Friday, January 10, 2014

Little children and the work of science

This last summer, I had the great opportunity to work with a bunch of kids, 5 - 8 years old, on "something sciency." What an experience! I wrote up the results in what we call a "document panel," a focused narration of an extended engagement with children. Writing this document showed me again what sophisticated thinkers and workers children can be, if we know how to work with them. Below is the beginning and ending of my "doc panel."
I BELIEVE children have an intellectual curiosity and a native ability to analyze their world to find what is important in a situation, to see abstract patterns in the world, to make models of the world, and to make good, pursuable guesses about how things work. However, they need practice, opportunities, and knowledgable mentors to guide them as they build trustable knowledge about their world. Schooling should provide these things. ...
... I was surprised at how little it bothered me that they fidgeted and squirmed around so. They’re human. They’re kids. They seemed to be listening and interested and they would engage. I needed less and less to direct them. What could I do with this group if I had more weeks? More days in a week? Several opportunities a day? Looking back, it feels like we were just laying the foundations for larger explorations.
The group was just getting an esprit of work, facility with tools, a sense of self-efficacy—the ability to complete tasks and reach goals. What else could this “technical team” tackle?
Timing how things dropped from a roof? Measuring the height of a stairway? Figuring out how fast they can run or the speed of a car? How could I go back to using numbers to compare inertial and forced motion? Can we build on the idea of “arguing with a bowling ball”? WHAT WOULD THEY be interested in trying? I would love to explore the way building teams like this could pan out over time.
What materials did I use? Half-filled ketchup bottles. The automatic door in the building. Paper, pencils, and crayons. A clipboard. An old bowling ball I had. A long hallway. A tape measure and some timers I borrowed from the physics guys here on campus. The biggest resource was time and personnel. I worked for hours at a time with 5 - 8 kids and one assistant. The document panel is really cool, full of great pictures and little moments. It's available in my Downloads section.

Tuesday, December 7, 2010

Language and the sense of self

In the book Choice Words, Peter Johnson describes how the words we use can lead our students down very different paths. In an early example, he lists three likely responses a teacher might give to a conflict in the classroom:
Each of these responses says something different about “what we are doing here,” “who we are,” “how we relate to one another in this kind of activity,” and how to relate to the object of study (p.6).
Each response might quiet the situation, but saying “get back to work or you’ll stay after” makes implicit that students are laboring at their task and work slavishly under threats from authoritarian control, while saying “Whoa. This is not like you. What’s up? What do we need to do here?” implies that students and teachers are collaborating, momentarily distracted from their progress by an aberration from otherwise respectful and admirable people.

Language is tricky

Johnson emphasizes that language “is not merely representational; it is also constitutive. It actually creates realities and invites identities”. How does this abstract thought apply in the classroom? I don't want to create the sense of simplicity and certainty where there is neither. I don't want to invite the student to be a technician who applies formulas without understanding. I want to invite them to be one who invents and judges tools of analysis.

Looking at even a simple physical experience and representing it according to physics is a complex activity, full of subtly, false starts, and roadblocks. Real scientists do this in collaboration with one another. They create simple representations of the world and try to use them. I think that inviting students into this collaboration and respecting their struggles and advances is the proper way to teach that.

It would be very easy to fall back to a standard pedagogic tack: “which equation do we need to use here?” This language positions the student as a technician at best, someone who knows without understanding, or does without inventing. However, saying “what model fits best here?” “Do you think it should be linear right now?” “What does your data say?” “How much do you trust that?” reinforces the collaborative and mentoring nature of being a teacher and positions the student as one who creates order out of a messy situation and judges its utility based on their understanding of the way science works.

What do we want out of the classroom?

If we wish to develop students and eventually citizens who understand how science builds trustable knowledge, I believe we need to rethink both the exercises we give them, the pedagogy, and the language that we use every day in class. Giving parsing and modeling exercises, explicitly teaching the separation of formal and formalization skills, and inviting students in as collaborators in messy problem solving, I believe, are good starts down this road.

Saturday, July 10, 2010

Pythagorus's theorem doesn't work.

Is a circle that's not a perfect circle still a circle?

The difference between models and reality is a complete muddle in school. In class, I try to get students thinking about models vs. reality by working up a confusion about what we mean by 'a circle' and then about how we can use something if we don't know what it is or whether it even exists.

I draw a figure on the board and ask the kids to tell me what it is. So far, they've always said it's a circle. Look at my circle, the first one, below. It has problems, so I redraw it, but still it's not a 'perfect circle' we all agree, so I improvise a compass and try to construct a 'more-perfect' circle, but usually give up half-way through. What about one drawn by a computer, is that a perfect circle?

How 'perfect' of a circle can we get?

What is the definition of a circle? What did you learn in geometry? A standard textbook definition, a 'formal' definition, of a circle is typically something like
those points in a plane which are equidistant from a given point called the center (wikipedia)
They remember some version of this, albeit pretty dimly. Do any of my pictures match the definition?

Be honest.

After thinking about it a while, in class we realize that it is impossible to draw such a thing, as all paper has an irregular surface; all pencil lines have a thickness and therefore 'points' on the outer edge of the line are farther away then points on the inner edge. Even computer screens have their own granularity, their pixel size. Anyway you try, no physical shape matches the definition.

There are no circles.

Have you been lied to all these years?

That's what I ask my kids. How would you answer? Aristotle, thousands of years ago, coined the term 'equi-vocal', or same-sound for words that really have different meanings depending on the context. Science and math education is full of equivocal words. These are one of the hardest things to overcome, I think, and create a near hopeless muddle.

Mathematics and the world are separate. Their terms are equivocal. There are no circles, but we use them all the time. This is the essence of modeling. We look at a messy shape but think of an idea that has a formal meaning. Then we pretend.

A formal meaning? I borrow the term from computer science and linguistics. The main idea is that a formal meaning is a set of rules. The rules are ideas, not experiences. Lines on a piece of paper do not obey these rules. Two thousand years ago, Euclid took the messy idea of 'circle' and made it into a rule, as above. Four hundred years ago, Descartes took Euclid's idea and turned it into an equation. Soon after, Newton used these.

There's the thing, and there's the ideal thing. Which are we talking about?

When the kids say 'perfect circle' they mean the formal idea, the model, yet 'circle' is a fuzzy experience to them, hard to define, but they can point to it. If you asked them to measure the diameter of a 'circle,' they would eventually pretend it was a 'perfect circle' and measure across it and come up with a single number to represent its size. If you asked them to find the perimeter, they would again pretend it was a 'perfect circle' and use a formula. If you ask them to measure the perimeter, they are typically confused, and return to the formula.

What's fun is to ask them to measure the perimeter and calculate it, that is predict it using a formula, and compare the two. This is where we can begin to unmuddle the muddle.

Monday, June 28, 2010


I borrow the term ‘parsing’ from grammar:
To break (a sentence) down into its component parts of speech with an explanation of the form, function, and syntactical relationship of each part (The Free Online Dictionary, 2009).
In the sciences, the thing being parsed is not a sentence, but an experience or event. In class, I ask the students 'to draw what is going to happen next;' then I throw a piece of clay at the wall. It usually sticks. Representing this on paper is hard, very confusing, but very open and creative.

How would you do it?

Parsing would break such an experience into meaningful and analyzable 'components' or time intervals. Think about it a second: how would you draw everything that happened? Truth is, you can't. No one could. So, what do you do? You pick out the important parts, find where things change significantly, start making distinctions, make judgements about what to throw away. Bingo, the beginnings of scientific thinking.

However, most textbooks just jump right to the fictions: the ball instantly goes a constant velocity and then instantly stops. In truth—and students can easily see this—there is a period of acceleration while the ball is still in my hand, there is a period where it falls freely as it crosses the room, and there is some period of deformation. It might seem like this complicates the discussion, but I find that it clarifies it, makes it simpler to talk about, and the kids get quite engaged.

Passing along the tricks of the trade

The students can see where things change, but they grope for a way to capture it, to make it simpler. If they don't come to it themselves, I introduce the idea of ‘timestamps’ to parse the chronology of the event, and a standard tool of physics, ‘free-body-diagrams’, to help parse the relationships between objects. Math then becomes a great friend, a helpful language that really streamlines the conversation and enables thinking. We talk about who's pushing what, what direction, how hard, for how long. Student's typically find this kind of thinking fun, and excellence becomes breaking the world down into parts that are meaningful and can be analyzed with the math we know at that point.

The simplest things become fascinating

Another simple example of parsing for a class is analyzing the experience of pushing a toy cart a few inches with your hand and watching it slow down to a stop. I find this is very difficult for students, even the advanced AP Physics kids. Yet the exercise is very fruitful. They begin to bring up contradictions in their own thinking and begin to form real, workable questions. The 'granularity' of their experience becomes finer, and they realize they don't understand things as well as they would like to. Bingo: get curious, play, form a good question.


Pedagogy: the principles and methods of instruction. The main pedagogic difficulty is in keeping the teacher out of it while the students grapple with how to analyze the experience, and then, where to step in and give aid. Open-ended questions like “why did it slow down?” or “what made it move?” are baffling to students, but they tend to really enjoy the effort of figuring them out. They talk to one another. They get frustrated, confused.

'That's great.' I tell them. 'It's fine. That means you're really thinking.'

‘Formal skills’ vs. ‘formalizing skills’

This is completely muddled up in the science class, and I think it is disabling students… and beating them up.

What do I mean? Formal skills are what dominate learning and testing in school. The formalizing skills are where real inquiry begins. Formal skills are what Dewey called "dead knowledge, other people's knowledge." Formalizing skills are natural, born again in all children, and are the essence of engaging the world. Formal skills let us work within an idealized world. Formalizing skills let us build one and use it.

This diagram gives a schematic distinction between them:

The world of experience vs. an ideal world.

Why the two circles with the line between them? Scientists—all people to some degree—work in two worlds at once: an ideal world they represent with symbols, and the real world that they engage and experience.

The formalizing skills are parsing, modeling, and interpreting. These let people move between the two worlds. Parsing and modeling let a person create an ideal world, and interpreting lets a person use the ideal world to understand the real world.

Teaching people how to read the world as math

The hardest part in teaching science is getting a student to be able to look at nature and see mathematics. This is the bread-and-butter of any math-based science or engineering. In my opinion, it is not being taught in secondary science classes.

My position

I believe we can only teach this process, what I will call ‘formalizing experience,’ by presenting students with messy, fuzzy problems embedded in real, physical experience, and then by having them struggle through creating a mathematical formalism, a ‘model,’ to the problem. I believe teachers need to be careful in the language they use: they need to explicitly differentiate between the physical world we are describing and the mathematical models we use to describe and analyze it. Their choice of language can be disabling to their students, sending them back to scramble for the magic equation to fit the word problem.