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:

• 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:

• 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.