Monthly Archives: May 2017

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:

prodlinesonly

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:

prodlinesoutline

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.

prodlinesplot

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:

prodlinesworksheet

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.

prodlinesthree

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:

prodlinesextensions

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.

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Catching Cheaters

Ben Orlin wrote a while back about the reasons students cheat–or, rather, the many reasons people offer for why students cheat. I’m mostly uninterested in that. What’s important, I think, is that they know you know.

Back in late March, I returned the first “Why is it Black Lives Matter?” history test back to the students and gave them this talk.

“So I was grading the papers, and pleased that people were doing well generally. Some kids saw tremendous improvement. But I was perplexed on one point. Several students were doing okay throughout the test, but getting slaughtered on the “identify key individuals” question, getting almost all of them wrong. I couldn’t figure that out. If you thought John Calhoun was an abolitionist who used to be a slave and Julia Ward Howe was at Harper’s Ferry,  then I would figure the finer points of the 1856 election were well beyond your ken. But you all did well on the 1856 election question, while a decent chunk of the class was telling me that Calhoun escaped from his owner in Maryland. This was confusing.”

“Then I had a horrible thought. As you know, because I mentioned it several times during the test, I created two versions of the assessment. I swapped the order of questions, and I swapped the order of some answer choices throughout the test. That would cause occasional problems on the True/False questions, but would be catastrophic on the “identify the right people” question, since the answers were chosen from ten options.”

“So I pulled all the tests that had the identify bloodbath, and sure enough, all of their answers matched perfectly to the other test.”

(Here’s the two test versions, and one example of student work)

I could tell that many students were experience a klong, a massive rush of shit to the heart1, that feeling you get when you realize far too late that you’ve done something very embarrassing and there’s no way to undo the action.  Like, say, cheating in such a way you’ve lost any hope of plausible deniability.

“Well. I’m somewhat new to teaching history. It’s pretty easy to spot cheating in math. You have a kid who screws around all the time, never doing any work, and suddenly he’s become so good at math he can do the work in his head. The test has a bunch of right answers and no work. The kid has a cousin who’s really good at math in the same class. I connect dots. Or you see two or three tests with the identical mistake on their tests, so the only challenge is to determine who originated the error so you can correct the misconception before you yell at her for giving others the answer.”

“History’s different, though, because there’s no work shown, and it’s not impossible that you could do really poorly on in-class activities yet be able to recall facts. A really quiet kid who has failed three tests and has taken utterly incomprehensible notes on several different activities could, theoretically, study really hard and how could I prove that she’d copied? That’s why I create two tests, with subtle differences in them that aren’t easy to spot.”

“I usually deal with cheating on an individual basis, but this is widespread. Out of the thirty-eight kids in this room, I have eight of you dead to rights–every single answer is from the other test. Another 6 cheated on at least a few of the answers–I can tell you knew most of them, then lifted the rest. That’s close to half the class caught–and I only caught them because you were unlucky enough to have the other test. How many cheated using the right test?”

“Then there’s the problem of who gave you the answers? I created the test yesterday. I only teach one section. The answers were very nearly all correct.  The questions were ordered differently, numbered differently, on different pages on each test.  This wasn’t opportunistic looking. This was collusion on a grand scale, probably involving cell phones.”

“I can’t figure out who provided the answers. But that person is in this class, listening to me now. So to that person, let me say three things. First, while I agree the Republicans of that era were nationalist, the party was formed in specific opposition to slavery, so I’d intended that answer to be “A”. It’s arguable, though. Second, you are very, very lucky that you aren’t the only top student who made that particular mistake, or we’d be having an extremely uncomfortable conversation. Third, you were cheating. You might think otherwise, since you were giving answers instead of receiving. But I call it cheating. The administrators would call it cheating. When you get to college, they’ll call it cheating, too.  If I could turn you in so that administrators could check your phone and see who you sent answers to, I would.”

“But without that information, my knowledge is incomplete. Some of you cheated with the same test. There are three students in particular who suddenly did exceptionally well. I was pleased. Now I’m just suspicious.”

“There’s some bright spots to all this, though. For example, when I first started grading I was really annoyed at Eddie, because he left half the test blank. But now, I’m kind of happy because this means Eddie didn’t cheat!”

Big laugh. Eddie stands up, arms held up in victory.

“Yeah, Eddie, you moved off the bottom of my disapproval list! But if you turn in a half-empty test again, I’m going to make you eat it. Other bright spots: Maria, who did well, Kevin,who did less well than usual, and Nero, who is doing not great for reasons I can’t figure out, all had tests that bear no relation to the cheating. They made mistakes that no one else made, so I know they weren’t involved.”

“Most of the copied tests seemed to be restricted to that one question. Since that question was worth close to twenty percent of the test, I decided the bloodbath of wrongness would do sufficient damage to your grade. No one who copied got higher than a 65. I’ll also write ‘Copied’ on the top of the test, so you won’t have to wonder if I know.”

“But two of you clearly copied almost every answer on the test and got nearly all of it right save the bloodbath. I don’t know how. But you’re not getting a grade. Since you’re both already failing the class, I don’t have to think about the best way to penalize you.”

“Before I give these tests back, I want to make a couple things very clear. I realize many of you see this class as pointless, and that there’s no harm to cheating. But I don’t care in the slightest whether you value this class or not. It’s my job to assess your knowledge. Cheating stops me from doing that, so I have to stop you from cheating. I’ve got a few security measures in mind. I don’t want any arguing about it, either.”

“Those of you who have me for math know that I’m pretty matter-of-fact about cheating. Just last week, I had exactly the case I described above: a near perfect test from a kid who can rarely focus on work in class, with no work shown and nothing but right answers. I called him outside, told him I was absolutely certain he’d copied the answers from someone, and that I was giving him a D because I figured that was where his knowledge was at. If he wanted to argue, he could do two or three of the problems in front of me. He didn’t want to argue. He took the D.”

“I won’t say I never hesitate about calling out cheating. For example, I’m pretty sure the three people who normally get Cs and Ds but did well on this test cheated and just got lucky having the right test. But I’m going to hold back this close to the semester. I’ll be watching you closely.”

“If I’m wrong, I’m wrong. No one is happier than I am when that happens. Over the years, I’ve had a couple students get really pissed off at the accusation, show me their work and what they know, and I’ve always been relieved. We go right on. No harm, no foul. Other students haven’t protested at all, just look chagrined. Same response either way.”

“It’s not personal. I might have my own moral judgments about cheating, but I’m not going to demand my students live by my morals. So if you get a test back that says “Copied”, don’t think I’ve put a black mark by your name, assume you may as well give up, and quit trying. Do exactly the opposite.”

“At the same time, I’m used to charging cheating with no evidence at all. Here I’ve got a rock solid case. I’m certain that some people cheated. So don’t fake outrage. If you want to talk to me, fine. Just don’t pretend and don’t waste my time.”

“But remember how I reassured you all when the class began that I wouldn’t fail anyone? Show up, do your best, and you’ll pass? Yeah. That’s off. This is your only warning. Cheat in my class again and you will sit on your butt in summer school. I’ll make sure of it.”

“Questions?”

There were none.
 

****************************************************
140 years ago, long before I had any interest in politics, I first learned the word “klong” from Full Disclosure, a William Safire novel about a president, blinded during an assassination attempt, fighting off a 25th Amendment attempt to remove him from office. Props to Noah Millman for being the only person other than me to remember the book whilst all around are calling for it, although his thoughts on Douthat’s madness were annoying. Safire properly credits Ben and Josh’s dad Frank Mankiewicz for the invention. Safire’s example of klong is enjoying a play then suddenly remembering you’d made dinner plans for the same evening on the other side of town, but that’s a tad civilized. A similar feeling is often experienced by murderers in Christie novels.


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.

LASlavegrowth

Student work: Louisiana slavery growth, 1810-1860

GASlaveGrowth

Student work: Georgia slavery growth, 1790-1860

NCSlaveStats

Student work: North Carolina slavery growth, 1810-1860

ALSlaveStats

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.”

Anyway.

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.

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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.