What do you mean by 'play with the equation'?

My professional obsessions lately have focused on using technology to turn traditional processes of learning into more inquiry based ones. A really big part of this is purely in the presentation.

Here's the traditional sequence:

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  • Calculate the force between the Earth (mass 5.97 x 10^24 kg) and an apple (100 g) at the surface of the Earth (which has a radius of 6.37 x 10^6meters.
  • If the Earth somehow doubles in mass, but keeps the same size, what would happen to the force of gravity?
  • Find at what distance from Earth's center the gravity force would be equal to 0.5 Newtons.

My observation in this type of sequence is that the weaker algebra students (or students that use the presence of mathematics as a way out of things) will turn off the moment you say calculate. The strong students will say 'this is easy', throw these in a calculator, and write down answers without units.

A modified presentation would involve showing students a spreadsheet that looks like this:
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  • Calculate the force between the Earth (mass 5.97 x 10^24 kg) and an apple (100 g) at the surface of the Earth (which has a radius of 6.37 x 10^6meters.
  • If the Earth somehow doubles in mass, but keeps the same size, what would happen to the force of gravity?
  • Find at what distance from Earth's center the gravity force would be equal to 0.5 Newtons.

I would argue that this is, for many students, just as much of a black box as the first. In this form, however, students are compelled to tinker and experiment. Look for patterns. Figure out what to change and what to keep the same. In the last question, students will likely guess and check, which may get tedious if the question changes. This tedium might motivate another approach. A mathematically strong student might double click on the force cell and look up the function, or look up Newton's Law of Gravitation on Wikipedia and try to recreate the spreadsheet on his or her own. A weaker student might be able to play with the numbers and observe how doubling the mass doubles the force, and feel like he or she has a way to answer these questions, albeit inefficiently. Both students have a path to wrestle with the question that forms the basis of physics: how do we model what we observe in our universe?

This approach makes obvious what it means to play with a mathematical object such as an equation. Playing with an equation is something that I've admittedly said to students before in a purely algebraic context. I know that I mean to rearrange the equation and solve for the variable that a given question is asking for. Students don't typically think this way in math or science, or any equation that we give them. If they do, it's because we've artificially trained them to think that this is what experimentation in math looks like. I think that this is primarily because the user interface of math, which has been paper and pencil for thousands of years, doesn't lend itself to this sort of experimentation easily. Sure, the computer is a different interface, and has its own input language that is sometimes quite different from mathematical language itself, but I think students are better at managing this gap than we might give them credit for.

Moving in circles, broom ball, and Newton's cannonball

In physics today, we began our work in circular motion. I started by asking the class three questions:

  • When do you feel 'heaviest' on an elevator? When do you feel 'lightest'?
  • When do you feel 'heaviest' on an airplane? When do you feel 'lightest?'
  • When do you feel 'heaviest' on a swing? Lightest?

We discussed and shared ideas for a bit. I tried my hardest not to nudge anyone toward thinking they were right or wrong, as this was merely a test for intuition and experience. We then played a few rounds of circular 'book'-ball, a variation of the standard modeling curriculum activity of broom ball from the modeling curriculum in which students use a textbook to push a ball in a circular path on a table. The students could not touch the ball with anything other than a single book at one time. A couple of students quickly established themselves as the masters:

I then had students draw the ball in three configurations as well as the force and velocity vectors for the ball at those locations. Students figured the right configuration much more quickly than in previous years:
8C4384A1 - image I think some of our work emphasizing the perpendicular nature of the normal force on surfaces in previous units may have helped on this one.

We then took a look at some vertical circles and analyzed them using what we knew from the last unit on accelerated motion together with our new intuition about circular motion.

We finished the class playing with my most recent web-app, Newton's Cannonball. We haven't discussed orbits at all, but I wanted them to get an intuitive sense of the concept of how a projectile could theoretically go into orbit. This was the latest generation of my parabolas to orbits exploration concept.

A new year in the cold: how can we play with this?

My wife and I spent our winter break exploring two places: Tibet and Harbin, China. Harbin is the location of the Harbin Ice and Snow Festival, where they do some impressive work building with ice and snow to show off for the rest of the world.

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We spent our first day in Harbin's sub zero temperatures wandering around on the frozen river. The locals have confronted the fact that their local river freezes over with a simple question: How do we play with this?
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They have a number of simple answers.

From bumper cars, to ice slides, to team-driven ice sleighs...
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...to ice bicycles, complete with a hilariously ineffective brake pedal:
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The locals in Harbin have taken the brutal reality of their sub-zero temperatures (in both Fahrenheit and Celsius) and created some creative channels through which to enjoy that cold environment in ways that are enjoyable, cooperative, and unique.

The question "how do we play with this?" has become the organizing question for lessons in my classroom. I want to give students chances to explore, have fun, and work together in the process. Though I don't always do so, I think the pursuit is a worthy one.

This blog has been silent for a while, not for any negative reasons, but because the realities of organizing for my classroom and school have compelled me to put my energy in places other than structured reflection. I hope to do more sharing, more reflection, and give more appreciation toward the members of my personal learning network in the latter half of this year. I wish everyone a productive, enjoyable, and satisfying 2014.