Algebra tile template

The algebra tile template I've used earlier has a minor error in it - the last row is too wide, so that the last two x²-pieces are not square and are different from the first two. Here is a revised version, with a separate template for negative terms.

Models, moderation, and climate change

How would the climate change debate change if it focused more on behaviors and less on ideas? Ilan Chabay visited the University of Oslo two weeks ago ¹ to talk about “improving how we learn for sustainability,” emphasising the value and necessity of models to deal with complex issues. There was a pronounced exasperation in the audience at large at the unwillingness of much of American society to seriously address humanly induced climate change, and much of the discussion centered on cognitive shifts needed toward a greater respect for scientific results.

Being a middle school teacher, I kept expecting the conversation to turn toward the teaching of sustainable behaviors, but this was dealt with only in a closing comment. Strictly, however, beliefs or models are hardly what cause pollution, depletion of resources, or climate change. Actions, purchases, behaviors are what cause these things.

And is there really that much reason to believe that modifying the former will result in changes in the latter? As far as I can see, people who are more educated in general, who are more knowledgeable about models of climate change, are no less - and are frequently more - inclined to fly frequently, enjoy foods imported from distant places, and consume electronics whose production involve environmental hazards. On the other extreme, the Amish, while likely not overly receptive to the science of climate change, probably also contribute less to the problem than does anyone who uses an electronic device to read this entry. The connection, then, between understanding of science on the one hand, and sustainable practices on the other, is not straightforward.

I do see that groups who are outright hostile to science and who also engage fiercely in politics do thwart necessary political efforts. However, I am not convinced that insisting on a common understanding on the way the natural world works, an agreement on metaphysical issues, is the only common ground worth seeking. Could some basis for agreement on careful and moderate consumption be found elsewhere? Could there be some convergence on how we will behave without agreement on why we would do so?

I would like to think that devoting resources to the development of measured skepticism, to the appreciation of the strengths and weaknesses of models of climate change, would result in more rational decision-making and in behavior directed toward limiting the problems. But I don’t know that there’s overwhelming evidence that this would be more effective an approach than eclectically building on whatever various motivations for moderation that one might find.

¹And due to that delightful arrangement, the Fall Break in Norway, I could attend! Yay for whoever decided we’d continue to have a week off in October even though most kids don’t need to help their families harvest potatoes at that time anymore.

Two years of silence and scrambling

I've started my third year at a Norwegian middle school (that is, my students are in grades 7 - 10, so some are in the "high school" age range - things are divided up differently here). This year I'm teaching 10th grade math and science, a course on nutrition and cooking for a 9th grade class, a small elective on electronics, and special ed math group for some 7th and 8th grade kids.

What we teach changes every year here. Last year I had a lot of 9th grade science and math, and the topics were mostly different from what they are this year. To some extent we follow classes we teach from year to year, which is nice in that we get to know the students better. Lots of random factors also affect what courses we're assigned to teach too, though. My first year I had several sections of remedial English. The cooking/nutrition and electronics courses are new to me this year.

A sweet thing about this system is the variety of interesting new things a teacher gets to learn every year. Last year I learned new things about the brain, the neural system and the endocrinological system, because those topics happened to be in the 9th grade curriculum. We dissected fish heads to find the brain and sensory organs, and I did a few practice runs before the actual class, having never taken biology before myself, not even in high school. The other major topics that year, electricity and chemistry, were more familiar. This year I'm learning about genetics and cytology, human reproduction, possibly something about the petroleum industry, while also teaching more familiar stuff on kinematics, gravity, organic chemistry, and optics. Science up to and including the 11th grade here is always integrated science, and not in the sense of every year having an integrative theme, but in that a wide range of disparate topics are covered.

A major drawback of this approach will be evident to any teacher familiar with the American system, where a teacher is responsible for a relatively small number of courses that is usually taught every year. Here, all my lessons have a kind of first draft quality to them. I do find all these various science topics to be intensely interesting, and might have become bored with the US approach of doing Algebra over and over and over again, but I do often feel bad about how little topic-specific pedagogy I know. In math, of course, typical pitfalls are easy to predict now that I have taught the same material a number of times, and I know a lot about what to do about them. But I really have no idea what will go wrong when we start working on DNA replication - I'll just have to try it out and see. We muddle along, learning as we go.

Teaching this way rarely results in any kind of polished approach that seems worth writing up for publication on a blog for other teachers. Nor does it leave much time for writing. This entry has been penned in the middle of our Fall Break, a Norwegian arrangement that originally was put in place to accommodate farmer kids who needed this time for harvesting potatoes. Now, I find it a good time for regrouping and for replanning new courses, after six weeks of getting a sense of what they are about. And perhaps for writing a little.

By the end of this school year I'll have completed a three-year science cycle, having had one group of kids from 8th through 10th grade in math and science. Then (depending on what teaching assignment I get) I just might teach a science course for the second time around.

I'd do 8th grade science very differently, though. The unit on evolution, for example, really didn't work at all.

¹ These courses aren't scheduled every day, though. In fact, two of them meet only once a week, and several meet only twice weekly. That's what makes planning it all remotely possible.

A negative times a negative

Math people are discussing how to explain to non-mathematicians that a negative times a negative is a positive. I mostly follow Mr. K's approach in teaching this, but sometimes project up an image of a mysterious, intense magician and talk about how this guy controls the weather in the imaginary country of Fictionia. He has a huge black cauldron filled with hot and cold cubes, and if he adds 4 hot cubes 5 times, the temperature rises by 20 units. He can also increase the temperature by removing 4 cold cubes 5 times, and there's our negative times a negative, or at least a four times repeated subtraction of a negative.

Neither of these approaches make signed numbers immediately easy for the kids, I find, but they're both interesting enough for a conversation of some length. We return to the mathemagician of Fictionia now and again during subsequent Openers/Do Nows, so that we start a number of classes with an informal discussion of what options the guy has for, say, decreasing the temperature to some comfortable level, or increasing it just a little for some imaginary event, and gradually the ideas become more natural to the kids.

I learned this story from a co-teacher at San Quentin who was himself a Physics grad student at Berkeley but had heard the tale from either his father or one of his early math teachers.

Deci and Dog Training

Or: Building Internal Motivation with Clicks and Treats.

It's the most exciting thing when it turns out that thinkers who appear to be oceans apart in terms of specialization or school of thought turn out to be saying strongly congruent things. I read Karen Pryor's Don't Shoot the Dog during Spring Break and then again during the Summer. It was one of those mind-changing books for me, and I think part of its eerily convincing quality came from the overlap of its advice with what I might have expected from an entirely different source.

Pryor is an animal trainer who details how to shape animal behavior by using a marker signal - typically the sound of a plastic clicker - that the animal has previously learned to associate with a treat. She strongly recommends extensive positive reinforcement in teaching new behaviors. Deci and Ryan, in contrast, elaborate on how we do what we do in order to meet deep psychological needs for autonomy, competence and relatedness. They warn against using rewards to promote learning. This does look like one huge disagreement if they even are talking about the same thing.

However, Pryor's approach is to strongly emphasize the role of the token reward as feedback rather than simply as an item traded for a certain behavior. This way, the animal being trained can be seen as being motivated precisely by the autonomy that the token reward affords, because

Reinforcement is information – it’s information about what you are doing that is working. If we have information about how to get the environment to reinforce us, then we control our environment, we are no longer at its mercy. … So subjects like to learn through reinforcement not for the obvious reason – to get food or other rewards – but because they actually get some control over what is happening. p. 160.

Pryor makes a similar point by noting that from the cat's perspective, the trainer is the one being manipulated - to dispense treats in response to specific behaviors by the cat. The function of the marker signal is that it

puts control in the hands, paws, fins, whatever, of the learner. After a while the subject no longer just repeats the behavior; the subject exhibits intention. "Hey! I made you click! Watch me, I'm going to do it again! (p. 16)

Further attacking the idea that in effective training, the reward is simply something traded for a certain behavior, Pryor provides evidence that depriving an animal of food so that it will need the reward more does not lead to better learning. If anything, the contrary is the case:

It is my suspicion now that trying to increase motivation by using deprivation of any sort is not only unnecessary but deleterious. Reducing the normal levels of food, attention, company, or anything else a subject likes or needs before training begins – and solely in order to make the reinforcer more powerful by making the subject more needful – is just a poor excuse for bad training. Maybe it has to be used in the laboratory, but in the real world it is good training that creates high motivation, not the other way around (emphasis added). p. 141

Pryor even suggests that the real motivator here can be the sheer joy of learning, with the marker signals guiding the animal in an inherently rewarding process. She narrates the story of a puppy who, on "grasping" the role of the clicker in telling it what it was doing right, overnight learned

sit, roll over, come, a super "high five" ... and the beginnings of a retrieve. All on cue, rapid-fire, correct, and in any order. The puppy, furthermore, was electrified, a totally different dog, attentive, full of fun, muscles all engaged - ready for life. p. 72.

I'm thinking that sounds like a dog whose basic needs for competence are being met :)

Finally, Pryor writes about aspects of training that sound, if anything, like answers to the need for relatedness, and here - despite all her hard Behavioral Science training - she sounds almost mystical:

To be using reinforcement is to be involved in a process of continual change, of continual give-and-take, of continual growth. One becomes aware of the dualistic, two-way nature of this communion. One becomes more aware of others and, inevitably, more aware of oneself. It could be said that training is a process that requires one to be both inside and outside of one’s own skin at the same time. Who is the trainer and who is trained? Both change and both learn. p. 163

For me, this was an astonishing book. One of my earliest exposures to behaviorism was through the exposition in Alfie Kohn's Punished by Rewards, which I read after my first semester of teaching, in thrashing around for better advice on classroom management. His book convinced me that my gut feelings about the efficacy of teacher praise, escalating punishment and predictable "consequences" were basically sound, that behaviorism was therefore bankrupt, and that there were better ways of eliciting cooperation from teenagers. I consequently approached behavioristically flavored management suggestions with much skepticism (rejecting many beneficial approaches in the process) and decided that if I couldn't get teaching to work without becoming awfully consistent at meting out predictable punishments I'd just find some other occupation. I did find other ways - always limited and never surefire, but nevertheless ways - but reading Pryor has expanded my toolbox with many items that due to sheer prejudice had been unavailable before. Her warnings against using punishment are as forceful as any around, and in this regard she is immediately in line with Kohn. However, in recasting rewards as feedback she goes further and establishes a language for a rewarding dialogue between traditions that often have been seen as polar opposites.

Fall Break in the Science Room

A colleague and I share the task of managing the collection of science equipment. It's been some years since this was taken care of in any systematic way, and there's a lot to do. We spent all day today in the lab, pulling out drawer upon drawer of dissection needles, microscope slides, magnets, batteries, rubber tubing, and similarly miscellaneous equipment. We sorted and labeled and cleaned until well into the evening, filled the staff room dishwasher with chemistry glassware, and left school with optics and electricity stuff still lying all over the floor. We'll finish tomorrow, and start next week with an up-and-running lab.

We found some pretty old items, in particular some pieces of complex glass tubing that I couldn't identify, not having had chemistry beyond high school myself. The pieces were all slightly different, so clearly made by hand. I'm wondering if such equipment is still made one piece at a time by a craftsperson, or whether they're all made in identical molds now? My colleague, who has no formal training in science but has a master's in the area of history of technology, gets excited about the same kinds of questions. We chatted about the aesthetics of old laboratory instruments, which were made carefully of dark wood, polished brass, and hand blown glass, and agreed that modern day plastic equipment just doesn't compare.

This week-long Fall Break has been incredibly useful. After six weeks of school we've gotten to know the kids and the courses well enough to know what it would take to meaningfully prepare for them, and so we've been at that for most of the week. I remember this time of the year in California, with no time for catching one's breath until Thanksgiving, and I'm thinking it would be in everyone's interest to cut the summer vacation one week shorter and insert a break right here. Teachers get a chance to regroup, avoid exhaustion, and clean up their planning, and the schools get lots of free work from their teachers because at this time of the year a week off will tend to involve a lot of time being invested in preparing for better work anyway.

Certainly, there's no way that our science room would have been fixed up before a major break came around. I'm excited about the prospect of having this project more or less completed when the kids come back on Monday.

On ravens and writing desks

How is learning vocabulary similar to learning fractions? What might a predominantly Muslim public school have in common with a private Catholic school? What struggles do Norwegian and Californian teachers share? And what - okay, fine, fine, and how is a raven like a writing desk and so on or something.

I moved back to Norway this summer, as we'd planned to do all along once my guy finished his studies in California. I now teach at a junior high school (that's grades 8-10 here), mostly Math and Science, and then two sections of English as a foreign language. It's all interesting and challenging and very busy, as might be expected. We're on Fall Break here though, which no Californian reader would have expected (sorry... I do feel for everyone bracing for the October stretch without such a chance to regroup) which leaves time for thinking about writing. And maybe even for actually writing.