## Wednesday, February 1, 2012

### On Teaching By Learning

One of the things that I have learned over the years is to let go of any preconceptions I have about how a problem should be solved.  I have methods I prefer, but my students need to develop their own.  Never has this been more obvious than it was today.

We are beginning to play around with non-linear functions and so I gave my class the following problem:

You're going  to build a garden and need to build a fence around it.  If you have 120' of fencing, how would you set it up in order to have the biggest garden?

...or something like that.

I didn't specify rectangle although I figured most kids would default to that.  I didn't mention the barn in the back of the property that could be used as one of the sides.  I just kept the prompt as simple as possible so I could see where they took it.

Many assumed a square right off the bat. And one group felt pretty ambitious and looked up some shapes in their school planner only to settle on the typical decagon.  (Good luck calculating the area of that one, fellas.)

So, I'm walking around and seeing groups all proud of themselves by defining x as one side and y as the other and they're arguing about whether 2x + 2y = 120 or x + y = 60 is the better equation.  I ask a few questions like, "so how will that equation help you determine the largest garden" to which they reply, "we'll find the maximum."

"Mmm-kay," I say as I mutter to myself, "I'll be back..."

Then I walk up to a group of three boys who usually push the envelope when it comes to creative problem solving and I see the equation x + y = 60.  I think to myself, "oh, no not you too."

Then I look a little closer and I see this other thing they're working on.

xy > 900

I double take.

"Wait, what?"

Before any of them even address my incredulous look, one of them says, "let's go put it in GeoGebra."

"So how'd you come up with that?"

"Well we figured that x + y = 60 tells us how much fencing we have.  And since a square would give us 900 ft2, we want to know if there is an area out there that's greater than 900."

"Uh, yeah.  That's, uh, yeah, that's exactly how I'd do it."

Not really.  This is what I really said.

These kids used a system.  Not in a million years would I have ever considered using a system to solve this problem and three 13 year-olds set me straight.

Man, I love this job.

Lizzy-Sensei said...

That is so cool. I had a similar experience when I asked my students to make a picture out of functions for a final assessment to a quadratics unit and a student figured out that he could tilt an absolute value graph on its side by rewriting it as x=abs(y). He even figured out how to shift it around even though it was all backwards from what we'd learned in class. That was not something I'd anticipated them learning at all.

Allison Krasnow said...
This comment has been removed by the author.
Allison Krasnow said...

Now that's CooL! And I love how they didn't want to talk to you, what they wanted was to see how it looked in GeoGebra and THEN talk about it. These are the moments that make our work so fun. I have student work sitting here on my desk from a similar experience this week in my geometry class when I asked students to draw a 30-60-90 triangle with a protractor and then without measuring anything figure out the relationship between the side. What two groups came up with completely blew me away. More on that in an upcoming post. So what I'm always left wondering is what I can do or say differently in my low tracked class to have these moments occur. Pedagogically, I try to teach as similarly as possible in my 8th grade pre-algebra class and my 8th grade geometry class. I want my mind to be blown with student creativity in both classes. I am working on it. But it's a work in progress.

Rebecka Peterson said...

This is amazing. I teach College Algebra and the half day we get to spend on optimization problems always seems to wear on my students. The way the book teaches it (and consequently the way I've been teaching it) just doesn't seem to cut it. I'm going to see if I can incorporate you students' thought process the next time I teach that lesson. Thanks for sharing!

David Cox said...

@Lizzy
I'm constantly amazed by what students will do when we get out of their way.

@Allison
Full disclosure: the class in question is an advanced algebra class and 90% of the student's were with me last year. We have a pretty god thing going with regards to my posing a question and them using me as a limited resource. However, there is still a small population of students who default to the "just show me how t do it" mentality.

@Rebecka
Be sure to blog about what happens. I'd love to read it.

Allison Krasnow said...

David--yeah, i teach a similar class in a similar way (8th grade geometry and i was their 7th grade algebra teacher last year). I wish I could have 2 years with struggling students to try to develop a similar rapport.

Fawn Nguyen said...

Love, love, love! I really have to make time for my algebra kids to use more GSP. But my geometry kids (8th grade) use GSP regularly and I learn so much from their crazy creative thinking. Kids are smart!! Thanks, David.