One of the biggest insights I’ve learned from embracing modeling more fully this year is pulling back on my favorite lessons so that they can become my students’ lessons. You know, all that guide on the side, not the sage on the stage stuff I’ve heard since my first year of teaching, but never paid enough attention to.

One of my most favorite lessons is tied up in one of my most hard fought understandings, one I never realized as a undergrad. The normal force—what the heck is it and how does it work?

I’m going to tell you how I helped my kids take control of this lesson a bit later, but let me start by setting the stage.

Think about the normal force for a second—how can you calculate it? There’s no formula in any textbook, and it seems to be crazy. Start by looking at these 3 free body diagrams for a person in the elevator:

Free body diagrams for a person in an elevator

Figuring out the gravitational force is easy $F_g=mg$, but getting the normal force requires some Newton’s second law jujitsu, and most importantly, it really doesn’t say why the normal force must get bigger when the elevator accelerates upward. I mean, it’s not like the floor has eyes and can see you pressed the button for floor 21, can it?

So this is the question I set my kids up with all along as we build up our understanding of Newton’s laws—how is the table so smart? It can calculate the normal force better than you will ever be able to, even if you go on to get a PhD in physics and win the Nobel Prize. The video below will help you to see how I build up this mystery for the kids.

Two bricks balanced on a meter stick

Now it used to be that all this would be a lead in to a day long lecture (of course, I’d call it a “discussion”) where I’d carefully start by asking the kids how springs know to pull harder when you place a larger mass on them. The spring can’t know? And how do you know it’s force changes anyway? Soon enough the kids see that the spring must stretch more, and that the reason the spring stretches more is because the mass causes the spring to stretch. So what does this have to do with our so called smart tables?

Let’s start with a flimsy table. The flimsiest of tables. A meter stick supported between two chairs.

Two bricks balanced on a meter stick

What happens when you place a mass on this meter stick? It bends. Why? Well, initially, the normal force on the mass is very small, and so the mass accelerates downward, pushing downward on the meter stick. And if we think of the middle part of the meter stick as an object, it suddenly has a large downward force from the mass that was just placed atop it, so it accelerates downward too. This creates tension at the ends, pulling upward on this “middle part” and the mass and meter stick get pushed together, so the contact force between them (a N3 pair) increases. This continues until the meter stick bends enough that the normal force of the meter stick on the mass is large enough to match the gravitational force of the earth on the mass (this is the equilibrium point). Actually, at this point the net force on the mass would be zero, and since the mass was traveling downward just before, it would continue to do so, bending the stick further past this equilibrium point, and the normal force would be larger than the gravitational force, so that the the acceleration points upward, causing the mass to slow down, and eventually move back upward toward equilibrium. And so this continues with the mass overshooting equilibrium, the normal force being too small, the mass slowing and moving back down, and repeating the oscillation again and again until it finally comes to rest at equilibrium when all of the energy is damped from the system.

Next we’d try the same thing with a bit more sturdy of a table—like a whiteboard supported by two chairs, and see the same behavior.

A whiteboard "table" supporting 3 bricks

But then the mystery came—if you try this for a real table, you don’t see the table bend at all. How can this be? Usually, someone says that maybe the bending is just too small to see. Then I show them a laser pointer, and how tiny movements of my hand lead to big changes in the position of the beam’s spot across the room. So if we place a small mirror on the table, and bounce a laser off it so that it crosses the room, any small changes in the surface will cause the mirror to change orientation slightly, leading to a visible shift in the location of the beam. Here’s a video showing the basic setup:

So the bend is just microscopic. Which pushes us to think about a microscopic model. We saw that when springs stretch or compress they exert more force—can we think of “springs” living inside our table? Sure—I show them this PASCO Matter Model:

And we talk about a new ball and spring model, where the balls are the atoms and the springs are the bonds holding the atoms in place.

So that’s my discussion, lecture, as I used to present it. I loved it, and I can vaguely remember once or twice a kid saying “cool.” It was great.

But this year I decided to do something different. Continuing my exploration of Ellena’s idea of seeding investigations with capitalism, I put this together this assignment:

View this document on Scribd

That’s it. I didn’t do a song and dance about springs, weak tables, or lasers. Instead, I put all that stuff in a pile on a lab table and let the kids try to figure it out. Along the way I offered hints, like asking them to build a model table, showing them the matter model and discussing it with them, or showing them how little movements in my hand holding the laser pointer led to big changes in the position of the spot.

I did hear at least half a dozen kids say “cool” this time, and I’m far more certain that they developed most of these understandings on their own. And as soon as I can I’ll post a sample of their work.

1. November 24, 2010 5:04 pm

Don’t dismiss your “lecture” too easily. What you did with students and “crazy eddie” vs the “lecture” is probably pretty similar – it is just that you changed the size of the group you were teaching to. In the “lecture” (which clearly wasn’t a lecture in the traditional sense) you taught to the whole group, but still had the kids doing much of the thinking. In the Crazy Eddie version, you still provided the same kind of scaffolds, but to individual groups of students. I wonder to what extent giving your “lecture” would be of value after the students have had these more autonomous experiences to help them fill in gaps.

• November 24, 2010 5:56 pm

This is a good point. Maybe I should try to put together a “flipped” classroom video that would show my explanation, and then assess their understanding of the idea with some sort of group assessment (I’ve been dying to do one of those).

• November 25, 2010 1:20 am

NOOOOOO. Now that your students have some background experience have a discussion in which the students can share their learning with each other. When they explain their ideas to each other you’ll be able to gauge their understanding and they’ll be hearing ideas in language they understand from peers. If you record your explanation and have them watch, you’ll undermine the students’ feeling of ownership.

The discussion is a consolidation of the open-ended experience.

2. November 25, 2010 2:17 am

This makes sense. It’s what we normally do after paradigm labs, but didn’t have time for this time.