Friday, May 27, 2011

And Now a Word From Elisabeth

Elisabeth decided to define her own project.  She became interested in the Golden Ratio.  She came to me with a boatload of questions she wanted to tackle ranging from: 1) How can I create the Golden Spiral? to 2) Why is the Golden Rectangle aesthetically pleasing?

I said, "yeah, that's some good stuff there.  I trust you.  Do it."

Two days later, I sit down with her as she's working and watch her think.  She has this perplexed look on her face as she is shuffling through her notes.  She has about a half-dozen different rectangles sketched with different dimensions along with more questions than what she started with.

She looks up and says, "The more I look through all this, I'm wondering, 'what's my question?'"

That's right, Elisabeth.  Don't ever forget that.

Thursday, May 26, 2011

Something Different

This year, I decided to take a much more hands-off approach when it came to student projects. There were some homeruns, but there were too many swings-and-misses. Some students opted not to even step to the plate. I suppose that's what happens when students are offered more autonomy. But, I didn't do enough to prepare them to make decisions in such an open ended environment. I think I was too hands-off.

For the final project, I gave my 8th graders seven choices; one of which was to determine the angle that would maximize the distance traveled by a projectile.

What they knew:
  • Linear motion model.
  • Vertical motion model.
What they didn't know:
  • Vertical and horizontal motion do not affect one another.
  • How vertical and horizontal motion work together to determine the path of a projectile.
  • Trig ratios

Last year I had students do an investigation on trig ratios prior to working with projectile motion. But due to a shortened school year and the fact that all of my students will be taking geometry next year, I had to cut something.

It took a few short conversations for the group to get the fact that horizontal and vertical work together to determine the path and that they needed to use the vertical motion model to determine how long the ball would be in the air. From there, they could figure out how far it would go.

But there was one problem: they didn't know how fast the ball was travelling which made it impossible to determine the vertical and horizontal components.

The Process

Q: How fast is the ball travelling when it is hit?
A: I didn't specify, did I?

This led to a nice conversation on how we need to eliminate as many variables as we can.

Solution: Pick a velocity and work with it. They chose 100 ft/sec.

Q: So how fast is the ball travelling vertically and how fast is it travelling horizontally?
A: That depends.
Q: On what?

So we took turns pushing Joey around the room from behind and the side simultaneously. Each time one person pushed harder than the other.

Conclusion: If the person from the back pushes harder, Joey goes forward more. If the person from the side pushes harder, Joey moves to his left more.

Then we talked about how the velocities can be modeled using vectors and we can use what we know about triangles. Since the forces are perpendicular, we have a right triangle.

Q: If all we know is the hypotenuse of the right triangle, how do we find the other lengths?
A: Is that really all you know?

Solution: They settled on using a 45-45-90 since that is the only way they could figure out the other two sides.

Q: But what do we do for other angles?
A: Yeah, that's kinda tough, huh? Why don't you use a protractor to draw the angle you want, build the triangle you want and measure.
Q: Can we use GeoGebra?
A: Or that.

They used an applet with a fixed hypotenuse of 100 and gathered data on the other two sides.

Q: Is there an easier way?
A: Yeah. It's called sine and cosine. See how these ratios don't change as long as the angle remains constant? (it took a little longer than that, but you get the point)

They were off and running.

  • 45 degrees maximizes distance.
  • Complementary angles yield the same distance.
  • Oh, and this:

I think you physics folks would say something like this:

Tuesday, May 17, 2011

( )conceptions

Preconceptions?  Misconceptions?  Heck, I don't care what we call them.  All I know is that I have kids coming to class and making decisions with their heart and not their head.  Intuition is great.  Inductive logic is great.  But it just isn't enough.  Back it up.  Verify it.  Embrace the conflict that arises when what you thought was true turns out to be, well, not so much.  

I've taken to putting these (   )conceptions front and center.  Put them out there for kids to wrestle with. Plug in some numbers.  Argue.  Get frustrated.  And then walk away with a little more understanding than they did before.

Today's episode centered on the equation:
(2/3) (3x + 14) = 7x + 6 and students were asked to multiply both sides by 3.

And, of course, they came up with 2(9x + 42) = 21x + 18.


Well because, naturally, a(bc) = (ab) (bc).

So what do you do?

The younger me would have said something profound like, "You don't distribute multiplication over multiplication.  I'll say it again slower for those of you taking notes.  You. Don't. Distribute. Multiplication. Over. Multiplication."

There ya go. My finest pedagogical moments summed up by slowly repeating a negative definition of a property they obviously don't fully understand.

The older, wiser, 5-kid-having self is a bit more patient.

Up on the board goes:

True or False

2(3 · 4) = 2(3) · 2(4)


2( 3 + 4 ) = 2(3) + 2(4)

Most kids said that their gut told them that both equations were true.  In fact, many said, "true" before the virtual ink had dried.

"But what does your head tell you?  Verify that both equations are true."

Oh, no they're not both true.  

"Ok, good.  So now you have a conflict.  What you think should be true is different than what you know is true.  Why?"

This is why I have been calling these things preconceptions.  Students bring something to the task.  Always.  They never come empty handed.  These responses that #needaredstamp are usually a right idea used at the wrong time.  It's like a kid who has never played sports before goes from learning basketball to soccer.  Coach says dribble and the kid picks up the ball and bounces it as he runs down the pitch.  Right rule; wrong application.

I've had kids tell me that they do certain operations on a problem because "it just felt right."  I'm not sure how to address that other than to put them in a position for their feelings to betray them and help them deal with the disappointment in a constructive way.

Next weeks episode:  Why Love Isn't an Emotion

Sunday, May 8, 2011

A Tale of Two Teachers

My day begins at the door where I greet my students with handshakes and fist bumps.
Her day begins with five little boys climbing into bed and dog-piling her.

My lesson planning is done sitting at my desk.
Her lesson planning is often done at the bottom of that pile.

My lessons are informed by pacing guides, practice tests and proficiency levels.
Her lessons are informed by bugs in the backyard, bicycle tires and brotherly conflict.

My students use manipulatives to learn counting techniques.
Her students count out baby carrots as they make Dad's lunch.

I use rabbits to teach about exponential growth.
She uses a persuasive essay so her students can decide if they really want a rabbit.

My students solve fraction problems about baking and measuring.
Her students cut recipes in half, measure and bake.

My students eat lunch and go out to a treeless field for recess.
Her students eat lunch in the tree.

My classes end when the bell rings.
Her classes end at bedtime.

I use formative assessment to shape lessons.
She uses formative assessment to shape lives.

My students call me "Mister."
Her students call her "Mommy."

One of us teaches. The other pseudo-teaches.

Friday, May 6, 2011

Well, Since You Asked

I had a copy of John Van de Walle's book on my desk the other day when a student asks, "Mr. Cox, what are you reading?"

"Oh, this book on how to teach math.  It's pretty interesting."

"Oh, yeah?  What's so interesting about it?"

"Well, I like this chapter on Teaching Through Problem Solving.  It mentions three different ways problem solving can be taught:  Teaching for problem solving, Teaching about problem solving and Teaching through problem solving."

"What's the difference?"

"Well, teaching for problem solving is when students will learn a certain set of skills and then later be asked  to solve a problem using those skills.  Teaching about problem solving is when students learn about particular strategies for solving problems.  And teaching through problem solving is when students are given the problem first and then they figure out how they want to solve it.  It's kind of the opposite of teaching for problem solving."

"Hmm. So teaching for problem solving means the teacher shows us how to do stuff first?"

"Yeah, pretty much."

"They don't think we can think for ourselves?  That's kind of offensive."