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The Product of Two Lines

I can’t remember when I realized that quadratics with real zeros were the product of two lines. It may have been
this introductory assessment that started me thinking hey, that’s cool, the line goes through the zero. And hey, even cooler, the other one will, too.

And for the first time, I began to understand that “factor” is possible to explain visually as well as algebraically.

Take, for example, f(x)=(x+3) and g(x)=(x-5). Graph the lines and mark the x-and y-intercepts:


Can’t you see the outlines of the parabola? This is a great visual cue for many students.

By this time, I’ve introduced function addition. From there, I just point out that if we can add the outputs of linear functions, we can multiply them.

We can just multiply the y-intercepts together first. One’s positive and one’s negative, so the y-intercept will be [wait for the response. This activity is designed specifically to get low ability kids thinking about what they can see, right in front of their eyes. So make the strugglers see it. Wait until they see it.]

Then onto the x-intercepts, where the output of one of the lines is zero. And zero multiplied by anything is zero.

Again, I always stop around here and make them see it. All lines have an x-intercept. If you’re multiplying two lines together, each line has an x-intercept. So the product of two different lines will have two different x-intercepts–unless one line is a multiple of the other (eg. x+3 and 2x+6). Each of those x-intercepts will multiply with the other output and result in a zero.

So take a minute before we go on, I always say, and think about what that means. Two different lines will have two different x-intercepts, which mean that their product will always have two points at which the product is zero.

This doesn’t mean that all parabolas have two zeros, I usually say at this point, because some if not all the kids see where this lesson is going. But the product of two different lines will always have two different zeros.

Then we look at the two lines and think about general areas and multiplication properties. On the left, both the lines are in negative territory, and a negative times a negative is a positive. Then, the line x+3 “hits” the x-axis and zero at -3, and from that zer on, the output values are positive. So from x=-3 to the zero for x-5, one of the lines has a positive output and one has a negative. I usually move an image from Desmos to my smartboard to mark all this up:


The purpose, again, is to get kids to understand that a quadratic shape isn’t just some random thing. Thinking of it as  a product of two lines allows them to realize the action is predictable, following rules of math they already know.

Then we go back to Desmos and plot points that are products of the two lines.


Bam! There’s the turnaround point, I say. What’s that called, in a parabola? and wait for “vertex”.

When I first introduced this idea, we’d do one or two product examples on the board and then they’d complete this worksheet:


The kids  plot the lines, mark the zeros and y-intercept based on the linear values, then find the outputs of the two individual lines and plot points, looking for the “turnaround”.

After a day or so of that, I’d talk about a parabola, which is sometimes, but not always, the product of two lines. Introduce the key points, etc. I think this would be perfect for algebra one. You could then move on to the parabolas that are the product of one line (a square) or the parabolas that don’t cross the x-intercept at all. Hey, how’s that work ?What kinds of lines are those? and so on.

That’s the basic approach as I developed it two or three years ago. Today, I would use it as just as describe above, but in algebra one, not algebra two. As written,I can’t use it anymore for my algebra two class, and therein lies a tale that validates what I first wrote three years ago, that by “dumbing things down”, I can slowly increase the breadth and depth of the curriculum while still keeping it accessible for all students.

These days, my class starts with a functions unit, covering function definition, notation, transformations, and basic parent functions (line, parabola, radical, reciprocal, absolute value).

So now, the “product of two lines” is no longer a new shape, but a familiar one. At this point, all the kids are at least somewhat familiar with f(x)=a(x-h)2+k, so even if they’ve forgotten the factored form of the quadratic, they recognize the parabola. And even better, they know how to describe it!

So when the shape emerges, the students can describe the parabola in vertex form. Up to now, a parabola has been the parent function f(x)=xtransformed by vertical and horizontal shifts and stretches. They know, then, that the product of f(x)=x+3 and g(x)=x-5 can also be described as h(x)=(x-1)2-16.

Since they already know that a parabola’s points are mirrored around a line of symmetry, most of them quickly connect this knowledge and realize that the line of symmetry will always be smack dab in between the two lines, and that they just need to find the line visually, plug it into the two lines, and that’s the vertex. (something like this).

For most of the kids, therefore, the explanatory worksheet above isn’t necessary. They’re ready to start graphing parabolas in factored form. Some students struggle with the connection, though, and I have this as a backup.

This opens up the whole topic into a series of questions so natural that even the most determined don’t give a damn student will be willing to temporarily engage in mulling them over.

For example, it’s an easy thing to transform a parabola to have no x-intercepts. But clearly, such a parabola can’t be the product of two lines. Hmm. Hold that thought.

Or I return to the idea of a factor or factoring, the process of converting from a sum to a product. If two lines are multiplied together, then each line is a factor of the quadratic. Does that mean that a quadratic with no zeros has no factors? Or is there some other way of looking at it? This will all be useful memories and connections when we move onto factoring, quadratic formula, and complex numbers.

Later, I can ask interested students to sketch (not graph) y=x(x-7)(x+4) and now they see it as a case of multiplying three lines together, where it’s going to be negative, positive, what the y-intercept will be, and so on.


At some point, I mention that we’re working exclusively with lines that have a slope of positive one, and that changing the slope will complicate (but not alter) the math. Although I’m not a big fan of horizontal stretch outside trigonometry, so I always tell the kids to factor out x’s coefficient.

But recently, I’ve realized that the applications go far beyond polynomials, which is why I’m modifying my functions unit yet again. Consider these equations:


and realize that they can all be conceived as as “committing a function on a line”. In each case, graphing the line and then performing the function on each output value will result in the correct graph–and, more importantly, provide a link to key values of the resulting graph simply by considering the line.

Then there’s the real reason I developed this concept: it really helps kids get the zeros right. Any math teacher has been driven bonkers by the flipping zeros problem.

That is, a kid looks at y=(x+3)(x-5) and says the zeros are at 3 and -5. I understand this perfectly. In one sense, it’s entirely logical. But logical or not, it’s wrong. I have gone through approximately the EIGHT HUNDRED BILLION ways of explaining factors vs. zeros, and a depressing chunk of kids still screw it up.

But understanding the factors as lines gives the students a visual check. They will, naturally, forget to use it. But when I come across them getting it backwards, I can say “graph the lines” instead of “OH FOR GOD’S SAKE HOW MANY TIMES DO I HAVE TO TELL YOU!” which makes me feel better but understandably fills them with apprehension.

Not Really Teaching English

The last time I wrote about my ELL class, I had six students: two from Mexico (Marshall and Kit from the story), two from China (Julian and Sebastian), one from Africa (Charlotte), one from India (Amit).

For the first ten weeks of school, my little gang fell into a routine. Monday, they worked in their Newcomers book, a “consumable” (new word for disposable) book that really added some structure to learning vocabulary–the chapters have interesting pictures wrapped around a particular content idea (going to the doctors office, colors, office furniture, math, etc). Tuesday was the online reading program. Wednesday was conversation day–I’d pick a topic and we’d go back and forth. Thursday, I’d find some short reading passages with questions, so I could test their understanding. Friday, maybe more of the same or a movie.

The Newcomers books were in the ELL classroom I used. Someone told me to use the Edge series, but the kids just weren’t ready. The room had tons of material–books, dictionaries, workbooks–but much of it was just at the wrong level, or too arcane, or simply uninteresting.

Charlotte is fairly fluent, but has a special ed diagnosis that will pretty much doom her to full English immersion for as long as she stays in high school, despite her teachers’ protests. (We did manage to get her sped support, at least.)  Sebastian has made no progress.  Amit has decent verbal fluency but his reading level is very weak, his written skills even worse. Marshall and Kit were my bright spots; they’ve been acquiring vocabulary and fluency at an exponential rate.

A week or two later after my last post on the class, in early October, Julian left for another school in the district, one with a higher Asian population than ours. Juanita, from Mexico, showed up at about the same time. Juanita is utterly uninterested in learning English or coming to school.

So had you asked me how I liked teaching ELL in early November, I would have talked about Marshall and Kit’s progress and how cheering it was, or my concerns about Juanita. I would have vented about Charlotte’s limited options, given state law. I’d have talked about Sebastian and Amit’s failure to progress and why. Amit was alert for every opportunity to gain approval. Sebastian was determined to get the right answer. They did not, alas, connect approval or the right answer to the goal of learning English. (How does Sebastian get right answers without learning any English? I asked the senior ELL teacher the same question. “He’s Chinese. It’s in the genes,” she said. But that’s okay. She’s Chinese, too.) I’d have bragged about the group cohesion–they have a Facebook page, and talk via Messenger.

But then things got crazy.

Between early November and Christmas break, six new students showed up. Four from Afghanistan (three siblings and a single), one from Mexico, one from Salvador.

In January, seven more: four from China, one from the Philippines, one from India, one from Vietnam.

My class size tripled. But there are no more Newcomers books. “We don’t use that curriculum any more.”

The ability range has also expanded, on both ends.

So my class now has three distinct levels, except I don’t yet have the expertise to run three classes, the way I did once in my all algebra year.

The first class would be for those who have little to no English. This became my most immediate problem. I couldn’t isolate the four kids who knew very little English, restricting their access to others fluent in their native languages. Elian, who arrived in November with nothing but “please”, “thank you”, and “soccer”, hasn’t progressed anywhere near as quickly as Marshall and Kit did  because Juan, Marshall and Kit are there to translate. He’s working, though, which puts him ahead of Juanita, who missed one to two classes a week for several months, and at this writing hasn’t been in class at all for two weeks. Ali and Monira are able to get translations from their older brother.  They’d progress more quickly in a more focused environment without friendly crutches. Juanita might feel like the course was designed for her needs and show up more.

Just for good measure, I’d put Sebastian and Amit in this class, which would be an enormous blow to their pride and dignity. But I’d remind them regretfully of the many times they’d done the wrong assignment, utterly failing to understand my instructions and being too proud to ask for help.But that’s ok, I’d tell them. They could be the class leaders and maybe, if they work harder, they’ll get moved up.  (Can you tell how attractive I’d find all this?)

Then the middle class of Aarif (Ali and Monira’s brother),  Huma (their fellow Persian-speaker), Marshall, Kit, and Amita (also from India),  the ones who are respectably fluent in English, but still need varying levels of finishing time to read and write in mainstream classes.

The four  from China (Anj, Song, Mary and sister Sara), the Filipino (Nancy), the Salvadoran (Juan), and the Vietnamese (Tran) are a real puzzle. They aren’t just verbally proficient, but can write and read reasonably well, with respectable vocabularies, better than all but the top 20-30% of my history class.  I can’t even begin to conceive why or how they were placed in ELL, much less the lowest level ELL class.  ( No one screwed up. ELL rules are what they are.)

The other teachers didn’t see anything odd about the wide range of abilities, but then the primary teacher, the ELL expert, has what I consider absurdly high standards. By her estimate, none of the kids were fluent. While I saw an enormous gulf between Elian and Tran, she saw two kids who couldn’t write an essay to her standards. I was relieved my responsibility to the class would be ending in late January, when the semester ended, and all this linguistic diversity would be Someone Else’s Problem and I wouldn’t need to try and argue about the various ability levels.

Then, just a week before the semester ended, I learned the replacement had turned down the job. The English department was about to be short yet another teacher, as a new one walked off the job with no notice four weeks into the second semester. Her classes have a long-term sub. The only plan B was me.  (Let me observe one more time how at odds the public conventional wisdom is with, you know, reality. Firing bad teachers is a trivial itch compared to the gaping maw of We Need More Teachers Now.)

Keeping my EL class required an enormous reconfiguration of the schedule, as my dance card for the second semester was already full (no prep).  My first block Trig course needed a teacher, and no other math teachers had a first block prep. Per my request, they reconfigured the schedule so that my closest colleague, who I’ve mentored since he arrived, got the class.

And so the linguistic diversity was now officially My Problem.

By early February, two of the Chinese students left–Song to the same school Julian absconded for, Sara to another city. I asked Mary why she wasn’t going with her sister?  Mary said Sara wasn’t her sister. Why would I think Sara was her sister? I reminded her they’d been introduced as sisters, had described themselves as sisters when they first arrived, and that I had referred to them as sisters several times to their acknowledgement. She looked vaguely panicked, tried to backtrack, and I told her to stop lying and drop it. Did I mention that Sebastian is supposed to be eighteen, but hasn’t hit puberty?  There’s a whole lot of birth certificate fraud going on in these Chinese visas. But I digress.

First problem: no more Newcomer books. I reached out to the language specialist: Any books like this? Hey, she remembered seeing  a bunch of books in a spare room. Would I be interested? Next day I had boxes and boxes of what  I considered two different publications–Read 180 and System 44–that are, apparently, the same program. I have no idea how this works, and that’s not because I didn’t take time and energy to look through them. Any connection must be found in the expensive training they want you to pay for. In any case, Read 180 was very writing-focused, with longer passages. Probably good for my middle group now; I may look at it again. But I was desperate for beginning texts and System 44 was a decent substitute for Newcomers.

So by late February, I had cobbled together an approach ensuring that my motivated beginners had the resources to improve their English. Fatima, in particular, made tremendous progress. Even Elian was at least showing more signs of comprehension, if he wasn’t speaking English at all.  Ali is moving much more slowly, but at least not backwards.

Marshall, Kit, and the rest of the middle group are continuing to benefit from the materials I have, plus our many class-wide discussions. I am constantly reassured by Kit and Marshall, my benchmark duo, showing constant improvement.

But the last group, I couldn’t figure out how to adequately challenge. Anything I came up with to do in the mixed class was too easy, but anything more difficult would require more support and attentiveness than I was giving.

One Monday in late March, I was driving to work bucking myself up about the coming week, thinking it was just a couple weeks until break, not to worry, don’t have such a bad attitude….and I stopped myself, because why the hell was I bucking myself up? I love my job. Really. I’m not a teacher who counts the days to spring break, normally.

So I went through all my classes: Trig, going great, really exciting work. For the first time, I was working with a like-minded colleague to build curriculum, common tests, a day by day approach. Wonderful stuff.  Mentoring an inductee, fun. Staff work, really promising. The upper math teachers were making real progress in settling our religious wars about coverage and depth by creating a federalist structure. My history class is a joy.  I was the adviser for a prominent after-school math-science program that succeeded beyond all expectations. Yes, I was busy, but I wasn’t particularly tired. I’d recognized the burnout signs last November and had successfully staved off an attack by taking it easy, resting more, traveling less. So why the motivation problem? My ELL class flashed into my mind and I felt an instant sense of….tension, dislike. Not quite revulsion, but definitely distasteful.

Until that minute, I hadn’t understood how much my ELL classs was pulling on my psyche, affecting more than just my feelings about that class. For the first time, I acknowledged that I was avoiding any sort of planning or development. Nothing felt enough, so I just avoided thinking about it outside class. I’d do whatever came into my head that morning. Head down, plowing through to the finish.

That very day, I walked into first block, and changed things up, created a wider range of activities, started coming up with more ideas, stopped just hoping it would be over when the year ended.

It worked.  I had more ideas for class-wide activities, more thoughts on how to differentiate. I could see the stronger kids were more engaged, learning idioms, thinking through grammar.  I’ll try to write more about these little activities in subsequent posts.

I’m not at all sure the kids notice any difference. I know the administrators and language specialist don’t–they already thought I was doing a good job.

I still don’t feel as if this is really teaching English. But I’m teaching better. I’m continuing to develop, rather than feeling stalled out. And that feels better.

Statistics of Slaves

I vowed to spend May documenting all the curriculum I’ve built that’s kept me from writing much. But writing up lesson always takes forever, so I don’t know how much I’ll get done.

I’ve revamped a lot of my history course since I first taught it in the fall of 2014,  but this lesson has remained largely unchanged. I was looking for data, originally for a lecture, on the growth of slavery after Eli Whitney went south for a visit.  I found this report with a most gruesome title. After spending an hour or four attempting to capture the information, the horror of it, in a lecture, I suddenly realized how much better it would work to let the kids capture and represent the data themselves.

So after a brief lecture on cotton ginning, before and after, the students get the second page of the report, with the slave census data from 1790 through 1860. I always assign states by group–the eleven eventual confederate, the four border, and New Jersey for contrast, so usually each group gets four states.I then go through a brief review of Percent Change (“change in value over ORIGINAL value”, please) .

The assignment: For each state, calculate the percentage change each decade. Create Create a column graph showing the real change each decade, with the percentage change shown at the top of the column.

Once all four states in the groups are graphed, compare the growth rates.

The work so far has been done on whiteboards. Some of the whiteboards are small, for personal use. In other cases, the students did the work directly on my whiteboard walls.


Student work: Louisiana slavery growth, 1810-1860


Student work: Georgia slavery growth, 1790-1860


Student work: North Carolina slavery growth, 1810-1860


Student work: Alabama slavery growth, 1800-1860

Right about now, the students realize it’d be much easier to compare the growth rates if they’d used a common scale. Meanwhile, I’d found it difficult to group the states in such a way that each group got a representative sample of growth rates.

In prior years, I’d just lectured through some examples. But my class was much more manageable this year, and for some reason I realized Oh, hey. A teachable moment.

Their “statistics of slavery” handout was doublesided with graph paper. After everyone had finished their group of graphs, I took pictures of any small whiteboard graphs and displayed them on the smart board.

The assignment: quickly graph a line sketch representing the slavery trends in each states using CONSISTENT AXES.  x is year, with  1790 as x=0, or the y-intercept. y is the number of slaves, using 100K chunks through 500K.  No need to capture specific percentage growth, but the graph should reveal it. Something in between “graph every single point” and “just connect the beginning and end value.”

They did really well. A few of them forgot what I said about consistent axes–and mind you, I said this some EIGHTY TIMES but no, I’m not bitter.

SDAInconsistent SDAConsistent

Happily, most compilations got the full 5 of 5, just like the kid on the right (you can see where I corrected his first two).

So these graphs really allowed for informed discussion. (A couple students said “Wow, I actually get slope now.”)  The students were able to identify states that saw tremendous growth vs states with slow or static growth.

Why would states have different growth rates? I reminded them of the national ban on slave trade. Where would slaves come from? And so to the domestic slave trade, another cheerful topic. Unlike the Caribbean slave population, slaves in North America increased their population through natural increase. States that cultivated tobacco exhausted the soil and, as Thomas Jefferson put it  in a letter to Washington, “Manure does not enter into this [soil restoration], because we can buy  an acre of new land cheaper than we can manure an old one.”  People just up and moved, or bought more land, when the productivity dropped, and so the state populations declined. Virginia, Maryland, Kentucky, and North Carolina, tobacco states all, sold their excess slaves to the cotton states.

Interesting note 1: Washington and Madison were both passionately interested in saving Virginia’s soil. Washington abandoned tobacco early, converting to wheat and other less damaging crops. He consulted with many English experts on best practices in soil management. Madison tried to spearhead agricultural reform, but ran up against the southern dislike of centralization.

Interesting note 2: Virginia was a southern agricultural powerhouse despite its reduced tobacco crop, but its primary product was wheat, produced primarily by non-slaveholders in Shenandoah Valley, not tobacco or cotton produced by slaveholders. (Remember, Jimmy Stewart’s Anderson clan wasn’t interested in fighting for the Confederacy.)

Studying slavery reminds me of how seemingly obvious goodness probably wasn’t. So, for example, the south had constraints on manumission. Slaveholders couldn’t even free their slaves if they wanted to! Slave states didn’t want them setting a bad example! Except the constraints existed in no small part because slaveholders dumped older slaves incapable of work, putting indigent elderly slaves  with no family and no means of supporting themselves out on the street. Most of the manumission laws specified age and remuneration requirements, and most didn’t ban the emancipation of young, healthy slaves. So manumission constraints were at least in part about protecting elderly ex-slaves. But would a slave  rather be free, even if impoverished, than living as property?

Or the debate about ending the slave trade, during the Constitutional Convention, when George Mason gave a fine speech, accurately laying out the arguments against slavery–it discourages free labor, gives poor people a distaste for work done by slaves, turns slaveowning men into petty tyrants.  And then General Pinckney says, yo, fine talk from a Virginian, whose huge slave population instantly gets more valuable if we stop bringing in new ones.

What was Pinckney saying? The kids were mystified.

“Why would Virginia’s slaves get more valuable?” asked Eddie.

“Well, remember, this is banning slave trade. Not slavery. The Constitution didn’t give the federal government the right to ban slavery. So if slavery still existed, but no new slaves were being imported, the only slaves being created would be here in America.”

“Yeah, but I don’t see what makes them more valuable?” Jia was confused.

I paused. “Think about supply and demand. What would banning slave trade do to supply?”

“It would go….down.” Jun.

“Right. But demand isn’t decreasing. South Carolina, Kentucky, Georgia, they need slaves.

“They won’t be able to get anymore, though, because there won’t be any more slave trade,” offered Lee.

I stopped moving, wait until eyes are on me. (Teaching’s all about the performance.)

“There will be more slaves. The slaves themselves are having children, right?” I had barely gotten the words out when Lee figured it out, and he literally gasped.

“Yeah. It’s horrible. When the federal government banned slave trade, Virginia had more slaves than any other state. And thanks to lousy farming practices, its land wasn’t much good for tobacco. But as slaves met, married, and had children, lo! the Virginians had a ready made product for sale.” More kids got it and groaned.

“That’s where the phrase ‘sold me down the river’ came from. The phrase means to betray someone. But originally, it referred to a slave whose Virginia or Kentucky owner sold them to the cotton plantations in the deep south, Mississippi or Alabama.”

“So banning slave trade was done to increase the value of slaves?”

“I’m…pretty sure that’s not true. Remember that before the cotton gin came about, many of the founding fathers really did seem to think slavery would fade out, although they were fuzzy on how that would happen. But certainly, South Carolinians would be the ones to identify the market opportunity for another state.”


Another little data analysis activity, done earlier than the slavery stats above: read a series of Wikipedia entries to determine when Northern states freed their slaves, then create a timeline with color-coded data. “I” was banning importation, B meant banning slavery, (“g” meant ban was gradual).    All students had to color code the dates for importing bans and slavery bans. This student came up with the idea of an identifier for those states that gave blacks the vote, and those that restricted the right to vote, particularly after the fact.

Anyway, I wanted the students to realize that organizing data can lead to insights. In this case, the bulk of the Northern states banned importation and slavery in the same 20 year cluster. New York and New Jersey stand out in sharp contrast. Another oddness: Rhode Island banned slavery earlier than it did imports, for the obvious reason that Rhode Island was the epicenter of the slave trade.


I never liked all the stories about slaves quarters, and jumping the broom, and so on. Not that they aren’t interesting, but they don’t carry the weight of data, of seeing the huge numbers. Of realizing that manumission might be a way to dump non-productive workers, or that ending slave trade might be a business move to increase property value.

It’s too much like Anne Frank, or the Anne Frank that her loving dad created. Whenever I hear kids say “Oh, I identified with Anne sooooooo much!” I want to smack something. She lived in an attic for two years. She was then sent to a concentration camp where she held onto life for six month and then died of typhus, her body crawling with lice, just a month or so before liberation. Identifying with that level of suffering is well-nigh impossible, so spare me your virtue signaling, you teen drama queen. Hrmph.

Corrupted College

I try  to take the long view on education policy.  In the long run, education reformers, education advocates, and policy wonks are wasting their time trying to change the underlying reality.  They’re paying their own bills and wasting taxpayer dollars. Nothing else.

But every so often, I worry.

Check out this Edsource story on the  California State University system’s announcement of its intent to abandon the “strategy” of remedial courses.

At last! I thought. CSU was finally telling low-skilled applicants to attend adult education or community college. Hahahaha.  Five years of education policy writing just isn’t enough time to become properly cynical.

CSU is not ending its practice of accepting students who aren’t capable of college work. CSU has ended its practice of remediating students who aren’t capable of college work. It makes such students feel “unwelcome.” Students who aren’t capable of doing college work are getting the impression that they don’t really belong at college.

And so, CSU is going to give students who can’t do college work college credit for the classes they take trying to become ready for college.

Understand that the CSU system has been accepting these students for over 30 years. CSU used to offer unlimited remediation until 1996. After taxpayers protested, CSU passed regulations reducing remediation efforts to one year and vowed to ultimately eliminate all remediation by 2001. But alas, when 2001 came along,  ending remediation would dramatically reduce black and Hispanic enrollment, so the deadline was extended to 2007. (Cite ) But 2007 came along and things were even worse. After that, well, California ended its high school exit examination  and retroactively awarded diplomas to all the students who hadn’t been able to pass it. Why bother? CSU was accepting students who didn’t have the diploma anyway.

So, CSU decided on a new “strategy”, defining “college readiness” as “student is earning us tuition dollars”. They’re even looking at ending any sort of reliance on California’s version of the Smarter Balanced test, the Early Assessment rating that California has used for years to guide high schools towards getting their students ready for college.

Loren J. Blanchard, CSU executive said  that remedial education represents a deficit model that must be reformed if we really hope to achieve our equity and completion goals.” James T. Minor, a “senior CSU strategist for Academic Success and Inclusive Excellence” says that purely remedial or developmental classes “is not a particularly  good model for retention and degree completion.” Jeff Gold “emphasizes” that all the new program does is offer “extra help and services”, that rest assured, academic quality shall continue undisturbed. The CSU just wants to make sure that students who can only do middle school work “belong here” at CSU. CSU trustee chairwoman Rebecca Eisen is “thrilled” to hear about this change, as more students will “feel this is something they can do” and stay in college for longer.

Reporter Larry Gordon accepts all this at face value. He doesn’t push Blanchard to explain why students who can’t do college level work aren’t, by definition, a deficit model. Or why students who couldn’t pass an 8th grade math test should be retained long enough to complete a degree.

Nor does Gordon  observe that CSU has been offering extra help and services for thirty years.  In the current model, the help and services were not counted towards graduation. In the new model, they will be. That’s the change. Giving college credit for colleges that an advanced eighth-grader could complete is a reduction in academic rigor.

And note that Rebecca Eisen, at least, knows that Jeff Gold is lying. The remedial students were leaving because they couldn’t do the work. The change will make the students stay. Because the classes will be made easier and the students will get credit for them in this reduced academic environment.

Edsource checks in at Cal State Dominguez Hills, which has already been converting its remedial courses to “co-requisite” courses in statistics and algebra and that remedial students taking the co-requisite courses are passing at roughly the same rate as those who aren’t remedial.

Left unmentioned is that Cal State Dominguez Hills’ converted SAT averages has a 75th percentile SAT score of 450.  Everyone at CSUDH is remedial by a “typical” college’s standards–and by CSUDH’s standards, eighty percent were remedial in both math and English, which gives a small hint as to why the college might want to end remediation.

While Gordon reports the news without any context on the student ability level, he hastens to assure readers that ignoring remedial status is a public university trend. “Several other states, such as Tennessee, reported success in putting students in so-called corequisite courses starting in 2015. The City University of New York is taking similar steps by 2018 and also is starting to allow math requirements to be fulfilled by statistics or quantitative reasoning classes, not just by algebra.”

Meanwhile, this  decision “dovetails” (read: is driven by)  the CSU Graduation Initiative, which is a plan to increase the four-year completion rate from 19 to 40 percent.

So in 1996, California wanted to completely end remediation by 2001. Now, in 2017, California wants to give students college credit for remedial courses so that in eight more years two out of every five students will graduate in four years.

I once wrote an essay calling for a ban on college remediation.  But events are just getting way ahead of me. Anticipating that colleges would start giving degrees to people with middle school skills was something I foolishly rejected as implausible.

But as bad as this is, my dismay and disgust is deepened a thousand-fold by this fact: high schools aren’t allowed to teach remedial courses.

We can’t say hey, this kid can only read at the eighth grade level, so let’s give him more vocabulary and leveled reading. Heavens, no. In fact, you see education advocates arguing that giving kids reading above their ability level is going to improve their reading (something unestablished at the high school level). In practice, this means that all but the most severely deficient readers are expected to read and thrive on Shakespeare and Sophocles.

We can’t say hey, this kid can’t do pre-algebra, much less algebra, and at his current knowledge and interest levels, he can’t possibly succeed at the three or four years of math past algebra that high schools require for graduation. No, we have to  teach second year algebra concepts to kids who aren’t entirely sure what 6×8 is because we know they’ll graduate before they end up in pre-calc.  High schools with diverse student populations can’t offer courses for the entire range of abilities encountered. Schools with entirely low-ability students can just lie.

Thanks to the education reforms of both the right and left, high schools are under tremendous pressure to force all their students into advanced courses and not given any options for students who aren’t ready. There is no “ready” but college-ready.

It’s gotten so idiotic that many high schools have started “dual enrollment” programs for their at-risk students. The best students are taking demanding high school courses. But the at-risk kids are going to college to get the remediation their high schools aren’t allowed to give them.  They shade the truth, of course, mouthing nonsense about giving kids a taste of college. But read between the lines and you’ll see that the students are getting remedial courses. So high schools are paying tuition for low-level kids to take middle school courses at their local college.

But why? I’ve asked, time and again. Colleges are allowed to remediate. Why not let high schools provide the remediation, get kids closer to college ready? Any remediation we do will reduce the burden on colleges.

Ah, but that’s where the idiocy gets intense. The same public universities offering (or ending) remediation require that all students take advanced courses in high school.   CSU application requirements include algebra 2. If CSU remedial students were even approaching second year algebra ability, the university system wouldn’t be ending remediation.

But CSU, and all the other colleges with admissions requirements well above the ability of the bottom 30% of their student population, know this. So why?

I’ve thought and thought about this, and can only come to one conclusion. Colleges are desperate to give opportunities to black and Hispanic students in a public atmosphere with no tolerance for affirmative action. They’ve tried every way they can think of. Standards have already been lowered. Course demands have been almost entirely eliminated–top-tier public schools will issue bachelor degrees with no additional math courses (after the remedial course, that is).  This is just the next step.

The public discourse has become almost entirely bifurcated. At one end, we see education reformers hammering on high standards while suggesting, tentatively, that perhaps everyone isn’t really meant for college. We see learned professors opining that of the two proposed methods of improving low-income kids’ academic achievement, “no excuses” is better than integration because at least “no excuses” won’t hurt suburban schools.

Meanwhile, the actual colleges are lowering standards dramatically to the point that we will now routinely see people–primarily but not all black and Hispanic–with bachelors degrees despite reading at the eighth grade level and minimal math abilities. What makes anyone think that actual achievement is going to matter?

I haven’t seen any education reformers discuss the constant push to end or limit remediation, which has been going on for five years or so. They aren’t terribly interested in college policies. Education reformers want to kill teacher unions and/or grab public funds for essentially private charter schools, and this doesn’t help.

So now our public universities will accept anyone with a transcript spelling out the right courses. They’ll just put them in middle school courses and call it college. Education reformers, college professionals, all the middlebrow pundits opining on our failed education system won’t care–they send their kids to more expensive schools, the ones whose diplomas won’t be devalued by this fraud.

I’d put this insanity into the bucket of “Why Trump Won”, but does Betsy DeVos even care? She’s too interested in using federal dollars to push choice to win disapproval  denying federal dollars to colleges who want to “improve access”. She’s the worst of both worlds: a committed voucher advocate who wouldn’t be bothered by the destruction of public universities. But then, a  Democrat EdSec wouldn’t give a damn–in fact, a Clinton or Obama presidency would probably pressure colleges to lower standards even more. No one seems to actively try to change these policies.

But public colleges like CSU and CUNY are what bright kids from less well-connected families, kids whose parents don’t have the social capital to get into the “right” schools, were once able to use to get ahead. These schools have already done themselves a lot of damage, making it harder and harder for anyone, no matter how qualified, to get through in less than six years because of the time, resources, and expense involved educating the near-illiterate–and, of course, paying for  vice-chancellors of gender sensitivity and diversity awareness by accepting loads of Chinese students who prepared for college by committing fraud on the SAT.

If this doesn’t stop, America will have a much more serious problem than failed college students with huge college debts and no diploma. We’ll have thousands of college grads who got their diplomas with no better than eighth grade reading and math skills.

I’m not a high-standards maven.  Nor am I patient with the pseudo-cynical idiots who think they’re in the know, smirking that college degrees have been worthless for years.

No, they haven’t. But they’re going to be.

Meanwhile, people should maybe read more David Labaree.












The Sum of a Parabola and a Line

For the past two years, my algebra students have determined that the product of two lines is a parabola, which instantly provides a visual of the solutions and the line of symmetry.  For the past year, they’ve determined that squaring a line is likewise a parabola, and can be moved up and down the line of symmetry, which is instantly visible as the line’s x-intercept. In this way, I have been able to build understanding from lines to quadratics without just saying hey, presto! here’s a parabola. I introduce them to adding and subtracting functions, and from there, it’s a reasonable step to multiplying functions.

Typically, I’ve moved from this to binomial multiplication, introducing the third form of the quadratic we deal with in early high-level math, the standard form. (The otherwise estimable Stewart refers to the vertex form as standard form, to which I say sir! you must reconsider, except, well, he’s dead.)

At some point in teaching this, you come to the “- b over 2a” (-b2a) issue. That is, teachers who like to build on existing knowledge towards each new step are a bit stuck when it comes to finding the vertex in a standard form equation.

(For non-mathies, the standard form of an equation is ax2+bx+c and the vertex form is a(x-h)2+k.  The parameters “a” “b”, and “c” are often just referred to by letter. Vertex form, we’re more likely to talk about the x and y values of the vertex, just like  when we talk about lines in the form y=mx+b, we don’t say “m” and “b” but rather “slope” and “y-intercept”. But teachers, at least, often talk about teaching different aspects of standard form operations by parameters: a>1, a<0, to say nothing of the quadratic formula.  So the way to find the vertex of a parabola in standard form is to take the “a” and “b” term and use the algorithm -b2a to find the line of symmetry,  which is the x-value of the vertex. Then”plug it in”, or evaluate, the x-value in the quadratic equation to find the y-value for the vertex.)

The only way I’ve found until now of building on existing knowledge to establish it is setting standard form equal to vertex form to establish that the “h” of vertex form is equal to the -b2a of standard form, something only the top kids really understand and don’t often enjoy. (they’re much more interested by pre-calc.)

Last year, I was putting together a worksheet on adding and subtracting lines, and on impulse I added a few that involved adding a simple parabola with its vertex at the origin with a line, mainly to add a bit of challenge for the top kids. I could see that adding a line and a parabola doesn’t provide the instant visual “hook” that multiplying or squaring lines does.


It’s obvious that the y-intercept of the sum will be the same as the y-intercept of the line. One can logically ascertain that in this particular case, the right side of the y-axis will only increase—adding two positives. The left side, therefore, as x approaches negative infinity is where the action is. But not too much action, since the parabola’s y is galloping towards positive infinity at a faster clip than the line’s is trotting towards negative infinity. So for a brief interval, the negative of the line will offset a bit of the positive of the parabola, but eventually the parabola’s growth will drown out the line’s decline.

All logically there to construe, but far less obvious at a glance.

This year, I decided to explore the relationship further, because deciphering standard form is where my weakest kids tend to check out. They’ve held on through binomial multiplication, to hang on, at least temporarily, to the linear term so that (x+3)2 doesn’t become x2 + 9. They’ve mastered factoring quadratics, to their shock. They understand how to graph parabolas in two forms. And suddenly this bizarre algorithm that has to be remembered, then calculated, then more calculations to find “y”, whatever that is. Can you say “cognitive load“, boys and girls? Before you know it, they’re using the quadratic formula for linear equations and other bad, bad things that happen when it’s all kerfluzzled in their noggins. That’s when you realize that paralysis isn’t the worst thing that can happen.

Could I break the process down into discrete steps that told a story?  Build on this notion of modifying the parent function ax2 with a line to shift it left or right? Find Raylene a new kidney now that her third husband discovered her affair with the yoga instructor and will no longer give her one of his?

My  first thought was to wonder if the slope of the line had any relationship to the graph’s location. My second thought was yes, you dweeb, “b” is the slope of the added line and b’s fingerprints are all over the line of symmetry. No, no, the other half of my brain, the English major, protested. I know that. But is there some way I can get the kids to think of “b” as a slope, or to link slope to the process in a meaningful way?

(This next part is probably incredibly obvious to actual mathematicians, but in my own defense I ran it by three teachers who actually studied advanced math, and they were like hey, wow. I didn’t know that.)

What information does standard form give? The y-intercept, or “c”. What information do we want that it doesn’t readily provide? The vertex. Factors would be nice, but they aren’t guaranteed. I always want the vertex. So if I graph the resulting parabola of the sum of, say,  x2 and 6x + 5, how might the slope be relevant?

The obvious relationship to wonder about first is the slope between the y-intercept, which I have, and the vertex, which I want. Start by finding the slope between these two points. And right at that point I realize hey,  by golly, that’s the rate of change(!).


The slope–that is, by golly, the rate of change(!)–is 3. The line of symmetry is -3. The vertex is exactly 9 units below the y-intercept, or the product of the rate of change and the line of symmetry. Heavens. That’s interesting. Does it always happen? Let’s assume for now a=1.

Sum Slope from y-int
to vertex
Line of
units from y-int to
y-value of vertex
x2 – 4x – 12 -2 x=2 -4 (2,-16)
x2 – 10x + 9 -5 x=5 -25 (5,-16)
x2 – 2x – 3 -1 x=1 -1 (-1,-4)
x2 +6x + 8 3 x=-3 -9 (-3,-1)

Hmm. So according to this, if I were trying to get the vertex for x2 +12x + 15, then I should assume that the slope–that is, by golly, the rate of change(!)– from the vertex to the y-intercept is 6. That would make the line of symmetry is x=-6. The y-value of the vertex should be 36 units down from 15, or -21. So the vertex should be at (-6,-21). And indeed it is. How about that?

So what happens if a is some other value than 1? I know the line of symmetry will change, of course, but what about the slope–that is, by golly, the rate of change(!). Is it affected by changes in a?

Sum Slope from y-int
to vertex
Line of
units from y-int to
y-value of vertex
2x2 – 8x – 5 -4 x=2 (-4/2) -8 (2,-3)
-x2 +2x + 4 1 x=1 (-1/-1) 1 (1,5)
-2x2 +14x +7 7 x=3.5 (-7/-2) 24.5 (49/2) (3.5,31.5)
4x2 +8x -15 4 x=-1 (-4/4) -4 (-1,-19)

Here’s a Desmos application that I created to demonstrate it.  The slope–that is, by golly, the rate of change(!)–from the vertex to the y-intercept is always half of the slope of the line added to the parabola–that is, half of “b”. The rate of change is not affected by the stretch factor, or a. The line of symmetry, however, is affected by the stretch, which makes sense once you realize that what we’re really calculating is the horizontal distance (the run) from the vertex to the y-axis. The stretch would affect how quickly the vertex is reached. So the vertex y-value is always going to be the rise for the number of iterations the run went through to get from the y-axis to the line of symmetry, or the rate of change multiplied by the line of symmetry x-value.


Mathematically, these are the exact steps used to complete the square but considerably less abstract. You’re finding the “run” to the line of symmetry and the “rise” up or down to the vertex.

Up to now, I’ve been describing my own discovery? How to explain this to the kids? As is always the case in a new lesson, I keep it pretty flexible and don’t overplan. I created a quick activity sheet.sumparabolalinehandout

The goal here was just to get things started. Notice the last question on the back: “Do you notice any patterns?” I was fully prepared for the answer to be “No”, which is good, because it was. We then developed the table similar to the first one above, and they quickly caught on to the pattern when a=1.

I was a bit worried about moving to other a values. However,  the class eventually grasped the basic relationship. The slope from the vertex to the y-intercept was always related to the slope of the line added  to the parabola. But the line of symmetry, the distance from the y-axis, would be influenced by the stretch. This made intuitive sense to most of the kids. They certainly screwed up negatives now and again, but who doesn’t.

Good math thinking throughout. I heard a lot of discussions, saw graphs where kids were clearly thinking through the spatial relationship. Many kids realized that when a=1, a negative b means the slope of the line from the y-intercept to the vertex is also negative, which means the vertex must be to the right of the y-intercept. A positive “b” means the slope is positive which means the vertex is to the left. Then they realize that the sign of “a” will flip that relationship around. he students start to see the “b” value as an indicator. That is, by making bx+c its own unit, they realize how important the slope of the added line is, and how essential it is to the end result.

All that and, you might have noticed, they get an early peek at rate of change concepts.

Definitely no worse than my usual -b2a  lesson and the weak kids did much, much better. This was just the first run; the next time I teach algebra 2 I’ll get more ambitious.

So I can now build on students’ existing knowledge to decipher and graph a standard form equation rather than just provide an algorithm or go through the algebra. On the other hand, the last tether holding my quadratics unit to the earth of typical algebra 2 practice has been severed; it’s now wandering around in the stratosphere.

I don’t mean the basics aren’t covered. I teach binomial multiplication, factoring, projectile motion, the quadratic formula, complex numbers, and so on. But the framework differs considerably from my colleagues’.

But if anyone is thinking that I’m dumbing this down, recall that my students are learning that functions can be combined, added, subtracted, multiplied. They’re learning that rate of change is linked directly to the slope of the line added to  the parabola, and that the original parabola’s stretch doesn’t influence the rate of change–but does impact the line of symmetry. And the weaker kids aren’t getting lost in algorithms that have no meaning.

I could argue about this, but maybe another day. For now, I’m interested in what to call this method, and who else is using it.

In Which Ed Explains Induction

So I’m at a Starbucks with my mentee, Bart. Bart looks like  Jared Leto playing Jesus. Many piercings, tattoos, big puppy dog eyes, long brown hair. We have been friends since his first day as a teacher, when I showed up in a (successful) effort to offer assistance, and I’m now mentoring him in his second year of induction (third year as a teacher.)

Some context: it is 6:15 pm. We both began our day at 7:15 am for a mandatory  75-minute staff development meeting, and not the sort where you’re surreptitiously grading papers while listening to required procedural instructions you’ve heard eight years in a row. No, this is intense department negotiations on curriculum and pacing. Interesting, but high intensity, and no checking out. Then our normal day.  Then we supervised our twice weekly, 90-minute sessions with about twenty kids working on science projects. Now we are at Starbucks, working on Bart’s induction project.  I don’t normally do the “teachers work long days” whine, but it had, in fact, been a long day.

Bart’s a great teacher, much adored by his students. He has his own idealistic values, like he still assigns homework because he wants kids to want to do it. I smile indulgently at such foolish romanticism. The guy spends hours working on lesson plans, writing extensive notes, building meaningful lessons and assessments. Not too much time–he’s not silly about this stuff–but he is a thoughtful person developing his practice, and he is in fact a really good teacher.

Induction is designed to engage and encourage new teachers to think productively about their practice. Bart and I had, up to this time, spent many hours in fruitful conversation, valuable to both of us, designing a year-long induction plan that interested him and would deepen his teaching experience.  He turned in his plan early, asking for feedback. I was pretty confident he’d be praised–my last mentee had done far less work under a different system and had done very well.

But alas, it was not to be. The induction administrator returned Bart’s plan politely, saying it showed real promise, but required a bunch of nitpicky changes.  In many cases, her changes expected Bart to be very detailed about the results of analytical or exploratory work that hadn’t yet happened.

I was very concerned. Bart thought the whole thing was absurd. So we were spending a few hours retooling his plan so that the wording pretended to comply with her demands. My years in corporate America have given me a thorough grounding in this task as well as an acute fear of failure; Bart has no such protection.

“What is the point of rewording all this?”

“Satisfying a bureaucrat without, you know, sex or money or drugs involved.”

“But why? I mean, why do we even have this induction nonsense?”

“Well, it all started with the achievement gap.”

“Induction will fix the achievement gap?”

“Of course not. Nothing will fix the achievement gap. So while there were some early successes, things mostly stalled out about twenty-thirty years ago.  Meanwhile, we started spending far more on education–bilingual education, increased academic requirements, special ed. Increased teachers–while our pay is about the same, we’ve had way more growth in teachers than in students. Many people noticed we had nothing to show for it, but no one seemed to notice that we are making far more demands on our students.”

“Completely unrealistic demands!”

“Of course. ” (Note: my original history here: The Fallacy at the Heart of All Reform on this topic is still one of my favorites.)

“But what does this have to do with this crappy makework?”

“Well, back in the 80s, when the Nation At Risk declared that we were destroying our country and Russia would win…”

“A Nation at Risk?”

I sighed. “That’s right, you went to one of those online ed schools. It was this huge report written by conservative Repulicans arguing, basically, that American high schools are destroying the country by making school too easy. So that began a wholesale upgrade of required high school courses–except, of course, many kids weren’t capable of learning advanced material. Schools tried tracking, but they were sued out of it in diverse districts, leading us to try things like differentiation and group work and resulting in the wide range of abilities you see in your classroom today.”

“Anyway, back in the 90s, it finally began to occur to folks that not all kids were ready for this material, but rather than change the requirements, they started a big push for “readiness” at the middle school and elementary school level. This is where charters had a lot of success; it’s how KIPP made its bones. Turns out  that if you cream highly motivated kids of average ability and push testing, you can bump test scores, and back in the 90s, everyone screamed that oh, my lord, this is proof that our public schools are disasters and teachers are morons.”

“Did they have success in high school?”

“No, but of course higher test scores in elementary scores would lead to  better high school performance.”


“That’s idiotic. High school is much more difficult. So is that when credential tests began?”

“Well, high school teachers have had difficult credential tests going back to the 70s, a fact conveniently ignored by reformers. High school teachers are well-qualified, so we already knew that boosting teacher cognitive ability doesn’t lead to higher student test scores. But what means these pesky facts in face of enthusiasm and certainty? It’s when credential tests for elementary and middle school teachers began, though. (You can read all about it here.)”

“But induction isn’t a credential test.”

“Yeah, I’m getting there. Because, as you’ve no doubt anticipated, a wholesale increase in teacher cognitive abilities didn’t have the desired result–although it did result in a huge decrease in black and Hispanic teachers, once the fraud ring was discovered and broken up.”

“Fraud ring? Like taking tests for teachers?”

“Yep. Long story. Never mind that, while the evidence for smarter teachers getting better results is fuzzy,research shows a much stronger link for achievement if teacher and student race match…”

“Teacher and student race? You’re kidding.”

“Nope. Particularly low achieving blacks. Sucks, huh.”


“Where was I? Oh, yeah. Anyway, at some point in there progressives and conservatives found something they could agree on. It was ridiculous to assume that teachers could just….teach. They sit in ed school, which is widely agreed to be a waste of time…”

“Mine was.”

“…and do a few weeks of student teaching, and suddenly, shazam. They’re teachers! Once all the professionals sat and thought about that, they decided it was stupid. After all, these professionals had insanely great test scores and got into terrific schools, but teachers, who have our nation’s kids’ future in their hands!–go to crap schools, have low SAT scores, and then we just put them in a class. This has to change. Some of them are terrible. Some quit. Let’s  invest in their success!  Give new teachers more support. Improve student achievement.Blah blah.”

“Ah. Here’s how induction comes into it. But hasn’t it always been that way? I mean, we’ve always just put teachers into a classroom. Were they smarter? I’ve heard that in the old days teachers were smart women who couldn’t get other jobs, and now we’re all idiots.”

“In fact, teacher ability has been pretty constant. While it’s true that fewer really smart women become teachers, a whole lot of reasonably smart men did, along with the existing reasonably smart women.”

“And you’re right. It has always been this way. In the very early days, teachers were taught content. But for sixty years or more, prospective teachers have spent a year or so thinking and reading about pedagogy, six to ten weeks student teaching, and then entered the classroom.”

“All so America could invent the Internet and go to the moon.”

“Win World War II, outlast Communism, make AIDS a manageable disease, and elect a black president. But yeah, faced with the choice of accepting cognitive ability or pretending that teachers are ludicrously unprepared for the classroom, it’s an easy pick: spend billions on a useless training program for new teachers.”

“And so here we are.”

“Well, be happy Linda Darling Hammond didn’t get her way. She wants teachers train for three years after graduation before getting a job. And she’s a liberal!”

“What the hell? Here’s what I don’t get. Teaching isn’t that hard…well, it is hard. But it’s not hard in a way that training helps. It’s incredibly difficult but….exciting.”

“Well, of course.  Teaching is a performance job. Teachers have an audience. And as any actor can tell you, facing a hostile audience is a hellish proposition. Facing a hostile audience every day, eight hours a day, can’t long be borne. Facing a hostile audience of 30 or more children? Sane people run screaming if they can’t do the job.”

“So teaching has its own quality control built right in.”

“Exactly. If you are completely inept, you will quit or be fired in the unlikely event you made it past student teaching.”

“But you’re not saying everyone is a great teacher.”

“No. Everyone who continues teaching is at least an adequate teacher. And beyond adequate, no one can agree on the attributes of a great teacher. Manifestly, great teachers aren’t necessary. Adequate to good teachers are sufficient.”

“But we could do better. I mean, I would have loved to have talked to you before I started work, to get a good idea of what I was facing.”

“You wouldn’t have believed me. In fact, you didn’t believe me! Remember when I gave you that assessment test to give your kids the first day, and you were shocked because it was pre-algebra? These were geometry kids, you said. They’d finish it in 20 minutes. Um, no, I said, they’d need at least 45 and my guess more. You were polite, remember? Like who is this crazy loon.”

Bart was chagrined. “My god, you’re right. I doubted you back then. And then the test took them an hour and the average score was thirty wrong.”

“You still doubt me! You shouldn’t, of course, but teaching is hard to believe until you do it. Which is why induction is a waste.”

“Well, at least they pay you to do this. I do it for free!”

“Yep. Teaching is pay to play. Anyway, it’s seven. Let’s send this off and hope it pleases the bureaucrat.”


(It didn’t. The bureaucrat demanded more nonsensical changes. I wrote a cranky note.)



Realizing Radians: Teaching as Stagecraft

Teaching Objective: Introduce radian as a unit of angle measure that corresponds to the number of radians in the length of the arc that the angle “subtends” (cuts off? intersects?).  Put another way: One radian is the measure of an angle that subtends an arc the length of the circle’s radius.  Put still another way, with pictures:

How do you  engage understanding and interest, given this rather dry fact?  There’s no one answer. But in this particular case, I use stagecraft and misdirection.

I start by walking around a small circle.

“How far did I walk?”

“360 degrees.”

“Yeah, that won’t work.” I walk around a group of desks. “How far did I walk?”

“360 degrees.”

“Really? I walked the same distance both times?”

“No!” from the class.

“So what’s the difference?”

It takes a minute or so for someone to mention radius.

“Hey, there you go. Why does the radius matter?”

That’s always an interesting pause as the kids take into account something they’ve known forever, but never genuinely thought about before–the distance around a circle is determined by the radius.

“Yeah. Of course, we knew that, right? What’s that word for the distance around a circle?”


“Yes. And how do you find the circumference of a circle?” There’s always a pause, here. “OK, let me tell you for the fiftieth time: know the difference between area and circumference formulas!”

“2Πr” someone offers tentatively.  I put it up:


“So the circumference is the difference between this small circle” and I walk it again “and this biiiiigg circle around these desks here.” Nods. “And the difference in circumference comes down to radius.”


“Look at the equation. 2 Π is 2 Π. So the only difference is radius. The difference in these two circles I walked is that one has a bigger radius.”

“So the real question is, how does the radius play into the circumference?”

“Well,” it’s always one of the better math students, here: “The bigger the radius is, the farther away from the center, right?”

“So then…you have to walk more around…more to walk around,” some other student will finish, or I’ll ask someone to explain what that means.

“Right. But how does that actually work? Can we know exactly how much bigger a circle is if it has a bigger radius?”

“A circle with a radius of 2 has a circumference of  4Π. A circle with a radius of 4 has a radius of 8 Π. So it’s bigger.” again, I can prompt if needed, but my class is such that the stronger students will speak their thoughts aloud. I allow it here, because they can never see where I’m going. See below for what happens if they start with spoiler alerts.

“Sure. But what’s that mean?”


I pass out pairs of circles, cut from simple construction paper, of varying sizes, although each pair has the same radius.

“You’re going to find out exactly how many radius lengths are in a circle’s circumference using the two circles. Don’t mix and match. Don’t write annoyingly obscene things on the circles.”

“How about obscene things that aren’t annoying?”

“If you can think of charmingly obscene comments, imagine yourself repeating them to the principal or your parents, and refrain from writing them, too. Now. You will use one of these circles as a ruler. All you have to do is create a radius ruler. Then you’ll use that ruler to tell me how many times the radius goes around the circumference.”

“Use one of the circles as a ruler?”

“You figure it out.”

And they do. Most of them figure it out independently; a few covertly imitate a nearby group that got it. Folding up one of the circles into fourths (or 8ths) exposes the radius.


Folding up one circle exposes the radius.

It takes most of them a bit more time to figure out how to use the radius as a ruler, and sometimes I noodge them. It’s so low-tech!


Curl the folded circle around the edge of the measured circle. 

But within ten to fifteen minutes everyone has painstakingly used the “radius ruler” to mark off the number of radius lengths around the circumference, and then I go back up front.


“Okay. So how many times did the radius fit into the circumference?”

Various choruses of “Over six” come back, but invariably, someone says something like “Six with and a little bit left over.”

“Hey, I like that. Six and a little bit. Everyone agreed?” Yesses come back. “So did everyone get something that looks like this?”


“Huh. And did it matter what size the circle was? Jody, you had the big two, right? Samir, the tiny ones? Same difference? Six and a little bit?”

“So no matter the circle size, it appears, the radius goes into the circumference six times, with a little bit left over.”

No one has any clue where I’m going, usually, but they’re interested.

“‘Goes into’ is a familiar term, isn’t it? I mean, if I say I wonder how many times 2 goes into 6, what am I actually asking?”

Pause, as the import registers, then “Six divided by two.”

“Yeah, it’s a division question! So when I ask how many times the radius goes into the circumference, I’m actually asking…..” The pause is a fun thing. Most beginning teachers dream of using it, but then get fearful when no one answers. No. Be fearless. Wait longer. And, if you need it:

“Oh, come on. You all just said it. How many times does 2 go into 6 is 6 divided by 2. So how many times the radius goes into the circumference is…”

and this time you’ll get it: “Circumference divided by the radius.”

“Yeah–and that’s interesting, isn’t it? It applies to the original formula, too.”


“Cancel  out the radius.” the class is still mystified, usually, but they see the math.

“Right. The radius is a factor in both the numerator and denominator, so they can be eliminated. This leaves an equation that looks like this.”


“The circumference divided by the radius is 2Π. Well. That’s good to know. Does everyone follow the math? Everyone get what we did? You all manually measured the circumference in terms of radius length–which is the same as division–and learned that the radius goes into the circumference a little bit over six times. Meanwhile, we’re looking at the algebra, where it appears that the circumference divided by the radius is 2Π.”

(Note: I have never had the experience where a bright kid figures it out at this point. If I did, I would kill him daid, visually speaking, with a look of daggers. YOU DO NOT SPOIL MY APPLAUSE LINE. It’s important. Then go to him or her later and say, “thanks for keeping it secret.” Or give kudos after the fact, “Aman figured it out early, just two seconds before figuring out I’d kill him if he spoke up.” Bright kids learn early, in my class, to speak to me personally about their great observations and not interrupt my stagecraft.)

And then, almost as an aside: “What is Π, again?” I always ask it that way, never “what’s the value of Π” because the stronger kids, again, will answer reflexively with the correct value and they aren’t the main audience yet. So the stronger kids will start talking yap about circles, and I will always call then on a weaker kid, up front.

“So, Alberto, you know those insane posters going around all the math teachers’ walls? With all the numbers?”

“Oh, yeah. That’s Π, right? 3.14.”

“Right. So Π is 3.14 blah blah blah. And we multiply it by two.”


That’s when I start to get the gasps and “Oh, MAN!” “You’re kidding!”

“….so 3.14 blah blah times 2 is 6.28 or…..”

“SIX AND A LITTLE BIT!” the class always shouts with joy and comprehension. And on good days, I get applause, too, from the stronger kids who realized I misdirected them long enough to get a deeper appreciation of the math, not just “the answer”.


So a traditionalist would just explain it, maybe with power point. I don’t want to fault that, but I have a bunch of students who would simply not pay any attention. They’ll take the F. I either have to figure out a way to feed them the math in a way they’ll remember, or fail more kids than I’m comfortable failing.

A discovery-oriented teacher would probably turn it into a crafts project, complete with pipe cleaners and magic markers. I don’t want to fault that, but you always get the obsessive artists who focus on making a beautiful picture and don’t care about the math. Besides, it takes forever. This little activity has to be 15-20 minutes, tops. Remember, there’s still a lot to explain. Radians are the unit measure that allow us to talk about circles in terms akin to similarity in polygons–and that’s just the start, of course. We have to talk about conversion, about the power that radians gives us in terms of thinking of percentage of the entire circle–and then actual practice. I don’t have time for a damn pipe-cleaning activity.

As I’ve written before somewhere between open-ended, squishy discovery and straight discussion lecture lies a lot of ground for productive, memorable teaching. In my  opinion, good teachers don’t just transmit information, but create learning events, moments that all students remember and can use as hooks for further memories of learning. In this case, I want them to sneak around the back end to realize that  Π is a concrete reality, something that can actually be counted, if not exactly.


Teaching as stagecraft. All the best teachers use it–even pure lecture artists who do it with the power of their words (and an appropriate audience).  Many idealistic teachers begin with fond delusions of an enthralled class listening as they explain math in terms that their other soulless, uncaring teachers just listlessly put up on the board. When those fantasies are ruthlessly dashed, they often have no plan B. My god, it turns out that the kids really don’t find math interesting! Who do I blame, myself or them?

I never had the delusions. I always ask my kids one simple question: is your life better off if you pass math, or if you fail?  Stick with me, and you’ll pass. For many, that’s a soulless promise. To me, that’s where the fun starts. How do you get them interested? How do you create those moments? How do you engage kids who don’t care?

It’s not enough. It’s never enough.

But it’s a good way to start.

This Great Election

This is the first election day since 1992 that I’ve really enjoyed. 1992’s election was exhilarating and in many ways a set up for this one. Bill Clinton back then gave a master class in how far a politician could go if he lacked shame and had a message the voters cared about. In 2000, I thought Gore ran a poor campaign over the summer, and the recount was a little too much evidence that our court system is just a reinforcement of our political system. I was just pleased it was close.

2008 radicalized me. I didn’t mind Hillary much back then (she was against driver’s licenses for illegal aliens, remember that quaint old restriction?), and the media’s anvil on the scale for Obama in both the primaries and the general was just nauseating.

I quit watching or reading about politics from late October 2008 to the Obamacare fights of 2009. And when I came back to it, I stopped trusting any media. Going on Twitter in 2012 further reinforced my understanding that even the ones who write in a seemingly neutral and unbiased style are, in fact, predictably liberal with tremendous disdain for half the electorate. For a news junkie living squarely in the mainstream, this comes as an unhappy shock.  (This time around, Sean Trende and Jack Shafer, two of my favorites, have been the most disappointing re the disconnect between the bias in their tweets and their carefully cleaned up columns, Josh Kraushaaer the one I still have illusions about so dammit Josh, don’t screw it up. Michael Goodwin, Mickey Kaus, and Byron York have, in their various ways, been solid gold treasures.)

Anyway. One thing I did learn from 2008 was that outside of progressives, white voters aren’t very interested in the presidential election issues. It’s been clear to me for a while that the public, particularly the GOP base, was not getting the candidates or the issues they wanted. Two elections in a row, I thought it likely that white voters were staying home, not bothering. Two elections in a row, I thought that the GOP was ignoring its voters in favor of ideas that no one really wanted–from immigration to education to social issues to entitlements. (I never thought of trade, sorry.)

Then came the 2012 autopsy, in which the GOP said hey, we need outreach to Hispanics in order to win back the presidency. Not to blacks. Noooo, the much-vaunted Party of Lincoln didn’t even think of blacks, didn’t think to find the common ground between their base of working class whites and the many blacks (and non-immigrant Hispanics). No notion of using immigration restriction as a uniter. Nope. Their money men wanted cheap labor, and they all figured that the 2012 loss could be used as rationale to argue against the base’s desire for restriction.  “See, we’d love to end H1B visas and implement e-verify, but we gotta do outreach!”  Because that’s how you grow the economy, with lots of businesses making money off of cheap labor. Good for the stock market. Meanwhile, of course, the GOP wanted to double down on blaming schools for failing to educate kids–that’s why they need immigrant labor, because teachers suck!

So I wasn’t excited about 2016, what with all the talk about another Bush, hints of returning to the autopsy plan, even after Rubio got his ears pinned back.

And then came Trump, down that damn escalator.

He never had to win to make me happy.  I wanted the message out there.  I wanted another politician to defy conventional wisdom, to refuse to step down or apologize, to insist that the people be given their choice. I wanted someone to show the popularity of issues the media and elites considered completely unthinkable, to force them into the debate. The Overton window has shifted feet–yards, even–back in the direction of sanity.

But GOP elites are trying to bargain their way out of reality. They  think fondly of a world where Rubio–the GOP’s version of bland, teleprompter-ready Obama–could have won if Kasich and Christie had dropped out because golly, he gave a good speech. Or Cruz–whose voice is so awful I change the channel when he shows up–could somehow win over enough swing voters.  Or they blame the media for giving Trump air time, forgetting that the airtime was devoted to blasting Trump for insensitivity, for “racism”, and demanding the public share their opinion. Instead he won more votes every time he refused to back down.

If you want to rebuild the GOP, start by asking a Trump voter what the key moment in his success was. Most will point to his refusal to apologize for his June 16 announcement. NBC dumped him. Univision fired him. And he didn’t back down. He didn’t play the game. He didn’t apologize, mend fences with the media. That was……well, huuuuge in the world of Trump’s base.  He snarled back, and got more popular.

What we’ve needed in America is someone willing to defy the media and the elite. Someone who had the money and message to succeed despite blasted disapproval. This forced the media and the GOP leadership to realize that all of their power relied on their ability to shut off the microphone. Take that ability away, they got nothing.

I don’t lionize Trump. I think he tried for years to win approval from the same elites who despise him now. I’m glad he chose to run. I’m glad he showed them, through the people, how wrong they were.

Because unless the polls are dramatically wrong in Clinton’s favor, Trump is not going to get destroyed. If he loses, it will be be a margin less than McCain, possibly less than Romney. With few ads and even fewer experts to advise him–the experts being the one class who still needs elite approval.

All he had was a message.

Next steps: win or lose, Trump voters need to see that class, not race, is the way to grow their ranks. This Sheryl Stolberg story on the decimated black working class that see no hope from Hillary but hate Trump–they’re the first step. I believe that African Americans can be convinced that our immigration policies are incredibly harmful to their interests: in jobs, in education, in reducing their political viability. Working class Hispanics, those of long-standing in this country, are also a great opportunity for actual outreach.

I’m not sure where it goes from here, because very few Republicans in media or leadership have any interest in rebuilding. Most of them believe that surgical removal of Trump voters is not only necessary, but simple. Laugh at them.

It’s all the meme these days for the media to talk about how horrible this election has been, how dispiriting it’s been to true believers in democracy and American greatness. That, again, is one reason why we all hate the media and elites, for failing to realize how exciting many of us are by the opportunity to vote our issues.

To all of you out there in Trumpland, I hope you share my sense of joy in this campaign. Watching everyone in power realize they had no power to stop Trump and his message.

If our side loses, it wasn’t because the media won the narrative. Entire publications were dedicated to convincing the public of Trump’s evil nature. They failed. They weren’t able to frame this election, because in their framing, Trump is unthinkable, a fascist racist misongynist who’ll start nuclear wars. But “unthinkable” doesn’t include close to half the country’s support.

If we lose, we’ll lose because we don’t yet have enough votes. Trump’s important qualities are alienating. I believe they were also essential. There was no moderating, no winning approval, that wouldn’t likewise end his ability to sell his message. And the conservative wing of the party has had it their way for so long that they can’t conceive of voting for a candidate they aren’t crazy about. That, too, was a non-negotiable constraint.

But moving forward, I believe this can be fixed. I believe the media  and the GOP will find it impossible to shut down these issues. I believe we’ll get more compelling candidates. I believe we’ll find a way to win more support.

If not, well, at least we had the chance to try.  That’s more chance than I ever expected.

Go Trump!

A Clarifying Moment

This semester has had several  unmitigated professional plusses: (1) my schedule is now ELL, trig, algebra 2, and pre-calc. (Cue Sesame Street.)  Last year, I briefly (and oh so irrationally) considered resigning because I only had two preps. Four is better. (2) I’m actually helping the school out in a pinch by taking this ELL class. Feels noble and self-sacrificing….(3) well, no, scratch the self-sacrifice, given the  33% pay bump for the fourth semester in a row, with next semester the fifth. You would be shocked to learn how much I make extra a month. Score. (5) I’m getting a new professional experience with no risk.

On the other hand, I’ve set a new benchmark for exhaustion. Work rarely tires me out. But for the first time in memory I’m mentally zonked by my schedule. Enjoying it, yes. But not only am I finding myself thinking longingly of Saturday and sleep,  but I’m often teaching my fourth block from a chair. I’ve been puzzling over the cause, because nothing about four preps should in and of itself be so draining (for me). As I wrote this,  I suddenly realized that club adviser should be added to the list. Then I’m an induction  mentor. And oh, yeah, an administrator voluntold me to co-lead a science/engineering after-school program, which is getting kind of ridiculous. I don’t do science.

The after-school program gave me some insight into my state of mind. I’d been MIA for the first few meetings, for good reasons. I’d done the several hours of weekend training, met with my co-lead (also my mentee), but had just not gotten dialed into the weekly sessions.  I’d been mentally shying away from even thinking about that two afternoon commitment, on top of everything else. But once my first meeting started, I was hooked and charged, working with the kids.

I suddenly realized that this is how I’m facing every single class, every obligation (save the induction meetings, which take place at a local liquor store with a great beer bar): mentally shying away from each instance until I’m in the moment, when it’s an electric shock of fun and joy. Which, for me, is a sign of incipient burnout. I have cancelled one road trip entirely over Thanksgiving, and am rethinking the best way to achieve two others. I may even fork out plane fare, which is a big concession. Semester two will be better, just two preps.



Yesterday, Friday afternoon, just minutes from beer and sushi, I was waiting for some pre-calc students to finish a test when in walked

“Hui! My lord, I haven’t seen you since…” and I stopped there, just jumping up to shake hands, because the last time I’d seen Hui, nearly three years ago, he’d been choking back tears as he told me his SAT scores.

Hui had been a junior in my first pre-calc class, where he struggled. (Based on my results with him and other similar stories, I slowed down instruction dramatically in subsequent precalc courses.)  He wasn’t a student I was particularly close to, but the next year, he stopped by and asked if I could give him advice about the SAT.  I wasn’t sanguine. He tested terribly in math, and he spoke, read, and wrote English at perhaps a fifth grade level. A top state university was his goal. Asians with impeccable scores and transcripts face routine discrimination by college admissions staff; the notion of an underprivileged Chinese lad whose abilities weren’t best captured by standardized tests simply does not compute in that world. I tried as gently as possible to prepare him for this likelihood, but didn’t push the issue, and twice a week, he came to my classroom after school for half an hour or more,  steadfastly working through test sections and trying to make sense of the questions.

After his test date, Hui asked me if I’d look at his personal statement. I gave him several tutorials in self-promotion.  Hui’s weak English suddenly became a remarkable achievement  when considered in the context of five years in America and two parents with limited education and less English. He was reclassified quickly (probably too quickly), which allowed him to take a normal schedule and qualify for admission to a state campus. Play up that achievement, I told him, and put your scores in context.  Hui had started a new draft when he came to my room one day, devastated: he’d received his SAT scores and they were as low as I’d feared.

His despair has remained a memory I flinch from–although at least in this case the recoil wasn’t for my poor handling of things. I didn’t try to console him, didn’t point out the local community college was very good (it is).  Hui accepted my heartfelt sympathy as best he could, nodding tightly, eyes filled with tears. He left my room, and I don’t remember another conversation, although I’m sure we ran into each other in the hallways.

“So how’s college?”

“Good. I want to get a degree in economics. I’m planning a transfer, getting everything in order, and…” Hui paused.

“Oh, hey. You didn’t just come by to say hi!”

Grin and a ducked head. “I’m want to apply to the same school as….. as last time. Could you look at my personal statements? They are short answer questions, so it won’t be one big essay.”

“Sure! You’ve got a good shot at transferring. I’m glad you’re trying again. You want to mail the responses?”

“They’re on my Google Drive. Do you have time?”

I sighed. “I do, but only until these last three are done with their tests, because then I have beer awaiting.”

I flipped through the short passages. “Hey, your writing has improved tremendously.” That wasn’t empty praise; his writing was still obvious an product of an English Language Learner, but the deficiencies now were….well, not infrequent, but not constant, either. Far fewer grammar errors, allowing me to focus on style issues.

Passage one needed a complete rewrite; Hui focused entirely on describing courses in his desired major. I told him to branch out. Passages two and three were nicely done, with only a few grammar and style edits. Passage four….

Passage four, in response to “what significant obstacle have you faced and how has it affected your academic progress” or something like that, was a lovely little explanation of the struggle he faced as a child who came to America at the age of ten, with two parents who still, to this day, speak no English.  Not just vague assertions, either, but entertaining, brief comparisons of verb tenses and articles that presented tremendous challenges to Chinese speakers, and finishing up with his constant efforts to remedy his gaps with books and films.

I looked over at Hui, who was watching me closely, and don’t tell anyone, but I was choked up. “You kept my notes from last time.”

“I didn’t need to. I remembered them. They really helped to think of my English as…something I’d achieved, rather than just something I do really bad at.”

“You should finish with a sentence to that effect.”


He left after wangling my phone number out of me, but promised to try email first. A student finishing up his test said “So can I come back to you for college admissions help after I graduate?”

“You better.”

I tell this story for two reasons. First: I write quite a bit about Asian immigrants , the corruption that China is introducing into US college admissions, the continual obsession with grades   and resumes with little interest in underlying knowledge, the pressure the parents put on the kids, and the  my concerns that they’re not here to become Americans, but to take advantage of a system not set up to defend against them. Inevitably, someone takes offense and argues that “they aren’t all like that”. Yes.  Even the ones who are like that….aren’t. I know that better than most.

But I tell this story in large part because I didn’t instantly think to write it up. I was just sitting around last night thinking of the three posts I have in the hopper, and trying to get the energy to finish one of them, when the events of the day popped into my mind and I thought it might make a good story. Then I realized it made a great story. Then–in the moment of this essay’s title–I realized the reason it didn’t instantly present itself as a great story is because this happens to me all the time.

A month ago, I was sitting in a Starbucks when I noticed the kid sitting next to me was a trig student from last year, now attending a graphics arts program. We were chatting when his pals showed up, all past students, and they sat down for half an hour and told me about their lives, exchanging funny stories about my classes. Two ex-students came back just this month asking for some help in their college math course. Every year, a few students make coffee dates, just to chat. Still others just stop by my classroom and say hi.

What a tremendous, amazing job I have. Teaching feeds my love of drama, my ability to think on the fly, and my love of intellectual challenges–and gives me tremendous independence. Then, it turns out, I live in my students’ memories.  I am Chips, not Browning.

In Clan Teacher, pay is substituted in part with ego gratification–and don’t think it’s not a fair trade. I’m a cranky introvert–you don’t think it matters to me that I send kids out into the world with Memories of Me? Good memories, of course–and yes, like all teachers, I worry about the damage, the memories I might cause through a careless word or ill-considered retort. But  I don’t demand perfection from my own performance. I am satisfied. I can try to do better.

So I’m not telling this story because it revived my flagging spirits, reversed my burnout. I’m telling you about Hui because it’s a glorious part of business as usual.

Which means I have to rest up, take this mild burnout seriously. Maybe take next summer off. (Yes. Laugh.) Get home earlier, particularly when I feel too tired to get up from my desk.

Because I never want to lose the sense of joy I get when remembering they actually pay me for this gig.







ELL isn’t Language Instruction

I’ve only taught English once in a public school (a humanities class), but I’ve been teaching private instruction English for a decade. Language instruction it’s not. I took French for a few years, and vaguely remember having to study verbs, and verb forms. Something about subjunctives. Unlike my father, I’m terrible at all new languages that don’t tell computers what to do.

I thought teaching English as a language was more structured.  Start with common verbs, the “persons”–I eat, you eat, he/she eats, they eat. Then common nouns. Then put things together? Isn’t that how it works? In other languages?

But then, French teachers speak English. Or Russian. Or whatever their students’ native language is–and a French teacher’s students only have one native language. You don’t see French teachers in American classrooms playing to a class of Punjab, Chinese, Spanish, and English students. Nor is the French teacher expected to be utterly ignorant of Punjabi, Mandarin, Spanish and English–yet still teach the students French.

Yet here I am with six students, only two of whom have even minimal conversational English, with four native languages. I’m not supposed to teach them English like a French teacher teaches French. Nor am I supposed to teach them English or anything else in Spanish, Punjabi, Chinese, or French as it’s spoken in the Congo.

American schools have never taught the English language.  Many education reform folk–and most non-experts–glorify immersion, our original method of handling language learners. Dump kids in, let them learn the language. That worked, right? Well, maybe not. Lots didn’t learn.  They just dropped out. As Ravitch the historian (not the advocate) observed, America’s past success educating immigrants has been dramatically overrated. (The immigrants’ children did well, but why we can’t expect that today is a tad Voldemortean for this essay.)

Giving additional services to non-English speaking students  became a public education mandate with Lau vs. Nichols.  But after the Chinese Lau, the case history shows that all major bilingual court cases involved Hispanics.

First, the Aspira case built on Lau, as  New York City signed a consent degree to provide bilingual education to limited English Puerto Rican students until they could function in regular classes. This led to a de facto mandate for nationwise bilingual education, and created the infrastructure of support. Not the curriculum, of course. (Ha, ha! Heaven forfend!)

One of those court cases was also one of the heads of the hydra known as US vs. Texas , which has a long, controversial history much of it not involving bilingual education. But at one point presiding judge  observed that the “experts” were appalled that Hispanic ELL students had only to reach the 23rd percentile in order to be reclassified as fluent.  The kids would only be doing better than 1 in 4 kids, wrote the judge, which simply wasn’t enough to perform adequately in mainstream classrooms. The judge never considered that black students aren’t given all this additional support, despite similar or worse test scores. We still don’t.

Anyway, as a result of that court case,  many if not all of states require ELL students to be proficient on achievement tests before they can be reclassified.  Proficient.  Often above average. Not basic. Different states have different procedures, different standards, but “proficient” is usually mentioned. And remember that ELL is only nominally concerned with teaching non-English speakers, since ELL students are primarily citizens.   Kids are asked  if  English is the only language spoken at home. Those who say “no” get tested, and if they don’t test proficient, they get tagged ELL and stay ELL until they do.  Schools don’t care–arent’ allowed to care–if the student came to America yesterday, a decade ago, or through a womb.

As I’ve written before, in math as it is in English, elementary school “proficiency” is much easier to acquire than the skill required for high school. It is thus much easier to test out of  ELL elementary school, regardless of original language, than high school. Most elementary ELL students test out after two or three years. Those who don’t make it out are categorized “long-term ELL”, meaning they’ve been ELL for over five years and never made proficient. Left unsaid is that kids need a certain cognitive ability to hit those test scores.

Thus by high school, over half the long-term ELL students are US citizens, split evenly among second and 3rd generation Americans who consider English their native language but have  lower than average cognitive ability or some specifically verbal processing issues. These are the kids who weren’t able to meet the relatively low elementary school proficiency standards. The other 44% are foreign born kids who couldn’t test out in the first five years.  It’s unlikely that either group is going to escape ELL in high school.

Consider: the primary reason for sheltering ELL learners once they’ve achieved functional fluency is to avoid kids being stuck in long term ELL. But there’s no solution to the “problem” of long-term ELLS, save accepting it as an artifact of an entirely different attribute.

If you’re following my dispirited trail of musings, you might be wondering if the elementary school proficiency levels are so low, then shouldn’t some of the kids who escape ELL status early run into trouble in high school?”   And to quote Tommy Lee Jones: Oh wow. Gee whiz. Looky here! Many Reclassified ELLs Still Need English-Language Support, Study Finds and points out that this finding is consistent with past research.

If you aren’t following my dispirited traill of musings, you’re thinking this has nothing to do with my assigned task of teaching English to one African, two Chinese, two Mexican, and one Punjabi student.

Sorry, I’m just explaining why I don’t teach English language instruction in an English class of kids who don’t speak English.

ESL and bilingual education from its earliest days was never intended to instruct students in the English language. It was actually a means of directing funding to close the Hispanic achievement gap for English speaking Hispanics which–it was believed–was due to inadequate academic instruction in English.   ELL’s purported objective is to provide support to non-English speaking students until they are proficient. Its actual  purpose is, first, to define a category that reports the academic achievement of  primarily Hispanic US citizens of lower than average cognitive ability–the better to beat our schools up with. Second, the classes gives the kids something to do until immersion gives them enough English to be mainstreamed, or at least into a higher ELL class.

So just as before, ELL teachers don’t provide English language instruction. Kids don’t come to America with a six word vocabulary and take English 1, followed by English 2, then English 3, and then AP English because hey, now they’re fluent.

When I express the concern   that I’m not teaching the kids English, I’m just giving them vocabulary and grammar enrichment in a sheltered English class, other ELL teachers and the admins nod their heads approvingly and say “You’re doing a great job!” Because ELL is not about teaching the English language.

Then I look at these six kids–and really, they’re terrific. In an ideal world, I’d never question my assignment. They’re a joy to teach and I’ll do my best for them. But only one of them is a citizen. Collectively, they are consuming one third of three English teachers’ schedule–that is, one full-time position at our school is dedicated to giving language enrichment to five non-citizens. All across America you’ll find thousands of these sheltered classes, for kids who just got here and instantly given free and guaranteed access to small classrooms and support in lessons that may or may not teach them the language, but gives them something to do in school until their English gets good enough for academic instruction. Which will–again–happen outside these classes, because lord knows, we’re not involved in language instruction.

I think of the millions of citizen kids. Of the bright high schoolers who could use challenging enrichment, maybe digging in deep to a Milton sonnet because they have the ability to do something more than fake their way through interpretation in carefully modeled  Schaffer chunks.  Of the many citizen students from the bottom half of the cognitive scale who didn’t check the “another language spoken at home” box and thus are not given additional time and money….not to get higher test scores, but just spend time with a teacher reading them a story and talking about vocabulary and context at a level they can enjoy. Every day. Of the many citizens from the bottom half of the cognitive scale who are told for their entire k-12 education that their native language isn’t, in fact, their native language.

Of course, whether or not we should be spending this kind of money on non-citizens never comes up. All we ever debate is whether we should use immersion or follow Krashen’s dictates and instruct every 1 in 20 kids in their native language. See, dedicating one full English position to six kids is the cheap version, the one favored by conservatives and most taxpayers. Bilingual advocates want native language instruction, which would further reduce class size from six to one or two, in every language we run into in our public schools.  Of course, we don’t have enough qualified teachers in each language, but since we can’t have perfection, at least  it’s a great way to boost employment in immigrant communities. So not only do we spend more resources on the kids, but the schools often provide more employment to the communities. As for citizens, well, you know, being bilingual is important. You should have studied more.

The entire debate about bilingual education vs. immersion is a canard. Of all the many education debates that aren’t as they seem, none wastes as much time,  money, and resources as that of the ludicrously named English Language Learner.

No one is asking whether we should be doing this at all. Well. I am. But then, I’m no one.

Someone, somewhere, will furiously argue that I’m “pitting brown students against each other”.  No. That’s what ELL does. And not just to kids of color, either.

Cynical? Scratch the surface of any ELL program and see how far off I am. Don’t listen to what they say. Go look at what they do.

Not sure if this piece has a point.  In math, I don’t have to think of this too often.

At the end of the day, I remind myself that I like the job, the boss folks like what I’m doing, and regardless of what you call it, this is a hell of a lesson.