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:
   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.”

A mom 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 dad nodded.

Original dad: “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 the first mom.

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

I smiled at his 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 dad’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.

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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.