Thoughts about questions

I have chosen not to structure my blog posts around the guiding questions that were given to us at the start of the quarter because I thought it would be boring (for me and you) to loop through them again and again. Perhaps it wouldn't have. But what I did instead was talk about what is on my mind, trying always to relate it to the discussions in class and the broader issues related to teaching.

Questions lie at the heart of education. How a teacher asks and answers a question reveals what she thinks about learning and education. It is the crucial pivot in all communication, and teaching is communication.

To ask a real, not merely trivial, question is to tell your students that you care what they think. It is letting them know that, if they want to invite you into their minds, you are willing to go there. When a student asks a question, often she is asking you to step out of your own mind so that you can see things from her perspective. I had a math teacher recently who was brilliant at explaining his own ideas, but really couldn't be bothered to interpret his student's questions in order to uncover the sense that was in them. People make sense of the world in different ways, and the skill of the teacher is to keep trying to think the way other people do so that you can find a common language.

Recently, I had a good experience in Karen's Adoloscent Development class. We were in small group discussions about Lawrence Kohlberg's theory of moral development. Because of my academic background (philosophy), I had more developed opinions about the issue than my other group members. I decided to be the informal moderator of our discussion simply by asking a bunch of focused questions in a way that, I hope, drew people out and helped to focus their thinking. I was able to connect with people and instigate a pretty meaningful discussion without making a single declarative statement of my own opinions.

science

I've been off the reservation for a few weeks, but I'm back! Going to do a few more posts before this quarter is through.

I was deeply impressed by Jeff's (was it Jeff? John?) presentation on astrobiology. Honestly, I felt humbled and a little embarrassed at my general ignorance of science. I was I could still explain to someone what oxidation is, for example. I took college level physics and chemistry, but after a dozen years it has mostly washed out with the tide.

I stopped at 3rd place books on the way home and bought a couple science books. One of them is "The Brain Rules" by John Medina, a neuroscientist who lives in Seattle. He has some pretty interesting ideas about learning and education based on his understanding of the brain. He believes that brain development is so unique in each person that we really ought to de-standardize education as much as possible. Also, he argues that multitasking is largely a myth: we can walk and chew gum at the same time, but two or more activities that each require focused attention, such as driving and texting, or reading and listening, simply cannot be performed simultaneously. Some people learn to switch back and forth very quickly, but switching is extremely taxing for the brain and tends to decrease overall efficiency.

That was a bit of a digression, but it does relate to some discussions we've had this semester.

More to come later.

theoretical frameworks

I wanted to reflect on what was said last week about theoretical frameworks. My academic background is in philosophy, not the social sciences, so I'm still learning the lay of the land. One thing I know about the program at UWB is that constructivism is the official theoretical framework for what we do. As well as I understand it, I think constructivism is basically the right idea about how people learn. However, I am eager for an explicit discussion about what exactly this theory is, and what it is not. It seems that many who write as constructivists think of it as a fairly specific doctrine about how teachers ought to teach, ie, a pedagogy, rather than simply a theory about how people learn. Thus, it is said that small group discussions, or hands-on activities, are constructivist activities. But as far as I can tell, constructivism as a theory of learning does not entail that one type of classroom activity is inherently superior to others. Although I think there are sound reasons for the view that too much listening and book work in the classroom is an ineffective way to structure classroom experience, I think an equal case can be made that too much small group work and/or computer work is also ineffective.

Anyway, I am somewhat suspicious of the idea of claiming a theoretical framework, except in a broad way, when it comes to taking an intellectual stand on something. It is always preferable to state what exactly your assumptions are about a subject, rather than to bat around labels that are ambiguously used in the literature. Obviously, the discourse should not degenerate to the level of cheap political dialogue, in which saying "I'm a Democrat" somehow justifies or explains your view of a particular question. Too often, labels are a way to avoid real thought.

Visualization Techniques

This is an ugly phrase to describe what I think is an important aspect of learning math. A lot of math skills go beyond reasoning, logic, and memorization. Sometimes a student needs to be able to visualize a concept or a relationship in picturesque terms. Here is an example.

Suppose you wanted your students to understand why cos(theta) and sec(theta) are even functions, while the other four trig functions are odd. There are several ways to show them, but an especially effective way is to get them to imagine two radii of the unit circle sweeping away from theta = 0 at the same time, so that the first angle is always the opposite of the other. They should be able to grasp more or less directly that the cosines of the two angles will always be identical, and therefore cos(-theta) = cos(theta), ie, the cosine function is even. The same visualization technique will illustrate why the other trig functions must be odd.

Hence the value of math teaching software like fathom and sketchpad, which allow you to make the visualization precise and immediate. However, I think that good teachers need to be able to think of visualization tricks on the spot, to explain them in words, so that the students have to construct the visualization in their heads from time to time.

tech toddlers

Today I went to the public library and observed some young kids, probably 2-3 years old, using the computer terminals labeled "preschool." The machines themselves are brightly colored things, set low. A few weeks ago, I was there with my son Ben (age 3), who was instantly mesmerized by the bright gooey graphics of "Dora the Explorer." On that occasion, I acted as though he was looking at an aquarium with especially dangerous inhabitants, holding his hand so that he wouldn't get close, but allowing him to get his fill. I guess I wasn't ready to put him in front of a computer. It feels like an enormous step, and at any rate I know that my techno-reluctant wife would be angry if I had done it.

Today there were 2 older toddlers, a girl and a boy, both accompanied and guided by their mothers. The girl was experienced, and knew how to use the mouse once the program (perhaps Dora, or something else) was up and running. Her mother quickly escaped into her novel, first while standing just behind her daughter, and later settling into a chair ten feet away. Both kids had on headphones. The boy was curious about the light shining from the mouse, but he didn't get to explore too much since his mother (or guardian) was controlling all of the action. She (the mother) seemed excited and engaged.

I was struck by how mesmerized the kids were, just as Ben had been earlier. Interacting with a cartoon character would have enthralled me at that age too. I was also struck that the girl, not more than 3, was using the mouse to move images around in a basic but very competent manner.

In reflecting on the implications of this activity for learning, I realize that I don't know where to draw, or whether one can or should draw, the line between entertainment and learning. The program was interactive and seemed to have the character of a puzzle, but at the same time it contains a narrative and a visually seductive virtual reality. The contrast between the behaviors of the two moms was also interesting. Computers can become a surrogate interlocutor/teacher, or they can be something that the teacher (parent) explores along with the student. The first arrangement is obviously the end goal and the more natural state of affairs. The boy whose mother assisted him will no doubt be using the program by himself within weeks or months, and his experiences and actions in the digital medium will become private in the de facto sense.

Their minds are engaged, but also isolated from the people around them.

1=2

When I was in high school, I was told that you can't divide by zero, but never shown why. "It's undefined," they tell you, but that's just a restatement of the rule. I recently came across (on some blogs called "mathematics learning" and "spirit of mathematics") a fun and effective way to justify this rule to your students. You simply prove that 1 equals 2, as in the following:

1. A = B let A equal B
2. A^2 = AB multiply both sides by A
3. A^2 - B^2 = AB - B^2 subtract B squared from both sides
4. (A+B)(A-B) = B (A-B) factor both sides
5. B = A + B divide both sides by (A-B)
6. B = 2B substitute B for A, since A=B
7. 1 = 2 divide both sides by B

Dividing by zero (A-B) in step 5 is what leads to the result that 1=2. Since every other operation is valid, this is a valid and effective (albeit informal) reductio ad absurdum. I think teenagers could get it, since it is so simple and powerful.

Here is another version with simpler operations (no factoring):

1. A = B
2. A^2 = AB
3. A^2 + A^2 = A^2 + AB
4. 2(A^2) = A^2 + AB
5. 2(A^2) - 2AB = A^2 + AB -2AB
6. 2(A^2) - 2AB = A^2 - AB
7. 2(A^2 - AB) = A^2 - AB
8. 2 = 1

How do you explain to your algebra students why they can't divide by zero?

weekly thoughts III

In last week's class we learned to use sketchpad, and similar tools. It got me thinking about the importance of play, experiment, touch, and action in the process of learning mathematical concepts. I am new to these types of software, and I struggled to keep up with many of the tasks simply because I couldn't get the application to do what it was supposed to.

What all this tells me is that technology in the classroom can be either rewarding or debilitating, depending on (a) how well we can navigate it, and (b) how well we can cope when it breaks down. There is no question that these are serious obstacles for the digital teacher, and I have to face up to them if I am going to use digital teaching tools effectively.

One additional issue with putting computers in front of students is that attention and discussion become massively fragmented. Sometimes that's okay. But it's still valuable to have integrated discussion with one's class and to focus everyone's attention on the same thing.

better

Well I got over the flu. Lots of rest and fluids, and some luck.

For those readers who might be associated with the Education & Technology course at UW Bothell, I can't resist giving a shout out to my sister Jess who (with her husband Stephen) founded and mostly directed the OnRamp Arts project in Los Angeles back in the early years of this decade. This project was cited on page 27 of the paper by Mr. H. Jenkins, which we read for class. Jess now teaches at Parsons/The New School. Small world. She hadn't heard of this paper and was pleased when I told her.

The latest bit of technology in my life is a thing called the Amazing Slow Downer, which takes an mp3 file or CD track and allows you to adjust (independently) the speed and pitch of the recording. This is a huge help if you are trying to learn a tune by ear, especially one played at top speed by (insert musician). I hear from those who grew up with records that they would have to play the turn table at half speed and learn the tune an octave lower.

one flu over the google nest

Today I got the flu... Not fun. I had to miss adolescent development.

Anyhow, I read today in the New Yorker a talk of the town piece about the H1N1 virus and vaccine. I came across this not-so-great example of how to use twitter if you are a celebrity:

Last week, the political pundit Bill Maher dispatched a communiqué to his fifty-six thousand followers on Twitter: “If u get a swine flu shot ur an idiot.”

Not so great Bill, on form or substance. This virus is a real threat, and I have little patience for those who fearmonger about the vaccine. But it's also an obnoxiously juvenile way to talk about an important issue.

He could have said it on his TV show as well. But I wonder if Twitter in some way encourages people to be punchy, blunt, crude, etc. just because of the 140-word constraint. I'm not trying to slam twitter, and I think we've said enough about it. But given our penchant as Americans to shun dialogue and discussion about complex issues, I can't help feeling that Twitter enables this unfortunate tendency.

music ed

My son Ben is almost three. His exposure to music is about 100 times better than mine was at his age, both in quantity and quality. The ease of CD technology makes a huge difference. Toddlers love repetition, and I can repeat a song in the car as often as he wants, or program the player at home to repeat a song automatically. When I was his age, all we had were cassettes and LPs--and lazy parents.

Here is Ben the musician.

serious amusement online

I will begin blogging about technology in my daily life by focusing on what I do online. If I'm not emailing or reading political news/blogs, I am usually playing chess. I got passionate about chess about 8 months ago and now it is a major hobby (sounds more dignified to call it a 'passion'). Technology has fundamentally changed the nature of chess (a game that's about a millenium and a half old) as more and more people play online, and as computer engines are increasingly seen as superior analysts of the game.

I mostly play at chess.com, where I can play against people from Mongolia to Mexico, chat with them, and enjoy the full aray of interactive practices (video, forums, expert commentary, etc.). Although I sometimes go to the Seattle Chess Club on Friday nights to play "over the board," I am increasingly fond of playing online. The psychological dimension of the game is less of a factor, and time pressure is not much of an issue. I hope that I will keep going to the club, just because I continue to believe that "real" chess takes place over the board, and that there is a value to playing face to face. I am a little worried that my motivation to do so will succumb entirely to the ease of playing online.

"Technology"

About yesterday's discussion led by Carrie, I conclude that "technology" in the context of this class means the practices and tools associated with the digital revolution of the 1990s: web, cell phones, pdas, ipods, and other connectibles. In a dictionary, or in an anthropology class, "technology" means something more basic. No doubt, it also means something larger from the perspective of a general theory of education. But that is not what we mean in the context of this class.

weekly thoughts I

I apologize that my submission is late.

a. What was the most significant thing you learned in class this week?

I began to think in a bigger way about the significance of digital media for education. I realized that there is a lot to think about on this subject, and that I would probably find this class more stimulating than I expected.

b. What questions do you have and what do you want to learn more about?

I worry a lot about students getting distracted by devices in the classroom. How does a teacher engage students at the digital level, but also at the person-to-person, conversational level that is (imo) the foundation of genuine learning in the classroom?

c. What applications do you see to classroom practice based on what you learned?

I can easily imagine using a wiki with my students, or an online forum where we can discuss math issues.

Twitting

I think the most interesting aspect of the twitter assignment was that I got to know my classmates, and reveal something about myself to them, in an online format. I really don't know anyone yet, so for me it was a new way to start a relationship. (Never got into online dating.) Usually I deal with either strangers, or people I already know, when I go online.

I like that twitter requires everyone to get to the point, and also that no one gets to dominate the discussion. Each tweet is as valid or meaningful as any other. On the other hand, it seems dangerously easy to say something inappropriate, or socially iffy. I initially wrote something in response to someone's question about David Letterman, and then deleted it because I just don't want to go out on a limb when it comes to a controversial topic like adultery. Part of it is just that I'm joining a new group of colleagues, so maybe it's not relevant to twitter per se.

This exercise has shown me at least the positive potential of a web 2.0 tool in the context of an already defined educational community. I can't imagine I would get into twittering with the world at large, but who knows.

About the title of my blog

Don't be upset: I don't really want you to eat a dead rat. That's what my son Ben (almost 3) tells me to do several times a day, so the phrase has a certain resonance for me. Just thought it would make a catchy title.