Beauty of the Algorithm
"The theory of computation is replete with deep theorems. Among the most beautiful are those showing that, in most cases by far, there exists no shorter means to predict what an algorithm will do than to simply execute it, observing the succession of actions and states as they unfold. The algorithm itself is its own shortest description. It is, in the jargon of the field, incompressible."
- Stuart Kauffman, At Home in the Universe

Perhaps that's why it's so tempting for math teachers to teach using examples and non-examples of algorithms. Particularly when time is such a pressing factor.

But if you take a constructivist view of how people learn, individuals may in fact need a longer description, an exploration, in order to appreciate what the concept actually is that the algorithm enacts.

(Note to myself - I'm wondering if this might be a way of dealing with the chaos that is the emergence of group understanding)

The challenge of changing practice
Although it's always felt to me like the new year really begins on Labour Day, I'm trying to make this January one of new starts, one of which is the way in which I teach math.

This isn't a new resolution. Ever since I started to do my own reading on teaching math, it's been clear there is a divide between how math is usually taught in classrooms and the best practices recommended by research.

Well okay, can I really say how math is usually taught in the typical classroom? No not really - how many other classrooms have I observed since I became a full-time teacher? I rely on what others tell me about what they do, and what they say sounds a lot like the way I myself was taught. A math class typically consists of some combination of the following, not necessarily in this order:
- homework check
- drills for practice
- examples provided by the teacher of successful algorithms
- seatwork/homework to practice these algorithms.

To be honest, this is much the same way that I run my own classes, although, and more about this in a moment, every year I make a push in at least one topic to break away from it.

Why is this "traditional" way so tempting?

#1. You teach what you know. If this is way you were taught, and you felt you were successful in math, why change what works? This method also fits the comfort level of many of your students' parents because it too is the way they were taught.

#2. When you're an elementary or middle school teacher, and you have other subjects to prep but not a lot of time, it's easy to fall back on the textbook and teacher's guide as a way to get through the topic. No gathering resources, no extra photocopying, just "Open your textbook to page 34 and do # 1- 35," and you've bought yourself some time to work with the school librarian to set up the social studies research project on Mayan civilization.

(More on the "traditional" method another day - a book by Liping Ma comparing Chinese and American elementary school math teachers makes it clear that it's the teacher's level of mathematical understanding that ultimately makes or breaks this or any other method.)

Every year I've tried to change some aspect of how I teach math. One year, since I had an extra block with each of my three math classes during each of the three terms, I worked with the ideas of George Tsuruda and introduced "Problem of the Week" as a way of highlighting problem solving. Another year, noting how difficult students were finding fraction operations and using the ideas in one of John Van de Walle's books, I made a big push to use fraction strips and drawings as visual and kinesthetic entry points to the mathematical meaning behind the operations. Last year, I borrowed an idea from one of my MiddleWeb listserv colleagues, about tying negative and positive emotions to negative and positive numbers, as a way of helping students understand how the subtraction of negative numbers work. And, of course, more general methods like using games, think-pair-share discussions, and encouraging students to come up to the overhead projector to explain their own methods.

This past Friday I tried another push towards creating a more inquiry-based classroom and I'm not quite sure yet how well it fared. Normally when I assign a Problem of the Week to do, I read it aloud to the class, answer any general questions in order to clarify the information presented in the problem, and then give the students 20 minutes or so to work on the question with whom they please. This time I gave students more than an hour to discuss and solve the problem, with the spur that they needed to complete their write-up before dismissal. My goals, I told them, were:
a) to have them learn to do a proper write-up explaining their thinking clearly (about 1/2 of them were still handing in miniscule write-ups that said what the answer was but little to nothing about how they'd figured it out); and
b) to get them thinking about ways they could prove their answers to themselves and to their classmates, rather than running to me to see if they were "right."

About 25 minutes into their work, I interrupted the class and asked volunteers to go to the overhead to explain their ideas so far. One group of volunteers had developed the "right" answer (although I did not indicate this to them nor to the class) but despite a couple of tries they were not able to explain their ideas clearly to their classmates. Another group's ideas were explained fairly clearly by one boy but there were mistakes in their logic. Again, it was difficult for the class to figure out what they meant.

The class continued working. A few students were obviously frustrated that I wouldn't tell them if their answer was right or if they were on the right track - "Does your answer make sense?" apparently was no help - making me wonder where the line lies between providing scaffolding and reinforcing the idea of me the sole mathematical authority. I was surprised that very few students travelled between groups to compare answers - a couple of students were quite vocal in their insistance that their solution belonged to their group alone! Some group discussions I overheard were quite promising but I wasn't able to listenfor long as I had to keep prompting a couple of students who'd given up without trying to understand what the question was asking. On the whole though, almost all of the students appeared to be engaged with the problem right until dismissal time.

Although I haven't yet read their written solutions, I have a clearer vision of what ideally the class's working behaviour could be like, and some thoughts about what I might do as a teacher to prompt changes in this direction. I also have a lot of questions about the reality of implementing any of this and wish I knew of someone in my school who was trying something similar and with whom I could discuss ideas. (Perhaps there is someone - as I mentioned before, when was the last time I was in someone else's room while they were teaching math?)

Yes a new start, and a long way to go...

Watching the puppies play
Since (for the first time this year!) my lessons were already planned and ready to go, I was able to relax a bit during this last week of classes and just plain enjoy my students.

There were all kinds of school holiday activities planned, and the first one of note was an intramural activity called the "Snowperson Decoration Challenge." Each homeroom had a team that consisted of a kid who would act as the snowperson, and four others who would be the decorators. The toilet paper (the "snow") was supplied, and the kids were to bring their own decorations. Since a number of students in my class wanted to participate, we chose names from a hat and it proved to be a nice mix of kids. D., a very quiet bookish boy with a sly sense of humour, was our snowperson and he seemed delighted by the attention. Our decorators were E. (D's best friend) equally bookish but much more outgoing; M. a quiet ESL girl who has recently started to bloom socially; T. an outgoing girl who is among the "popular" crowd; and B. an aspiring comedian who has unfortunately spent most of the year in Success Club because he can't seem to get any schoolwork done on his own. I was delighted that B. was not only organized enough to bring all the decorations, but he was the leader of the decorating. I don't think we actually won any prizes, but the sight of D, stoically standing there completely wrapped in toilet paper and draped in tinsel and lights, with two straws in his mouth in case he needed to breathe, with the others sitting at his feet grinning up at him, well, that was enough to win my heart.

Then there was the hallway Garland decorating contest. Our class's first idea, which was to take digital photos of everyone and make decorations from them, had already been used by another class. So, taking inspiration from a colleague I did secretarial work for years ago whose favourite phrase seemed to be "Go big or go home," I proposed that we get some rolls of paper, each student get a friend to trace them on the paper, cut the shape out and decorate it. Admittedly a couple of the girls thought this was a bad idea (and being grade 8's they bluntly told me so), but the rest were game and within a day or so of taking over the hallway and making a huge mess (my thanks to our fine custodians who patiently cleaned up whatever we missed) we had, well, the strangest looking Christmas garland I've ever seen. But we won the school prize for Most Original, and since that meant the Student Council provided us all with candy canes and chocolate, everyone was pleased with the results.

And finally our class Christmas party. I was a little anxious about this - at my old school I was able to keep the last day of classes pretty structured, but I wasn't able to do that this year and I was afraid of what could happen. But my students were great. We had a pizza lunch (which they earned by bringing in 176 donations for our Christmas food hamper drive), they exchanged Secret Santa gifts, then it turned out a bunch of them love to play cards and were content to do that, while another group was content to sit and chat and argue about what radio station we should listen to. K. let himself be decorated with make-up that one of the girls got from her Secret Santa, S. instigated a cola-chugging contest (fortunately we ran out of pop before the idea spread too far), and later S. started a spinning contest with some other girls at the front of the room (how many times can you spin around before you fall over). Yes, a good time was had by all.

Mostly I sat by my desk and observed (a habit left over from my lifeguarding days I guess), but I also found out about Ko.'s father's pool-playing abilities and how a good pool-cue is constructed, I learned from S. more than I wanted to know about Jackass the movie (and now I NEVER plan to see it), I heard about a few students' holiday plans, I watched some of them delightedly play with their Secret Santa gifts, and I thought to myself that yes, I'm glad I'm a teacher.

If only if I could figure out a way to get rid of all the marking... ;)

Happy Holidays!

Greetings from the gerbil wheel
There's just something sick about this time of year..

For the past month or so I've felt like I've been running on a perpetually spinning gerbil wheel. Everything's been about school - when I'm not actually there I've been at home pondering it, preparing for it, and endlessly dreaming about it. It's bad enough that it's pitch-black when the alarm clock goes off, but do I have to be dreaming about a math lesson too?

Then, this week the wheel has turned into one of those clear plastic gerbil balls - you know, put your gerbil in it and let him travel around your living room and bounce off furniture whenever he tries to get under the couch. (I'm just speculating here - maybe they don't try to hide, maybe gerbils like to feel exposed in the middle of a big bright room) But I'm no longer running. No, now I'm plastered face down in the ball and getting dizzy from spinning around out of control.

But it's okay. There are little things that keep me sane.

- Today I watched my classes from the point of view that they're not grade 8 students, they're actually puppies. This makes it a lot easier to take when one of them suddenly falls out of a chair, or decides it would be fun to try to snort pepper.

- Realising that even when I'm sure it's obvious I don't know what I'm doing, the kids really can't tell. For instance, one student (bless her) kept saying "How do you do that?" whenever she asked me what a word was in French and I answered her. "And last week you were reading out that French to us. That's so amazing." I got C's in French all through high school - the irony that I now teach it still tickles me.

- And yes, I'll admit the big reason - there are only 7 teaching days left...

What if there were no centre?
Consider the healthy human body. A system of organs, all with various functions, all working together, adapting to each other's actions and changes.

Focus closer. Each organ comprised of a group of different cells working together, acting, reacting. And closer. An individual cell, with different organelles working away. Closer still. Different molecules acting, reacting, combining, transforming.

Now move to a wider view. A classroom. A group of people, working away, acting, reacting, transforming. Discoveries are made, ripples of reaction and interpretation spread outwards.

Notice something about these situations. All of them are ecological systems, working to stay in balance. All parts are dependent on one another; what affects the individual affects the collective. And notice something else. In the classroom situation, there is no differentiation between teacher and student.

Lately I've been reading articles written by Brent Davis of the University of Alberta and others (Jo Towers, Elaime Simmt, Thomas Kieren, etc) about complexity theory as it may be applied to teaching, learning, and understanding. It's slippery stuff, difficult for me to grasp without several rereadings, but fascinating. And incredibly valuable for rethinking about the average classroom as a learning system.

Up til now I'd heard of teacher-centred classrooms and student-centred classrooms, and neither side of the dicotomy really fit with the vague construct I had in my head of what learning and understanding are all about. The expert lecturing in the front of the room as while everyone else soaked in Knowledge didn't feel true to me - my memories of learning had as much to do with what the students around me said and did as with the teacher. The other extreme, with each student charging away on her/his own individual project, seemed impossible for me as a teacher to manage, never mind guide and assess.

I kept thinking back to the years when I was a swim instructor, particularly to when I worked with 3 to 5 year olds. Things seem to have changed - my local pool's swimming lesson guide shows that they now have a multi-levelled learn-to-swim program - but back in my day all 3 to 5 year olds were lumped together. Maybe because none of them were tall enough to touch the bottom of the pool with their toes and breathe at the same time. And from what I remember, our course objectives were few and, frankly, unrealistic for that age level. What the kids and I ended up doing was hanging out together.

So, depending on how comfortable the kids were with being in a swimming pool, I'd have 4 kids either perched on my hips, or orbiting around me wearing water wings, and off we'd go exploring. I guess I had a purpose - keep them moving so they wouldn't get cold, keep them interested, and increase their ease in the water. But every group was different, and with in every group we developed different routines. Some groups liked traditional games. Some liked to chase toys around the shallow end. Some liked simply standing on me and splashing my face.

One group I remember in particular (because it was so exhilarating and tiring working with them) liked to play a version of "House" in a loud fast-forward way. I'd call out "What are we doing now?" and they'd call back "WE'RE EATING BREAKFAST!" and then we'd all splash ourselves in the face and pretend we were eating. "What are we doing now?" "WE"RE BRUSHING OUR TEETH!" More water in the face. "WE"RE WASHING OUR HAIR!" And so on, for 1/2 an hour.

The thing was, in any of these classes, I wasn't in charge. Sure I was the tall one who knew how to swim, how to tell time, and could physically carry everyone around. But I didn't come up with most of the ideas, and I didn't decide how long each idea would last. The group did, and I was part of the group.

What Davis and his colleagues argue in various articles is that a classroom situation functions in very much the same way. Knowledge/understanding is not an object/reality transmitted from the teacher to the learner. The locus of learning is not situated in a solitary individual. Learning and understanding occur within the classroom collective, which "unfolds from and is enfolded in learners and teachers" This point of view "undercuts many of the binary oppositions that are so often used to characterize learners and schooling - most obviously, perhaps, the common contrast of teacher-centred and student centred instruction. There are no centres to complex systems."

These complexivist notions are both liberating for me as the teacher (less perceived responsibility) and a bit frightening (less perceived power). It will be interesting to see how folding back to the way I used to see teaching 3-5 year olds in the pool will affect how I now perceive teaching 12-13 year olds in the classroom.

* Quotes are from "Teachers' Mathematics: Curious Obligations" by Elaine Simmt, Brent Davis, Lynn Gordon, and Jo Towers but I'm not sure yet what the name of the Journal is.

Why don't we trust experience?
"What he told us was what *experienced* teachers already know - although it's always good to be reminded."

Last Friday was a professional development day, and our school district held a mini-conference focusing on middle school issues, featuring three "experts" running the workshop sessions. The paraphrased quote I've opened this blog entry with is from an e-mail written by a friend and colleague of mine, who's been teaching for over 20 years. Her words got me thinking: why do we need "experts" to tell us what our experience shows us?

Of course, often these experts are former teachers, with lots of experience behind them, who have chosen to study a specific area of education. And sometimes they're experts in different areas, whose ideas might be applicable to what goes on in our classrooms.

But still, if Dr. Neufeldt tells me that children may be acting up because they haven't formed an emotional attachment with me, and my experienced teacher friend tells me the same thing, why would I privilege his words over hers? Are his quantitative data worth more than her qualitative experiences in the classroom? Is it simply hard versus soft science?

Perhaps there's more to it. I think of the actions of our provincial government, which took over the College of Teachers and filled the board with its own appointees, the insinuation being that teachers can't be trusted to oversee their own members. I read the occasional "those damn teachers" letter in the paper - you know, the "they work from 9 til 3 and have the summer off so why are they complaining" type. And I wonder why teachers aren't trusted, and I wonder if that mistrust may sometimes subtly flavour teachers' attitudes towards one another.

Or maybe it's just that we need to rework our notions of what an expert is. Chances are she/he could be in the classroom next door.

Introvert as Performer
Early last summer, there was a feature published in our local daily newspaper, the Vancouver Sun, that I cut out and kept because it was so personally reassuring.

It was about Brian Little, originally from Victoria but now a popular psychology professor at Harvard. The reason he's so popular? He simply loves to teach, managing, as he put it, that "creative tendency between whimsy and gravitas that is something that professors who truly profess need to strive for."

He describes his improvisational method as being like a symphony: "There's a slow movement and there's an allegro movement. You can be allegro vivace for three lecturs and then largo."

The improvisational metaphor is a fascinating one, and one I'll no doubt beat to death in a future entry. But what really fascinated me when I first read about Prof. Little was his personality. You see, like me, he considers himself an introvert.

I didn't return to university to become a teacher until I was in my early 30's because, although I loved teaching swimming and did so for 12 years (despite the funny-coloured hair and constant whiff of chlorine about my person that it generated) I never thought I was the right type of person. I'm not talkative, I don't enjoy being in the spotlight, I don't like telling people what to do: now where I came up with this stereotype in the first place I don't know, but I fully bought into it.

So this was what was reassuring about reading Prof. Little's profile. He doesn't fit the stereotype either. As an introvert what he has learned to do is turn it "on" in order to perform. And then, after a lecture, he seeks solitude to recharge himself, even going as far as hiding in a washroom cubicle. Apparently, he's not alone - when he confessed this to CBC radio-broadcaster Peter Gzowski during "Morningside," Gzowski admitted that he did much the same thing.

"It's a fine balance, [Little] says, because introverts do make good professors - or radio hosts - attuned as they are to other people's cues, constantly scanning the room to make sure they haven't lost anyone. If they have introverts will adjust their style mid-lecture, throwing in a different example, elaborating a particular point.

'The risk is that they can burn out, because in a sense you're acting out of character. That's why we really need those restorative niches.'"

It makes sense then that some of Little's academic research has to do with what he calls "personal projects." These are core projects that are so meaningful to the individual that they can make one act out of character. Little says, "'One project for me is to profess with passion.... So, that as a professor, I am 'on' as a pseudo-extrovert.' This does not make it wrong or phoney [sic], he says. But it does help to understand what is occurring, so that you can take time to re-balance and reduce the risk of burnout."

I used to feel guilty that I prefer to spend my lunch break quietly at my classroom desk rather than chatting with my colleagues in the staff room, or that I spend most of my weekends by myself, just milling around in a semi-meditative (vegetative?) state. But really, my behaviour makes sense given who I am.

A couple of years ago, at a June staff farewell function I gave a very energetic presentation, acting pretty much as I normally do in front of my students, and surprising most of my colleagues who have never been in my classroom.

Afterwards, one of them, a wonderful woman and excellent teacher - and, may I add, a complete extrovert - came up to me and said "You shouldn't hide your light under a bushel so much!" After reading about Brian Little and his theories I now know that isn't true. I'm not hiding anything; I'm just recharging.

[All quotes are from the Saturday, July 5th edition of The Vancouver Sun]

Specialist or Generalist?
"I teach kids."

That's supposed to be the definitive answer to give when someone asks you what you teach. In reality when asked, people either describe the level they teach, the subject(s), or both.

And then there's middle school.

Last year my answer would have been "I teach grade 8 math/science/French/drama" and, having taught those subjects for five years, it would have been an answer with which I was pretty comfortable.

Then I changed schools. I still teach grade 8, I still teach math and French, but now CAPP (which I haven't taught in five years), Social Studies (which I haven't taught in six), and Art (which I've never ever taught, although I took oil painting lessons as a kid and know my colour wheel inside and out dammit) are now supposedly among my repertoire.

This, I admit, scares me. I hate to represent myself as something I'm not - the fact I'm teaching French after posting straight C's through my high school French career still bothers me. I am a teacher, I've taken methods courses in these subjects, but I'm hardly a specialist (although I'm very slowly working towards becoming one in math). Is that fair to the kids?

One could argue that it's not, that I lack the requisite knowledge to lead the more capable kids to "exceed expectations" (as the current jargon goes). And it's true - I find myself saying "I don't know" more often than I like or, the tested-and-true "Why don't you look that up?" dodge.

On the other hand, one could also argue that you can know everything in the world about a subject but if you can't get it across to a bunch of rangy 12/13 year olds then what's the point? Six years ago, I had little to no experience teaching math or science. Every summer I did a whole bunch of reading and research about the curriculum topics and ways to teach them. Every year I tried new things - some worked and some practically burst into flames before my eyes. And last spring, when I collected and read my students' anonymous evaluation forms about the past year's work, I realized that the majority of them not only liked math and science but that their interest had increased because of our work together.

So, when I sit down, as I did this very afternoon, wondering "What on earth will we do in Social Studies this week and what's the most effective way to do it?" I need to remind myself that with time it will come. And what matters most is what I really teach: kids.