Illustrating Functions

Function definitions aren’t usually tested on either the SAT or the ACT and since I never worked professionally with math, functions were something I’d barely considered in algebra a billion years ago. So for the first few years of teaching, I kind of went through the motions on functions: unique output for each input, vertical line test, blah blah. I didn’t ignore them or rush through them. But I taught them in straight lecture mode.

Once I got out of the algebra I ghetto (which really does warp your brain if that’s all you do), I accepted that a lot of the concepts I originally thought boring or unimportant show up later. So it’s worth my time to come up with the same third way activities and lessons for things like functions or absolute value or inverses that I do for binomial multiplication and modeling linear equations and inequalities.

So every year I pick concepts to transfer from pure lecture/explanation to illustration. Sometimes it’s spur of the moment, other times I plan a formal curriculum change. In the case of functions, the former.

Last year I was teaching algebra II/trig and–entirely in passing–noted a problem in the Holt book that looked something like this:

and had two simultaneous thoughts: what a boring question and hey, I could really do something with that.

So the next day, I tossed this up on the board without comment.


I’ve given these instructions three times now–a2/trig, trigonometry, algetbra 2–and the kids are always mystified, but what the heck, the activity seems simple enough. No student ever reads through the entire list of instructions first. They spend a lot of time picking the message, with many snickers, then have fun translating the code twice.

But then, as they all try to translate someone else’s message using the cell phone code, bam. They realize intuitively that translating the whole-alphabet code would be an easy task. And with a few moments of thought, they realize why the cell phone code doesn’t offer the same simple path. They don’t know what it means, exactly. But the students all realize that I’ve demonstrated a difference that they’d never considered.

From there, I graph the processes, which is usually a surprise as well. The translation process can be graphed?



At this point, I can usually convince kids to remember the Vertical Line Test, which they were taught in algebra I. At that point, I go through the definitions for relation, function, and one-to-one function, using a Venn diagram (something like this with an addition inner circle for one to ones). Then I go back through what the students vaguely remember about functions and link it to the correct code example.

Thus the students realize that translating the message into code is a function in either code key. I hammer this point home, because the most common misconception kids get from this is that all functions must be one to one. Both are functions. Each letter has one and only one number assigned, and the fact that one translation key puts several letters to the same number is irrelevant for its determination as a function. Reversing the process, going from numbers to letters, only one of them is a function.

Then I sketch parabolas and circles. Are they both functions? Are either of them one-to-one functions?

In Algebra 2, I do this long before the inverse unit. In Trig, I introduce it right before graphing the individual functions as part of an overview. In both classes, the early intro gives them time to recognize the significance of the difference between a function and the more limited case of the one-to-one function–particularly in trig, since the inverse functions are very limited graphs for exactly the reason. In algebra II, the graphs reinforce the meaning of the Horizontal Line Test.

I haven’t taught algebra I recently, but I’d change the lesson by giving them a coded message and ask them to translate with the cell phone code first.

This leads right into function and not-function, which is all they need in algebra I.

I have periodically mentioned my mixed feelings about CPM. Here’s a classic example. The CPM book introduces functions with the following example.

Okay. This is a terrible example. And really boring. Worst of all, as far as this non-mathie can tell, towards the end it’s flat out wrong. A relation can be predictable without being a function (isn’t that what a circle is?). But just looking at it, I got an idea for a great test question (click to enlarge):


And I could certainly see some great Algebra I activities using the same concept. But CPM just sucks the joy and interest out of the great starting ideas it has.

Anyway. I wanted to finish up with a push for illustrations. What, exactly, do the students understand at the moment of discovery in this little activity? I’ve never seen anyone make the intuitive leap to functions. However, they do all grasp that two tasks that until that moment seemed identical…aren’t. They all realize that translating the message in the whole-alphabet code would be a simple task. They all understand why the cell phone code translation doesn’t lend itself to the same easy translation.

I look for illustrative tasks that convince kids to think about concepts. As I’ve said before, the tasks might kick off a unit, or they might show up in the middle. They may demonstrate a phenomenon in math, or they might be problems designed to lead the students to the next step.

The most common pushback I get from math teachers when I talk about this method is “I love the idea, but I don’t have enough time.” To which I respond that pushing on through just means they’ll forget. Well, they’ll probably forget my lessons, too, but–maybe not so much. Maybe they’ll have more of a memory of the experience, a recollection of the “aha” that got them there. That’s my theory, anyway.

There’s no question that telling is quicker than illustrating or letting them figure it out for themselves. Certainly, the illustration should be followed by a clear explanation with much telling. I love explaining. But I’ve stopped kidding myself that a clear explanation is sufficient for most kids.

That said, let me restate what I said in my retrospective: The tasks must either be quick or achievable. They must illustrate something important. And they must be designed to lead the student directly to the observations or principles you want them to learn. It’s not a do it yourself walk in the park. Compare my lesson on exploring triangles with this more typical reform math example. I resist structure in many aspects of my life, but not curriculum.

In researching this piece, I stumbled across this really excellent essay Why Illustrations Aid Understanding by David Kirsh. I strongly recommend giving it a read. He is only discussing the importance of visual illustrations, whereas I’m using the word more broadly. Kirsh comes up with so many wonderful examples (math and otherwise) that categorize many different purposes of these illustrations. Truly great mind food. In the appendix, he discusses the limitations of visually representing uncertainty.


On reading this, I felt like my students did when they realized the cell phone message was much harder to translate: I have observed something important, something that I realize immediately is true and relevant to my work–even if I don’t yet know why or how.

Education: No Iron Triangle

I came from the corporate world, which invented the project management triangle. (“Fast, Good, Cheap: Pick Two.”)

Education has no triangle.

Money, of course, doesn’t work. Just ask Kansas City. Or Roland Fryer, who learned that kids would read more books for money but couldn’t seem to produce higher test scores for cash. Increased teacher salaries, merit pay, reduced class size are all suggestions that either don’t have any impact or have a limited impact….sometimes. Maybe. But not in any linear, scalable pattern.

“Good”? Don’t make me laugh. We don’t have a consensus on what it means. Most education reformers use the word “quality” exclusively to mean higher test scores. Teachers do not. Nor do parents, as Rahm Emanuel, Cami Anderson, Adrian Fenty and Michelle Rhee have learned. Common Core supporters have had similar moments of revelation.

So until we agree on what “good” is, what a “high quality education” means, we can’t even pretend that quality is a vertex of education’s triangle, even if it existed. We could save a whole lot of wasted dollars if people could just grasp that fact.

Time is an odd one. We never use the word directly, but clearly, politicians, many parents, and education reformers of all stripes believe we can educate “faster”. Until sixty years ago, calculus was an upper level college course. Once the high school movement began, fewer than 3% of students nationwide took trigonometry, between 10-20% took geometry, and the high point for algebra was 57%–over one hundred years ago–then declining to 25%. (Cite.) One of the little noted achievements of the New Math movement was to alter the math curriculum and make high school calculus a possibility. At first, just kids with interest and ability took that path. Then someone noticed that success in algebra I predicted college readiness and everyone got all cargo cult about it. By the turn of the century, if not earlier, more of our kids were taking advanced math in high school than at any point in our history.

And that was before kids started taking algebra in seventh grade. Sophomores take now take honors pre-calculus so they can get a second year of AP calculus in before graduation. Common Core has gone further and pushed algebra 2 down into algebra I.

Yet 17 year old NAEP scores have been basically stagnant for the same amount of time our high school students have been first encouraged, then required, to take three or more years of advanced math.

Not only do we try to educate kids faster, we measure their gain or loss by time. Poor kids of uneducated parents lose two months learning over the summer. CREDO, source of all those charter studies, refers to additional days of learning. Everyone comparing our results to Singapore always mentions the calendar, how much earlier their kids start working with advanced math. These same people also point out that Singapore has a longer school year. Longer school years don’t appear to work reliably either.

Except maybe KIPP, whose success is mostly likely due to extended school hours. KIPP focuses on middle school and has not really been scrutinized at the high school level. Scrutiny would reveal that the program doesn’t turn out stellar candidates, and while more KIPP alumni complete college than the average low income black or Hispanic student, the numbers are reasonable but not extraordinary when compared against motivated students in the same category who attended traditional schools. Particularly given the additional support and instruction hours the KIPP kids get.

So KIPP’s “success” actually adds weight to the NAEP scores as evidence that time–like money and quality–doesn’t respond to the project management constraints.

Kids learn what they have the capacity to learn. Spending more instruction hours will–well, may–help kids learn more of what they are capable of learning in fewer school years. But the NAEP scores and all sorts of other evidence says that learning more early doesn’t lead to increased capacity later. And so, we’ve moved 1979 first grader readiness rules to preschool with considerable success, but that success hasn’t given us any traction in increasing college readiness at the other end of childhood. Quite the contrary.

I probably don’t have much of a point. I was actually thinking about the increasing graduation rates. It’ll be a while until part 2. I’m swamped at work, moving again, writing some longer pieces, and really would like to post some math curriculum rather than detangle my mullings.

But the triangle thing is important. Really.

Take note: under 1000 words. Hey, I have to do it every year or so.

Group Work vs. Working In Groups

I sit my kids in groups. But I don’t like “group work”.

No, that’s not a paradox. Sitting in groups isn’t “group work”.

Group work is an activity that falls under the larger rubric of “collaborative learning”, an organizing bubble to collect techniques and strategies like “Think Pair Share”, jigsawing, peer tutoring, and the like. The most fully-realized collaborative learning pedagogy is probably complex instruction, which was developed by Elizabeth Cohen. (That’s CI, not CISC.) To illustrate, CPM curriculum is based on complex instruction, whereas Everyday Math is not.

Complex Instruction had been in development for over 20 years, but really caught on during the early 90s, when detracking was all the rage, thanks to the Demon Goddess Jeannie Oakes and her book Keeping Track, a synthesis of the arguments against tracking developed since the late 60s, when the feds and the Supreme Court decided by god, they were serious about this integration business, enforcing busing and other means of insuring that no schools were too white or too brown.

In Keeping Track, Oakes accused parents and schools of racial discrimination, and a good chunk of the 90s was wasted as districts and states desperately tried to win her approval. Fortunately, they all ultimately learned it was easier to disappoint her.1

Complex Instruction was also developed by tracking opponents, but opponents who nonetheless cared about learning. It’s explicitly designed to give schools a tool for the havoc that results when kids with a 3 to 8 year range in abilities are put in the same room, and thus was grabbed at by many schools back in the early 90s. Many CI concepts are also found in “reform math”—Jo Boaler’s Railside study on San Lorenzo High School was all about Complex Instruction. Carlos Cabana and Estelle Woodbury, who just co-authored Mathematics for Equity, a book on teaching math with Complex Instruction, both worked at San Lorenzo High School during Boaler’s study.

So start with the theory, articulated here by Rachel Lotan, the late Cohen’s key associate. You should watch this, or at least fast forward through parts, because Lotan clearly articulates the admirable goals of complex instruction minus the castigation, blame, and fuming ideology. Or, Complex Instruction’s major components in written form:


Both Lotan and the writeup offer much that is problematic. Reducing the ability range: not good. Creating busywork tasks (writing down questions, getting supplies) to let everyone feel “smart”: not good.

The write up mentions “status problems”. Lotan gives a great account of an absurdly pretentious term, “mitigating status” that is something every teacher in every classroom–no matter how they are seated—should take seriously. Lotan does a better job of explaining it, but since many won’t listen to the video, here’s a written version:

CI targets equity and, in particular, three ideas: first, that all students are smart; second, that issues of status—who is perceived as smart and who is not—interfere with students’ participation and learning; and third, that it is teachers’ responsibility to provide all students with opportunities to reveal how they are smart and develop/recognize new ways of being smart. The complex instruction model aims to “disrupt typical hierarchies of who is ‘smart’ and who is not” (Sapon-Shevin, 2004) by promoting equal status interactions amongst students so that they engage with tasks that have high cognitive demand within a cooperative learning environment.

(emphasis mine)

Ed schools wanting to hammer home how putting kids in groups doesn’t by itself address status usually show this video, but brace yourself. I tell myself that the ignored kid is probably a pest all the time, that everyone in the class is tired of his nonsense, that we’re just seeing a carefully culled selection to maximize the impact of exclusion and of course, race. It doesn’t matter. It’s still hard to watch.

And the video also reinforces the practical message that CI advocates are pushing, as opposed to the theory. In theory, status can be unearned by anyone of any gender or color. In practice, most CI advocates expect teachers to shut down white males. In theory, kids learn that everyone is smart. In practice, kids still know who’s “smart” and who’s not.

But then, CI advocates have their own frustrations. In theory, they’d put teachers in PD designed to indoctrinate them into realizing the error of their racist ways. In practice, teachers who haven’t already drunk the Koolaid either politely fake it until they can find an exit or get really annoyed when they’re called racists, as an excerpt for Mathematics for Equity makes clear:

Cite: Mathematics for Equity1

Complex Instruction done well is pretty interesting and often thought-provoking. Cathy Humphreys is a long-time advocate of “reform math” and complex instruction. She was at the center of one of those “rich educated parents” meltdowns that you saw over reform math back in the 90s. Humphreys represented the reform side, of course, and further endeared herself to parents by proposing to get rid of tracking at a Palo Alto, CA middle school. That went over like a water balloon down a balcony, she quit, worked as a math coach for a while, and then taught for years at a diverse high school in the Bay Area that had ended tracking. She also teaches at Stanford’s education program, according to her bio. Carlos Cabana, one of the co-authors of Mathematics for Equity, has also been teaching complex instruction for a long time; he’s one of the teachers at Railside, Jo Boaler’s pseudonym for San Lorenzo High School.

You can see both Humphreys and Cabana here at a website put together by the Noyce Foundation to promote the 8 essential practices. (Notice the link between “reform math” and supporting “common core”? As Tom Loveless says, Common Core is a “dog whistle” for reform math. Humphreys and Cabana are teaching high school math in the videos. You can also see Humphreys teaching at what I assume is the middle school that melted down. Humphreys and Cabana are much better demonstrations of complex instruction than the absurdly flashy promos that Jo Boaler puts out.

When I began teaching, I thought sitting kids in groups was absurd. I remember being pleased one of my mentoring teachers put kids in rows. But my primary student teaching assignment required me to sit kids in groups, as we were using CPM, a reform text that requires groups. I adjusted and liked it much more than I thought I would, especially when I took over the class and could group by ability. But my first year out, I happily put my desks in rows, thinking that groups were good, but now I could finally run my class the way I wanted.

Four weeks later, I put the kids in groups. It just….felt better. Year 2, I was teaching all-algebra, all the time, and thought rows would make more sense. The rows lasted 2 weeks and since around September of 2010, the only time my kids sit in rows is for tests.

I have….mixed feelings about CI. When promoted by the fanatic adherents, it’s both Orwellian and despicable. Teachers have to squelch kids who know the answer, force kids who understand the concept to explain, endlessly, to the kids who don’t, and then grade the kids who know the answer not on their demonstrated knowledge but on the success of their explanation and their willingness to do so. Teachers have to pretend to their students that asking a good question or taking notes is just as important as understanding the math (no, say the fanatic adherents, teachers aren’t pretending. These tasks are just as important!).

But while no student is ever going to believe that everyone is smart, “issues of status” do absolutely impact a students’ willingness to participate. Let the “smart kids” talk, everyone thinks, and sits back and zones out.

However, in my opinion and experience, CI methods often achieve exactly what they are defined to avoid, precisely because of their Orwellian insistence on ignoring reality. Kids know who is smart. They shut down if the smart kid is in their group, and go through the motions when the teacher walks by.

Ironically, I “mitigate status” by violating Complex Instruction’s most sacred tenet. Complex Instruction holds that student groups must be heterogeneous. Organization can’t be based on the rigid, academic version of “smart”. But I group my kids by ability as the most effective way of “mitigating status”.

I don’t want the weakest students in my class feeling as if any success short of an “A” is irrelevant. I also don’t want to try and convince them they’re just as “smart” as students who don’t struggle with the same material. That way, my students know that they can talk about math, what they need to know, what questions they have, knowing that other students probably have similar issues.

I don’t want to make it sound as if “mitigating status” is the only reason I sit kids in groups. Groups allow me to differentiate tasks slightly (or extensively) and enables me to quickly give help or new tasks. Groups allow kids to work together, discussing math, developing at their own speed with peers who have similar abilities.

But whether it’s status or some other curricular reason, when I sit them in groups, they start working and talking about math. They discover they are working with peers who won’t make them feel stupid, and they start to have discussions. Should we do this or this? They call me over to adjudicate. They try things. They check their notes, engage in all those excellent student behaviors. Not always, of course. But many times. They are less likely to sit passively and wait until I come by to personally tutor them through problems.

Moreover, because they are working with students of their own ability, they don’t feel alone or stupid. They work to improve. Maybe not great, maybe not good. But better.

Sitting kids in groups is not group work. But sitting kids in groups based on ability and giving them achievable tasks makes them more likely to work, and as math teachers often know, that’s no small thing.

1 I was thinking crap, I don’t want to have to look up the whole history of the ebb and flow of tracking and then went hey, Tom Loveless has to have something on this and by golly he does: The Resurgence of Ability Grouping and Persistence of Tracking covers the whole era, Oakes included. I would only quibble slightly with this sentence: Although the call to detrack was not accompanied by conventional incentives—the big budgets, regulatory regimes, and rewards and sanctions that draw the attention of policy analysts—detracking was, in a field famous for ignored or subverted policies, adopted by a large number of schools.

Loveless appears to forget the biggest incentive of all: lawsuit avoidance. Detracking lawsuits were the rage in this time period. Unlike new curriculum or teaching styles, detracking is achieved by executive fiat by district superintendents. No training, no carrots needed. Shazam! But leaving aside that minor quibble, a great piece documenting the move to and then the move away from heterogeneous classrooms (de-tracked).

Teaching: My Retrospective

Okay, I’m rolling along on my task of drawing clear lines of demarcation between my particular brand of squish and traditional progressive education (heh–traditional progressive. Get it?). First up was my new no homework policy.

I then decided to take on sitting my kids in groups (as opposed to group work), which led me to look back at some old post, which forced me to look back at my practice over the years, and that’s been a trip. So much of a trip that I decided to do the retrospective first.

The introspection kicked off when I reread one of the first posts I ever wrote on this site, over 3 years ago, halfway through my third year of teaching. Some key observations:

  1. I focused almost entirely on classwork, even then. The essay doesn’t even mention homework which, at that time, I assigned in much the way I describe in my last essay.
  2. At that time, the school I worked at used a traditional schedule of 60 minute classes, so the 3 day span per lesson is about two days at my current school. Additional evidence I was focused primarily on what kids learned in class, although as I said, my original homework policy goes back even further than this post.
  3. Here’s a real change. Me on low ability students three years ago: lowabilstds3yrs
    I’m so cheered to realize how much I’ve improved. I had good student engagement back then, but in rereading this I can remember how many students I had to nudge endlessly, how I had to constantly pick up pencils and hand them to kids to get them to work. Recall I was teaching algebra and geometry, and had just begun what is now my bread and butter class of Algebra 2. So my experience at the time of writing those words was with a lower level of math class, which will always mean lower engagement. Nonetheless, that simple paragraphs reminds me of the struggles I had to get total engagement. I’ve come a long way. Yay, me.

  4. Interesting to see my off-hand mention of EDI. No one seeing my teaching would think of me as using the direct instruction mode, but in fact I always, at some point, give kids specific, explicit instructions on the concept at hand.
  5. While I talked about differentiation and my need to challenge top students, I have actually moved away from different assessments for different students. At that time, I was just three months of teaching out from year two, all-algebra I-all-the-time, and I basically taught 4 different classes. I’d tentatively planned on continuing this approach, but learned that year (and confirmed in later years) that this wouldn’t work for any class but algebra I.

I wrote this post on January 8, 2012, at almost exactly the same time I began an experiment that utterly transformed my teaching. I speak, of course, of Modeling Linear Equations, which I’m amazed to realize I wrote just one week after the “How I Teach” post. So shortly after I began this blog and described my teaching method, I started on a path that took my existing teaching approach–which was pretty good, I think–and gave it a form and shape that has allowed me to grow and progress even further.

I haven’t really read this post in over two years—I tend to link in Modeling Linear Equations, Part 3, written a year later (two years ago today!), when I’d realized how much my teaching had changed. So reading the original is instructive. I talk about the Christmas Mull, something that stands very large in my memory but don’t remember quite as described here:


The part that’s consistent with my memory: Christmas 2011, I was depressed by the dismal finals in my three algebra II classes. In the first semester, I had gone through all of linear and quadratic equations, including complex numbers, at a rate considerably slower than two colleagues also teaching the course. Yet the kids remembered next to nothing. Every single person failed the multiple choice test–the top students had around half right. I had experienced knowledge fall-offs in algebra and geometry, but nothing that had so sublimely illustrated how much time I’d wasted in three months. So I came out of the Christmas break determined to reteach linear and quadratic equations, because to continue on teaching more advanced topics with these numbers was purely insane. And I wasn’t just going to reteach, but come up with an entirely different, less structured approach that allowed my students to use their own understanding of real-life situations.

What I hadn’t remembered until reading this closely was my rationale for ignoring the regular curriculum requrements. At the time, Algebra 2 was considered a “terminal” class; students weren’t expected to take another course in the college-prep sequence. This has changed, of course–these days, algebra 2/trig is, if anything, experiencing a fall-off in favor of a full year of each course. But at the time, I justified my decision to go off-curriculum based on the student needs. These students’ primary concern, whether they knew it or not, was what happened to them in college. How much remediation were they going to need? Could the best of them escape any remedial work and go straight onto credit bearing courses? This, of course, still remains my priority–I’d just forgotten how linked it was to my initial decision to try something new.

Also interesting that I described this approach by the specific method I used for linear equations–using “inherent math ability”. That’s not how I describe my approach these days, but I can see the germination of the idea. At the time I wrote this, I had no idea I would go beyond linear equations and use this approach consistently throughout my instruction.

I think the best description I’ve come up with for my approach is modified instructivist, which comes in one of two forms: “highly structured instructivist discovery, and classroom discussions with lots of student involvement”.

As for the latter: I don’t lecture, with or without powerpoints. When I do explanations, they are classroom discussions, and you can see this demonstrated in all my pedagogy posts. However, I am constantly migrating my classroom discussions to structured discovery.

What’s structured discovery? Imagine a teacher and students on a cliff, with a beach below. There’s a path, but it’s not visible.

In a traditional lecture or classroom discussion, the teacher shows them the path and leads them down to the beach.

In a discovery class, the teacher doesn’t even tell them there’s a path or even a beach. In fact, to the discovery/reform teacher, it doesn’t matter whether there’s a path or not—the kids will all find their own way down. Or maybe they’ll just find some really cool flowers and stop to examine their biology. Or maybe they’ll just kick back and have a picnic. It’s all good, in reform math. (sez the skeptic)

In what I call structured discovery, the kids are given a series of tasks that use their existing knowledge base and find the path themselves. They may not yet know there’s a beach. They may not know what the path means. But they will find the path and recognize it as a consistent finding that makes them go “hmm”. In some cases, an interesting finding. In other cases, just something they can see and understand.

Sometimes the path they’ve found is the concept–for example, modeling linear equations or exponential functions, or finding gravity in projectile motion problems.

In other cases, the model just introduces an inevitable observation that leads to the new concept. For example, I teach my kids about function operations when we do linear equations–adding and subtracting are good models for simple profit and loss applications.

So I kick off quadratics by asking my students to multiply linear functions, which they can see clearly as an extension of adding and subtracting them. This is an activity they can start off cold, with no intro (I haven’t written it up yet). I designed this because parabolas just don’t have a natural “real life” model other than area, which gets kind of boring. Plus, I need to cover function operations anyway, so hey, synergy. In any event, the kids are seeing an extension of a concept they already know (function operations) and seeing a new graph form consistently emerge. Then we can talk about factors (the zeros) and realize that we are looking at products of two lines. Could a parabola exist without being a product of two lines? Well, this is algebra 2 so they are fully aware that parabolas don’t have to have zeros. But what does that mean in terms of multiplying lines being factors of parabolas? Well, they must not have factors. So are all parabolas the product of two lines? And we go from there.

Understand that my classes still have lots of practice time where kids just factor equations and graph parabolas, learn about the different forms, and so on. But rather than just saying “now we’ll do this new thing called a parabola”, I give them a task that builds on their existing work and leads them into the new equation type. I don’t define the path. But nor do I let them go off on their own. I give them something to do that looks kind of random, but is in fact a path.

And all of this came from the results of the Great Christmas Mull. The previous Christmas had been productive, too–it’s when I came up with differentiated instruction for my algebra class.

So what can I say about my teaching, 5.5 years in? What’s consistent, what’s changed?

  1. I never lectured. I always explained, with increasing emphasis on classroom discussion.

  2. I have always been focused on student work during class, emphasizing demonstrated test ability above everything, and minimizing (or now eliminating) homework.
  3. I have always tried to move the student needle at all ability levels, from the no-hopers to the strugglers to the average achievers to the top-tier thinkers. I’m not always successful, but that’s consistently my stated priority.
  4. I have always designed my own curriculum and assessments.

  5. My teaching was transformed Christmas of 2011, when I realized I could introduce and teach topics using existing knowledge, forcing students to engage immediately with the material and start “doing” right away, increasing engagement and understanding. I have evolved from a teacher who mostly explains first to a teacher who only occasionally explains first. And that is a huge change that takes a lot of work.
  6. The observer might think that this change makes my classes student-centered, but I disagree. My classes are definitely teacher-centered, and let’s be clear, I’m the star of my teaching movie.
  7. Thanks also to the Great Christmas Mull, I’ve become far less concerned about curriculum coverage than I was in my first two years of teaching.
  8. I have always been a teacher who values explanation. It’s the heart of my teaching. I’ll explain through discussion or demonstration, but I’m not a reformer letting kids “construct” the meaning of math. I’m there to tell them what it all means.

I have plenty of development areas ahead. I’m working on tossing in the occasional open-ended instruction, just to see if I can come up with ideas that don’t waste hours and have some interesting learning objectives. I still have many concepts waiting to be converted to a “path to the beach”. And I’m now teaching something other than math, which gives me new challenges and more opportunities to see how to construct those paths without running off the cliff.

I Don’t Do Homework

Our school had its second Back to School Night. Attendance was spotty. I don’t judge. As a parent, I rarely attended.

But boy oh boy, could four sets of parents generate some excitement. I had a genuine culture clash.

It all began when I was going through my brief dog and pony show for my second trig class.

“Student grades are 80% tests and quizzes, 20% classwork. But I don’t grade classwork. Students get a B or A- just for showing up and working, which bumps their grade slightly.”

Until recently, I weighted homework for 10% and classwork for 15%–but not really. More accurately, if a student did most of his homework in a relatively timely manner, he’d get a little more of a boost. He couldn’t get the boost by “making up” missed homework; nor could he get the boost for just a couple homework completions. But if he didn’t do the homework at all, no harm no foul.

A few of my students got the boost, and they came from all points on the ability spectrum. I always remembered to assign homework through the first semester, then I’d fall off. For my first five years of teaching, homework had always completely stopped at some point in the third quarter.

“But last term, I suddenly realized that the end of the first semester was weeks away, and I hadn’t been assigning homework for a very long time.”

Remember my mentioning it had been a busy first term? Well, yeah.

“Most of my kids don’t do homework. So this realization just reinforced my awareness that I was only engaging in the homework ritual because I didn’t want to stray too far off the beaten path in comparison to my colleagues. But once I’d given up homework by accident, it seemed natural to make it official.”

The fact that I got that glorious tenure email and didn’t have to worry too much if my colleagues complained may have played a teensy, tiny part.

“So if you’ve got one of those kids who gets an A on tests but pulls his grade down by ignoring all homework, he–and it’s a usually a he–has probably mentioned it by now, and worships at my feet. I accept Starbucks cards or sixpacks of Diet Coke in tribute.”

One parent raises his hand.

“But don’t you find that homework ensures the students will get more practice? They need practice, just as we did when we were kids. I think it’s best for students to genuinely learn the math with practice.”

Uh oh. I take a deep breath.

“My students have always been graded overwhelmingly by what they do in class and the learning they demonstrate on tests. Homework was always optional, and I didn’t assign enough of it for students to practice fluency.”

“But I want my son to have practice material.”

“Well, I use the book pretty regularly, and there’s plenty of relevant practice material in there.”

“But do you think that’s how we all learned math?”

“Well, we weren’t all required to take advanced math. Look, I want to be clear: my method is the ultimate in hippy dippy squish.” Two parents laughed.

“I’m not trying to pretend that it’s normal for a math teacher to abandon homework. The whole homework ediscussion is basically a religious issue–and I don’t mean Muslim, Christian, and Jewish. People have strong ideological beliefs about the best way to achieve academically. However, the research on the intellectual impact of homework is very weak. But no research has shown that doing homework is the cause of comprehension.”

Another parent spoke up with a, er, very pointed tone. “I am so happy that you grade based on their work in class. So much better than to have them confused with nothing more than busy work after school. They can’t ask questions, they feel lost, and then they get discouraged.” Another parent nodded.

Original parent: “But the confusion is part of learning. Then they can come in the next day and ask for help.”

“They learn in class. If I take the bulk of one class to explain something, then they spend the next day working on that concept. I ensure students demonstrate their understanding, to the best of their ability. They won’t be able to copy the work from someone else; if I spot them not working, I work with them until I can see them understand it. If they’re talking or goofing around, they move to a different seat. My kids work math while they’re here. And ninety minutes of working or thinking about math is plenty.”

“But shouldn’t the students be practicing at home? Couldn’t you go through the course much quicker if they did?” the original parent is not to be discouraged.

“Again, they are welcome to work additional problems of their choice. But in my experience, students forget a lot of what they ‘go through’. My goal is to ensure that if they do forget material in this course, at least they really did understand at the time, rather than just follow through on some algorithms.”

“Exactly. I want them to understand the math.” said another parent.

“One last thing: I follow my students’ progress in subsequent classes. For the most part, they are keeping up and doing fine. I teach some of those subsequent classes, and so am able to compare my students to those given a more traditional course, and they’re doing fine. Many of my students go to junior college or local public universities, and I track their placement results as well. They, too, are ending up just as I’d expect. The weakest ones need some small amount of remediation, but most are placing in college credit courses. Meanwhile, they have far more accurate GPAs and weren’t forced to retake courses and slow down their progress simply because they didn’t do homework.”

And….the bell rang. Saved!

The original parent came up to me and asked, “You will assign my son additional homework?”

I smiled at the dad and the son. “All he has to do is ask.”

(He hasn’t.)

I decided to describe my policy change thusly because, well, the story happened and it was fun. All parents were respectful; I did not feel insulted or bothered by the first parent’s concerns. If I have in any way seemed contemptuous of the parents involved it’s unintentional. That said, ethnic stereotypes will prove helpful in deciphering the anecdote. The reason for the change is as described—I was busy, suddenly realized I had stopped assigning homework, decided it was time to cut the cord.

I usually just pick holes in everyone else’s arguments, but math homework is a teaching issue I have strong feelings about. Grading homework compliance is hurting a lot of kids, and all it does for those who comply is give them higher grades, not better academic skills.

Administrators understand this more than most, as they’re the ones putting additional math sections on their master schedule to accommodate all the kids with reasonable test scores who nonetheless flunked for not doing their homework. That’s the impetus behind all those stories you read of a district limiting homework’s percentage on the grade.

So as I wave goodbye to homework, let me take this opportunity to urge my compatriots to consider a similar policy, particularly if their classes look something like this:

The class opens with a warmup, designed to either review the previous material or introduce a new concept. Teacher reviews the warmup problem, then lectures or holds a class discussion on a new concept, works a few problems, has the class work a few problems, assigns a problem set, and those problems are called “homework”. Your basic I tell, I do, we do, you do.

The kids have the rest of the period to work on the problems, while the teacher is available to answer questions. If they finish in class, no “homework”! If they don’t work in class or do work for some other teacher, no big deal. It’s just time-shifting. They’ll turn in the work tomorrow, maybe do it with their tutors, maybe just copy it from friends who did it with their tutor.

Or they won’t do the problem set, either because they don’t understand, can’t be bothered, or just forget. The teacher will encourage them to come in and ask for help, or go to after school tutoring. Some of them will. Many of them won’t show up. Then they’ll get a zero, or turn it in late for a reduced grade, or stop doing homework altogether until they flunk. Or maybe their parents will call a conference and the teacher will be persuaded to accept a bunch of late homework to help the student pass the class.

How many high school math classrooms does this describe, with the occasional variation? A whole lot.

Notice that it’s only “homework” for those who can’t finish the work in class. The kids who don’t understand the material have to struggle at home. The students who really understand the material and could use more challenge get the night off.

High school teachers borrowed this method from colleges fifty years ago or more, a method designed for highly ambitious 20-somethings with demonstrated ability and interest. Today, our well-meaning education policy forces everyone into three years or more of advanced math, regardless of their demonstrated ability and interest. The college model is unlikely to work well with many students.

So go ahead and sneer at me for being a softie who skips homework, but understand that my students work to the bell. More often than not, my introduction is 10-20 minutes or even less, so the students are working the entire class period, taking on problems of increasing challenge. On those occasions where I have to explain something complicated, they focus on the relevant concepts for another day or more. But all my students are getting 60-90 minutes each day actively thinking and working about math, and my student engagement level has always been high. Strong students who finish early just do more problems. The student who treats my class as a study hall for her other homework because she has a tutor will experience teacher disapproval, often for the first time, and I’m a cranky cuss. She rarely makes the mistake twice.

When I did assign homework, I didn’t just continue from the same classwork problems, but created or selected much easier problems, designed for students to determined if they understood the basics of that particular concept.

Most education debates are tediously binary and thus wholly inaccurate. And so the math homework debate becomes “teachers who want to challenge their kids assign demanding homework” vs. “teachers who want to coddle their kids neglect their responsibility to prepare kids for college.”

In my classroom, kids are working pretty much non-stop, usually much harder on average than in the classrooms where kids are left to their own devices to finish their work. But somehow I’m the squish because I don’t engage in the great morality play known as homework. Are there teachers who don’t assign homework and also allow their kids to discover their pagh? Sure. That’s why the binary debate is a waste of time. The reality of classroom activity requires many additional points on a compass–not a bi-directional spectrum.

Finally, none of this really has anything to do with the actual teacher quality. Many teachers are doing a great job explaining math in those I do, etc lessons. Nor would any observer consider me hippy dippy or squish, which is why the comment always gets a laugh.

I was going to end with a joke about being a Unitarian in a Calvinist world. But hell, that plays right into the wrong sterotype.

Troubling Students

My classes are easy to pass, hard to do really well in. I’m a pushover for a D, but think three or four times about giving out an A. I didn’t fail a single kid last year. Save for Year Two, All Algebra All the Time, I’ve failed fewer than six kids a year, and even Year Two I had the second lowest fail rate of the math teachers.

I teach mostly math at a comprehensive high school, and the previous paragraph is very near heresy. Some math teachers cheer me on as a brave, admirable soul, but I spot them making the Mano Pantea while they walk away, just in case the Overlord is Watching. Others think I’m What’s Wrong With Education Today. These teachers hold as gospel that math standards could be upheld if we teachers were just willing to fail 60-70% of our students. In contrast to Checker Finn, who thinks teachers like me are spreading out two years of math content over three years of instruction because we can’t be bothered, these folks don’t think I’m lazy. They think I’m soft. They think I’m damaging their ability to cover all the course content they could get through if there weren’t all these kids who shouldn’t be there.

I became a lot less conflicted about my high pass rate–not that I ever lost sleep over it–after teaching precalc and discovering that a third of the kids had forgotten how to graph a linear equation and half couldn’t graph a parabola. These were kids that those other teachers had, teachers who had covered everything. Meanwhile, my kids do well in subsequent classes, so I’m not doing any harm.

But I digress. The students who trouble me aren’t the strugglers. I can take a kid who hates math, doesn’t want to be in class, and get him (it’s usually a him) to try. I can get that kid to attack a projectile motion problem and, even while making multiple small mistakes, beam with pride because by god, he kind of gets this and who ever would have thought? Kids like that, I can pass with nary a qualm.

The worrisome ones pretend they understand, but don’t have a clue. They cheat whenever they can, and not just on tests. They copy classwork in the guise of “working together” or “getting help”, and do their best to sit next to strong students. I group students by ability and, unless they can cheat on my assessment test, they are outed and placed up front, where I can keep an eye on then. They will then ask if they can sit next to John, or Sally, or Patel, their friend, because “they explain it so well”. I say no.

But if they cheated on the test, they can sometimes escape notice for a while. I circle constantly, watching kids work, changing seating when I see too much “consulting” with little discussion. Still others are more clever, and it takes a while before I realize they’ve been cheating not only in classwork, but on the tests–even when I create multiple tests. As a new teacher, I would sometimes miss these kids through the first semester. My success rate at pegging them early has improved.

This isn’t a big group, thank god. I might run into one or two a year. They have a telltale bipolar profile: for example, failing English entirely one year, and passing it the next year with Bs. Passing algebra with straight As, failing geometry completely–and failing the mostly pre-algebra and algebra state graduation test with a spectacularly low score. They aren’t fooling all of the teachers all of the time.

These kids are not your Stuyvesant cheaters, conspiring with others to satisfy demanding parents and create a fraudulent resume to get into a good school. Nor are these the low achievers who just want to get a passing grade in these time units called classes organized into a larger time period called school that others apparently view as a place of learning but they see as little more than a community network in which they have invested considerable social capital.

In fact, they’re almost worse than identified low incentive low achievers, cheating or otherwise. These kids almost seem incapable of learning. I can’t get them to slow down. They often resist help from me. Typical conversation:

Me, stopping by: “Okay, let’s start this again. You’ve plotted these points….”

Student: “Oh, yeah, I see.” Frantically erases.

Me: “Well, hang on, I want to be sure…”

Student: “I got it I got it I got it.” Starts to plot a point, then pauses.

I realize the student is waiting for me to say where to plot it in order to say “Yes, I know, I know.” So I wait. The student takes a deep breath and plots the point then lifts his pencil. “No, that’s not right, duh…”

Me: “You aren’t sure how to plot points.”

Student: “Yes, I am.”

Me: “Great. Plot (7,-7).”

Student plots (-7, -7).

Me: “Stop there.” I go grab a handout I have specifically for these situations, a simple handout that explains plotting points with some amusing activities to drive the point home.

Student: “I don’t need this. I know how to do it!”

Me: “Great. Then it should just take you a few minutes.”

At this point, I get a variety of reactions. Some students become furious. Others get sulky. Still others do the handout, making many mistakes, all the while assuring me that this is easy. I obligingly correct the mistakes, make them do it correctly. The ones that get furious, I shrug and let them continue.

Regardless, within a day, they are making the same mistakes. Nothing sinks in. Don’t get overly focused on plotting points; the problem could be anything–factoring, solving multi-step equations, working with negatives, exponential properties, fractions, whatever. Or a new concept. They have absolutely no clue, and can’t do much of anything.

Yet they don’t have the profile of a low ability student. Test scores, yes. Profile, no. They often have As, win praise from teachers for their teamwork and effort. They are heavily invested in appearing “normal”. Serious control freaks. Sometimes, but not always, with parents who expect success. More often, but not always, Asian. All races. Both genders.

I haven’t taught freshmen since oh, lord, fall 2012.1 I teach relatively few sophomores these days, running into them only in Algebra 2.

That matters because when I taught freshmen and sophomores, I would go full-scale intervention. I might talk to a counselor to see if they should be assessed for a learning disability. I would insist that they stop lying to me and themselves. I had no small success at getting some of them to acknowledge their desperate attempts at fraud, get them to work at their actual level, deal with the discomfort. They didn’t make much progress, but it was real progress, and they had skills to move forward. I ran into some of them again the next year, and we could start on an honest basis and make additional progress. Those who didn’t acknowledge their issues were among the few students I failed.

But that’s a lot harder to do when dealing with juniors taking trigonometry or, god forbid, precalc. Should I fail them? They will probably do better in a class with teachers who give “practice tests”, study guides that have exactly the same questions as the eventual real test but with different numbers. They will definitely do better with teachers who actually grade homework and count it as 25% of the overall.

A small problem. This approach turns my grading policy into: work hard and honestly acknowledge your ignorance and I’ll pass you. Lie and do your best to cheat with similar ignorance and I’ll fail you. I’m comfortable with holistic grading at the bottom of the scale, but I don’t like morality plays.

Then I remember that kids who honestly acknowledge their inability in a trig or pre-calc class are usually seniors, off to junior college and a placement test that will accurately put them in remedial math. I’m only ensuring they are learning as much as possible for free before paying. If they are juniors, I always have a talk with them about their next steps, telling them not to take the next course in the sequence but maybe stats or something else that will keep them working math, but not out of their league.

The kids who cheat and fake it in trig and precalc are usually juniors, and they will be going onto another course. They will not listen to me when I tell them under no circumstances should they continue into pre-calc or, god forbid, calculus. I might be teaching that course, which just gives me the same problem again. Or they’ll be cheating their way through with another teacher–or, that teacher will do what I should have done and flunked them.

This quandary doesn’t make any sense unless you realize that in my view, these kids are pathologically terrified of facing reality, the sort of thing that some of them, forced to face up, might not survive in good form. These aren’t blithe liars gaming the system to look good. Then I remind myself that they’ve been caught before, they’ve flunked other classes, they’ll survive. But I still don’t like the quandary, because these are kids who literally can’t learn. (And remember, I’ve seen them in my non-math classes, too). By junior year, given their denial and fear, does it do any good to make them aware of this? They’re going to be able to point to any number of teachers who disagree with my assessment, and have all sorts of excuses for why they got those Fs. Besides, they just don’t test well. It’s always been a problem.

At times like this, I envy my colleagues who never notice the cheating, or who focus purely on achievement and aren’t interested in the distinctions I’m making.

But these are the students who trouble me.

1Holy Crap. That’s an amazing realization. New math teachers doing your time in the algebra/geometry trenches, take heed. If you want variety, it will come.

Wearing Anonymity

I wear my anonymity loosely. It’s mostly fine if you know who I am. It’s Google I want kept in the dark.

“Mostly” in that sure, there are people out there who would be very happy to see me lose my job, and I’d just as soon those people didn’t have the opportunity to put together a campaign to get me fired. While I have just recently obtained tenure (whooohoo!), I’m not at all sure that tenure would protect me in this circumstance. Despite all the whines, teachers with tenure are fired all the time. The administrator just has to want it. Just ask Natalie Munro, a tenured teacher who blogged about her “lousy” students and was gone within two years. I despise Munro’s behavior, but I believe her over the administration when she says she had no problems before her blog.

For the record, my school administrators think I’m terrific, and I admire their work. I have never knowingly said anything offensive or critical about my co-workers, bosses, or students. Even when I’ve disagreed with them, my disagreement has been couched as “choices are hard”. I love my current school and I’ve always loved all my students at every school.

But we teachers aren’t guaranteed first amendment protection, and the rules on blogging are very fuzzy. My administrators know about my blog; I hope they check in on it periodically, although that’s unlikely. None of that would save me if there was the wrong kind of fuss.

For this reason, I don’t tell people who I am without asking that they not disclose this information online. Gender, location, name, all left out of the discussion. Every person I’ve informed of my identity has complied with this request. The bulk of the people I’ve told were journalists. The rest were mostly professors or policy wonks. And this number is very, very small–no more than 15-20 people.

That means if someone out there in the wide world of the internet says “Ed Realist is Mark Murgatroyd from Chicago” or “Ed Realist is a San Francisco-based teacher who hates Asians” or “Ed Realist also posts as Lance Jackson” or “Ed is one of those rare women who speaks honestly about race and IQ”, that person did not get this information from me. In some cases, they believe they have guessed my identity but are speaking of it, wrongly, as a fact. In others, they read this information at another site from another person who did not get this information from me. In still other cases, they may have heard the information second-hand offline from someone who did get it from me, although I doubt that last one. I’m not important enough to discuss offline.

I’m not commenting about the accuracy or inaccuracy of the information. Nor do I want anyone to go out there and build a case for me being person X or person Y. I’m not saying “nyah, nyah, you can’t catch me, coppers!” My blog has gotten much, much bigger than I ever dreamed. I would have kept age, parental status, and a few other details back had I known. Anyone who wanted to build a logical case to strongly suggest that person X is me could probably manage it.

For this reason, I try very hard not to be coy, give hints, or deny. Someone claims I live in Location Y, I respond I’ve never mentioned my geographic area online. Someone claims I’m a man or a woman, I respond that I’ve never mentioned my gender online. Someone claims that I’m teacher X, I respond that I’ve never identified myself online. I like to think that’s why I’ve managed three years of anonymity, but then maybe no one has ever cared enough. I hope I’m still unimportant enough that this post won’t lead to speculation about my identity.

I would appreciate reader consideration when characterizing me and my work. I’m a teacher. I used to be a tutor and test prep instructor. Anything else I mentioned on my blog you are free to use, but try not to overstate.

If you’ve read someone comment about my gender, location, or identity, please remember they did not get this information from me. No reason to get into a pissing match, but a link to this statement would be appreciated.

If you think you know who I am: You might be right. So what? What is it you hope to achieve by posting about your guess? If you’re wrong, you could be hurting another teacher. If you’re right, then you could be putting me at risk of losing my beloved job. If that’s what you want, well, then I guess I can’t stop you.

But you didn’t get the information from me.

What You Probably Don’t Know About the Gaokao

I didn’t intend to write about the gaokao, or Brook Larmer ‘s profile of 18-year-old Yang and his family inside Chinese test prep factory. I just started out googling, as is my wont, to find out more information than the article provides. I certainly did that.

The novice might find Larmer’s article emotionally draining. Anyone with even a rudimentary understanding of Chinese academic culture will notice a huge, gaping hole.

I noticed the hole, which led me to an observation, which led me to a better understanding of how the gaokao works, which is almost exactly the opposite of its presentation in the American press.

The hole: In a story dedicated to students preparing for the National Higher Education Entrance Examination (aka the gaokao) Larmer never once mentions cheating. This would be a problematic oversight in any event, but given the last anecdote, the omission strains credulity.

When Larmer returned to the town for his second visit, the day before the gaokao, Yang’s scores, which had been dropping, had not improved. As a result, Yang had kicked out his mom and brought his grandfather to live with him in Maotanchang for the last few weeks of prep. While Larmer drove into town with Yang’s parents, the grandfather refused to let Larmer accompany the family to the test site. Grandpa was afraid the family might “get in trouble” for talking to a reporter, according to “someone”.

Yang does exceptionally well, given his fears—“his scores far surpassed his recent practice tests”. Sadly, his friend Cao tanks because he “had a panic attack”.

Yang’s scores were considerably beyond what his recent performance had predicted. Yet it apparently never once occurred to Larmer that perhaps Yang and Grandpa prudently got the New York Times reporter out of the way before they arranged a fix. Maybe Yang wanted more aid than could be provided with “‘brain-rejuvenating’ tea”, or Gramps didn’t want Larmer to see Yang wired up for sound, or that he’d really put in some money and paid for a double.

Yang’s performance might have been entirely unaided, of course. But any article about the gaokao should address cheating, even with Gramps banning access.

When I realized that Larmer hadn’t mentioned cheating, I read the piece again, thinking I must have missed it. Nope. But that second readthrough led to an observation.

I got curious—just curious, nothing skeptical at this point—about the school’s gender restriction on teachers. Was that just for cram schools? What was the gender distribution of Chinese teachers?

I couldn’t find anything. No confirmation that the teacher were all male, no comprehensive source on cram schools, no readily available data on Maotanchang. I couldn’t find anything at all about the school’s business practices online. So I went back to Larmer’s paper to look for a source for that fact—and nothing.

And so, the observation: In his description of the school’s interior and practices, Larmer doesn’t mention interviews with school representatives, other journalism, or a Big Book of Facts on Chinese Cram Schools.

The earliest detailed description of Maotanchang online appears to be this August 2013 article in China Youth Daily, a Beijing paper, which created quite a furor in China and largely ignored here because we can’t read Chinese. Rachel Lu, senior editor at Foreign Policy magazine, restated some key points for those folks who don’t read Chinese, which is nice of her, because what idiot would copy and paste the Chinese piece into Google Translate?

Yeah, well, I’m an idiot. I won’t bore people with the extended version, but a lot of the details that Larmer didn’t seem to personally witness show up in the Chinese story: same school official quoting management theory, teachers using bullhorns, Maotanchang’s 1939 origins, bus license plates ending in 8, burning incense at the town’s sacred tree, teacher dismissals for low scores.

The excitement over the China Youth Daily article generated more interest, like Exam Boot Camp, also written in August 2013, happily in English, which profiled a female student and her mother who provide data points like higher prices for lower scoring students ,lack of electrical outlets, and surveillance cameras in the classroom.

Am I accusing Larmer of lifting tidbits from these other stories? Well, I’d like to know where he got the information.

Leave that aside, though, because reading through these stories looking for sources led me to all sorts of “new things” to learn about the gaokao. These “new things” are readily available online; in fact, anyone can find most of the information in the Wikipedia entry. But you will rarely read these not-in-fact new things, but well-established facts, explicitly laid out by any major media outlet (although now that I know, I can see hints). I don’t know why. I can’t even begin to see how any reporter wouldn’t trumpet these facts to the world, narrative or no.

China’s supposedly meritocratic test is a fraud.

To begin with, Larmer, like just about any other reporter discussing the gaokao, describes it as a “grueling test, which is administered every June over two or three days (depending on the province), is the lone criterion for admission to Chinese universities.”

Wrong. The test score is, technically, the sole criterion for admission. But in China, the test score and the test performance aren’t the same thing.

Testers get additional points literally added to their scores for a number of attributes. China’s 55 ethnic minorities (non-Han) get a boost of up to 30 points , although the specific number varies by province. Athletic and musical certifications appear to be in flux, but still giving some students more points, even though the list of certification sports culled from 70 to 17. Children whose parents died in the military and Chinese living overseas get extra points, and recently the government announced point boosts for morality.

Remember when the University of Michigan used to give students 20 points if they were black, and 12 points if they had a perfect SAT score? Well, imagine those points were just added into the SAT/ACT score. That’s what the Chinese do.

But even after the extra points are allotted, test scores aren’t relevant until the tester’s residence has been factored in. Larmer: “The university quota system also skews sharply against rural students, who are allocated far fewer admissions spots than their urban peers.”

I first understood this to mean that colleges used the same cut scores for everyone, but just accepted fewer rural students, without grasping the implications: city kids have lower cut scores than rural kids.

Xu Peng, the only Maotanchong student to make the cut off score for Tsinghua, where the “minimum score for students from Anhui province taking the science exam was 641.”

Two years earlier, the cutoff score for Tsinghua for a Beijing student was somewhere under 584.

Rachel Lu again:” the lowest qualifying score for a Beijing-based test-taker may be vastly lower than the score required from a student taking the examination in Henan or Jiangsu. [rural provinces]. ”

A joke goes:

Of course, don’t make the mistake, as I did, of thinking the cut scores mean the same thing for each student.

Curious about the nature of the studying/memorization the students do (another vague area for Larmer’s piece), I tried to find more information on the gaokao content. The actual gaokao essay questions are usually published each year and they’re….well, insane.

When I finally did find an an actual math question:


it seemed surprisingly easy and then, I realized that it was only for the Beijing test:


Then I went back to the essay questions and it sunk in: the essay questions differed by city.

The gaokao isn’t the same test in every province. Many provinces develop their own custom test and just call it the gaokao.


At which point, I threw up my hands and mentally howled at Larmer, my current proxy for the mainstream American press: you didn’t think this worth mentioning? Or didn’t you know?

If all this is true, then the wealthier province universities use a lower cut score for their residents. But just to be sure, some provinces make an easier test for their residents, so that the rural kids are taking a harder test on which they have to get a higher score. Please, please, please tell me I’m misunderstanding this.

Consider Larmer’s story again in light of this new information. Larmer can’t say definitively who had the best performance without ascertaining whether Yang or Cao got extra points. Both Yang and Cao might both have outscored many students who were admitted to top-tier universities. Cao may or may not have “panicked”, and may not have even done poorly, in an absolute sense. None of this context is provided.

In my last story about Chinese academic fraud, I pointed out that so much money was involved that few people have any incentive to fix the corruption. All the people bellyaching about the American test prep industry should pause for a moment to think about the size of the gaokao enterprise. The original China Youth Daily story focused on Maotanchang’s economic transformation, something Larmer also mentions. Parents are paying small fortunes for tutoring, for cheating devices, for impersonators, for bribes for certificates. All of these services have their own inventory supply chains and personnel. Turn the gaokao into a meritocratic test and what happens to a small but non-trivial chunk of the Chinese economy?

But I’m just stunned at how much worse the Chinese fraud is than I’d ever imagined.

Sure, well-connected parents could probably bribe their kids into college. Sure, urban kids who had better schools that operated longer with educated teachers would likely learn more than those stuck with “substitutes”. Sure, the content was probably absurd and has little relationship to actual knowledge. Sure, the tests were little more than a memory capacity game, with students memorizing essays as well as facts that had no real meaning to them. Without question the testers were engaging in rampant cheating.

But not once had I considered that the test difficulty varied by province, that some kids got affirmative action or athletic points added directly to their score, and worst of all, that a kid from Outer Nowhere who scored a 650 would have no chance at a college that accepted a kid from Beijing with a 500.

Once again, I am distressed to realize that my cynical skepticism has been woefully inadequate to the occasion.

The gaokao isn’t a meritocracy. Millions of kids who live in the wrong province are getting screwed by a test whose great claim to fame is that it will reward applicants strictly by merit. And of course, the more kids who apply to college, the more cut scores and test difficulty will increase–but only for those students from those wrong provinces. Meanwhile, the kids from the “right” provinces have a (relatively) easy time.

In this context, the 2013 gaokao cheating riot takes on a whole new light. If you really want to feel sad, consider the possibility that Yang’s friend, Cao, now working as a migrant, might have scored higher on a harder test than a rich kid in Shanghai.

By the way, could someone alert Ron Unz?

*Note: in the comments, someone who understands this is (bizarrely, to me) fussed over my use of the “rural/urban” paradigm. I was using the same construct that Brooke Larmer and others have. The commenter seems to think it makes a difference. My point is simpler, and I don’t think obscured for non-Chinese readers. But I caution anyone that I’m utterly unfamiliar with Chinese geography.

2014: Half a million satisfied page views

Yes, I have half a million page views. Not bad for someone who only has 650 Twitter followers.

My page views increased from last year, but not by a whole lot. I had 42% more views in the first half of the year, but was down 22% for the second half. As I mentioned, I had an insanely busy first semester, teaching two brand new classes (one not math) and mentoring two teachers. I only had 3 posts in November, and one lonely post in October. I’d hoped to write 72 posts (6/month); in fact I averaged just fewer than 4 posts a month, at 45. That accounts for most of the drop off.

But I also didn’t have the huge posts that I had last year. At the bottom of this post is a list of my top posts overall (1500 views or more).
Here are the top posts I wrote this year (over 1000 views):

Just a Job 2831
The Dark Enlightenment and Duck Dynasty 2527
Strategizing Horror 2027
Encylopedia of Ed, Part I: Things Voldemortean 1802
Ed Schools and Affirmative Action 1776
The Available Pool 1721
Timothy Lance Lai: Reading Between the Lines 1588
College Confidential and Brain Dumping the SAT 1575
SAT’s Competitive Advantage 1392
Reading in the Gulag of Common Core 1236
Finding the Bad Old Days 1224
A Talk with an Asian Dad 1156
Memory Palace for Thee, but not for Me 1128
Multiple Answer Math Tests 1086
Parents and Schools 1067
Math Instruction Philosophies: Instructivist and Constructivist 1022
Why I Blog 1016
Advanced Placement Test Preferences: Asians and Whites 1008

In all, 41 posts out of the 244 got over one thousand views in 2014 alone (not counting views from prior years).

Compared to last year, I had far fewer big posts. Compared to posts written in prior years, this year’s posts did far less business. Also, the disappearance of both Who Am I and About from my top posts means I had far fewer new readers.

I’m not bothered by this. First, I chose a bunch of esoteric topics. Fox, dammit, not hedgehog. Second, as I said, I had an incredibly busy second half of the year.

Third, when I did have time to write, I spent all the time researching. These pieces consumed well over hundreds of hours of googling and reading:

Only three of them made my top posts. Meanwhile, I knocked out The Dark Enlightenment and Duck Dynasty in 2 hours one very late evening and it hits second place. Again, I’m not complaining. If Steve Sailer or Charles Murray isn’t interested in a post, it’s unlikely to get big numbers on the first viewing.

I also didn’t spend much time on pedagogy this year, and that’s something I vow to change in the upcoming year. I have all sorts of topics that I don’t think of as much because I’m teaching advanced math. The following pedagogy posts got at least 1000 views, got more readers this year than last, despite being over 2 years old, and three of them made my top posts for the year:

Multiple Answer Math Tests, written this year, also got over 1000 views, and a lot of my older curriculum work gets close to 1000 views.

This reinforces a pattern I’ve seen for over two years: Google likes my blog, and teachers like my curriculum. Teachers are not a big part of my regular reader base, but they seem to find my work and if they didn’t like it, google would know somehow. I can’t tell you how pleased I am that teachers might be finding my pedagogy useful.

I am also reminded that the teacher tales, which I consider some of my best work, are not google friendly. Teachers really like my stories, but since they aren’t part of my regular base, they don’t often stumble across my work. I’m not sure how to address this—I mean, how often does someone think “Hmm, I want to google some fun teacher stories!”?

In the meantime, I thought my Teacher Tales from this year were very good. Hey. Maybe I could do a page. Huh.

I will update my Encyclopedia of Ed pages pretty soon–it’s clear they are getting some use, which is nice.

Finally, the second half of this year did see some disillusionment on my part. Not with teaching, or with writing, but with the realization of just how many people in education reform are poseurs, and yet are treated as experts simply because they’ve got an employer claiming they are. I thought I was cynical to begin with, but at this point I’ve become exhausted realizing just how many people are just flat out regurgitating opinions that their employer pays them to have.

On to year 4.

Posts getting over 1500 views this year:

Asian Immigrants and What No One Mentions Aloud 8577 2013
Homework and grades. 3590 2012
The Dark Enlightenment and Me 3058 2013
Binomial Multiplication and Factoring Trinomials with The Rectangle 2524 2012
SAT Prep for the Ultra-Rich, And Everyone Else 2490 2012
Algebra and the Pointlessness of The Whole Damn Thing 2419 2012
Core Meltdown Coming 2317 2013
The Dark Enlightenment and Duck Dynasty 2527 2014
The Gap in the GRE 2213 2012
College Admissions, Race, and Unintended Consequences 2151 2013
Strategizing Horror 2027 2014
Philip Dick, Preschool and Schrödinger’s Cat 1818 2013
Encylopedia of Ed, Part I: Things Voldemortean 1802 2014
Ed Schools and Affirmative Action 1776 2014
The Available Pool 1721 2014
Teaching Algebra, or Banging Your Head With a Whiteboard 1640 2012
Timothy Lance Lai: Reading Between the Lines 1588 2014
Kicking Off Triangles: What Method is This? 1554 2012
College Confidential and Brain Dumping the SAT 1575 2014

The SAT is Corrupt. No One Wants to Know.

“We got a recycled test, BTW. US March 2014.”.

This was posted on the College Confidential site, very early in the morning on December 6, the test date for the international SAT.

Did you get it?

Get what?

I mean how do you know it was a recycled Marhc test? Do you have the March Us test?

Oh, no. I just typed in one of the math questions from today’s test and the March US 2014 forum popped right up.

And of course, the March 2014 test thread has all the answers spelled out. The kids (assuming it’s kids) build a Google doc in which they compile all the questions and answers.

This is a pattern that goes on for every SAT, both domestic and international. The kids clearly are using technology during the test. They acknowledge storing answers on their calculators, but don’t explain what allows them to remember all the sentence completions, reading questions and even whole passages verbatim, much less post their entire essay online. Presumably, they are using their phones to capture the images?

They create a google doc, in which they recreate as many of the questions as can be remembered (in many cases, all) and then they chew over the answers. By the end of the collaboration, they have largely recreated the test. They used to post links to openly with any request. But recently the College Confidential moderators, aware that their site is being exposed as a cheating venue, have cracked down on requests for the link, while banning anyone who links to the document.

So floating out there somewhere in the Internet are copies of the actual test, which many hagwons put out (and pull them down because hey, no sense letting people have them for free), as well as the results of concentrated braindumping by hundreds of testers.

For international students, “studying for the SAT” doesn’t mean increasing math and vocabulary skills, but rather memorizing the answers of as many tests as possible.

And those are just the kids that aren’t paying for the answers.

The wealthy but not super-rich parents who want a more structured approach pay cram schools–be they hagwons, jukus or buxiban–to provide kids with all the recycled tests and memorize every question. No, not learn the subject. Memorize. As described here, cram schools provide a “key king”, a compilation of all the answer sequences for sections, using all the potential international tests. They know which ones will be recycled because the CB “withholds” these tests.

Of course, the super-rich parents don’t want to fuss their kids with all that memorizing. Cram schools have obtained copies of all the potential international tests by paying testers to photograph them. Then they pay someone to take the SAT in the earliest time zone for the International, and disseminate the news via text to all the testers. They just copy the answers from the pictures. Using phones. Which they have told the proctors they don’t have, of course.

I don’t know exactly how all this works—for example, are the cram schools offering tiered pricing for key kings vs. phoned in answers? Do different cram schools have different offerings? I’ve read through the documented process provided by Bob Schaeffer of FairTest (a guy I don’t often agree with), and it seems very credible. He’s also provided a transcript of an offer to provide answers to the test. Valerie Strauss got on the record accounts of this process from two international administrators, Ffiona Rees and Joachim Ekstrom.

Every so often Alexander Russo complains that Valerie Strauss shouldn’t do straight education reporting, given her open advocacy against reform.

Great. So where’s all the other hard reporting on this topic? The New York Times, whose public editor Margaret Sullivan just encouraged to “to enlighten citizens, hold powerful people and institutions accountable and maybe even make the world a better place”, bleeds for the poor Korean and Chinese testers anxious for their scores and concerned they’ll be tarred with the same brush. Everyone else just spits out the College Board press release–if they mention it at all. While most news outlets reported the October cancellation, few other than Strauss reported that the November and December international tests scores were delayed as well.

At the same time Strauss reported the College Board is stonewalling any inquiries as to how many kids were cheating, how many scores were cancelled, or what it was doing to prevent further corruption, an actual Post “reporter”, Anna Fifield, regurgitates a promotional ad for a Korean SAT equivalent coach.*

Well, you can understand why. The millionaire Korean test prep coach-called-a-teacher story is one of the woefully underreported stories of the 21st century. I mean, we only had one promo put out by the Wall Street Journal the year before, and another glowing testimonial CBS a few months later (even mentioning the tops in performance, bottom in happiness poll). But really, only one or two a year of these stories have been coming out since 2005.

So you can see why the Post felt another story on a Korean test prep instructor making millions required immediate exposure, if not anything approaching investigation or reporting.

These stories are catnip to reporters who get all their education facts from The Big Book Of Middlebrow Education Shibboleths. First, unlike our cookie cutter teacher tenure system, Korean teachers work in a real meritocracy where kids and their parents reward excellence with cash. Take that, teachers!

Then, unlike American moms and dads, Korean parents care about their kids and put billions into their education. Take that, parents!

And oy, the faith Anna shows in her subjects. Cha is a “top-ranked math teacher” who “says” he earns a “cool $8 million last year.” Cha says he’s been teaching for 20 years, but refuses to give his age and there’s no mention of the topic or school he attended for his PhD, or if he ever got one. But he’s got a really popular video, so he must be great!

Some outlets are less adulatory. The Financial Times points out that the Korean government is cracking down on hagwon fees and operating hours, and preventing them from pre-teaching topics. Megastudy, the company in the 2005 story linked in above, just went up for sale because of those government changes. Michael Horn of the Christiansen Institute is doing no small part to alert people to the madness of the Korean system. The New York Times, despite its tears for the Korean and Chinese testers, has done its fair share to report on the endemic cheating in Chinese college applications.

But when it comes to the College Board and the SAT, everyone seems to be hands off the international market. At what point will it occur to reporters to seriously investigate whether a large chunk of the money spent on cram schools is not for instruction, but for “prior knowledge” cheating? When will they ask the Korean cram school instructors if they are fronts for an organized criminal conspiracy, if the money they get is not for tutoring, but for efficient delivery of test answers on test day? And how many of those test days are run by the College Board?

People think “well, sure, there’s some cheating, but so what? Some kids cheat.” Yeah, like I’d be writing this if it were a few dozen, or even a few hundred kids. Asian immigrants cheating on major tests in this country is in the high hundreds a year. Maybe more. In China and Korea? I suspect it’s beyond our comprehension, us ethical ‘murricans.

One of the depressing things about the past three years is that I start looking into things more closely. I never really trusted the media, mind you, but I did assume that journalists skewed stories because of bias. I fondly imagined, silly me, that journalists wanted to investigate real wrongdoing. Yes. Laugh at my foolish innocence.

Consider what would be disrupted if public American pressure forced the College Board to end endemic international student cheating. First, the CB would lose millions but weep no tears, it’s a non-profit company. hahahahah! Yeah, that makes me laugh, too.

But public universities increasingly rely on international student fees and the pretense that they are qualified to do college work. After all, the thinking goes, we accept a lot of Americans who aren’t prepared for college work—may as well take in some kids who pay full freight. Private schools, too, appreciate the well-heeled Chinese students who don’t expect tuition discounts.

So suppose public pressure forces the College Board to use brand new tests for the overseas market, require all international testing to be done at US international schools, use different tests at different locations. The College Board might decide that the international market profits weren’t worth the hassle for other than US students living abroad (as indeed, the ACT seems to have done for years). Either way, a crackdown on testing security would seriously compromise Chinese and Korean students’ ability to lie about their college readiness and English skills.

A wide swath of public universities would either have to forego those delightful international fees or simply waive the SAT requirement, but without those inflated test scores it will be tough to justify letting in these kids over the huge chunk of white and Asian Americans who are actually qualified. No foreign students, more begging for money from state legislatures. Private universities would have a difficult time bragging about their elite international students without the SAT scores to back thing up.

Plus, hell, we changed the source country for zombies because we didn’t want to piss off China. Three years ago, the College Board wanted to open up mainland China as a market. 95% of the SAT testers in Hong Kong are Chinese. Stop all that money flowing around? People are going to be annoyed.

At this point, I start to feel too conspiratorial, and go back to figuring that reporters just don’t care. I’ve got a lot of respect for education policy reporters—the Edweek reporters are excellent on most topics—and most reporters do a good job some of the time.

But the SAT is basically corrupt in the international market. I’ve already written about test and grade corruption among recent Asian immigrants over here, particularly in regards to the Advanced Placement tests and grades.

Yet no one seems to really care. Sure, people disapprove of the SAT, but for all the wrong reasons: it’s racist, it’s nothing more than an income test, it reinforces privilege, it has no relationship to actual ability. None of these proffered reasons for hating the SAT have any relationship to reality. But that the SAT is this huge money funnel, taking money from states and parents and shoveling it directly or indirectly into the College Board, universities, and the companies who have essentially broken the test? Eh. Whatever.

The people who are hurt by this: middle and lower middle class whites and Asian Americans. So naturally, who gives a damn?

enlighten citizens, hold powerful people and institutions accountable and maybe even make the world a better place

Sigh. Happy New Year.

*In the comments, an actual SAT prep coach making millions–no, really, he assures us, millions!–simply by being a fabulous coach with stupendous methods is insulted that I insinuated that the Washington Post story was on an SAT prep coach, rather than the Korean equivalent of the SAT. I knew that, but at one point referred to the guy as a SAT prep coach. I fixed the text.


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