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.

"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.

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.

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.

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.

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. 

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.