Sunday, December 8, 2013

Want to make math interesting? Try this!

This week's mandatory hell-in-a-handbasket piece is the New York Times's "Who Says Math Has to Be Boring?" gives several clear, cogent recommendations for improving math education in the United States:  move to curricular models more flexible than the traditional Algebra-Geometry-Algebra/Trig-Precalculus sequence; improve math teacher preparation in mathematics (now there's a chicken-and-egg problem if I ever saw one); expose kids to numbers (not just numerals) and numerical relationships earlier, before first grade if possible; give students exposure to math in the real world.  But despite the title--and the early admonishment that "Finding ways to make math and science exciting for students who are in the middle of the pack could have a profound effect on their futures," it leaves off what is to me, the most important point. Here it is: to make math interesting, do interesting math, what I would call "actual math".

Now, I'm not taking sides in the perennial "pure-versus-applied" debate, although Dan Meyer's blog last week includes the important point that the new-traditional view "kids are interested by applications, not abstractions" is simply false:

Traditional (false) view of the relationship
between engagement and abstraction
Actual relationship between
engagement and abstraction

What makes an activity engaging isn't how "real-world" it purports to be.  It's a combination of factors, including:
  • Challenge level
  • Immediacy and quality of feedback
  • Stakes--low but nonzero is better, especially in the short-term
  • Visibility of progress towards ultimate mastery
  • Novelty
  • Familiarity
(Yes, I did mean both of the last.  Things that are completely new are often off-putting; things that are the same every time are drudgery.  So you need to provide novelty within familiar frameworks, or something like that ... but that's another blog post.)

Math that's been reduced to algorithms for students to memorize and apply with perfect precision on tests whose scores "will follow you for the rest of your life" has none of these qualities.  So of course it's not interesting.

What is interesting is thinking about problems, trying to carve them into meaningful abstractions, attacking them again and again with slight gains, etc.  Doing problems that seem familiar, but with a twist: this time I'm not asking for the hypotenuse given two legs, but for a leg; or for a right triangle with integer sides and a particular leg; or for the dimensions of a rectangle, given its area and the fact that its sides and diagonal are all integers ....  Or this time I'm wondering:  suppose I approximate the square root of 2 as 1.414.  How bad an error will I get for the hypotenuse of a 10m-10m right triangle?  100m-100m?  1000m-1000m?  If it takes a minute to walk 100m, how long do the legs have to be before the error is five minutes of walking?  

All those are abstract problems, with varying degrees of "real-world" relevance.  I'm not claiming they're great questions--I just came up with most of them right now--but they avoid most of the pitfalls above, and if done in class, in a problem-solving environment (i.e. where the kids have been exposed to the fact that math problems don't always yield on the first attempt, several times), would be a heck of a lot better than a page full of Pythagorean theorem practice problems.  Some of them lead to deep questions:  are there numbers that can't be sidelengths of a right triangle whose other sides are integers?  Are there numbers that can only appear once as members of pythagorean triples?

Math is at its heart a process of abstracting and connecting ideas through logic and generalization.  We worry that most kids can't do these things, and so we often do our best to avoid engaging in such practices overtly, or even requiring kids to engage in them at all.  But that's exactly the wrong approach.  Engaging kids in the process of actually abstracting, connecting, reasoning, generalizing, conjecturing, and applying--rather than just practicing spitting out formulas--now that would make math class interesting.  It would make math class about math.

Friday, November 29, 2013

"Irresponsible Teenagers": an oxymoron?

Dubois then turned to me. "I told you that `juvenile delinquent' is a contradiction in terms. `Delinquent' means `failing in duty.' But duty is an adult virtue -- indeed a juvenile becomes an adult when, and only when, he acquires a knowledge of duty and embraces it as dearer than the self-love he was born with. There never was, there cannot be a `juvenile delinquent.'" Robert A. Heinlein, Starship Troopers
I hear a lot from my colleagues about making teenagers be responsible, and indeed I think that's a really important (primary?) goal of high school. But the thing we as teachers often fail to realize is that teenagers aren't responsible.  They're not really capable of planning ahead long-term, they often make poor decisions for reasons that, more and more, we understand as weaknesses in brain development, a mismatch between the complexity and long-term consequences of what kids can do on the one hand and their brain's inability to think through complex decisions with long-term consequences on the other.  (See here for one of many scholarly articles on the subject.)

So when we give kids a long-term project that we don't help them break down into pieces--and I think there's a huge distinction between handing it to them, all sliced into pieces, and walking through the planning process with them--or make passing or failing a single test a huge piece of their grades, or create any other single point of failure, we're really playing into the thing that we know they can't do.  And then when they don't do it, we call them irresponsible.

The thing is, if kids can't really be responsible (yet), they can't really be irresponsible, either.  It only makes sense to talk about irresponsibility in the context of something that we can reasonably expect someone to be responsible for.  Kids can be responsible for lots of things with short-term consequences, and they can be taught (helped) to see the connection between lots of short-term decisions and long-term consequences.  But those lessons are hard ones to learn, and the process often mirrors a saying attributed to Mark Twain:  "Good judgment comes from experience.  Experience mostly comes from bad judgement."  What that saying suggests is that we need to create and preserve opportunities for kids to fail safely and learn from those failures, rather than making those failures catastrophic.  And we need to be right there so that the kid can connect the dots between what he or she did or didn't do, and the negative consequences that resulted.

So often, when I hear a teacher talking about a kid's being irresponsible, I wonder two things:

  1. What did you, as the responsible adult, do to bring about this situation?  More importantly, what did you, as the responsible adult, do to avert it?
  2. What could we have reasonably expected this child to do in this situation?  Why didn't he or she do the "responsible" thing?
#2 is shorthand for my exasperated "Of COURSE he was irresponsible -- he's a child!" But it doesn't really help kids to throw around this moralistic label -- it only makes them feel cruddy about things that happened in the past, rather than accepting the consequences and doing better in the future.

One last pitfall.  We all know kids who are remarkably responsible--who manage to pull it together and keep it together despite an insane array of pressures, conflicts, and demands.  But that phenomenon is just the end of a bell curve of development and personal characteristics.  We all know--or know of--six-foot-two seventh graders, or freshmen taking Calculus.  Yet we don't hold those kids up as examples against which other kids are judged.  No matter how much we wish that kids were more "responsible" than they often are, blaming them for being irresponsible -- especially ordinary-kid-kind-of-irresponsible -- isn't any more reasonable (or responsible) than blaming them for their height, or for "only" being in Algebra.

Tuesday, November 19, 2013

Objectives and Experiences

And we turn him into an anecdote, to dine out on, like we're doing right now. But it was an experience. I will not turn him into an anecdote. How do we keep what happens to us? How do we fit it into life without turning it into an anecdote, with no teeth, and a punch line you'll mouth over and over, years to come: "Oh, that reminds me of the time that impostor came into our lives. Oh, tell the one about that boy." And we become these human jukeboxes, spilling out these anecdotes. But it was an experience. How do we keep the experience?
 John Guare, Six Degrees of Separation
When I was a starting teacher, it was as much as I could do to articulate what I wanted my kids to know at the end of the day, much less at the end of the week or month.  A key development for teachers in transitioning from that beginner level to something like "proficient" is learning to anticipate what's needed for instruction over the next week, month, and year: for example, knowing that you have to teach a bunch of chunking in Algebra II and Precalculus so that students can do integration by substitution in Calculus.

That's where standards come in: they tell you what you have to do in each year so that kids can go on and do what they need to do the next year.  Whether you're using the Common Core standards, Next Generation Science Standards, or an in-house list, this shift in perspective -- from "What am I going to do today?" to "What do kids need to learn?" is crucial if you're going to accomplish anything--and if your students are going to make any genuine progress.

But I've noticed that master teachers--like my co-blogger John, or my friend Doug O'Roark--ask a different question, not necessarily first, but early in the planning process.  This question is: "What experience do I want kids to have?"  I find that this question more than any other has changed my perspective about planning for classes.

What experience do I want kids to have?  Is the goal to give them the experience of discovering something?  Of exploring in a rich "sandbox" of cool math ideas, regardless of what they wind up conjecturing and proving?  Of solving involved numerical problems?  Of developing a set of ideas to describe a new situation?  Of applying old ideas to solve a new problem?  Or ... ?

Attending to the quality of my students' experience rather than simply on what I want them to learn leads me to new ideas--that, even in a mostly-remedial algebra class, it's important to have fun.  (The way Doug and I did this was to do magic tricks with Algebra.)  And it puts common teaching pitfalls into perspective. I mean, who would answer the question "What experience do you want the kids to have today?" with "I want them to watch a powerpoint for 30 minutes"?  And who would want kids to have the exact same experience, every day, for 180 days?

I think that one way to understand the Standards for Mathematical Practice section of the CCSS is very much in this vein:  they describe and, to a certain extent, prescribe the kinds of experiences we want kids to have while they are learning the content in the other standards.  For example, take SMP-1, "Make sense of problems and persevere in solving them."  The "standard" doesn't describe a set level of perseverance that kids are supposed to attain, or even clearly define what "making sense" of a problem is.  But it suggests that kids ought to be experiencing problems that are ill-defined, or at least initially resistant to mathematical analysis, and that these experiences should include trying more than one approach before being successful.  Thinking of the SMP in this "experience" way helps me reconcile the essentially-fuzzy nature of those standards to the others, and also helps me think about how to mesh the two: the point isn't to do one kind of standard and then the other, but to approach one (content) standard in the mode of one or more of the others.

And that "or more" leads my musings to a caveat.  At its best, mathematical experience is rich: a great class is one in which kids are spotting patterns, making conjectures, trying things out, having fruitful errors, using various technologies (from compasses to computers), all woven together into great mathematics.  It's a rich story.  What the CC-SMP provide us is more like a set of anecdotes.  Their list isn't exhaustive, and it isn't supposed to be exhaustive--but that's not my problem.  My problem is rather that boiling down one of these terrific class days into a set of three or four practice standards is, as Guare's character Ouisa warns us, turning an experience into a set of anecdotes.  When we do that, we lose--the math loses--something essential: by being just explorers, or cross-examiners, or number-crunchers, we stop being the rich, mathematical people we were when the class was going on--and the math stops being, well, mathematics.

Sunday, November 10, 2013

What do we learn from our students?

It's been a while--partly because of work, and partly because I just found out about the death two summers ago of one of my former students, tragically in the course of mourning the (more-recent) death of another former student.

I've been hesitant to write about him, but I haven't been able to write not about him either, so here goes:

Like many of my colleagues and friends, I went into teaching in large part to "make a difference" in the lives of young people.  And I have, but in some cases, I have to admit that the difference isn't necessarily the difference I intended to make.  This young man in particular started ninth grade as an angry, somewhat-alienated wannabe skate punk; I say "wannabe" because although he was an accomplished skater, he hadn't quite worked out the "punk" part besides just being angry a lot.  Something about him spoke to me: I wanted to be "that teacher" for him, the teacher who got what he was about, who saw what amazing, exciting gifts he had to offer, who helped him mediate between his anger and desires and the sometimes-irrational (and always irrational-seeming) system in which he lived and learned, who he'd talk about fondly as "the only reason I stayed in school".  But when I was done intervening, he had become an extremely angry, completely alienated fifteen-year-old on his way to dropping out of high school (which I believe he did two years later, shortly after moving to another city).  His final exam in my class was covered with obscenities.

I was "that teacher," all right.

I've gotten better about such interventions, and I share this "wisdom" so far as I can put it into words--although I think that the experiential way in which I learned is probably the only way to learn.

First and most important, I've learned that you can't make students trust you.  You can act to earn their trust, by being trustworthy in your actions, and by reminding them that you're there.  But all you can do is open the door.  To put in terms of my favorite joke:  it only takes one teacher to change the lightbulb, but it has to want to be changed.

On a more pragmatic front, there's a specific mistake I've decided not to make again: while I'm willing (indeed, in some sense, happy) to bust students with whom I've forged close relationships, I won't jeopardize those relationships by asking them to turn in their friends.  In fact, I've decided that that question--"who else was with you?"--is just not fair, unless it's literally a matter of life-and-death.  And in that case, I'd rather convince my student of the life-and-deathness of the situation rather than simply use my personal leverage to get the answer out of him.

Third (and my current or recent students might laugh at this), I've learned to use a lighter touch.  I'm not naturally subtle, and I've had to realize that anything I say -- in particular anything negative -- is effectively amplified many, many times (maybe I should call this the "multiplier affect"?).  Most of being "that teacher" is really about listening, and waiting, rather than talking.  And when talking, it's hard to underestimate the importance of being positive, positive, positive:  not untruthful, not unrealistic, but as relentlessly positive as possible given those two constraints.  Remember how insecure and terrified you were as a teenager?  That's what I'm talking about.

As teachers, much of what we teach is propositional--facts and ideas that can be put into words easily.  Much of the important stuff isn't so propositional--for example, how to approach a math problem, or ways to analyze a text--and arguably the most important stuff is the stuff we don't even think of as part of the curriculum.  (Ted Sizer's excellent book The Students are Watching is all about this last part.)  But when we think about "learning from students," I think we sometimes default to the propositional mode.  Sure, I'll never forget Dan's incredulity that I--who knew way more than he did about math--didn't know that hogs have to be walked.  And I learned a fact: that if you're raising hogs, you have to walk them.  But I also learned something much more important:  that students whose knowledge in one area is only a small subset of your own can be experts in areas about which you're totally ignorant.

I think the most important part of what I've learned from my students--slowly, painfully, extremely imperfectly--has to do with how to be more like the person I wish I could be when I'm teaching them.  And most of that's been learned the hard way.

For the record, after a couple of years of wandering in the desert--I'm reminded of Tolkien's line that "not all who wander are lost"--Constantine apparently found his way and what he was about.  Friends of his friends tell me that his last years were good ones, where his creativity and energy were valued and celebrated by the people around him.  I wish I'd been a part of that, of course, but even more, I wish I'd been able to see it for myself.  I'm sorry I couldn't be that teacher.

Sunday, October 6, 2013

The Homework Paradox

Robert Pondiscio's recent article in The Atlantic argues that while upper middle-class "gifted" kids may not need homework, for children in poor or less-well-educated households, homework can be an invaluable source of intellectual stimulation.  He's responding to people who--like me--wonder why our children's reading ability grows far more in two months of summer vacation than in nine months of education (the answer: she read something like a book a day when she didn't have homework to do).  These observations lead well-meaning liberals--like myself--to question the value of giving any homework at all.  As Pondiscio rightly points out, it's not fair or reasonable to impose on other people's children conditions (or, in this case, lack of conditions) simply because those conditions make sense for our own children.  He's right about that, and also when he carefully argues
The proper debate about homework – now and always – should not be “how much” but “what kind” and “what for?”  Using homework merely to cover material there was no time for in class is less helpful, for example, than “distributed practice”: reinforcing and reviewing essential skills and knowledge teachers want students to perfect or keep in long-term memory.  Independent reading is also important.  There are many more rare and unique words even in relatively simple texts than in the conversation of college graduates.  Reading widely and with stamina is an important way to build verbal proficiency and background knowledge, important keys to mature reading comprehension.
But I still worry about homework as an educational prescription for the poor, for a few reasons.

  1. First, I think the same factors that disadvantage the children Pondiscio says need high-quality homework also make it less likely that they will get it.  These children are less likely to have access to the high-quality teachers who assign the thoughtful and thought-provoking tasks that Pondiscio praises.  (For a prime example, go back to the Summer Homework Remix Challenge, or simply look at the books full of repetitive worksheets textbook publishers use to sell their series.) And they're less likely to have the academic and intellectual support at home to complete those tasks: parents who understand what a genuine science experiment is, or how to think about a complex text.  This isn't racism, or classism, but simple logic:  if the problem that homework is supposed to fix is inadequate intellectual stimulation at home, why would we expect those homes to provide adequate intellectual support for challenging tasks?
  2. Second, especially in high school, kids from poor backgrounds are likely to have circumstances that make it hard to get homework done.  Many of my economically disadvantaged students babysit for younger siblings so that their parent(s) can work, or for nieces and nephews whose own parents' income is essential for the functioning of the wider family.  Others work after school--not for money to blow on an iPad, but for essentials like winter coats or even the family rent.  Even those who don't have these responsibilities often lack the most basic requirement: a calm, reasonably quiet place to work.  So asking these students to do large quantities of homework isn't always reasonable.
  3. Items #1 and #2 point to a perverse Matthew Effect of the exact sort that Pondiscio wants homework to overcome:  poor students are often in circumstances that make it harder to get homework done, less likely that they'll get what they're supposed to out of homework, and more likely that they'll find homework just another unreasonable demand of an already-harsh educational system.  At my school, inability to get homework done well is often one of the leading causes--and symptoms--of a student's failure to make a real go of it at all.  So relying on homework widens the gap between the educationally/economically advantaged and the disadvantaged.
  4. Finally, I'll admit it: I'm a skeptic.  Not about homework, but about middle-class arguments that essentially boil down to saying "The poor need this, but not my children."  We've done a pretty crappy job educating other people's children differently from our own, and I guess my tendency is to err--pretty far--on the other side: if I wouldn't want a school, teacher, rule, or homework policy for my child, I'm hesitant to recommend it to anyone else.
Pondiscio raises good points, and I'm not--really, I'm not--saying that we should get rid of homework for everyone.  But I am wary of saying that homework is likely to solve the problem of socio-educational inequality, especially before we have a coherent way of ensuring that the students homework is supposed to help have equal access to the kinds of high-quality assignments, and homework support structures, that we'd want our own kids to have.

Sunday, September 22, 2013

Department Chairperson



Frank May hired me in 1969 to teach at Evanston Township High School. Frank was my supervisor for the first part of my career, observed my classes, advised me, taught me, encouraged me and was a role model for what mathematics teachers could accomplish.
It would be hard to find two people more different than Frank and I. Frank never raised his voice; I never lowered my voice. I continually tried to find different ways to bring mathematics to students; Frank stood at the board and lectured. Frank was careful and meticulous. No one has ever described me in those terms. Frank May was a master teacher, and I was lucky to have him as my mentor and advisor.
Frank taught me to pay attention to the details. In particular, he taught me to write complete mathematical sentences using equal signs, stating conclusions;. he took care to ensure students understood the underlying reasons for procedures and notation. He taught me the importance of using correct notation as well as language in the classroom. To this day, I can’t bring myself to describe congruent shapes as being equal.
It often strikes me, how much I learned from someone with a dramatically different personality and point of view. Part of it is that I always respected his knowledge, and part of it is that I am uncontrollably drawn to people who love mathematics.
His evaluation meetings after observing my class were more like math lessons than like criticisms of what I had done. It was clear that he had an abiding love for mathematics as well as a deep understanding of the subject and how students learned it. His quiet, soft-spoken manner allowed his beliefs and understandings to gradually sink into my brain without me actually realizing the impact he was having on my teaching.
He had a deep understanding of how topics related to each other and of why some things were difficult for students to understand. I often went to him when I had a difficult topic to teach, and he always started by agreeing with me that it was a difficult topic to teach and then uncovered two or three insights that helped me understand why students had trouble. He never told me what I should do, nor did he tell me what he always did. Instead, he made careful observations about why students found the particular topic difficult and sometimes made a suggestion or two as to what might be done to help them. He then let me figure out how to overcome student difficulties after he shed light on the nature of those difficulties. In short, he used exemplary teaching techniques to teach me how to become a better teacher. He would always check back to see if what I had done had worked better, and again he would offer an insight or two that would help me fine tune my approach, but he never insisted that I do it his way.
   At some point in my career I was asked to teach B.C. Calculus. It had been many years since I studied calculus, and I was not sure I ever really understood series. I touched base with Frank about one thing or another almost every day. He single handedly taught me how to teach Calculus.
School bureaucracy being what it is, Evanston eliminated the position of Department supervisor. Many of his fellow administrators retired or moved to other institutions. Only one went back to full time teaching, Frank.  He could have retired, but he wasn’t ready to stop teaching. He could have taken a supervisory position at a different school. Frank went back to the classroom and taught five classes a day for another ten or so years. I think he looked at the situation and determined what would be best for the students, for the school, and for him, and gracefully took a step down and finished his career doing what he did best: teaching students mathematics.  Of course he did more than that: he continued to mentor many of us even though it was no longer in his job description. He was still the person I went to when I needed math help, and he was still eager to talk math.
While he was a master teacher, he never boasted about his accomplishments, but he came close once. I complemented him on the incredibly high scores of his BC Calculus students one year. He told me that he had had the same group for pre-calc two years in a row and so was able to prepare them appropriately. I often wondered how successful students would have been if they had him for four years, or if they had someone as good as him for four years.

The reason I am writing this now is that Frank May passed away last week at the age of 89. There was no mention of his passing in the front page of The New York Times or The Chicago Tribune, because he was not a famous athlete, entertainer, or politician. All he did was  positively influence thousands of students and hundreds of teachers. Rarely a week passes when I don’t reflect on one of Frank’s insights about how people learn. He is the primary reason I have reverence for the Parallel Postulate, the Distributive Property of Multiplication over Addition, and the Differential. I learned from Frank that one can exhibit enthusiasm in a quiet, dignified manner, that it is not necessary to jump up on tables and throw things across the room to share with students how exciting mathematics is. Let this be one small instance of tribute to a great man. Thank you, Frank, for all you have done for our profession, and therefore for countless people. I am forever grateful for all that you did for me. 

Wednesday, September 4, 2013

My daughter's good, but don't call her smart!

This story on NPR reminded me -- again -- of how much it bothers me when we call kids smart.  In fact, calling successful kids smart is one of the worst things we can do: to (and for) them, and to (and for) other kids.

Many years ago I decided to stop using the word "smart" to describe my students, on the grounds that the word "smart" is so imprecise, using it is just an excuse for sloppy thinking.  There are a bunch of different ways a student can be "smart":
  • Catches on to new ideas and techniques quickly.
  • Doesn't forget things he or she has learned, even a long time ago.
  • Anticipates consequences and implications of new ideas and issues.
  • Sees generalizations; synthesizes readily.
  • Sees new applications for already-learned facts and skills.
  • Makes few, if any, mistakes in applying already-learned knowledge and skills.
  • Generates new, unusual ideas.
  • Is intellectually playful: likes wordplay, quasi-argumentative banter, hypotheticals.
After a year or two I backed off, but I still use lists like this when I'm writing college and scholarship recommendations: it doesn't help my students if my praise is essentially meaningless.

The NPR story took a different tack.  It compared Western and Asian parents' comments to their children, both when their children are successful, and when their children are unsuccessful.  This ground has been trodden many times -- notably by Po Bronson and Ashley Merriman, in articles and in their awesome book Nurtureshock.  I'll summarize typical comments in the chart below:


Western Parent
Asian Parent
Successful Child
“You’re so smart!”
“You worked so hard!”
Unsuccessful Child
“I wasn’t good at math either!”
“You must work harder!”

There are two major takeaways from an educational-policy perspective:
  • Teaching kids that success is a result of being smart sets them up for failure.  When these kids encounter genuine struggle, they often conclude that they are simply not talented enough to be successful.
  • Teaching kids that success is a result of working hard sets them up for further success, because attributing their success to the only factor that an individual can control (as opposed to talent, luck, and ease of task).
But I'd add a third issue--one that I've encountered with my students and my own child, and especially with talented girls who work hard.  These students are proud of their hard work, and when someone says "You're so smart" or "You're a genius," they don't experience those comments as praise.  Instead, they feel that their hard work has been devalued, because they've just been told, "Look, you're not successful because of anything you chose to do -- you're successful because you were born that way."  (The variation on this that drives my daughter bananas is "Of course you're good at math--your dad's a math teacher"--which is why she never lets me actually teach her math.)

So telling successful kids they're smart is bad for at least five reasons:
  1. It tells successful kids that they shouldn't get credit for their success, because that success isn't due to anything they actually did. So it's insulting.
  2. It sets successful kids up for failure, because it doesn't give them anything to fall back on when they encounter challenging tasks.
  3. It tells unsuccessful kids that they can't do anything to become successful, because the successful kids are the ones who are already smart.  So it's implicitly insulting to unsuccessful kids:  you're not doing well because you're dumb.
  4. It sets unsuccessful kids up for continued failure, because--in this worldview--there's nothing they can do to become smarter.
  5. It's an inexcusably-sloppy way for a teacher to describe a student, because it doesn't say anything about what the student actually does.
So please do recognize your successful students' (and children's) achievements--just don't call them smart.

Tuesday, August 27, 2013

Thoughts on Collaboration

It's the first week of school, and I put in some time rewriting my course policies to articulate why, when, and how to collaborate on problem sets.  This issue is particularly crucial in my advanced Geometry class, where students solve college-level problems, not just exercises. Feel free to use some version of this in your own classes, and -- even more important -- to let me know what you think!
------------------

Collaboration, Research, and Hard Problems

Contrary to stereotype, mathematics is best done as part of a community—not alone in a study.  I hope and expect that students will frequently discuss problems, ideas, and solutions in and out of class.  Some guidelines:

·         Make sure that when you are “working with” someone else, you are both really working and contributing.  Contributions can take many forms—clarifying, questioning, justifying, and restating, to name a few—not just coming up with “the idea”, but each of you should leave the collaboration feeling good about what you, and the other person, contributed to the session.

·         One test for how much you did in solving a problem is whether you can reconstruct the entire solution on your own afterwards.  If you need to look at your notes extensively, or get lost in the middle, you probably should have collaborated more actively.

·         One time when it is useful to be alone in your study is to check for your own understanding.  I recommend that students start problem sets by themselves, to see what they can accomplish and what ideas they can generate individually before working with other students.  I also recommend that students finish and write up problem sets by themselves, to make sure that they really understand the work they and their fellow students did together.

·         Give credit generously: it doesn’t subtract from the points you get (and in the real world of “karma points”, giving credit almost always adds to your own).  Write “Wolfgang suggested this auxiliary line” or “Credit to Seraphina for spotting the similar triangles.”  Taking ideas from other people without attribution is plagiarism; taking ideas with attribution is research.


·         Finally, unless specified in a particular project or assignment instruction, please DO NOT do research on the web (or in books, if you still remember those).  The essence of this course is learning to reason mathematically by solving problems and thinking through solutions, not regurgitating theorems and ideas you learned somewhere else.    

Monday, August 12, 2013

Summer Homework Remix Challenge

So I'm just getting back from three weeks teaching at HCSSiM (which in case you don't know about it, is an absolutely awesome summer math program for high school students), and one thing that really leapt out at me was the number of students trying to cram in several hours of summer homework on top of (literally) 45+ hours of mathematics each week.  (You read that right: four hours of class each morning, quasi-optional lecture at 5pm, three hours of problem session from 7:30-10:30pm.)

This situation is appalling.

First, the kinds of high-performing kids who go to HCSSiM are working harder than ever during the year; they need a break from regular work.  Second, the work that gets sent home is--in general--the worst sort of homework you can imagine.  A sampling:

  • Outline n chapters of a bio/history/government textbook, where n ≥ 3. My experience is that few kids, if any, are taught how to outline, so these wind up being lists of section headings.  Teachers typically don't even read or give feedback on this work.  I wonder, too: if a kid can get "enough" out of this type of learning experience, what do teachers think the are adding in the actual classroom?
  • Write out, by hand, 100 vocabulary words and definitions.  (My colleague and friend Erica, soon to be a teacher in a Massachusetts middle school, asked "If the work is so low-level that it's impossible to tell whether one kid has copied another's assignment, why would you even assign it?"
  • Read a 400 page, mostly stream-of-consciousness novel chosen by the teacher, with little or no guidance.
  • Fill out dozens of worksheets practicing math facts, or vocabulary from a foreign langauge, or ...

Assignments like these replicate the worst parts of the school experience--repetitive, low-level work with little attention to context or purpose besides "get it done"--without any of the other experiences that can make going to school worthwhile.  What's more, they put this school-sucked-dry experience into the summer, which kids typically enjoy.

Why do kids who enjoy learning prefer summer to school?  I can think of three reasons:

  1. During the summer, kids can choose what they will do and when they can do it.

    Of course, choice is more satisfying than constraint.  But study after study has shown that giving kids choices about what learning activities they do increases student engagement and the activities' effectiveness.  So taking the choice out of summer learning activities is a double-whammy.
  2. During the summer, kids do things that they find relevant.

    In actual summer academic programs--like the ones my kids and my friends' kids do--the learning activities are focused on things kids actually want to know about.  And kids don't choose to do things that they find irrelevant.
  3. During the summer, kids do things that are challenging.

    Have you seen a kid spend hours practicing a skateboard move, or throwing a football, or playing a videogame?  Kids don't do things they find easy--they find things that are at a "can't-quite-do-it-yet" level, and when they've mastered something, they move on.  
By contrast, the summer homework assignments listed above involve no choice, make little attempt to be relevant to kids' own interests, current issues, or anything else of interest, and are very much one-size-fits-all (in fact, because no teacher is available, they're usually way too easy--which reinforces the idea that they're mostly "get-it-done" work, not an opportunity to actually learn anything).

So here's my remix challenge, in three parts.

I.  Find an actual summer homework assignment passed out to kids in grades 6-12.
II.  Rework it to fit the three criteria--choice, relevance, adaptable challenge.
III. Post it here, in the comments, or via email to me at pjkarafiol@gmail.com.

Three guidelines:

  • Obviously, the made-over assignment must address many of the same objectives and issues as the original.
  • Obviously, the assignment must be one that students can complete, with some reasonable degree of success, on their own.  The assignment doesn't have to be easy, but (e.g.) feedback might be built-in or easy to obtain.
  • Third, the assignment shouldn't take more than 4-5 hours of work, 8 tops.  Really, guys, this is the summer.  If you want the kids to attend summer school, teach an actual class.

As extra credit, ask a student to suggest a summer homework "makeover" of their own.

Here are two to get you going:

1.  For the AP Biology Chapter 1 assignment on basic physics and chemistry: read two articles from one of the following periodicals (Scientific American, New York Times Science Tuesday, Discover Magazine) written within the last year about a discovery or problem in biology.  For each, identify what chemical or physical processes are described in the article, and be ready to give a short (3-5) minute presentation on the underlying chemistry or physics described.

2.  For the vocabulary list -- given the same list of words, find instances of 20 of these words in recent writing (last five years) on the web, in periodicals, or published books.  For each, give the surrounding paragraph, explain what the word means in context, and write a sentence or two evaluating whether the author should have chosen a less-esoteric word (with a suggestion).

Happy end of summer!

PS: Full Disclosure -- I havea a summer assignment of my own, but it involves choice, is not onerous, and shouldn't take more than 3-4 hours to complete.  Here it is:  http://www.wpcp.org/StudentLife/SummerAssignments.aspx

Monday, July 1, 2013

Quiet

I suggest adding the book Quiet by Susan Cain to your summer reading list.

I read it at the suggestion of a former student. It is well written, researched, and short.

It is not specifically about education but it is about assumptions our culture makes about quiet people. We all have quiet people in our classrooms. She challenged many of the assumptions that I had made about quiet people and forced me to rethink some of my long held beliefs about structuring my classes. She raises some interesting and profound ideas about things like group work, whole class instruction, brainstorming, problem solving, calling on students, and the role each student plays in the dynamic of the classroom.

I also found it insightful with regard to interactions with friends, neighbors and colleagues.

Thursday, June 27, 2013

The Four Most Important Words in Teaching

My co-blogger, John, often mentions that a crucial part of his practice is standing at the door, greeting students as they come in.  Although this practice started as a way to keep order in the halls, for him it's persisted because it gives him an opportunity to check in with students individually.  I don't have anything like that kind of discipline, but I have to agree that the 3-5 second "touch" is incredibly important, wherever and however you make it happen.  So my candidate for the four most important words is "How are you doing?"

As a math teacher, I don't "get" the opportunity to talk with my students about their personal lives; I have to make that opportunity.  But I think that the kids who most need to talk are often the most fearful of actually opening up; the biggest secrets are just below the surface.  "How are you doing?" is a low-stakes way of saying "I'm interested, and if you want to talk, we can."  When a student has already opened up to me, "how are you doing?" is really a statement: "I know it's been rough, and I'm concerned."  It doesn't demand an extended exchange.  "Not so great thanks" -- "I'm sorry; find me at lunch if you want to talk" is almost always as long as it gets: five to ten seconds.

It's easy to misjudge the amount of effort needed to care for our students' social and emotional health on the basis of that very small number of students whose drama is like a riptide, dragging in friend after friend and teacher after teacher.  But most students aren't like that.  And I find that especially when a student is in crisis, or just coming out of one, regularly asking "How are you doing?" makes a huge impact--probably more than an hour-long "session" would, at least with this nonskilled practitioner.

This impact was demonstrated to me by an unusual coincidence during this last week of school.  I said goodbye to two boys who had been going through rough times this year (one just for the summer, one who is graduating); both were practically in tears.  One wrote in my yearbook that my checking in with him had made it possible for him to finish school and graduate, and he meant it.  Literally--I promise you--no conversation with this student had lasted more than five minutes.  Then our graduation speaker was an alum who, five years ago, had told his entire class and their parents (he was the graduation speaker at his own graduation, too) that my "How are you?"'s had been a lifeline during difficult times.  (And again, at the time, I was bowled over:  none of these conversations lasted more than 3 minutes, and only a few were even half that long.)

When I first started teaching, I envisioned long, soulful one-on-ones with students about all the bumps and pitfalls of adolescence, of which I'd had my own share.  But I've come to realize that most students don't want those conversations most of the time, and that they can't be forced.  Two of my mentors, at Andover, pointed me in the right direction.  Craig Thorn (beloved house counselor and English department chair) told me his secret the day I arrived: just be around, in their rooms or wherever, so that they see you and know you and talk around you.  Doug Kuhlmann, who was the math department chair, said something like this: when you ask a student a personal question, you need to be aware of your own stake in the answer.  What he meant, roughly, was that when you ask a student a personal question, you need to be aware of your own reasons:  are you asking because talking will help the student, or for the emotional buzz of reinforcing your relationship with the student, or to validate your own self-image as "the teacher who cares." It's easy, he warned, to think of yourself as asking for the student's benefit, when really it's about you, which is problematic:  as an authority figure, you're in a position to demand a response, even when you shouldn't. 

"How are you doing?" largely avoids this pitfall: asked in the hallway, or when you're checking in homework, or outside the lunchroom, it doesn't demand much, if anything, of the student.  Just saying "Okay" is enough--in fact, it's standard protocol.  The first couple of times, I might follow up with "Really?  Doing okay?" -- to indicate that I am really asking, not just following the script.  But then it's up to the student.  So "How are you doing?" is empowering, not disempowering: it says "Remember, I'm here if you need me, but I'm not going to push it if you don't."  And it's a "touch" I can make in front of other students, who may or may not know the backstory.

With 140 students per teacher, it's easy to fret about how little we can do.  But what I take away is that it only takes a little.  The key is -- to circle back to John -- to keep asking, to make it a regular routine.  The kids who thank me for it later remember this:  I asked them every time.  And that regularity--and the care behind it--mean more than you'd expect.

Tuesday, June 11, 2013

Improvement

In my first several years of teaching, when things did not go as expected,  I thought about how to improve results. This [what] often meant adding another rule or expectation [to what], because my students were not doing what I perceived they needed to do in order to learn. After a few years, I had created a bureaucracy that was unmanageable for me as well as for my students. The worst part was that these rules and expectations had not helped improve instruction.

I came to realize that my job was not to tell students what to do. My job was also not to show them how to do a problem. My job was to create interesting situations where they could think about mathematics and learn from the work and discussions that followed. My job was to assist students in anyway I could to develop their own understandings of important mathematics.  I have documented these thoughts and processes many times in earlier posts. Here is a new thought for me.

This same process applies to teacher improvement. Those in charge of teachers at the local, state and national level are ill served by piling on more and more rules and mandates about how teachers should teach. If they really want to improve instruction, they must follow the same model that I followed in my classroom. The supervisors need to see that their job is to do whatever they can to facilitate teacher learning (therefor student learning), as opposed to requiring teachers to do certain things in a certain way. Teachers, with proper resources, will find ways to reach students. As things stand now, those in charge are working very hard to make a teacher's job as hard as possible.

Administrators need to observe teachers teaching. Administrators need to listen to what teachers have to say about the difficulties they have, and administrators need to work hard to help the teachers solve their problems. I have had to good fortune of working for a few such supervisors, and it makes more of a difference than I would have ever imagined. And the very best supervisors worked hard to help the top-level administrators understand that learning to teach well is a very difficult process, that it takes time, and that it requires support and nurturing. Learning to teach well does not require demands and punishment on the part of the Administration.

It is part of the job of experienced classroom teachers to facilitate this process with newcomers and to help administrators understand what they need to do to be effective.

The importance of Content knowledge

I was asked a while ago to write a blog for the National Council on Teacher Quality. I agreed and here it is. This is also posted on their website nctq.org/commentary/blog. This is re posted here with their permission.

During my forty-two years of teaching high school mathematics in Evanston, Illinois, I concluded that an essential ingredient for providing quality learning is that the teacher be well versed in the subject that the student is learning as well as the content that comes before and after the subject being learned. This may sound obvious, but it often happens that teachers have mastered what is in the textbook they are using without having knowledge far beyond. I believe this greatly inhibits their ability to help students make connections and often such teachers make poor choices about instruction because they fail to see the entire picture.

I taught a two semester Algebra 1 class, Empirical Geometry, Mathematics--A Human Endeavor, as well as Traditional Euclidean Geometry, Trig, Calculus, Multivariable Calculus and Linear Algebra. I found deep knowledge to be useful at all levels, all the time.
A teacher who has mastered the material well beyond the course being taught will understand why certain topics are presented the way they are and will anticipate what's next. A less prepared teacher will emphasize tricks and shortcuts that will get the students through Friday's test, but leave them ill-prepared for future courses.  For example, a student who learns to multiply binomials using FOIL (First, Outer, Inner, Last) instead of the distributive property of multiplication over addition may do well on the problems involving multiplication of two binomials, but will be hopelessly confused when multiplying more than two, or when one of the factors is a trinomial.
Part of good teaching involves understanding the importance of what is being taught and how it can be applied. Sometimes application of the content does not come until the student studies physics, or calculus, but a teacher who is not well versed in those subjects will not understand their importance. For instance, a teacher who is not familiar with Linear Algebra will not understand the importance of row-reduction of matrices and probably will not present it as the tool of choice for solving systems. In fact, many of those teachers will never ask their students to solve two equations with three unknowns because they do not see the big picture, limiting their students.
I have also observed that students can have remarkable insights into the subject at hand, but those insights may not be well formed. A teacher with deep content knowledge will be able to see the gem the student has noticed and clarify it for the rest of the class. A less prepared teacher will not. I found that by giving students a problem and walking around observing their work, I could find the teachable moment for the concept I was trying to teach, and I could make intelligent use of student work in bringing that moment to life in the class. This would have been very difficult if I was not confident in recognizing good and bad mathematical work.
To ensure that our students receive a rich math education rather than a string of rules, I think we should move forward by insisting that certified math teachers know a lot more mathematics than what they will be expected to teach and that they know it well. 


— John Benson

Wednesday, June 5, 2013

Directions

The Shanghai Metro website is terrific: you can look up stops, fares from point to point, in nearly-flawless English.  But while doing some browsing I came across the instructions you see at right.  In case you're having trouble reading the image, they are:

Take the Metro

  1. get into the station
  2. buy the tickets
  3. move to the platform through turnstile with ticket
  4. wait for the train
  5. get on the train
  6. get off the train
  7. move out of the platform through turnstile with ticket
  8. get out of the station
What struck me as funny about these instructions was that I couldn't figure out who they might be for.  I mean, if you can't figure out that after going through the turnstile, you have to wait for the train, what use is that instruction going to be?  I could imagine one of our students with autism using these kinds of instructions ... but also with lots of practice and review.  What I can't imagine is someone who really needs these instructions being able to go on the web, download them, and then use them to actually successfully navigate the subway.  Anyone who can do all that can probably figure out the subway.

To be fair, another page on the Shanghai Metro website gives really helpful, step-by-step directions with warnings and pictures.  It's still a little funny to me to think of someone needing to be told things like:


but I can imagine saying them, so I guess it's worthwhile.  (And the instructions for using the ticket machines are actually excellent.)

But I was left wondering how often we as teachers make this exact same mistake, namely, give directions that would only be useful for people who don't really need them.  For example, when we "teach" kids to write research papers ("teach" being a term I use loosely in this context), we often say things like:
Step 1: Identify a topic.  Pick something that interests you that you can write about.
Step 2: Research the topic.  Keep track of your sources so that you can footnote them in your text.
I'm being a little facetious here, but not very.  Instruction about selecting a topic might include some platitudes about not being too broad or too narrow, but how often do teachers actually sit with each kid and talk about the topic for 3-5 minutes to help the kid learn what is too broad or too narrow, and how to widen, narrow, or pivot the scope?  We teach the mechanics of how to research ("This is how you use the online database" "This is the card catalog") but do we actually model the process of finding a source and using it to find others, or to supply background knowledge, or questions for further inquiry?   Do we model the process of constructing a paragraph in which information from two different sources is combined in a synthetic way, so that students can actually see the difference between copy-paste and genuine research?

The same is true about other kinds of products.  I've never yet seen an elementary school teacher workshop students' written fiction.  Neither of my children has actually designed an experiment in science class.  These challenging processes require actual instruction--not just assessment--as much as any other.  We need to be sure that the directions we give are useful to the students receiving them.




Sunday, May 5, 2013

Questions are Expensive!

A passage from this great posting on standards-based grading caught my attention:
Am I tempted to include one question on my test to send the message “HEY! We spent a day on this in class and we had a homework assignment on it, so you better do it because I said it would be on the test!” Yah, I’d probably do this. But the real message it sends is “I use my tests to reinforce that you should be doing my homework for arbitrary reasons and to punish you when you don’t”
What the author realized can be summed up in the title of this post:  questions are expensive!

In my work on a test development committee, one of the first things I learned was that a standardized test  ("item") that appears on the ACT and SAT represents a substantial investment of cash and time.  It costs upwards of $1,000 a question (in fact, by some estimates, more than $10,000) to develop, vet, and pre-test a single item of the hundred-plus items that appear on a typical SAT.  In the context of test creation and administration, this fact makes test editors somewhat conservative: deciding to change (and re-vet) or throw out a test item in the late stages is actually a major financial commitment.  But what I'm suggesting here is that, as teachers, we should all be somewhat conservative about what we put on tests, because every test (or quiz, or homework, or project) item is expensive.

"Expensive how?" you ask.  The costs abound.  It takes time to write, proofread, and format the test.  It takes time for your students to do the test, time that could be spent in doing other questions or assignments or just (imagine!) having fun.  It takes time to check and grade the test, and then you have to figure out what to do with the scores and information about student performance.  Most important, every item you include on a test represents a decision not to include something else:  you can't give a class of fifth-graders a six-hour math exam.  Unless you teach the most boring class ever, the chances are that in the course of a single unit, you've had your students work on many different skills in literally dozens of tasks and contexts.  You can't rehash all of that on a test, so every item that gets on the test has pushed four or five or six more off.

So you need to be a little conservative.  By that I don't mean that you can only assign items that you've already reviewed in class, or that you should never change a test--quite the opposite.  You do need to choose your items carefully, thinking about them more as an incredibly expensive data sample -- or a trip to a very expensive gym or tourist destination -- rather than as simply a collection of objects that more-or-less mimics some of the things you've done in class.  You need to ask:

  • What will I learn from doing this about what my students know?  What skills and concepts does this item assess?  How are my students likely to respond to the item, and what will I learn about my students from those responses?
  • What will my students learn from doing this?  Is this an opportunity for them to grow and stretch in some interesting ways, or just a check that they can spit out what we've put in?  Will students come away from the experience with a better sense of what they themselves know and can do?  And will they come away with a better sense of what it is I'm trying to teach them?

My friend and mentor Diane Herrmann speaks sarcastically about the "sponge theory" of teaching:  you start the term with a dry sponge and spend the term pouring water into it.  At the end of the semester, you squeeze out the sponge into a measuring cup (graduated cylinder, whatever): the student's grade is the percentage of the poured-in water that you can successfully squeeze out.  I think that theory drives a lot of the garbage-y tests kids wind up taking--tests with 50 or 75 or 100 items to be done in 45 minutes, tests that ask similar questions again and again.  By contrast, if you think about teaching as developing a kind of mental fitness--with certain types of habits, strengths, and skills--then you realize that a test is not just a way to find out what a kid can do, it's a chance for you to provide the kid another growth experience.  And that's the real value added.

Tuesday, April 30, 2013

How Not to Return a Quiz

This is how not to return a quiz.
  1. Walk around the room handing each student his or her quiz while everyone else is coming in or sitting down doing nothing.  Even better, wait until after the class has done your opener (or "bell ringer") and fully settled down.
  2. Don't post the answers, either in class or online.  That way, students will have to follow your in-class explanations to learn how to fix their mistakes.  
  3. Go over every problem that anyone got wrong.
  4. When you go over a problem, make sure that you're the one giving the explanation.  If possible, give the same explanation you gave the first time.  Don't let students who got the problem right give an explanation at length, and if a student does start explaining a problem, make sure you talk over him or restate his explanation.  Don't give students who got the problem wrong the opportunity to explain their misconceptions to the rest of the class.
  5. After going over a problem, don't give students an opportunity to do a similar problem.  You already have an assessment of what they know and don't need another one.
There are lots of ways to return quizzes.  Some things I've learned:
  1. Get mailboxes for your room.  Use them.  Put papers in them before or after school and let kids pick up their work on their way into class.
  2. Post or hand out solutions; don't go over them in class. Suppose eight students got question #3 wrong.    Of those eight, two have errors they can see immediately from your corrections, and don't need further explanation.  Of the other six, whatever it was that they didn't get the first time, they're unlikely to get by having the same explanation a second time.  Either watching someone else do the problem doesn't work for them (does it work for anyone?), or they've got some underlying misconception that made the first time through not so effective.  Whatever the cause, it's unrealistic to expect more than half of them to actually correct their error when you go over it.  So you spend four or five minutes doing something that only benefits three students.  What's the point?
  3. If there's a problem that a majority of students couldn't do, briefly illustrate the main point or issue, then give a followup problem.  Or if students got wrong answers, post some popular wrong answers and have students explain what's wrong with them.
  4. Anything that's important enough to talk about in class is important enough to re-assess, sooner rather than later.

Sunday, April 21, 2013

Adolescence, and other mental "disease or defect"

My non-teaching friends who knew me "back when" are probably surprised that I love teaching adolescents as much as I do:  as a high school student, I often felt disconnected--even alienated--from my peers, and certainly my best friends were typically a few years older, not kids my age.  And yet here I am, loving being with kids: on field trips, I tend to sit in the back of the bus, with the students.

Part of that is probably my mild but ongoing sense of alienation from my "peers":  I'd often rather spend time with high school kids than with people my own age, partly because I don't necessarily like people my own age.  But the major shift is perspective: I'm able to enjoy the company of adolescents much more now than I did when I was an adolescent precisely because I'm not one of them.  The things that kids do that I found weird or incomprehensible or eneverating are now more curiosities than anything else.  I can understand and appreciate where they're coming from, even if it's not where I was coming from at that age.  (Every so often I'll say to kid X about kid Y, "I totally get that kid Y is driving you crazy, but ... ")  The grade-panics, the living from dramatic cliffhanger to dramatic cliffhanger, the sometimes near-total oblivion about the "big picture"--I now see these things as part of the pathology of adolescence, not personal defects.  They're not bad people, and I've come to realize that the many of the things they do and say--even to me--aren't really about me.  Those behaviors are about where the kid is at this moment.  There are some truly antisocial behaviors, but often I can do something about those.

In the big picture, adolescence doesn't bother me, because it's just adolescence.

I find myself thinking about this appreciation-with-distance as I think about teaching and working with students who have learning disabilities or mental illness.  A kid who can't get it together to get work done and in on time, who can't be relied upon to seek out extra help (even after multiple suggestions), or with whom I find myself having the same conversation for the fifth or sixth time--that kid can get under my skin.  As teachers, it's much easier for us to accommodate obvious physical disabilities.  I can teach a blind kid geometry: I provide raised-print materials and manipulatives, allow him to talk through problems that would be too hard to write out, give him a partner that can help with the manipulative stuff or describing diagrams.  (Actually, my two blind geometry students have been among my strongest, perhaps because they expend so much effort on retaining and adjusting a mental representation of the physical space around them.  But I digress....)  I know what's causing his problem, and I know what I can do to work around it.  But the etiology of these just-not-enough-effort-applied-in-the-right-way behaviors -- I don't know it, I can't see it, and I don't know what to do with it.  And that's frustrating.

I don't think I'm alone.  While the teachers at my school do a great job with students with autism, or with disabilities that come with clear-cut accommodations (written directions, access to technology or reference materials during tests, etc.), I find our hardest conversations revolve around kids who "just aren't doing it"--even when we know that those kids have processing disorders, or are struggling with major depression.

One reason is that these kids are capable of consuming an almost infinite amount of one-on-one resources, which are scarce at the best of times.  That scarcity comes from our basic teaching model:  somewhere between 20 and 35 kids together in a room, working on roughly the same thing at the same time.  (The fundamental inadequacy of this model is why I find resources like Khan Academy worth investigating and thinking about, despite their overemphasis on rote or procedural knowledge.)  So I have a choice about whether to spend fifteen minutes or half an hour with a single kid, often with no obvious results beyond the task at hand, or to spend that time doing something that will help the other 19-34 kids in the room, or the other 119-150 kids in my courses.  And so it's hard to put that time in with that one kid and not feel like I'm taking something away from everyone else.

But a more fundamental problem is that it's hard to see the student's disability--whether a learning/cognitive disability, or part of a mental illness--as a symptom rather than as a character flaw.  That same breakthrough that I've had about my adolescents' adolescence is harder to attain.  Part of that is background knowledge:  unless you've had or spent considerable time with someone undergoing clinical depression, it's hard to see how debilitating that condition can be.  But part is that we work so hard on communicating the news about these proactive student behaviors to our classes that when someone seems like they're "not catching on", we can't really understand why.  "What's the mystery?  Just get it done!" we say, although I'd never say to a kid struggling with quadratics "What's the mystery?  Just use the quadratic formula!"

Kids with these problems can be almost infinitely frustrating--I want to say "annoying", even though I know that's not fair.  And that frustration makes the whole problem harder, because it makes it harder for me to achieve that distance where I say "This is not about me, or the assignment, or even the kid.  It's about this disease."  It makes it harder for me to devote that time to that student without feeling like I'm (or he is) ripping my other students off.  And when I do that spend that time, two things tend to happen.  First, the student's behavior doesn't change much right away, and then it's hard not to take the continuing "apathy" personally:  I spent all this time on you, and you won't even meet me halfway.  Second, it's hard to know when or where to stop:  I find myself having the same conversations over and over again, without any evidence of progress.

I don't have an answer to this problem as a learning problem, but I do know this:  the kid isn't just the symptom, or the collection of symptoms.  Every kid wants to feel like a whole person.  So the one thing I can do is communicate that fact to the kid, that regardless of how they're behaving now, I still think they're a whole, valuable person; that I know they're not just this one set of symptoms; that I still love them.  I'm not sure how much saying that helps in the short term, or even in the medium term.  But over the long term, I think it's the only thing that can.