Common Opener

I really like where we are going here.  Our video production class has been broadcasting daily announcements live for a while and now we are really starting to take advantage of this time to establish some campus habits of mind. Recently, we started a series of "Common Opener" segments airing on Tuesdays and Thursdays focused on Flexible Thinking.  Math was up first and we used the Number Talk as the vehicle.

Basically, we are asking students to deconstruct and reconstruct numbers by looking for ways to find the answer that may be different than what they're used to.  We want them to leave their comfort zones just a bit.

Once the announcements are finished, teachers send their "exit slips" up to the office (or a student aide will pick them up) and I end up with a delivery of ~500 index cards/half slips of paper with student responses.  I quickly go through them and look for patterns.  It's been interesting to say the least.  Many students have started to look for new ways to work with numbers while a large majority still stick with the standard algorithms, but I'm optimistic that students will become more comfortable with the uncomfortable.

Hopefully, this acts as a vehicle to further the student/student, student/teacher and teacher/teacher conversations about learning on our campus.  What really encourages me is the opportunity we have to show how big ideas show up in all over the place.  Once the other content areas have a chance to interpret Flexible Thinking, we may see some real aha moments with our students.

You can find the announcements here.

Hypothetically...

What if we could design a project-based class for middle school students that integrated math and science-- that provided context for math and gave instructional minutes back to science?  What if this class capitalized on the natural correlation between the Practices For K-12 Science Classroom and the Standards for Mathematical Practice.  What if students were asked to design investigations, used modeling to move from the concrete to the abstract and then presented their findings for peer review?  What if they used technology to take snapshots of their learning along the way and kept a running journal of the process?

What if...

The Comma

If [this], then [that].  We talk a lot about [this] and [that] in the math classroom. Teacher supplies [this], student responds with [that]. They even have names:  hypothesis and conclusion.  But, what about the comma?   All the power of the entire process is summed up with a tiny little "," that is all too often ignored.

No more.  It's time to give the comma a voice.

We were getting ready to add rational expressions and I wanted my students do have a workable rule for adding fractions with like and unlike denominators.  My goal was to develop the idea that when adding fractions with unlike denominators:

A lot of students don't see this very clearly.  They do what they do to jump from [this] to [that].  And most of the problems end up looking a lot like

with no real understanding of what's taken place between the hypothesis and conclusion.  And up to this point, no one has really cared because Johnny was able to find the correct answer on multiple choice scavenger hunts with a great deal of accuracy added fractions like a champ in 6th and 7th grade. However, when Johnny gets to algebra, and sees

for the first time, you'd think he's never worked with fractions before.


Time to talk about the comma.  It was actually a pretty simple adjustment to a simple question, but the conversations it generated made all the difference in the world.

This quickly became

See, the beauty here is that the process became the outcome.  The numbers become the variables and we get a good grip on how one-third plus two-sevenths becomes thirteen-twentyfirsts.  The abstract isn't so abstract and the easy part is swapping out the 1, 2, 3, and 7 for a, b, c, and d.

Mission accomplished.  Now, lets hope they remember it tomorrow.

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.

Something's Wrong Here

Search Institute _[1] has compiled a list of 40 Developmental Assets that may protect youth from high-risk behaviors while promoting positive attitudes and behaviors.  I don't know how valid the research is, but I found the list to be interesting and the data to be fascinating.

Each of the 40 assets remain the same for each developmental stage while the application of assets change based on the age of the child.  They're worth a look. 

Early Childhood (ages 3 to 5)

Grades K-3 (ages 5 to 9)

Middle Childhood (ages 8 to 12)

Adolescents (ages 12 to 18)

I took a particular interest as I have a son in each of these developmental stages.  Suffice it to say, I have some reading to do.  

When I first saw this chart, I thought, "Man, I gotta get my kids 31 of these assets."

But then, when I saw this chart I thought, "Holy, crap.  We're doing something really wrong."

Figure it out in the comments, will ya?

_[1] I have no idea if this is even a credible organization.  I learned about this today at a meeting I attended and it was the one thing that didn't make me want to stab myself in the eye with a pencil.  

All I Really Need to Know About Teaching I Learned From...

Mike Krzyzewski: 

“The truth is that many people set rules to keep from making decisions. Not me. I don’t want to be a manager or a dictator. I want to be a leader—and leadership is ongoing, adjustable, flexible, and dynamic. As such, leaders have to maintain a certain amount of discretion.”

Look, it doesn't matter if we are talking about lesson design, assessment (formative/summative or whateverative), reporting, feedback or any other thing you can get yourself riled up about.  The bottom line is that leaders--decision makers-- will find a way to be successful.  They'll find a way to be successful because they realize that anything worth doing is about relationships. And relationships are dynamic. Relationships are messy.  They're frustrating and they sure as heck don't come in a box.   If you're looking for a program, system or formula to guarantee success for yourself or your students, STOP!  It doesn't exist!

That's all.

Iron Sharpens Iron

Based on the feedback I've received in the comments and on Twitter, I have an updated version of this applet.  Linda has also created a screencast that will probably prove to be way more cogent than my attempt at describing the process.





Thanks to @jk_herbert, @mrhodotnet, @MrPicc112, @mrautomatic and @mathhombre for the suggestions on Twitter.

Updated version addresses the following:

• Writing equation given slope and y-intercept places less cognitive demand on a student than writing the equation based on the graph. Levels 3 and 4 have been switched. 
• Points and answers can't be changed after answer has been submitted.  
• A running total has been added so student and teacher can view overall performance.  
• Correct answer shown once answer is submitted giving immediate feedback to students on all levels.

There were also a couple of suggestions that were already embedded in the applet.  There is a "reset" and "go to next level" button that can be accessed in the object properties.  Double clicking on the applet should open it in a separate window which will allow you to save as well as make any changes you'd like.  

Updated Applet

Update (12/6/11): Here is a version for student practice that includes both the "Reset" and "Next Level" buttons.