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.