teachers see as a feature. We trade promotions and pay recognition for job security and freedom from management that industry can only dream of.

Certain things just don’t make a dent in the teacher universe. When math teachers get together for beers, we don’t secretly bitch about how much more money we’d get if teaching salaries were determined by scarcity. Very few sigh for a world in which our pay is dependent on our principal’s opinion of our work. Many of us either aren’t fussed by system bureaucracy or—as if often the case—understand that the bureaucracy isn’t the underlying reason for whatever wall we face.

Given the utter lack of internal demand, teachers suspect, with much justification, that those calling for “leaders” are looking to install their mouthpieces in positions of authority over the rest of us. Call us cynical. Call us justified.

So the next time anyone calls for “teacher leaders”, please remember a few things. Any teaching community has leaders both official and informal. The official leaders are selected, often by management, sometimes by majority vote. The informal leaders are often sought out by colleagues, but occasionally self-drafted. Regardless of selection method, the relationships are many to many, not one to many. These leaders have little actual authority. They have influence. Sometimes.

Teachers don’t want leaders. We have management. We’re good, thanks.


2015: Turning a Corner. Maybe.

This chart may be complicated, but since these retrospectives mostly function as my diary, I’ll not worry about that.

blogpageviewcomparison

So here’s the last three years: Year 4 (blue, 2015), Year 3 (yellow, 2014), and Year 2(green, 2013). The last set of columns shows the cumulative traffic for each year. So Year 3 saw slightly higher traffic than Years 2 and 4. In 2015, I saw about 214K views compared to 215 in Year 3 and 223K in 2014, my high point.

If I wanted to, I could be bummed and think man, I’ve had no increase in growth over the past two years. But that’s why the jaggedy lines are there, to cheer me up. They remind me that if I want people to read me, I have to actually write. I put out a grand total of 5 posts in May, June, and July. Five posts! I only got two out in November, too.

Since I get a minimum of 200 page views just by actually posting, those three months cost me a lot of traffic. And since I’m not really in this for the overall traffic, it means I shouldn’t fuss about the lower overall number, and I won’t. Except to remind myself that I need to write more.

In 2014, I set myself the goal of writing 72 posts, and only managed 46. This year, I just managed 36. THIRTY SIX. That’s ridiculous.

Before I go onto the brighter side of last year, I want to write this down, to document my change in productivity. I wrote 108 pieces in my first year. Year 2, 2013, I wrote 61 essays. It’s like that math activity where the kids bounce balls and measure the height. My essay output is a decaying exponential function….Output year=.75 * Outputyear-1.

What the hell have I been doing with my time since? It’s not that I’m lazy, or that I did less research back then. Some of my best, most popular pieces were written in 2012, including 5 of the 18 pieces that saw over 1500 views just this year–or 6 of the top 20, if you prefer that method, but I don’t because it doesn’t allow like to like comparisons. Admittedly, many in that first year were short teaching stories I don’t do anymore (short? Me? Let’s all laugh.), but that just makes it more astonishing. I did some major research and throwaway posts. How?

I’m not out of ideas. I am more than occasionally frustrated by the utter nonsense I see bruited about confidently, by people paid huge heaps of money to be experts. But instead of writing about it, I get bogged down. That’s the problem. I try to do one big piece and cover everything. I need to create bite-sized chunks. The problem, alas, lies in my knowledge of the likelihood that I’d do the next chunk rather than move on to something else.

The problem isn’t the writing. I can knock out essays in relatively little time when I need to. I did On The Spring Valley High Incident in an evening (a very late evening, though), because I wanted my thoughts to be in the mix so timeliness was essential. I got the five political proposals and their bookends done in a month, a magnum opus of focus. (I suspect hocus pocus. Sorry.) The problem is in the organization and structuring, identifying the goals of the piece.

And yet, this studied consideration is often a strength. I spent nearly a month mulling the “explaining your answer” discussion and came up with the first “math zombie” piece, which contributed much more to the longer term discussion than whatever I would have written in the first week. Except, alas, I couldn’t get beyond thinking about it and so didn’t write anything else that month, killing the momentum I built up from August through October.

I read a P-J example in a Myers Briggs book somewhere. The president of a hobbyist club asked for a volunteer to put out a monthly newsletter. The volunteer who responded put out a charming newsletter, filled with fascinating and useful information, a real pleasure to read—but always put out on the third, rather than the first. In frustration, the president took it over herself, and put out a brief, functional newsletter right on time. Or the test question: “Do you think a meeting is successful when everyone leaves knowing what to do, or when every issue has been thoroughly explored?”

Which is not to say that everything done to task on time is always dry and boring. Mickey Kaus quotes someone else (I forget who) saying that he writes faster than anyone who writes better, and better than anyone who writes faster.

I will put more emphasis this year into improving my essay entry procedures, to stop putting off the challenging task of structuring a piece so that I can write it. Moreover, I was once able to write more than one piece at a time, putting out something simple and descriptive (say, on curriculum) while working on a larger piece. I need to get back to that.

I’m going to try to get back up to consistently four pieces a month. Wish me luck.

Now, on the bright side:

While 2014 saw the most consistent traffic, year 3 was also a relatively unpopular year. Of those 42 pieces, only 18 of them saw over 1000 views that year. My usual benchmark is 1500, and only eight made it over that mark.

This last year was much better. A full 24 of my paltry output of 36 pieces saw traffic over 1000 views; 11 made it over 1500. The most popular essay in 2014 was 2,800 views; this year my international SAT piece saw over 6,000 hits. My college remediation piece and the one on the gaokao got over 4,000.

These numbers are nowhere near my average essay popularity of 2013, the year of my all-time popular piece on Asians, as well as my Philip K. Dick piece on IQ, both of which went over 6,000–and that was just the start. I did some good work that year, and I’m pleased that 2015 was at least in the hunt.

So even though 2014, year 3, was the high point traffic wise, my new work received much more attention this year.

Some highlights:

  • My seven essay series on unmentionable education policies, most of which topped the 1500 mark–the rest just missed. I’m most proud that I gritted my teeth and followed through, devoting the entire month of August and not giving up.
  • What You Probably Don’t Know About the Gaokao highlights my ability to go deep and make a whole bunch of information digestible by the casual viewer–and got lots of traffic for my troubles.
  • I did carry through on my vow to write more about math, showing different aspects of my teaching and writing. Illustrating Functions is a nice pedagogy piece, while functions vs. equations sparked some tremendous discussions throughout the math community. I couldn’t have been more pleased. Jake’s Guest Lecture and The Test that made them go Hmmmm is an accurate representation of my classroom discussions. The zombie sessions with my private student capture my strength as an explainer. I also contined to build my series on multiple answer math tests, and what I can learn from the student responses.
  • I was offered a chance to write an op-ed in a major media outlet about my college remediation policy! I had to turn it down! The downside of anonymity. Although really, is it so terrible a newspaper publish an anonymous op-ed? They use anonymous sources and expect us to believe the journalists have used their judgment. (cough). So why not op eds? But still, it was great to be asked.
  • My Grant Wiggins eulogy—and may I say to Grant, wherever his spirit is, you’re sorely missed.

The pieces that didn’t get as much attention, but should have:

On a personal note, my granddaughter has a new baby brother! My next generation is expanding. And while I like to beat myself up for not writing more, I didn’t waste the time. I had a wonderful year of travel that took me to amazingly beautiful sights, multiple, happy family get-togethers necessitating time spent preparing fabulous food, and oh, yes, I taught a grand class or two. It was a fun year.

So let’s see if I’ve turned the corner on the productivity slump for 2016. Wish me luck.

Below are the pieces that had over 1500 hits.

Asian Immigrants and What No One Mentions Aloud 10/08/13 6,948
The SAT is Corrupt. No One Wants to Know. 12/31/14 6,329
Homework and grades. 02/06/12 4,491
Ed Policy Proposal #1: Ban College Level Remediation 08/01/15 4,360
What You Probably Don’t Know About the Gaokao 01/18/15 4,325
On the Spring Valley High Incident 10/27/15 2,948
Five Education Policy Proposals for 2016 Presidential Politics 07/31/15 2,944
Evaluating the New PSAT: Math  04/16/15 2,866
Algebra and the Pointlessness of The Whole Damn Thing 08/19/12 2,717
Binomial Multiplication and Factoring Trinomials with The Rectangle 09/14/12 2,699
Education Policy Proposal #2: Stop Kneecapping High Schools 08/02/15 2,253
I Don’t Do Homework  02/15/15 1,799
Education Policy Proposal #3: Repeal IDEA 08/07/15 1,708
Teachers and Sick Leave: A Proposal 05/26/13 1,629
SAT Prep for the Ultra-Rich, And Everyone Else 08/17/12 1,616
Education Policy Proposal #4: Restrict K-12 to Citizens Only 08/16/15 1,582
Kicking Off Triangles: What Method is This?  11/12/12 1,572
Functions vs. Equations: f(x) is y and more 05/24/15 1,514

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.

zombiecomp1

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:

zombiecomp2

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.

zombiecomp5

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.

zombiecomp4

(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:
zombiecomp3

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.

Homer got yelled at. I expressed (snarled) 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.

weakstudentwork

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).

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.


Understanding Math, and the Zombie Problem

I have been mulling this piece on the evils of explanations for a while. There’s many ways to approach this issue, and I highly recommend the extended discussion at Dan Meyer’s blog, as it captures experience-based teachers (mostly reform biased) with the traditionalists, who are primarily not teachers.

What struck me suddenly, as I was engaged in commenting, was the Atlantic’s clever juxtaposition.

All the buzz, all the sturm und drang about Common Core and overprocessed math has involved elementary school. The cute show your thinking pictures are from 8 year olds and first graders. Louis CK breaks our hearts with his third grader’s pain. The image in the Atlantic article has cute little pudgy second grade arms—with just the suggestion of race, maybe black, maybe Hispanic, probably male—writing a whole paragraph on math. The evocative image evokes protective feelings, outrage over the iniquities of modern math instruction, as a probably male student desperately struggles to obey meaningless demands from a probably female teacher who probably doesn’t understand math beyond an elementary level anyway. Hence another underprivileged child’s potential crushed, early and permanently, by the white matriarchal power structure unwilling to acknowledge its limitations.

And who could disagree? Arithmetic has, as John Derbyshire notes, “the peculiar characteristic that it easy to state problems in it that are ferociously difficult to solve.” Why force children to explain place value or the division algorithm? Let them get fluency first. Garelick and Beals (henceforth referred to as G&B) cite various studies finding that elementary school students gain competence by focusing on procedure first, conceptual understanding at some later point.

There’s just one problem. While the Atlantic’s framing targets elementary school, and the essay’s evidence base is entirely from elementary school, G&B’s focus is on middle school.

Percentages. Proportions. Historically, the bane of middle school math. Exhibit C on high school math teachers list of “things our students should know but don’t” (after negatives and fractions), and an oft-tested topic, both conceptually and procedurally, in college placement.

G&B make no bones about their focus. They aren’t the ones who chose the image. They start off with a middle school example, and speak of middle school students who “just want to do the math”.

But again, there’s that authoritatively cited research (linked in blue here):

gbquoteresearch

Again, all cites to research on elementary school math. The researched students are at most fifth graders; the topics never move above arithmetic facts. G&B even make it clear that the claim of “procedure without understanding is rare” is limited to elementary school math, and in the comments, Garelick discusses the limitations of a child’s brain, acknowledging that explanations become more important in adolescence—aka, middle school, algebra, and beyond.

G&B aren’t arguing for 8 year olds to multiply integers in happy, ignorant fluency, but for 14 year olds to calculate percentages and simply “show their work”. And in the event, which they deem unlikely, that students are just going through the motions, that’s okay because “doing a procedure devoid of any understanding of what is being done is actually hard to accomplish with elementary math.” Oh. Wait.

Once you get past the Atlantic bait and switch and discuss the issue at the appropriate age level, everything about the article seems odd.

First, Beals and Garelick would–or should, at least–be delighted with math instruction in 8th grade and beyond. Reform math doesn’t get very far in high school. Not only do most high school teachers reject reform math, most research shows that the bulk of advanced math teachers have proven impervious to all efforts to move beyond “lecture and assign a problem set”. Most math teachers at the high school level accept a worked problem as evidence of understanding, even when it’s not. I’m not as familiar with middle school algebra and geometry teachers, but since NCLB required middle school teachers to be subject-certified, it’s more likely they profile like high school teachers.

G&B don’t even begin to make the case that “explaining math” dominates at the middle school level. They gave an anecdote suggesting that 10% of the week’s math instruction was spent on 2-3 problems, “explaining thinking”.

This is the basis for an interesting discussion. Is it worth spending 10% of the time that would, presumably, otherwise be spent on procedural fluency on making kids jump through hoops to add meaningless detail to correctly worked problems? And then some people would say well, hang on, how about meaningful detail? Or how about other methods of assessing for understanding? For example, how about asking students why they can’t just increase $160 by 20% to get the original coat price? And if 10% is too much time, how about 5%? How about just a few test questions?

But G&B present the case as utterly beyond question, because research and besides, Aspergers. And you know, ELL. We shouldn’t make sure they understand what’s going on, provided they they know the procedures! Isn’t that enough?

Except, as noted, the research they use is for younger kids. None of their research supports their assertion that procedural fluency leads to conceptual understanding for algebra and beyond. We don’t really know.

However, to the extent we do know, most of the research available in algebra suggests exactly the opposite–that students benefit from “sense-making”, conceptual approaches (which is not the same as discovery) as opposed to entirely procedural based instruction. But researching algebra instruction is far more difficult than evaluating the pedagogy of arithmetic operations—and forget about any research done beyond the algebra level. So G&B didn’t provide adequate basis for making their claims about the relative value of procedural vs conceptual fluency, and it’s doubtful the basis exists.

I’ll get to the rest in a minute, but let’s take a pause there. Imagine how different the article would be if G&B had acknowledged that, while elementary school research supports fact fluency over sense-making (and fact fluency seems to be helpful in advanced math), the research and practice at algebra and beyond is less well established. What if they’d argued for their preferences, as opposed to research-based practices, and made an effort to build a case for procedural fluency over comprehension in advanced math? It would have led to a much richer conversation, with everyone acknowledging the strengths and weaknesses of different strategies and choices.

Someday, I’d like to see that conversation take place. Not with G&B, though, since I’m not even sure they understand the big hole in their case. They aren’t experienced enough.

Then there’s the zombie quote, where Garelick and Beals most tellingly display their inexperience:

Yes, Virginia, there are “math zombies”.

In high school, math zombies are very common, particularly in schools with a diverse range of students and thus abilities. Experienced teachers commenting at Dan Meyer’s blog or the Atlantic article all confirm their existence. This piece is long enough without going into anecdotal proof of zombies. One can infer zombie existence by the ever-growing complaints of college math professors about students with strong math transcripts but limited math knowledge.

I’ve seen zombies in tutoring through calculus, in my own teaching through pre-calc. In lower level classes, I’ve stopped some zombies dead in their tracks, often devastating them and angering their parents. The zombies, obviously, are the younger students in my classes, since I don’t teach honors courses. Most of the zombies in my school don’t go through my courses.

Whether math zombies are a problem rather depends on one’s point of view.

There are many math teachers who agree with G&B, who rip through the material, explaining it both procedurally and conceptually but focus on procedural competence. They assign difficult math problems in class with lots of homework. Their tests are difficult but predictable. They value students who wrote the didactic contract with Dolores Umbridge’s nasty pen, etching it into their skin. They diligently memorize the cues and procedures, and obediently regurgitate the procedures, aping understanding without having a clue. There is no dawning moment of conceptual understanding. The students don’t care in the slightest. They are there for the A and, to varying degrees, play Clever Hans for math teachers interested only in correctly worked procedures and right answers. Left as an open issue is the degree to which zombies are also cheating (and if they cheat are they zombies? is also a question left for another day). For now, assume I’m referring to kids who simply go through the motions, stuffing procedures into episodic memory with nothing making it to semantic, all to be forgotten as soon as the test is over.

Math zombies enable our absurd national math expectations. Twenty or thirty years ago, top tier kids had less incentive to fake it through advanced math. But as AP Calculus or die drove our national policy (thanks, Jay Mathews!) and students were driven to start advanced math earlier each year, zombies were rewarded for rather frightening behavior.

G&B and those who operate from the presumption that math can easily be mastered by memorizing procedures, who believe that teachers who slow down or limit coverage are enablers, don’t see math zombies as a problem. They’re the solution. You can see this in G&B’s devotion and constant appeal to the test scores of China, Singapore, and Korea, the ur-Zombies and still the sublime practitioners of the art, if it is to be called that.

For those of us who disagree, zombies create two related problems. First, their behavior encourages math teachers and policy makers to raise expectations, increase covered material, accelerate instruction pace. They allow schools to pretend that half their students or more are capable of advanced, college level math in high school while simultaneously getting As in many other difficult topics. They lead to BC Calculus pass rates of 50% or more (because yes, the AP Calc tests reward zombie math). Arguably, they have created a distortion in our sense of what “college math” should be, by pretending that “college math” is easily doable by most high school students willing to put in some time.

But the related problem is even more of an issue, because the more math teachers and policies reward zombies, the more smart, intellectually curious non-zombies bow out of the game, decide they’ll go to a state school or community college. Which means zombie kids just aren’t numbered among the “smart” kids, they become the smart kids. They define what smart kids “are capable of”, because no one comes along later to measure what they’ve…well, not forgotten, but never really learned to start with. So people think it really is possible to take 10-12 AP courses and understand the material (as opposed to get a 5 on the AP), and that defines what they expect from all top rank students. Meanwhile, those kids–and I know many–are neither intellectually curious nor even “intelligent” as we’d define it.

The Garelick/Beals piece is just a symptom of this mindset, not a cause. They don’t even know enough to realize that most high school math is taught just the way they like it. They’d understand this better if they were teachers, but neither of them has spent any significant time in the classroom, despite their bio claims. Both have significant academic knowledge in related areas–Garelick in elementary math pedagogy, which he studied as a hobby, Beals as a language expert for Asperger’s—which someone at the Atlantic confused with relevant experience.

Such is the nature of discourse in education policy that some people will think I’m rebutting G&B. No. I don’t even disagree with them on everything. The push for elementary school explanation is misguided and wasteful. Many math teachers reward words, not valid explanations; that’s why I use multiple answer math tests to assess conceptual knowledge. I also would love–yea, love–to see my kids willing to work to acquire greater procedural fluency.

But G&B go far beyond their actual expertise and ultimately, their piece is just a sad reminder of how easy it is to be treated as an “expert” by major publications simply by having the right contacts and backers. Nice work if you can get it.

And the “zombie” allusion, further developed by Brett Gilland, is a keeper.


Follow

Get every new post delivered to your Inbox.

Join 1,361 other followers