Tag Archives: curriculum

Curriculum Development: Not Work for Hire

I chopped off part of my last piece to expand more on teacher intellectual property, a topic near and dear1.

The conventional wisdom (which Stephen Sawchuk nicely outlines in the last part of this piece) holds that teachers are district employees, so any curriculum, lessons, or tests are considered work for hire . The teacher is paid specifically to develop the curriculum by the district, so the district owns the copyright and any subsequent profits from all of their teachers’ work—tests, worksheets, lesson plans, sequencing, whatever. .

In theory, my district could force me to pull down my posted curriculum from this blog—since I don’t own the copyright, I don’t have the right to give it away for free. Sites like Teachers Paying Teachers are illegal in this view, since teachers are making profits off their district’s property.

Originally, a teacher’s work was exempted from the work to hire rule, but in 1978 Congress didn’t include the exemption. Teachers’ unions have been trying to get the exemption reinstated.

Not for the first time, I’d argue the unions are going about this in exactly the wrong way. The exemption is unnecessary. Teachers aren’t hired to write curriculum. We are hired to teach. I’ve now outlined three well-established, time-honored practices that support this interpretation.

  1. Teacher contracts spell out their time commitments, which are the time in the classroom, staff and department meetings, supervisories, and mandatory professional development. No contracts hold teachers responsible for developing their own curriculum. A teacher is welcome to teach day by day from a provided textbook, or eschew a textbook altogether. They are not evaluated on the strength of their curriculum development in any way, nor can they be required to improve performance on this point. (More about this here.)
  2. While districts have begun to claim copyright, districts have never paid each other for teacher-developed curriculum. I have been in three districts. Like all teachers, I have a directory of my own curriculum, and I’ve carried it from school to school without any district ever informing me I couldn’t–much less demanding payment from my new district for use of their copyrighted curriculum.

    This practice, which has gone on for generations, clearly demonstrates that districts don’t consider themselves owners of the teacher curriculum. So if they want to ban a teacher from selling it, they need to start seizing the curriculum from teachers who developed it. Good luck with that.

  3. As I recently wrote, teachers given the extra duty of a class are paid purely based on the class instruction time, not the additional time (or not) needed to develop curriculum for that class. I’ve written before that teacher preps, or number of subjects actually taught, impact teacher workload. Teaching three different classes would be considerably more work, for most teachers, than teaching the same class four (or six) times. Teaching large classes also impacts workload. The teacher with multiple preps but a free period could have a student load of 150, while the teacher who works the prep could have 120 students (6 classes of 20). Unlikely, but theoretically possible. Doesn’t matter. More preps, more students, more outside work: irrelevant. What earns teachers a significant premium is the number of scheduled classes they are responsible for.

No one ever listens to me, but I’d advise unions to look for a good test case to challenge the work-for-hire idea, rather than argue for a change to copyright law, on the grounds that existing practice has acknowledged teacher intellectual property for decades. Certainly, the district should never be required to pay for the teacher’s work product in later years, should receive automatic use of anything developed during the teacher’s term of employment. But any rights in the curriculum we develop is our own.

I’ve often seen reformers–and other teachers—bemoan the notion of teachers who go home right after school everyday, clearly implying that the extra work developing lesson plans and curriculum is an element of our salary. But this simply isn’t true.

Besides, we don’t have any real idea of what makes a good teacher. Some of us work hours after school, some leave right after. No teachers who spend hours crafting curriculum, be it handouts, lesson plans, or tests, have any guarantee that they are getting better results. What they do know is that they are creating, creating without pay, and what they create should be theirs.

Here, again, acting works well as an analogy. Two actors are cast in a play, given supporting roles with an equivalent number of lines. They are both paid “scale” (whatever that is). The first actor spends six hours a day outside of rehearsal, practicing and perfecting the role, trying out different readings. The second actor barely makes it to rehearsal because he’s busy auditioning for a movie, doesn’t put any time into preparation.

They both would be paid scale for rehearsal and performance hours. The first actor wouldn’t be paid for the additional hours. The second actor might, in an unfair world, receive more acclaim and audience approval despite his lackluster approach.

But neither of them would be precluded from re-using aspects of their performance in later roles. The studied wince. The knowing sneer. The warm beaming smile, the turn and rapid delivery. Their performances were the result of work-for-hire. The script, like the textbook, belongs to someone else. The manner and method they use to deliver the performance are entirely theirs.

I ran into our union rep, an excellent English teacher, in the copy room. We began by chatting about class size (I’m teaching three massive A2 classes, which has given me some sympathy for the limits) and for various reasons (no doubt because this was on my mind), we got around to curriculum development.

“I wonder why the union doesn’t realize that we aren’t paid to develop curriculum? They don’t really need to change the copyright act to give teachers ownership of their work.”

“Or to give everyone ownership,” she said instantly. “There’s good reason to believe that no one’s work is truly original, that everything is derivative.”

Oh, lord. A CopyLeft fan. If our conversation had been Twitter based, I would have been properly contemptuous, but she’s a colleague and really very smart (she knew about the 1978 Copyright Act!) and besides, on this issue, I am actually seeking to persuade so I bite back my first response.

“Yeah, I ‘ve never agreed on that. But can we agree, at least, that whether teachers own their work or everyone owns their work, that the district doesn’t own our work?”

“Oh, absolutely. In order to give it away, we need the rights to it.”

So to the many loopy committed Creative Commons, Open Source, everything is derivative folks, can I just ask that we put aside our differences long enough to get the union to argue our case?

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1I’ve been writing about teacher IP and curriculum development for four years, as long as this blog’s been around–that’s in addition to many, many posts on my actual curriculum development. Here’s the primary pieces:

Teaching and Intellectual Property
Grant Wiggins
Developing Curriculum
Handling Teacher Preps
Math isn’t Aspirin. Neither is Teaching.


Content Knowledge and Reading Comprehension: Bold Talk and Backpedaling

Empty buckets seldom burst into flames. –Robert Pondiscio, Literacy is Knowledge.

People who push curriculum as a solution are generally pushing content knowledge, and they’re pushing content knowledge as a means of improving reading comprehension. Most of these people are in some way associated with Core Knowledge, the primary organization pushing this approach. They aren’t pushing it for money. This is a cause.

Pondiscio’s piece goes to the same well as E. D. Hirsch, who founded the Core Knowledge Foundation to promote the cause of content knowledge in curriculum, Lisa Hansel, the CK Foundation’s current Pondiscio, and Daniel Willingham, who sits on the board of Core Knowledge.

Pondiscio even borrows the same baseball analogy that Hirsh has used for a decade or so, to illustrate the degree to which content knowledge affects reading comprehension. Many Americans are unfazed by “A-Rod hit into a 6-4-3 double play to end the game”, but might be confused by “I’ll see how the wicket is behaving and then decide who are the bowlers I’ll use in the last few overs.”

We can understand content if we have the background knowledge, Hirsh et. al. assure us, but will “struggle to make sense” of reading if we’re unfamiliar because, as Pondiscio asserts, “Prior knowledge is indispensable”.

Let’s take a look at what some people do when they read without requisite content knowledge. (you can see other examples from my early childhood here).

Let’s pick another sporting event—say, the Kentucky Derby, since I don’t pay much attention to it. I googled, saw a headline at Forbes: “Final Kentucky Derby Futures Wagering Pool Opens Today”.

I don’t watch horseracing, I don’t bet, I know about futures because they were a plot point in “Trading Places”, but until that google I had no idea that people could bet on who won the Derby now, in advance. And now I do.

I was not confused. I didn’t struggle, despite my lack of prior knowledge. I constructed knowledge.

But Pondiscio says that any text on horse-racing is a collapsing tower of wooden blocks, “with each block a vocabulary word or a piece of background knowledge”, to anyone unfamiliar with horseracing. I have too few blocks of knowledge.

Robert Pondiscio would no doubt point out that sure, I could figure out what that Kentucky Derby headline meant, because I knew what the Kentucky Derby was. True. I’ve known what the Kentucky Derby was ever since I was 9 or so. I didn’t get the information from my parents, or my privileged life (I grew up decidedly without privilege). I read through all the Highlight articles at the doctor’s office and picked up a Sports Illustrated out of desperation (the internet is a glorious place; I just found the article) and then did exactly what Pondiscio suggests is impossible—read, understood, and learned when before I knew nothing.

I first knew “derby” as a hat, probably from an Enid Blyton story. But I had recently learned from “The Love Bug” that a derby was also a race. What did racing have to do with hats? But now I learned that horse races could be derbies. Since horses were way older than cars, the car races must have gotten the “derby” idea from horses. Maybe jockies got hats when they won horse races. (I learned many years later, but before today, that I was wrong.) I not only built on my existing knowledge base, I learned that the Kentucky Derby was a yearly horse race almost a century old and the results this year were upsetting. No one expected this horse to win, which probably was why people were upset, because just like the bad guy had a bet with the Chinese guy in “The Love Bug”, people made bets on who won. The article also gave me the impression that horses from Venezuela don’t always win, and that lots of horse races had names.

Pondiscio gives another example of a passage requiring background knowledge: the Dutch in New Amsterdam. Oddly enough, I distinctly remember reading just that sort of passage many years ago back in the fifth or sixth grade, about New Amsterdam first being owned by the Dutch, then control going to the English. I knew about Holland from Hans Brinker, which I’d found in someone’s bookshelf, somewhere, when I was six or seven. So New York was first founded by the Dutch–maybe that’s why they called the dad Mynheer in Legend of Sleepy Hollow just like they did in Hans Brinker, because according to the cartoon I’d seen on Wonderful World of Disney, Sleepy Hollow took place in New York .And then the English took it over, so hey, York must be a place in England. So when done, I knew not only that the Dutch had once been in the New World, but that other countries traded colonies, and that while we all spoke English now, New York had once been Dutch.

I didn’t carefully build content knowledge. I just got used to making sense of chaos, grabbing onto whatever familiar roadmarks I saw, learning by a combination of inference and knowledge acquisition, through haphazard self-direction grabbing what limited information I could get from potboiler fiction, magazines, and limited libraries, after gobbling up all the information I could find in schoolbooks and “age-appropriate” reading material. And I learned everything without prior knowledge other than what I’d acquired through previous reading, TV and movies as came my way. I certainly didn’t ask my parents; by age six I acknowledged their expertise in a limited number of topics: cooking, sports, music, and airplanes. In most important topics, I considered them far less reliable than books, but did deem their opinions on current events useful. Yes. I was obnoxious.

My experiences are not unique. Not today, and certainly not in the past. For much of history, people couldn’t rely on information-rich environments and supportive parents to acquire information, so they turned to books. Using vocabulary and decoding. Adding to their existing knowledge base. Determinedly making sense of alien information, or filing it away under “to be confirmed later”.

But of course, say the content knowledge people pushing curriculum. And here comes the backpedal.

E. D. Hirsch on acquiring knowledge:

Almost all the word meanings that we know are acquired indirectly by intuitively guessing new meanings as we get the overall gist of what we’re hearing or reading.

That describes almost exactly what I did for much of my childhood. But this is the same Hirsch who says “Reading ability is very topic dependent. How well students perform on a reading test is highly dependent on their knowledge of the topics of the test passages.” Nonsense. I scored at the 99th percentile of every reading test available, and I often didn’t know anything about the topic of the test passage until I read it—and then I’d usually gleaned quite a bit.

Pondiscio slips in a backpedal in the same piece that he’s pushing content.

Reading more helps, yes, but not because we are “practicing” reading or improving our comprehension skills; rather, reading more is simply the most reliable means to acquire new knowledge and vocabulary.

This is the same Pondiscio who said a couple years ago:

What is needed is high-quality preschool that drenches low-income learners in the language-rich, knowledge-rich environment that their more fortunate peers live in every hour of every day from the moment they come home from the delivery room.

Well, which is it? Do they think we learn by reading, or that we only learn by reading if we were fortunate enough to have parents who provided a knowledge-rich environment?

Take a look at the Core Knowledge promotional literature, and it’s all bold talk: not that more content knowledge aids comprehension, but that content knowledge is essential to comprehension.

I’ve likewise tweeted about this with Dan Willingham:

Me: Of course, taken to its logical conclusion, this would mean that reading doesn’t enable knowledge acquisition.

Willingham: if you have *most* of the requisite knowledge you can and will fill in the rest. reading gets harder and harder. . . ..as your knowledge drops, and the likelihood that you’ll quit goes up.

Me: The higher the cog ability, the higher ability to infer, fill in blanks.

Willingham: sooo. . . hi i.q. might be better at inference. everyone infers, everyone is better w/ knowledge than w/out it. yes?

So Willingham acknowledges that IQ matters, but that as knowledge and IQ level drops, engagement is harder to maintain because inference is harder to achieve. No argument there, but contrast that with his bold talk here in this video, Teaching content is teaching reading, with the blanket statements “Comprehension requires prior knowledge”, and attempts to prove that “If you can read, you can learn anything” are truisms that ignore content knowledge. No equivocation, no caveats about IQ and inference.

So the pattern: Big claims, pooh-poohing of reading as a skill that in and of itself transfers knowledge. If challenged, they backpedal, admitting that reading enables content acquisition and pointing to statements of their own acknowledging the role reading plays in acquiring knowledge.

And then they go back to declaring content knowledge essential—not useful, not a means of aiding engagement, not important for the lower half of the ability spectrum. No. Essential. Can’t teach reading without it. All kids “deserve” the same content-rich curriculum that “children of privilege” get not from schools, but from their parents and that knowledge-drenched environment.

And of course, they aren’t wrong about the value of content knowledge. I acknowledge and agree with the surface logic of their argument: kids will probably read more readily, with more comprehension, if they have more background knowledge about the text. But as Daniel Willingham concedes, engagement is essential as well—arguably more so than content knowledge. And if you notice, the “reader’s workshop” that Pondiscio argues is “insufficient” for reading success focuses heavily on engagement:

A lesson might be “good readers stay involved in a story by predicting” or “good readers make a picture in their mind while they read.” ..Then the children are sent off to practice the skill independently or in small groups, choosing from various “high-interest” books at their individual, “just-right” reading level. [Schools often have posters saying] “Good readers visualize the story in their minds.” “Good readers ask questions.” “Good readers predict what will happen next.”

But Pondiscio doesn’t credit these attempts to create engagement, or even mention engagement’s link to reading comprehension. Yet surely, these teachers are simply trying to teach kids the value of engagement. I’m not convinced Pondiscio should be declaring content knowledge the more important.

Because while Core Knowledge and the content folks have lots of enthusiasm, they don’t really have lots of research on their product, as Core Knowledge representatives (q6) acknowledge. And what research I’ve found never offers any data on how black or Hispanic kids do.

Dan Willingham sure seemed to be citing research lately, in an article asking if we are underestimating our youngest learners, citing a recent study says that we can teach young children knowledge-rich topics like natural selection. He asks “whether we do students a disservice if we are too quick to dismiss content as ‘developmentally inappropriate,'” because look at what amazing things kids can learn with a good curriculum and confidence in their abilities!

Of course, a brief perusal of the study reveals that the student populations were over 70% white, with blacks and Hispanics less than 10% total. Raise your hand if you’re stunned that Willingham doesn’t mention this tiny little factoid. I wasn’t.

Notice in that study that a good number of kids didn’t learn what they were taught in the first place, and then a number of them forgot it quickly. Which raises a question I ask frequently on this blog: what if kids don’t remember what they’re taught? What if the information doesn’t make it to semantic memory (bottom third of essay). What evidence do the curriculum folks have that the kids will remember “content” if they are taught it in a particular sequence? (Note: this essay was too long to bring up Grant Wiggins’s takedown of E. D. Hirsch, but I strongly recommend it and hope to return to it again.)

Like reformers, curriculum folk are free to push the bold talk, because few people want to raise the obvious point: if content knowledge is essential, instead of helpful, to reading comprehension, then no one could ever have learned anything.

But contra Pondiscio, empty buckets do burst into flames. People do learn without “essential” content knowledge. Even people from less than privileged backgrounds.

Here’s the hard part, the part too many flinch from: Smart people can learn this way. All anyone has ever needed to acquire knowledge is the desire and the intellect. For much of history educated people had to be smart and interested.

In recent years, we’ve done a great job at extending the reach of education into the less smart and less interested. But the Great Unspoken Truth of all education policy and reform, be it progressive, critical pedagogy, “reform” or curricular, is that we don’t know how to educate the not-smart and not-interested.


Math Instruction Philosophies: Instructivist and Constructivist

Harry Webb has been on a tear about discovery vs. traditional explanations. The hubbub has pulled the great god Grant Wiggins, originator of backward design, which is a bible of ed schools as a method for developing curriculum.

Now, let us pause, a brief segue, to reflect on those last two words. Developing curriculum. I’m talking about teachers, yes? Teachers, building their own unit lessons, their own tests, their own worksheets. As I’ve written, teachers develop their own curriculum and, to varying degrees, have intellectual property rights (I would argue) to their material. So when reformers, unions, politicians, or whoever stress the importance of curriculum, textbooks, and professional development in implementing Common Core, there’s a whole bunch of teachers nationwide snorfling at them.

So Wiggins and Jay McTighe wrote Understanding by Design, which describes their framework and approach to curriculum. It is, as I said, a bible of ed schools. I have a copy. It’s good, although you have to look past their irritating examples to figure that out.

(Note: See Grant Wiggins’ response below. I’ve reworded this slightly and separated it to respond to his concerns. Also throughout, I changed “direct instruction” to some other term, usually instructivism.)

The book clearly states that there’s no one correct approach for every situation, that arguing between instructivism and constructivism creates a false dichotomy. So I was jokingly sarcastic before, but my point is real: it’s hard to read Grant Wiggins and not think that, so far as K-12 curriculum goes, he leans heavily towards constructivist. As one example, in a text section that discusses the fact that there’s no one right approach, he includes this table on the activities dominant in each approach. When I look at this table, I see a clear preference for constructivist approaches. I also see it in this highly influential essay and much of his writing. But as Wiggins states in the comments, and in the book, he clearly denies this preference. However, Wiggins’ book is the bible of ed schools for a reason, and it’s not for its categoric embrace of all things instructivist. So put it this way: what he says are his preferences and what any instructivist would take away from his preferences are probably not the same thing. I say this as someone who periodically rereads his work because of the value I find in it once I shift my focus away from the trappings and focus in on the substance. I encourage anyone who agrees with my impression of Wiggins’ preference to read him closely, because he’s done a lot over time to inform my approach to curriculum development.

(end major edits–I put the original text at the bottom)

So Wiggins reads all this hooha, and comes out with this outstanding description of lectures and why they are a problem. I agreed with every word of this post (there are two others), so much so that I tweeted on it. (Note: I agree for math. History’s a different issue.) As I did so, I was vaguely disturbed, because look, while I don’t write a lot about ed school per se (and even defend it, slightly), I spent a lot of time in class naysaying. And if they’d been saying reasonable things like this about lectures, what had I been disagreeing about?

And then Harry comes through brilliantly, answering my question and pointing out a huge hole in Wiggins’ 3-part series:

Wiggins writes of a survey of teachers in order to support his view that different pedagogies are required to achieve different aims. Unsurprisingly, the teachers give the right answers; the ones that they probably learnt at Ed School. However, the survey response that is taken to represent lecturing is called, “DIRECT TEACHING Instruction on the knowledge and skills.” Now, although I do not recognise my practice in Wiggins’ definition of lecturing, I do recognise myself in this definition wholeheartedly. And so I think we are being invited here to see all direct teaching – dare I say direct instruction – as non-interactive lecturing that lasts for most of a period.

Hey. Yeah. That’s right! Wiggins naysays the lecture in his essay, but the overall debate is between instructivism, of which lectures are just a part, and it’s , and it’s instructivism that has a bad name in ed school, not solely lectures. Harry says that he explains in classroom discussion, but rarely lectures. Which may sound like someone else.

Harry scoots right by this, because he’s all obsessed about the fuzzy math and constructivist debate, and it occurred to me that this area needs elucidation, because most people—and reporters, I am looking at you—don’t understand this difference.

So here it is: not all explanation is lecture, and not all discovery is constructivist.

In an effort to not turn all my posts into massive tomes (don’t laugh), I’m going to write about this difference later. Here, I’m just going to show you the difference through different teachers.

Before I start: labels are hard. Roughly, the terms reform, student-centered, constructivist, “facilitative” (Grant Wiggins’ term) all refer to the open-ended investigative approach. Instructivist, teacher-centered, traditionalist, direct instruction are all terms used to describe the approach where the teacher either tells you how to do it or wants you to figure out the way (not a way) to do it. (Note: I left “direct instruction” in here, because I believe it’s still an instructivist approach.)

Very few math teachers are pure constructivist. We’re talking degrees. I have no data on usage rates, but I’d be pretty surprised if 80% of all high school math teachers didn’t use traditional instruction-based approach for 90% of their lessons. I speak to a lot of colleagues who dislike pure lecture and would like to teach a more modified instructivist mode, but they aren’t sure how it works. However, most high school math teachers are instructivists who lecture. Full stop.

Constructivist Approach (aka investigation, reform)

Dan Meyer: Dan Meyer’s 3-Act Meatballs
Fawn Nguyen: Barbie Bungee
Fawn Nguyen Vroom Vroom
Michael Pershan: Triangles and Angles (he calls this investigation. I’d personally characterize it as “in between”, but it’s his call.)
Cathy Humphries: Investigation into Quadrilaterals

This is a partial list. Dan’s blog has links to all his various projects, as well as other bloggers committed to the investigative approach.

(By the way, I am dying to do the Vroom Vroom one, but I’m not enough of a mathematician to understand the math behind it. Neither does Fawn, apparently. The math looks quadratic. Is it?)

I’m not a fan of the open-ended reform approach, but I like all sorts of the activities the constructivists come up with. I just modify them to be more instructivist.

Remember that both Meyer and Nguyen use worksheets, practice skills, and many other elements that are pure instructivist. Pershan rarely does open-ended activities. In contrast, Cathy Humphries is very close to pure constructivist math. Total commitment to reform.

Traditional Instructivism (Lectures)

Much MUCH harder to find traditionalists bloggers. I’ve included two of my “lectures” that have relatively little discussion, just to fill out the list:

Me: Geometry: Starting Off
Me: Binomial Multiplication and Factoring with the Rectangle

Dave at MathEquality is traditionalist, a guy who works hard to explain math conceptually, but does so for the most part in lecture form. However, it’s also clear he keeps the lectures fairly short and gives his students lots of in class time for work.

But he’s the only one I can find. Right on the Left Coast appears to be a traditionalist, but he writes more about policy and his disagreement with traditional union views. (Huh, I should have mentioned him in my teacher blogger writeup of a while back.)

In order to give the uninitiated a good idea of what lecture looks like, three google searches are informative:

factoring trinomials power point

holt math power point

McDougall Litell math power point

Many high school teachers build their own power point explanations. Others just take the ones provided by publishers.

Still others use a document camera or, if they’re extremely old-school, transparencies.

What they look like is mostly this:

Khan Academy: Isosceles Right Triangles

Many teachers are really, really irritated at the fuss over Khan Academy because all he does is lecture his explanations—and not very well at that.

The most vigorous voices for traditional direct instruction comes from people who don’t teach high school math. That’s not a dig, it’s just a fact.

Modified instructivist

I’m not sure what to call it. There’s not just one way to depart from instructivist or constructivist. The examples here generally fall into two categories: highly structured instructivist discovery, and classroom discussions with lots of student involvement.

Me:Modeling Linear Equations
Me: Modeling Exponential Growth/Decay
Michael Pershan: Proof with Little Kids
Michael Pershan: Introducing Polar Coordinates
Michael Pershan: The 10K Chart
Ben Orlin: …999…. and the Debate that Repeats Forever.
Ben Orlin: Permutations and combinations

For a complete list of my work, check out the encyclopedia page on teaching. I likewise recommend Pershan and Orlin’s blogs.

A question for Grant Wiggins, and anyone else interested: what differences do you see in these approaches?

A question for reporters: when you write about reform or traditional math, do you have a clear idea of what the fuss is about? And did these examples help?

Question for Harry Webb: You sucked me into this, dammit. Satisfied?

If you have good examples of math instruction that falls into one of these categories, or want to propose it, tweet or add it to the comments. I’m going to write up my own characterizations of this later. Hopefully not much later.

*******

Here’s what I originally said in the changed paragraph:

So Wiggins and Jay McTighe wrote Understanding by Design, which describes their framework and approach to curriculum. It is, as I said, a bible of ed schools. I have a copy. It’s good, although you have to look past their irritating examples to figure that out. The book clearly states that there’s no one right answer, that arguing between direction instruction or constructivism creates a false dichotomy, but then there’s this table on the activities dominant in each approach. Cough. Okay, no one right answer, but a strong preference for facilitative/constructivist.


Memory Palace for Thee, but not for Me

Should we teach kids how to memorize?, asks Greg Cochran. It’s a worthwhile question, and I have some thoughts, but got halfway through that post and hit some snags.

The comments, though, got me thinking about memory in general.

Back in high school, I used to write out all the acting Oscar nominees in order, lefthanded, to keep me at least somewhat focused in math class. During college, I’d write out the Roman emperors, English rules from William I to Elizabeth II, or the US Presidents, again, left-handed, to keep myself focused during college. I outgrew this habit at some point, probably when people asked me what I was writing; by my 40s, in grad school, I know I just doodled. I rarely set out to memorize things, and get no pleasure from the knowledge. What I do enjoy is the memories that come back with the data. So 1934 was It Happened One Night, with Clark Gable and Claudette Colbert, which got me thinking of Roscoe Lee Karns and Cliff Edwards and all the other reporters in His Girl Friday. Then the throwaway movies until Gone with the Wind, all hail Olivia (who outlived her feuding sister), and thinking about what a great decade the 40s was, oops, losing track of the lecture, get back to writing names. Recalling information keeps me focused, but the information itself doesn’t give me much enjoyment.

So if I know all the actors who’ve won best supporting Oscar and then best acting Oscar, and I didn’t set out to learn it, have I memorized that information? Huh. I realized I didn’t know, so I went and Wiki’ed up on memory. This was a helpful read, although I’m sure I have some of it wrong. Take my descriptions at your own risk.

If I understand all this, my echoic memory is much better than my iconic; both being short-term memory associated with a sense (hearing and sight). My students get a kick out of the fact that I look at geometry figures probably five times while drawing them. I can forget which way a right triangle faces in the time it takes me to turn 180 degrees from the book to the board.

In practice, this means my short-term auditory memory would be termed eidetic if it were vision-based, while I suck at that game where they give you 30 seconds to look at a tray of items. I’ve known this for a while, that hearing is extremely important to my short-term memory. I can easily maintain five or six conversations at once (very useful in teaching). However, five days later, my recall isn’t better and often worse than anyone else.

Fun example of this: a few months ago, Steven Pinker tweeted a language and memory study. I started to take it and then ran into a hitch.

The problem was, for me, that the practice wasn’t anything like the test. In the practice, up comes four groups of two letters situated around a cross. Then the screen blanks out, and occasionally you’ll see an arrow pointing to the position to remember. Some time later, a letter pair appears in the space and you indicate if it’s the same pair or a different one.

So I went through 20 or so practices, and did great, getting them all right. Then comes the actual test format, and I fall out of my chair, howling with laughter:

gamechinese

No more letters.

Until today, I didn’t even have the words to describe this. But in the practice, I literally vocalized the four pairs, saying “yn, qg, ds, hm”. The arrow would come up, and I’d say “okay, that’s qg”. Up comes “qg”, it’s the same. The whole test, I did with my short-term auditory memory, the echoic.

I guess if I were Chinese, I could do the real test that way. But the minute it came up, I realized I couldn’t rely on my auditory memory, and that’s game over. I’ve come back to the test since he closed it for research (didn’t want to screw up his results), and practiced two forms of memory. In the first, I say them aloud: “x on top, 7 on left, double T on right” and that helps, but I can’t say it fast enough and the image blanks out. Then I try it using my iconic memory (as I now know it), and if I focus really hard, I can usually see three of them before it blanks out. But it’s really hard.

I wondered if maybe the researcher planned it, but wouldn’t the practice be part of the actual test? I guess for most people it’s not a huge change.

Anyway, it’s a great example of how I use auditory memory instead of iconic. Auditory and visual long-term memory, if there is a such a thing, reverses in strengths for me. I can only vaguely recall the names of my high school history teachers, but their faces are quite clear in my mind. I likewise remember student faces much better than names, which is weird given that my short-term visual memory (iconic) is dreadful, but I guess they aren’t linked. For names, I need not only appearance but position—I can be completely discombobulated if a student changes seats. Periodically, I will randomly screw up names. I went a month calling an Anthony Andrew, despite having taught him two years in a row. So it appears that long-term, I rely more on visual than auditory, whereas short-term it’s reverse. When I think of the word “capybara”, I visualize the page of Swiss Family Robinson, kids’ edition, where I saw it at seven, the picture and the words on the page. The memories of books I recall in this essay are all strongly visual.

Long-term memory breaks down into into explicit/declarative memory and implicit memory (also known as procedural memory). Explicit memory is composed of episodic (autobiographical) and semantic (factual and general knowledge).

So my semantic memory is outstanding. My episodic memory, not anything special, particularly if it’s not autobiographical (there is some difference there).

Reading all this made something very clear that’s bothered me for thirty years or more: I have a terrible time with implicit memory if it involves moving parts not under my direct control. Specifically: driving, horseback riding, skiing, tying shoelaces. I didn’t drive until I was 22, because the act of learning was just so unnerving. Ran away like a ninny from skiing, never bothered with horseback riding. My younger brother finally shamed me into learning to tie my shoes, although which brother, and our relative ages, changes as the years go by, the better to embarrass me. If everything’s under my control, no problem: cooking, typing.

This gave me an interesting new way to think about how I learn. I’ve thought of my memory as a database since I first knew what a database was. Every so often the database goes wonky. Sometimes it’s a random switch of names. Sometimes I just can’t get the name. Once, on a contract, I remember complaining about the fact that I couldn’t remember the name of the Vietnamese PC guy, the one with the really heavy accent. “Which one?” asked my boss. “There are two?” “Sure. Pham and Tran.” “Crap. I had a duplicate data key.”

And every so often I just file away a false fact. For a good decade or so, my memory said that Richard Nixon had been governor of California. It’s not that I thought he was. I knew he wasn’t. But my memory thought he was, so if I were writing out the presidents who had been governors, Dick Nixon would be on the list. Took me years to find and fix that key.

In most cases, I effortlessly add information to my semantic database, creating links and keys between “data” fields and update references—that is “learning”, without really thinking of it as such. Until I was 25 or so, I had no idea how to learn if the process couldn’t be added to the database. As described in the “learning math” essay, I was completely helpless in those cases. When I was younger, I was often told I was exceptionally bright, and while I didn’t disagree, there was always this nagging concern—if I’m so bright, why are some things utterly incomprehensible to me?

Thanks to my first real job out of college, I finally figured out that, when the learning process wasn’t invisible, I had to learn through an insane series of trial and error tasks in which I traverse the landscape like Wile Coyote waiting for something to blow up in my face—this is how I learned programming and most of the math that I know.

So, put into my new terminology (probably inaccurately), if the new information has no link to my current knowledge, if I have no way to store and access it, I have to go out and acquire a metric ton of episodic memories to create a database table for my semantic memory, to build the connections and cross references. I have always known that this is somewhat unusual; I’ve watched many folks listen to lectures and get right up to speed while I’ve been sitting around unable to focus. I’ve tentatively concluded that my data fields have far more attributes—more metadata, if you will (but not metamemory, which is different), which makes the initialization more labor-intensive, but more useful in the long run.

Obviously, storage and recall is a whole field of memory study, and in some way everyone struggles with the process I describe. But for me, it was a very clear gap between the information I could or couldn’t learn, and I had no way of bridging that gap for the first 25 years of my life. For a long time, whole areas of learning weren’t under my conscious control and I had no way of anticipating what they might be. The trial and error, Wile Coyote process was a huge breakthrough that changed my life and expanded my career paths.

So when people talk about memory palaces, like in The Mentalist or Sherlock, I’ve got no frame of reference. My memory is not spatial, but associative. It’s a database that I retrieve, not a room where I put things. Do not tell me that it’s just a simple technique that anyone can learn. I couldn’t. Full stop.

End the investigation into Ed’s memory.

I imagine that during ed school, I read something about semantic memory. Lord knows Piaget must mention it somewhere. I probably dozed through a Willingham post on it, or read it in a book. But you know me, it doesn’t get put into semantic memory until I have a data table and some crosstabs.

I am not trying to become an expert on memory, nor am I unaware of the fact that there’s probably all sorts of reading I could do simply to discuss it more intelligently.

But all this leads me to a few observations/questions.

First, it seems that the myth of ‘they’ve never been taught’, the problem of kids forgetting what they learned, could be framed as the difference between episodic memory and semantic memory. That is, the kids I’m describing remember the topic as an episode in their lives, not as reference information, and since it wasn’t a very interesting episode they lose it quickly.

Next, I’ve remarked before, and will do again, that many smart kids (in my work, almost entirely recent Asian immigrants) can regurgitate facts and learn procedures to a high degree of accuracy but retain none of them and even while knowing them have no idea what to do with them outside a very limited task set. Whenever you see kids screaming “we have to have the test today” they are kids who know they will lose the information. So these kids may be relying entirely on episodic memory? If this is true, our reliance on test scores as a knowledge indicator becomes, er, unnerving, particularly since this behavior seems so strongly linked to one demographic here in the US. And I speak as someone who likes test scores.

Then the opposite case: in math, at least, I’ve noticed that fact fluency is not required for understanding of higher math, and that it’s not at all unusual to see kids who are fact fluent but can’t grasp any abstraction. It may be that these kids are filing math facts into implicit memory, as tasks. Or maybe that’s always the case in math.

It goes without saying that memory is linked to cognitive ability, right? Oops, I said it anyway.

I’ve also noticed that teaching, like police work, is a profession with limited need for semantic memory (the content fact base, rules of the job) and a tremendous need for episodic memory (what worked last time). This may be why teaching isn’t given much respect as a profession, and also why smarts, past a certain level, doesn’t appear to play much of a role in teaching outcomes.

And so, Greg Cochran asks whether we should teach kids to memorize.

Realize too that the memorization battles are just another front in the skills vs. knowledge debate. The skills side, touted primarily by progressives and, separately, many teachers themselves, emphasize the need for students to know how to do things—think critically, problem solve, analyze information. The knowledge side, headed by the great E. D. Hirsch, complain about the skills stranglehold and want to emphasize the need for students to know things—facts organized into a logical curriculum. Those pushing for memorization are squarely on the knowledge side of things, and often mock teachers for being too stupid to understand the importance of knowledge.

I have not entered this debate because until now I haven’t had a framework for my answer of “it depends”. Do we want to reward bright kids for memorizing content knowledge without a clue about what it means and little ability to use it, as is de rigueur in many Asian immigrant populations? I submit that we don’t. Do we want our kids of middle ability or higher to memorize math facts and general content knowledge, the better to improve their reading comprehension and understanding of advanced math? I submit that we do. And how much memorization, exactly, can we expect and demand from our low ability kids? I submit that the answer is “We don’t know, and are scared to find out”.

In other words, memorization requirements, like everything in education, is ultimately set by student cognitive ability, which we aren’t allowed to discuss in any meaningful way. But teachers like me, who are required to deal with a 3-4 year range in ability levels, with a canyon-sized gap in content knowledge from high to low, have to make decisions about skills vs knowledge debate every day. Those on the outside should realize that teachers have many good reasons for pushing back on the memorization point, given the students they teach and the expectations forced on them by those who don’t know any better.

Yeah, I said six essays a month? Feel free to laugh at me. But I have a part 2 on this one, and I’ll try to finish it soon.


Core Meltdown Coming

I’ve stayed out of the Common Core nonsense. The objections involve much fuss about federal control, teacher training, curriculum mandates, and the constructivist nature of the standards. Yes, mostly. But so what?

Here’s the only important thing you need to know about Common Core standards: they’re ridiculously, impossibly difficult.

I will focus here on math, but I’m an English teacher too, and could write an equivalent screed for that topic.

I’m going to make assertions that, I believe, would be supported by any high school math teacher who works with students outside the top 30%, give or take.

Two to three years is required just to properly understand and apply proportional thinking–ratios and percentages. That’s leaving off the good chunk of the population that probably can’t ever truly understand it in non-concrete situations. Proportional thinking is a monster. That’s after two to three years spent genuinely understanding fraction operations. Then, maybe, they could get around to understanding the first semester of first year algebra–linear equations (slopes, more proportional thinking), isolating variables, systems, exponent laws, radicals—in a year or so.

In other words, we could use K-5 to give kids a good understanding in two things: fractions and integer operations. Put measurement and other nonsense into science (or skip it entirely, but then remember the one subject I don’t teach). Middle school should be devoted to proportional thinking, which will introduce them to variables and simple isolation procedures. Then expand what is currently first semester algebra over a year.

Remember, I’m talking about students outside the top 30% or so (who could actually benefit from more proportions and ratios work as well, but leave that for another post). We might quibble about the time frames and whether we could add a little bit more early algebra to the mix. But if a math teacher tells you this outline is nonsense, that if most kids were just taught properly, they could learn all this material in half the time, ask some questions about the demographic he works with.

Right now middle school math, which should ideally focus almost entirely on proportions, is burdened with introductions to exponents, a little geometry, some simple single variable equations. Algebra I has a whole second semester in which students who can’t tell a positive from negative slope are expected to master quadratics in all their glory and all sorts of word problems.

But Common Core standards add exponential functions to the algebra one course load and compensate by moving systems of equations and exponent laws to eighth grade while much of isolating variables is booted all the way down to sixth grade. Seventh grade alone bears the weight of proportions and ratios, and it’s one of several curricular objectives. So in the three years when, ideally, our teachers should be doing their level best to beat proportional thinking into students’ heads, Common Core expects our students to learn half of what used to be called algebra I, with a slight nod to proportional thinking (and more, as it turns out. But I’m getting ahead of myself).

But you don’t understand, say Common Core devotees. That’s exactly why we have these higher, more demanding standards! We’ve pushed back the timeline, to give kids more time to grasp these concepts. That’s why we’re moving introduction to fractions to third grade, and it’s why we are using the number line to teach fraction numeracy, and it’s why we are teaching kids that whole numbers are fractions, too! See, we’ve anticipated these problems. Don’t worry. It’s all going to be fine.

See, right there, you know that they aren’t listening. I just said that three to four YEARS is needed for all but the top kids to genuinely understand proportional thinking and first semester algebra, with nothing else on the agenda. It’s officially verboten to acknowledge ability in a public debate on education, so what Common Core advocates should have said, if they were genuinely interested in engaging in a debate is Oh, bullpuckey. You’re out of your mind. Four years to properly understand proportional thinking and first semester algebra? But just for some kids who aren’t “smart”? Racist.

And then we could have an argument that matters.

But Common Core advocates aren’t interested in having that debate. No one is. Anytime I point out the problem, I get “don’t be silly. Poor kids can learn.” I point out that I never mentioned income, that I’m talking about cognitive ability, and I get the twitter version of a blank stare somewhere over my shoulder. That’s the good reaction, the one that doesn’t involve calling me a racist—even though I never mentioned race, either.

Besides, CC advocates are in sell mode right now and don’t want to attack me as a soft bigot with low expectations. So bring up the difficulty factor and all they see is an opportunity to talk past the objection and reassure the larger audience: elementary kids are wasting their time on simple math and missing out on valuable instruction because their teachers are afraid of math. By increasing the difficulty of elementary school math, we will forcibly improve elementary school teacher knowledge, and so our kids will be able to learn the math they need by middle school to master the complex, real-world mathematical tasks we’re going to hand them in high school. Utterly absent from this argument is any acknowledgement that very few of the students are up to the challenge.

The timeline isn’t pushed back for algebra alone. Take a look at Geometry.

Geometry instruction has been under attack for quite some time, because teachers are de-emphasizing proofs and constructions. I’ve written about this extensively (see the above link, here, and here). Geometry teachers quickly learn that, with extensive, patient, instruction over two-thirds of their classes will still be completely incapable of managing a three step proof. Easy call: punt on proofs, which are hard to test with multiple choice questions. Skip or skate over constructions. Minimize logic, ignore most three dimensional figures (save surface area and volume formulas for rectangular prisms and maybe cylinders). Focus on the fundamentals: angle and polygon facts (used in combination with algebra), application of pythagorean theorem, special rights, right triangle trig, angle relationships, parallel lines, coordinate geometry. And algebra, because the train they’re on stops next at algebra II.

Lowering the course requirements is not only a rational act, but a sound curriculum decision: educate the kids in what they need to know in order to succeed pass survive have some chance of going through the motions in their next math class.

But according to everyone who has never worked with kids outside that 30%, these geometry teachers are lazy, poorly educated yutzes who don’t really understand geometry because they didn’t major in math or are in the bottom third of college graduates. Or, if they’re being charitable—and remember, Common Core folks are in sell mode, so charity it is—geometry teachers are just dealing with the results of low expectations and math illiterate elementary school teachers.

And so, the Common Core strategy: push half of geometry down to middle school.

Here’s what the Common Core declares: seventh graders will learn complementary and supplementary angles and area facts, and eighth graders will cover transversals, congruence, and similarity.

But wait. Didn’t Common Core standards already shove half of algebra down to middle school? Aren’t these students already learning about isolating variables, systems of equations, power laws, and proportions and ratios? Why yes, they are.

So by virtue of stuffing half of algebra and geometry content into middle school, high school geometry, as conceived by Common Core, is a stripped-down chassis of higher-order conceptual essentials: proofs, construction, modeling, measurement (3 dimensions only, of course), congruence and similarity, and right triangles.

Teachers won’t be able to teach to the lowest common denominator of the standards, not least because their students will now know the meaning of the lowest common denominator, thanks to Common Core’s early introduction of this important concept, but more importantly because the students will already know the basic facts of geometry, thanks to middle school. The geometry teachers will have no choice but to teach constructions, proofs, logic, and all the higher-order skills using those facts, the part of geometry that kids will need, intellectually, in order to be ready for college.

Don’t you see the beauty of this approach? ask the Common Core advocates. Right now, we try to cover all the geometry facts in a year. This way, we’re covering it in three years. Deeper understanding is the key!

High school math teachers treat Common Core much like people who ignored Obamacare until their policy got cancelled. We don’t much care about standards normally: math is math. When the teachers who work with the lower half of the ability spectrum really understand that the new, dramatically reduced algebra and geometry standards are based on the premise that kids will cover a good half of the math now supposedly covered in high school in middle school, that simply by the act of moving this material to middle school, the kids will understand this material deeply and thoroughly, allowing them, the high school teachers, to explore more important topics, they will go out and get drunk. I did that last year when I realized that my state actually was going to spend billions on these tests. I was so sure we’d blink at the money. But no, we’re all in.

Because remember, the low proficiency levels we currently have are not only based on less demanding standards, but they don’t include the kids who don’t get to second year algebra by their junior year. That is, of the juniors taking Algebra II or higher, on a much harder test, we can anticipate horribly low proficiency rates. But what about the kids who didn’t get that far?

In California (I’ll miss their reports), about 216,000 sophomores and juniors were taking either algebra I or geometry in 2012-2013. California doesn’t test its seniors, but to figure out how many seniors weren’t on track, we can approximate by checking 2011-12 scores, and see that about 128,000 juniors were taking either algebra I or geometry, which means they would not have been on track to take an Algebra II test as juniors. That is, in this era of low standards, the standards that Common Core will make even more rigorous, California alone has half a million students right now who wouldn’t have covered all the material by their junior year. So in addition to the many students who are at least on paper on track to take a test that’s going to be far too difficult for–at a conservative guess–half of them, we’ve got the many students who aren’t even able to get to that level of math. (Consider that each state will have to spend money testing juniors who aren’t taking algebra II, who we already know won’t be able to score proficient. Whoo and hoo.)

Is it Common Core supporter’s position that these students who aren’t in algebra II by junior year are by definition not ready for college or career? In addition to the other half million (416,000 or so) California students who are technically on track for Common Core but scored below basic or far below basic on their current tests? We don’t currently tell students who aren’t on track to take algebra II as juniors that they aren’t ready for college. I mean, they aren’t. No question. But we don’t tell them.

According to Arne Duncan, that’s a big problem that Common Core will fix:

We are no longer lying to kids about whether they are ready. Finally, we are telling them the truth, telling their parents the truth, and telling their future employers the truth. Finally, we are holding ourselves accountable to giving our children a true college and career-ready education.

If all we needed to do was tell them, we could do that now. No need for new standards and expensive tests. We could just say to any kid who can’t score 500 on the SAT math section or 23 on the ACT: Hey, sorry. You aren’t ready for college. Probably won’t ever be. Time to go get a job.

If we don’t have the gumption to do that now, what about Common Core will give us the necessary stones? Can I remind everyone again that these kids will be disproportionately black and Hispanic?

I can tell you one thing that Common Core math was designed to do—push us all towards integrated math. It’s very clear that the standards were developed for integrated math, and only the huge pushback forced Common Core standards to provide a traditional curriculum–which is in the appendix. The standards themselves are written in the integrated approach.

So one way to avoid having to acknowledge a group of kids who are by definition not ready for career and college would be to require schools to teach integrated math, as North Carolina has done. That way, we could mask it—just make sure all students are in something called Integrated Math 3 or 4 by junior year. If so, there’s a big problem with that strategy: American math teachers and parents both despise integrated math. I know of at least one school district (not mine) where math coaches spent an entire summer of professional development trying to convince the teachers to adopt an integrated curriculum. The teachers refused and the district reluctantly backed down. Few people have mentioned how similar the CC standards are to the integrated curriculum that Americans have consistently refused. But I do wonder if that was the appeal of an integrated curriculum in the Common Core push—it wouldn’t increase proficiency, but would make it less obvious to everyone how many students aren’t ready. (Of course, that would be lying. Hmm.)

At around this point, Common Core supporters would argue that of course it’s more than just not lying to the kids! It’s the standards themselves! They’re better! Than the lower ones! That more than half our kids are failing!

And we’ll only have to wait eight years to see the results!!!

Eight years?

Yeah, didn’t anyone mention this? That’s when the first year of third graders will become juniors, the first year in which Common Core magic will have run its full reign, and then we’ll see how great these higher standards really are! These problems—they just won’t be problems any more. These are problems caused by our lower standards.

Right.

Or: As we start to get nearer to that eight year mark, we’ll notice that the predictions of full bore Common Core proficiency isn’t signaling. With any luck, elementary school test scores will increase. But as we get nearer and nearer to high school, we’ll see the dreaded fadeout. Faced with results that declare a huge majority of our black and Hispanic students and a solid chunk of white and Asian students are unready for career and college, what will we do?

Naw. That’s eight years out! By that time, reformers will need a next New Thing to keep their donors excited, and politicians will have figured out the racial disproportionality of the whole college and career ready thing. We barely lasted ten years with No Child Left Behind, before we got waivers and the next New Thing. So what New New Thing will everyone be talking about five to six years out, what fingers will they be pointing, in which direction, to explain this failure? I don’t know. But it’s a good bet we’ll get another waiver.

Is it at all possible that the National Governors Association thought up the Common Core as a diversion, an escape route from the NCLB 100% proficiency trap? It’s not like Congress was ever going to get in gear.

But it’s an awfully expensive trap door, if so. Much cheaper to just devise some sort of Truth In Education Act that mandates accurate notification of college readiness, and avoid spending billions on tests and new materials.

Notice how none of this is a public conversation. At the public debate level, the only math-based Common Core opposition argues that the math standards are too easy.

At which point, I suddenly realize I need more beer.


Polynomial Operations as Glue: Second Year Algebra

A couple years ago, I suddenly realized that my students rarely evaluated quadratic expressions. And when I thought about it, I could see why.

Create a table of values for y = x2 -6x – 16. Start with -3

evalbysub

These are kids who aren’t too great at working with negatives, yes? And it’s a whole bunch of work for a relatively small gain. Makes it tough to guess and check, to work velocity problems, and so on. I want something simpler.

Enter the Remainder Theorem: the remainder of the division of a polynomial f(x) by a linear polynomial x-a is equal to f(a).

We usually teach synthetic substitution when introducing with the Fundamental Theorem of Algebra, which is when we give advanced students the bad news—at a certain point, factoring higher-degree polynomials becomes guess and check. Here’s the Holt book, for example: Chapter 5, Quadratics, covers evaluation by substitution (aka, plug it in). Chapter 6, Polynomials (meaning degree greater than 2), covers polynomial division, synthetic substitution/division, remainder theorem, and factor theorem, leading up to the fundamental theorem of algebra. Notice, too, that the book is a tad soulless on two of the more remarkable theorems, as I write about here.

So this is screwed up. First, quadratics are polynomials, thankyouverymuch. Second, synthetic substitution/division solves the problem I started with: it’s brutal to evaluate quadratics if you can’t do it in your head—and most of my students can’t.

Then, there’s the fact that polynomial operations in algebra 2 are like kissing a sister; students don’t really learn the purpose for these operations until math analysis and calculus. Over half my students are in their last high school course and won’t be taking anything more advanced in college, but they will need knowledge of these operations for math placement tests. The other half will be moving on to math analysis, and need the skills.

Over the past two years, I’ve played with different ways of teaching polynomial operations, and different ways of introducing synthetic substitution for quadratics.

My algebra II/intermediate algebra class is comprised of four modeling units: linear equations (and inequalities), quadratic equations, exponential functions, and probability. I intersperse polynomial operations, inverses and logarithms between these four units. Logarithms fit organically with exponential functions; polynomial operations and inverses, not so much in a world where I’m not going on to the more rigorous parts of algebra 2. But inverses work as a good review of multistep equations, so the kids get some good practice in another skillset they need. Leaving polynomial operations as just….out there.

I haven’t been terribly unhappy with this, given the purely functional nature of the lessons, but I want my kids to know synthetic sub/div, dammit, and I want an organic way of introducing it. Right now, I go from linear equations to polynomial operations, ending with multiplication, which takes me into quadratics. That works, but not as smoothly as I want.

A couple days ago, I was pondering how to explain the synthetic substitution/division problem as a blog post, when I suddenly thought of a way to better integrate polynomial operations in and around my first two modeling units. I can use function operations as a method of introducing the transitions. Normally, I just introduce the function notation so they’re familiar with it. (Composites don’t normally show up on the test, and are covered again in pre-calc.)

This is just an outline, but remember that I have all the units done. All I’m describing, broadly (without any curriculum yet) is the transitions, the points at which I introduce and then return to polynomial operations.

After Linear Equations and Inequalities,

FuncAdd

I could start with a question like: “Part 1: Sami needs three more dollars to buy the new hoodie that he wants. Model a relationship between the money Sami has and the money he needs, and plot.”

Then, Part 2: “If Sami skips the hoodie, he needs just one more dollar to buy a ticket to the pizza feed on Friday. Model a relationship between the money he has and the money he needs, and plot.”

Part 3, starting as a discussion: “How much more money does Sami need if he wants both the hoodie and the ticket to the pizza feed?” My guess, although I’m happy to be wrong, is the kids will say that Sami needs four more dollars. And so how can they use the graphs to show otherwise?

So we can show graphically and algebraically that adding the two equations together will give us one equation that we can use to see how much more money Sami needs. At this point, I can introduce polynomial addition and subtraction in its simplest form. This will just be a couple days–one for addition, one for subtraction. But it allows me to reinforce linear graphing one more time, in addition to the new concept.

Then I can move from addition and subtraction to multiplication.

Introduction to Quadratics

I’ve always introduced quadratics with the modeling exercise above, then moved onto binomial multiplication. I really like the possibilities that come up after adding and subtracting linear functions, by asking the question (without the graph, at first):

FuncMult

“Okay, we’ve added two lines. What happens when we multiply two lines?”

In class discussion, I’ll point out the negative values, the positive values and the points at which one graph is positive and one negative. What’s going to happen when these are multiplied? (Hey, it never hurts to remind them about negative integer operations.) I haven’t completely thought through implementation—I definitely want them graphing this. Maybe give them the two lines at first, have them multiply the values.

I mentioned earlier that I’ve been looking for a better method of modeling quadratics. While this approach doesn’t involve situation modeling, it does organically introduce the shape of a parabola. It will also help them spot zeros.

And this leads in perfectly to my binomial multiplication unit, which I already extend to include higher degree polynomials. With the strongest kids, I can even give them three lines and have them determine what a cubic function looks like.

Factoring, Division, Remainder, Synthetic Sub/Div

funcDiv1

Then, when I’m moving from binomial multiplication to factoring, I can show a graph like the one at right and ask:

“So we multiplied the linear equation by another linear equation to get the parabola. What are the equations you see, and what’s the missing linear equation?”

which, of course, brings up function division, and allows me to introduce factoring as a variant of division–and, a month or so after we’ve done linear equations, they get to review the concepts. As I write this, I’m trying to think if it makes more sense to introduce long division and synthetic substitution at this point, or to work on factoring for a while and then bring up division. TBD.

If you’re not familiar with synthetic sub/div, take a look at long division and synthetic division side by side:

Evalbylongevalbysyn

Synthetic sub/div is far easier than substitution, even in quadratics. It’s also noticeably easier when evaluating fraction values for velocity problems.

After I’ve finished all of linear and all of quadratics, I can do a few days on polynomial operations and function notation, just to wrap up.

Again, this is very skeletal. I just had the idea because of the writing challenge. Thanks, blog! But I know it will work; I can feel it. I just have to be careful and think through the transitions thoroughly, make sure I’ve given the kids plenty of support. For example, I don’t want to overemphasize the function operations of this. I just want the kids to be comfortable with the notion of addition, subtraction, multiplication and division of equations. That will give me the entrance to teach them synthetic div/sub, as well as the reason for practicing polynomial ooperations.

Those of you who are thinking, “Hey. This is really algebra one.” well, welcome to my world. My kids learn a whole bunch of first year algebra in my algebra II and geometry classes. But I cover about 60% of the algebra II standards to kids with very weak skills, and the class is pretty conceptually interesting, I think. It’s definitely not just a rehash of algebra one.

I’ve also been thinking a lot about this post on curriculum mapping, which I found very interesting. I hope it’s okay that I borrow his image:

I was talking with Kelly Renier (@krenier), director at Viking New Tech, and we began discussing the concept of “power standards” or “enduring understandings” or “What are the Five Things you want your students to know when they leave your class?” then build out from there. However, we didn’t discuss building those Five (or whatever number) Things out into linearly progressing units, but rather concentric circles.

So this is absolutely how I teach, as regular readers may know. Teaching Algebra, or Banging Your Head with a Whiteboard covers, literally, the Five Big Ideas of algebra I. I also have them for geometry and algebra II (for my students, anyway). I thought the advantages of this approach were interesting in that I didn’t realize how many teachers don’t do this already. Again, quoting:

  1. Students get to revisit a general topic every few weeks, rather than a one-and-done shot at learning a concept.
  2. Students have time to “forget” algorithms and processes and when they see a scenario they have to fight their way through it accessing prior or inventing new knowledge, rather than relying on teacher led examples. Yes, I consider this a benefit.
  3. Teachers may formatively assess more adeptly.
  4. Students may see math as a more connected experience, rather than a bunch of arbitrary recipes to follow.
  5. It probably better reflects the learning process, which happens in fits and starts, and frankly, cannot be counted upon to be contained within a specified time frame.

This is a really good explanation of what I see as the advantages to my approach. I have never taught in anything approaching a linear fashion, probably because I used CPM, which spirals as a matter of course, in my first two years and was nonetheless shocked at how much kids forgot. So once their forgetting is shoved in your face, it’s hard to go back to the linear curriculum design.

I don’t obsess with getting every single connection made from the first time I teach the class. Sometimes I’ll just acknowledge, as I’ve done with polynomial operations up to now, “Hey, this is kind of an odds and ends thing you just need to know.” There’s nothing wrong with making clean breaks between some units—it doesn’t automatically turn the curriculum linear. For example, I make a very clean break between quadratics and exponentials, because the kids have never seen exponentials before. I show the connections between linear and exponential functions, but I also don’t just lead in. NEON SIGN: NEW EQUATION is a helpful way for kids to realize they’re getting something new. (Common Core says they’ll be learning this in Algebra I. Jesus. These people are friggin’ delusional.)

Going back to who I am as a teacher, I start with explanations. Not necessarily verbal explanations every time, but making sense of a concept before doing it is an essential element of my teaching.

That doesn’t mean I start with a lecture, which I rarely do, or an explanation. I often begin a unit or a concept with an activity. But if I’m asking my students to engage in an activity with no concept or prior understanding, then they can be sure it’s going to be simple, straightforward, and illustrative.


Developing Curriculum

Ed schools, particularly elite schools, preach the need for teachers to create their own curriculum. We don’t get any lessons on how to take a few pages of a textbook and break it down into explanation and practice, or how to select a good range of problems from a textbook to assign for classwork. Many ed schools use the Wiggins and McTighe text Understanding by Design (and its follow up on differentiating instruction), which argues that adherence to textbooks lead to the sin of coverage. No, not only shouldn’t we follow the book, said my ed school (and many others) but we should throw the book out and work backwards from out learning goals.

I thought this an idiotic idea. Open the book, give some examples with a good explanation, and have them work some problems. I explain things well, and any decent textbook has a wide range of problems to assign.

I had good instructors, particularly in C&I, and I expressed this opinion frequently and openly. I was not shot down—ed schools are doctrinaire at the administrative level, but at the instruction level, I found all my professors to be open to challenge. In fact, when I look back, I’m struck by how often my instructors reiterated what Wiggins himself says time and again: Understanding by Design is a framework, not a philosophy.

But it was impossible for me to believe that because the examples in the readings, and the openly progressive politics of ed school always sold curriculum design as a way to indoctrinate. The UBD books use, as an example, a history teacher who created an elaborate project for students to design their own constitution, reflecting the needs and interests of everyone in the community, not like those racist Founding Fathers. Or the instructor might describe a math teacher with an equally elaborate projects for students to “discover” transformations of functions when none of them have the skills to solve the functions in the first place.

Or there was the time we had to listen to an absolute idiot of an English teacher at an inner city charter school yammer on ignorantly about the “culturally whitewashed curriculum” that gives urban kids Robert Frost, who (to paraphrase) didn’t know trouble, didn’t know suffering, and wrote peaceful rural poems this teacher’s unfortunate inner city students couldn’t relate to, instead of the “real” poetry of Gwendolyn Brooks. There I was, paying huge chunks of money to listen to a brain dead jackwit present Frost as an out of touch white guy who wrote pretty poems about happiness and peace. Surely there’s a teacher out there who has considered a lesson for inner city kids helping them to see Frost’s hidden bleakness by contrasting it with Brooks’ open despair, and surely that would be the teacher invited to lecture at an elite university instead of this buffoonish hack? But I digress.

Ironically, I learned that textbooks could be a problem when, my first year out, I worked with a famously progressive, constructivist text known as CPM. I’ve used CPM to teach geometry and both years of algebra, and all the books had a few moments of interesting brilliance, way way WAY too much text, not enough practice problems, insufficient respect for the real priorities of any subject and an ordering approach that drove me crazy. I hated it, just like the good little progressive teachers hate their cold, formula-laden traditional text, and so I learned to ignore the book and develop my own curriculum.

Three years out, I have a very different view of textbooks. Good ones are great tools for ideas and problem sets that I can dip into as needed. But all textbooks fall short in some ways, many of which aren’t their fault.

The big problem: I often teach kids who won’t use them.They certainly won’t take them home and back to school (most of them just leave the books at home). They won’t use them as a resource. In fact, many kids actively prefer a worksheet to a textbook, as they get a sense of completion from finishing a page of problems. A textbook never ends.

Another problem is, of course, the size. At more than one of my schools, the kids don’t all get lockers. So the books are either going to stay home or stay at school. Last year, (2011-2012), I asked my kids to keep the books in the classroom, so I could use them on and off as needed. The textbook supervisor at the school got very upset at this, for good reason, but it worked. I was able to pull in the books as needed, particularly for my strongest kids, and ignore them the rest of the time. (Update: this year, at a different school, all my kids have lockers. It’s very convenient–I just write BOOKS in big letters on the board, and the kids go back to their lockers to get the books on the days I need them.)

Another problem is that textbooks are designed for one audience. Progressive texts, like CPM, are primarily designed for low ability kids in a constructivist classroom. Not enough problems, very few challenging problems, too much text, usually strained efforts to connect math to “real-life”, way too much indoctrination, and an exhaustive preference for explanations over answers. Spare me.

So when I got to my current school, using Prentice for algebra and Holt for Geometry, I was happy to shake the CPM dust off and use textbooks daily. Alas, I realized that these books were the Papa Bear to CPM’s baby—a fire hose for all but a fraction of my kids. Most textbooks are designed for students who are actually ready for the material, with low ability kids an afterthought. These textbooks cover material far too quickly. In my current geometry book, three pages in one section covers both 30-60-90 and 45-45-90 triangles. Sure, because kids pick it up just like that. These books often include different worksheets on a CD for lower ability kids, but at that point, you’re not using the textbook anymore.

I also find books are too limiting. They rarely provide teachers with useful illustrating activities–sometimes the book will sketch out an interesting possibility, but leave the details to the teacher. For example, the Holt Algebra II text introduces complex numbers as if the topic is little more than a walk to the drugstore. I mean, the numbers are imaginary, for chrissakes, and the text just spells it out in a sentence and moves on. For as huge as math books have gotten, publishers still haven’t used any of that space to lay out an explanation that works for low ability kids. Of course, the kids wouldn’t use it anyway, since they’d avoid the textbook, but at least it would give me something to copy so I didn’t have to create my own or steal a good start off the Internet.

Then, the book ordering is often insane. The year I taught all Algebra I, the Holt book introduced rate problems and work formula problems in Chapter 2. I laughed. My kids are still shaky on subtraction, and I’m going to cover high-complexity word problems in the third week. Sure. Only my top students got these problems, and then only at the end of the year.

When I realized how advanced the book was, I checked with a senior teacher, and she snorted. “Oh, I don’t use the books.” At our school, in algebra and geometry, relatively few teachers use the books on a regular basis. They develop their own lessons, their own tests, they borrow worksheets, and cobble together a curriculum that, in their view, meets their students’ needs.

I know my experiences aren’t universal. I know many teachers teach from the book, and many teachers work collaboratively to produce a class taught over 13-15 sections by multiple teachers. I student taught at a school that planned course-alikes collaboratively, were faithful both to the (CPM) text and a common schedule for all classes. The next three schools I’ve taught at gave me a solid grounding in “teacher as island”; everyone does their own thing. Given my druthers, I’d rather the latter. While I do wish I worked in departments that did more course-alike planning, I’m becoming increasingly sympathetic to the teachers who resist lockstep synchronicity. (Update: Right now, I’m dealing with a math teacher who insists that everyone teach trinomial factoring in exactly the same way. Um. No. Unless y’all want to use my way.)

What spun off this post was a review of The Tyranny of Textbooks, a purported expose of the textbook selection committee, with proposals to change and improve the process.

But that makes me laugh. Improve the process? Tons of teachers don’t even use the books! Why waste billions on textbooks that go home to serve as doorstops? Pick a few approved texts. Buy a few sets of each. Let the teachers who want to use them get a class set, or (in the case of advanced classes) check them out to the students. In all but a few cases, schools could save money by using class sets—except, of course, many states legally require districts to give every kid a book. Taxdollars in action, baby.

Here’s the really funny thing: I’ve described what I do, and what many high school teachers do—develop our own curriculum to cover the standards. But it’s clear from even a cursory overview of the education debate, that a million teachers planning their own curriculum is not what eduformers foresee as the future of education in this country. It’s also clear, however, that most education reforms never make it to the classroom and don’t have a clue how teaching actually happens, particularly in high school. (Eduformers believe that once elementary school is fixed, all will be well. They’re wrong.)

So if you don’t like teachers overriding local and national priorities by developing their own curriculum, and using the books, too bad. First off, progressives own ed schools and they’re always going to be pushing teacher curriculum development.

But more to the point, the range of student abilities, and the expectation that low ability students are to be taught a college-prep curriculum, pretty much mandates curriculum development at the district, school, or even classroom level. You want lockstep classroom curriculum? Bring back tracking and develop different texts for different ability levels. Let’s all laugh at that idea.

And now, a mea culpa: given how much misery I caused my ed school, I feel it only fair to acknowledge that my disdain for the progressive agenda and my dislike of constructivism was drowning out my instructors’ message about Understanding By Design: Here’s a framework for building your own. Take what works and toss what doesn’t. While they might approve of a particular agenda, the framework is ideologically neutral.

My last three plus years of teaching have done much to increase my approval of progressives. Yes, their agenda is still overtly political and yes, they still ignore ability in much the same way that eduformers do. But progressives in ed schools know far more about teaching than eduformers will ever know, and buried underneath their squishy curricular nonsense is a core of useful knowledge that I tap into quite often.

(Note: I first wrote this while at my last school; the updates were made in mid-September at my third school.)