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.