The Test that Made Them Go Hmmmm

So school has begun and despite my palpitations about the boredom of only two familiar preps, I’m pleasantly busy. Last year was a hell of a lot of work, and given the nosedive that my writing time took, I should maybe not be so eager for a less…familiar schedule. So instead of demanding new classes, I accepted the first semester, threw a minor temper tantrum when no one listened about second semester and all is well. Algebra 2 in particular is proving a delightful challenge, given my new emphasis on functions.

In no small part because of this planning breathing room (is anyone noticing I’m saying my panic was a total overreaction?), the senior Water Park Day registered in my awareness ahead of time. In prior years, I didn’t heed the warnings that half my class would disappear, and so would be forced to dump my lesson plan on the Day itself, when the smaller classes would just have a day to practice. But thanks to this old, familiar schedule that gives me more time, I anticipated the impact.

So for the first time, I was able to give serious thought to having a day to pursue math without regard to subject matter or schedule. I could have a “math day”! Then I remembered Grant Wiggins’ challenge to math teachers everywhere in the form of a conceptual knowledge quiz.

Grant proposed this as an actual test: I will make a friendly wager: I predict that no student will get all the questions correct. Prove me wrong and I’ll give the teacher and student(s) a big shout-out.

What math teachers think their kids would know the answers? I certainly didn’t. In some cases, they probably were taught, but in others, I doubt an elementary school teacher would ever think to bring them up. But even if all the concepts were taught by fifth grade, how many kids of that age could really appreciate the questions?

Most of the questions tease at the paradox….wrong word? tension? between the functional day-to-day applications of arithmetic, and the amazing truths that underlie them. John Derbyshire wrote, in Prime Obsession, that “arithmetic has the peculiar characteristic that it easy to state problems in it that are ferociously difficult to solve.” (I was rereading Prime Obsession last night; there’s tons of useful thought material for math teachers. I need to go get his book on algebra.)

Arithmetic looks easy. (And certainly in the last twenty years, the rush to shove everyone into calculus has led to a certain contempt for “basic arithmetic” classes.) But even if elementary school age children are capable of understanding its ideas fully (and most of them aren’t), they haven’t experienced several years’ utility of arithmetic. They haven’t had time to get bored of the routine rules that they are expected to remember (mind you, many don’t, but leave that for another day.) Yeah, yeah, invert and multiply. Yeah, yeah, you can’t divide by zero. Wait, what the hell do you mean multiplication isn’t repeated addition?

To really enjoy this test, to be fascinated by the underlying truths–or misconceptions–behind certain everyday math tools, requires familiarity with “the rules”. Time spent in the trenches of doing math just because.

That’s when a teacher can spend an enjoyable hour taking the kids back through a re-examination of the basics and what they really know. I’d much rather discuss these concepts with adolescents who have survived two or three years of high school math than try to force sixth graders to “demonstrate conceptual understanding” of dividing by zero.

I had no real expectations—no, that’s wrong. I had hopes. My sense was the students would be interested in the exploration, if I didn’t take on too much or dive in to the wrong end of the pool. But which end was the wrong end?

So for each of my four classes–two Algebra 2, two Trigonometry–I gave them the test and 20 plus minutes to write down their thoughts. I was alert to the possibility that kids would use five minutes to doodle and fifteen to giggle, but in each class the bulk of students asked for and got an additional five minutes to finish up. I collected their answers and will share some of them in later posts; they were often detailed and thoughtful.

After the writing time, the students had a few minutes to “share out” in their groups, so they could learn what questions puzzled their classmates—and also as reassurance that they weren’t alone in their befuddlement. Again, this seems different from Grant’s intent; he considered it a real test that the students would either answer correctly or leave blank in confusion. I listened in on many conversations; they were rich with exchange as the students realized they weren’t alone in their uncertainty.

But certain questions also sparked genuine debate and interest. More than a few students offered up multiplying negatives as an example of multiplication being something other than repeated addition. In every case I witnessed, their group members, who had written something to the effect of “isn’t it always repeated addition?” instantly recognized the roadblock that negative numbers posed to their definition. I came across more than one group arguing whether multiplying by zero counted as repeated addition (“yes, it does. If I have zero groups of five, I have zero!”). Interestingly, no one came up with the roadblock I was interested in, and I’d never once considered negative numbers until my students brought it up.

Their discussion time was about ten minutes. My goal wasn’t to have them determine the answers; rather, I wanted them all to have a shared experience before we discussed them as a class, and I gave them the “answers” (to the extent I knew them). That way, there’d be more of a sense of “we”–yeah, we thought of zero, too! yeah, we all have 3F=Y–that’s not the answer? yeah, we think dividing by zero gives you zero–it doesn’t?

So then we went through the answers as a group.

I had taken a subset of Grant’s list, ignoring the last three items. Doing it again, I would have swapped out question 2 for question 11 “appropriately precise”), because while question #2 is good, it really requires its own day. The rest of them are easily covered and discussed in at most 15-20 minutes each.

The questions I really wanted to spend time on, to explain in at least introductory depth, were 1, 3, and 5. From a practical standpoint, I wanted to be sure everyone understood why they got questions 4, 6, and 8 wrong, assuming most missed at least one of them. I was genuinely interested to see what they had to say about 7 and 9 but was going to take most of my lead from them. Question 10, I wanted to know if the trig students knew it; obviously, my algebra 2 students learn about imaginary numbers for the first time.

My trig classes are quite different in nature. Both are small, just 25 in each. Both are doing quite well; I have no kids who simply shouldn’t be there, as I did last year. My first block class is stronger, on average, but has more surly kids who mouth off. It’s very irritating, frankly, since the five or six kids giving me quite nasty sass are seniors who are doing relatively well (Bs and Cs), and who openly acknowledge that they think I’m a hell of a teacher. Two of the surlies had me last year for algebra 2, when they were much less trouble, and had been switched into my class because they were failing with another teacher. But these other teachers, who they didn’t like (and often failed, forcing them to retake a fake summer school course if they couldn’t switch to my class), didn’t get nearly the lip. I’m a tad flummoxed. My second block class has more kids who are amiable and interested but not taking the class as seriously as they should, so several more low scores on the first test. First block has a stupendous top tier, but it’s just three or four kids. Second block has a top tier of close to eight, but they aren’t quite as strong.

Anyway, I was expecting more interesting conversation from second block, and I had it backwards. First block was on point, even the cranky ones. They loved the test, wrote detailed responses, discussed it thoroughly in group, and were wildly participatory in the open discussion. Easily 90% of them came up with the correct response to imaginary numbers (and the ones from my algebra 2 class identified multiplying by i as 90 degree rotations in the complex plane, which was quite gratifying, thanks so much). Second block, the amiable, mildly uninterested ones pulled things down slightly, goofing around and making jokes while the stronger kids would have preferred more time to explore things. The conversation was still great, the students learned a lot and enjoyed the discussion, but I had the enthusiasm levels backwards.

My algebra 2 classes, I nailed in terms of expectations. Block three is a fairly typical profile, except I have a lot more sophomores than usual (which is due to our school successfully pushing more kids through geometry as freshmen). But still a good number of seniors who barely understood algebra I, a lot of whom are just hoping to mark time til graduation without ending up in summer school. (One of my specialty demographics.) And in between, juniors and seniors who are often thrilled to find themselves actually understanding math and succeeding beyond anything they’d ever hoped (another specialty of mine). Typically, many of the seniors were in class, as they lacked the the behavior or grade profile (and sadly, in some cases, the money) to go to the water park. So I expected conversation here to be a bit lower level, with less interest. Happily, everyone engaged to the best of their ability and many told me later how much they loved just “talking about math”. I spent much more time on questions 4, 6, and 8, and could see them all really registering why they’d made the mistakes they did. But they still were enthralled by questions 1, 3, and 5, which is great because it’s going to give them some memories when we review percentages in preparation for exponential functions.

Last up was block 4 algebra 2, a ridiculously strong class; only five students are of the usual caliber I expect. The seniors are all well above average ability level. Two of the kids are so skilled that I’ve already introduced three dimensional planes and the matrix, while still forcing them and the other really strong kids to deal with complex linear word problems (mixture questions! I usually skip them, so it’s a trip). They stomped all over the test, writing at great length, discussing it with their teams and then shouting out to other groups to see what they’d answered for multiplication. The class discussion took so long that I actually allowed it to continue for 20 minutes into the next day, when I invited one of my mentees to watch. He came away determined to try the test in his honors geometry class.

Look, the whole day was teacher crack. Take a day. Try the test. I’ll be discussing individual questions and my explanations in future posts, but this introduction is offered up as invitation. High school teachers working in algebra 2 or higher would be a good starting point. Honors classes in algebra and geometry would also benefit. Every math teacher can find links from this test to their math class—but then, that’s not the point.

As for me, I started out the day with hope, but also a determination to see it through as part of a way to honor Grant Wiggins, who felt very strongly that students needed to do more than just march through curriculum. I promised myself I wouldn’t abandon the effort even if it went wrong. It didn’t go wrong. Quite the contrary, the test sparked delighted interest and intellectual curiosity among students who are often hard to push into exploring mathematics in depth. So hey, Grant, thanks for the idea–and the inspiration.

Grant Wiggins

Curriculum is the least understood of the reform efforts, even though parents have more day to day contact with curriculum than choice or accountability. This is in large part because curriculum advocates don’t agree to the degree that accountability and choice reformers do, but also because teachers have far more control over curriculum than most understand. As Larry Cuban explains, curriculum has multiple layers: intended, tested, taught, and learned. Curriculum battles usually involve the intended curriculum, the one designed by the state, which usually creates the tested curriculum as a manageable subset. (Much of the Common Core controversy is caused by the overwhelming difficulty of the tested curriculum, but leave that for another time.)

But intended and tested curriculum are irrelevant once the doors close, and in this essay, I refer to the taught curriculum, the one that we teachers sculpt, whether we use “the book” (actually just pieces of the district approved book), use another book we like better, or build our own.

To the extent most non-educators know anything about curriculum advocacy, it begins and ends with E. D. Hirsch, otherwise known as “the guy who says what my nth grader should know”, author of a book series he eventually transformed into a curriculum for k-6, Core Knowledge. Hirsch offers one Big Idea: improving student background knowledge will improve their reading comprehension, because only with background knowledge can students learn from text. But, the Idea continues, schools ignore content knowledge in favor of teaching students “skills”. To improve reading comprehension and ongoing student academic outcomes, schools must shift from a skills approach to one dedicated to improving knowledge.

Then there’s Grant Wiggins, whose death last week occasioned this essay as an attempt to explain that we’ve lost a giant.

The media proper didn’t give Wiggins’ passing much notice. Valerie Strauss gave his last blog sequence a good sendoff and Edutopia brought back all their interviews with him. Education World and Education Week gave him obits. It doesn’t look as if Real Clear Education noted his passing, which is a bit shocking but perhaps I missed the mention.

Inside education schools, that world reformers hold in considerable contempt, Wiggins’ work is incredibly influential and his death sent off shockwaves. Since 1998, Understanding by Design has been an essential component in preparing teachers for the professional challenge of deciding what to teach and how to deliver the instruction.

Prospective teachers don’t always understand this preparation will have relevance to their lives until their first year in the classroom. Progressive ed schools would never say anything so directly as “You will be faced with 30 kids with an 8 year range in ability and the textbooks won’t work.” Their ideology demands they wrap this message up in hooha on how insensitive textbooks are to the diverse needs of the classroom. Then, their ideology influences the examples and tasks they choose for instruction. Teacher candidates with an instructivist bent thus often tune out curriculum development classes in ed school, rolling their eyes at the absurd examples and thinking keerist, just use the textbook. (Yeah. This was me.)

Usually, they figure out the relevance of curriculum instruction when they get into the classroom, when they realize how laughably inadequate the textbook is for the wide range of abilities and interests of their students. When they realize the book assumes kids will sit patiently and listen, then obediently practice. When they realize that most of the kids won’t bring their books, and that all the well-intended advice about giving consequences for unprepared students will alone result in failing half the class, never mind the problems with their ability. When they realize that many kids have checked out, either actively misbehaving or passively sitting. Worst of all the teachers experience the kids who are eager to learn, try hard, don’t get it, and don’t remember anyway. Then, even after they make a bunch of adjustments, these teachers realize that kids who do seem to be learning don’t remember much—that is, in Cuban’s paradigm, the learned curriculum is wildly different than the one taught (or in the Wiggins universe, “transferred”).

The teachers who don’t quit or move to charters or comprehensives with a higher SES may remember vaguely hey, there was something about this in ed school (hell, maybe that’s just me). So they go dig up their readers and textbooks and suddenly, all the twaddle about diversity and cultural imperialism fades away and the real message becomes legible, like developing invisible ink. How do you create a learning unit? What are your objectives? How will you assess student learning? And at that point, many roads lead to Wiggins.

Grant Wiggins was impossible to pigeonhole in a reform typology. In 1988, he made 10 proposals for high school reform that leaned progressive but that everyone could find some agreement with. He didn’t think much of lecturing, but he wrote a really terrific analysis of lectures that should be required reading for all teachers. (While I also liked Harry Webb’s rejoinder, I reread them in preparation for this essay and Grant’s is far superior.) He approved of Common Core’s ELA standards, but found the math ones weak. In the space of two weeks in 2013, he took on both Diane Ravitch and E. D. Hirsch, and this is after Ravitch flipped on Hirsch and other traditionalists.

Grant Wiggins was more than ready to mix it up. Both his essays on Hirsch and Ravitch might fairly be called broadsides, although backed with research and logic that made both compelling, (perhaps that’s because I largely agreed with them). His last two posts dissected Hirsch supporter Dan Willingham’s op-ed on reading strategies. While he listened and watched teachers intently, he would readily disagree with them and was rarely gentle in pointing it out. I found his insights on curriculum and instruction absolutely fascinating, but rolled my eyes hard at his more excessive plaints on behalf of students, like the nonsense on apartheid bathrooms and the shadowing experience that supposedly revealed the terrible lives of high school students—and if teachers were all denied the right to sarcasm, as he would have it, I’d quit. He didn’t hesitate to say I didn’t understand the lives that students lead, and I told him right back that he was wrong. More troubling to me was his conviction that most teachers were derelict in their duty and his belief that teachers are responsible for low test scores. But what made him so compelling, I think, is that he offered value to all teachers on a wide range of topics near to our needs, whether or not we shared all his opinions.

I knew him slightly. He once linked to my essay on math philosophies as an example of a “learned” teacher, and read my extended response (do I have any other kind?) and took the time to answer. Then, a few months later, I responded to his post on “teacher job descriptions” with a comment he found worthy of pulling out for a post on planning. He then privately emailed to let me know he’d used my comment and asked me to give feedback on his survey. That was a very big day. Like, I told my folks about it.

In the last week of his life, Grant had asked Robert Pondiscio to read his Willingham critique. Pondiscio, a passionate advocate of all things content knowledge, dismissed this overture and declared his posts on both Willingham and Hirsch “intemperate”. Benjamin Riley of Deans for Impact broke in, complimenting Grant and encouraging the idea of debate. The next day, Daniel Willingham responded to Grant on his site (I would be unsurprised to learn that Riley had something to do with that, and kudos to him if so). Grant was clearly pleased to be hashing the issues out directly and they exchanged a series of comments.

I had been retweeting the conversation and adding comments. Grant agreed with my observation that Core Knowledge advocates are (wrongly) treated as neutral experts.

On the last day of his life, Grant favorited a few of these tweets, I think because he realized I understood both his frustration at the silence and his delight at finally engaging Dan in debate.

And then Grant Wiggins died suddenly, shockingly. He’ll will never finish that conversation with Dan Willingham. Death, clearly, has no respect for the demands of social media discourse.

Dan Willingham tweeted his respect. Robert Pondiscio wrote an appreciation, expressing regret for his abruptness. If the general media ignored Grant’s passing, Twitter did not.

I didn’t know Grant well enough to provide personal insights. But I’m an educator, and so I will try to educate people, make them aware of who was lost, and what he had to offer.

Novices can find plenty of vidoes on his “backwards design” with a simple google. But his discussions on learning and assessment are probably more interesting to the general audience and teachers alike—and my favorites as well.

Reformers like Michael Petrilli are experiencing a significant backlash to their causes. Petrilli isn’t wrong about the need for parent buy-in, but as Rick Hess recently wrote, the talkers in education policy are simply uninterested in what the “doers” have to offer the conversation.

Amen to that. The best education policy advocates—Wiggins, Larry Cuban, Tom Loveless–have all spent significant time as teachers. Grant Wiggins set an example reformers could follow as someone who could criticize teachers, rightly or wrongly, and be heard because he listened. If he disagreed, he’d either cite evidence or argue values. So while he genuinely believed that most teachers were inadequate, teachers who engaged with him instantly knew this guy understood their world, and were more likely to listen.

And for the teachers that Grant found inadequate—well, I will always think him in error about the responsibility teachers own for academic outcomes. But teachers should stretch and challenge themselves. I encourage all teachers to look for ways to increase engagement, rigor, and learning, and I can think of no better starting point than Grant Wiggins’ blog.

I will honor his memory by reading his work regularly and looking for new insights to bring to both my teaching and writing.

If there’s an afterlife, I’m sure Grant is currently explaining to God how the world would have turned out better if he’d had started with the assessment and worked backwards. It would have taken longer than seven days, though.

My sincere condolences to his wife, four children, two grandsons, his long-time colleague Jay McTighe, his band the Hazbins, and the many people who were privileged to know him well. But even out here on the outskirts of Grant’s galaxy of influence, he’ll be sorely missed.

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.