Middle School Modeling: Integrated Math/Science

or  My Apologies to the Scientists, Polya and All the Modeling Teachers Out There

I decided to go with a process rather than specific content in this class.  I know stuff is going to be on the test and we need to cover it.  But, I also know that my students will one day leave and go be anything but a scientist or a mathematician.
So I settled on asking students  to question, think, plan, model/analyze and tell people about what they did. That's it.
Everything we did this semester followed this template.  I found the following questions/directives to be helpful when turning students loose on a problem.

1. What's the problem?
I think we call this "inquiry", but I really don't know anymore.  Does it count if I give the question?

2. What do you think the answer's going to be?
Props to Dan for making a guess be an explicit part of the lesson plan.  Something I should've been doing 10 years ago but somehow didn't.

3. What smaller questions will you need to answer first?
This is tough.  Students live in circular argumentation.  I mean, c'mon kid, give me at least a spiral argument once in a while.  The name of this blog should mean I have some grasp on the importance of questions, but I've never explicitly asked students to break larger questions into smaller manageable questions nor realized how badly students need help with this.

4. What's the plan for answering the smaller questions?
Two big take away here for students:
1.  a good plan = good data = good analysis
2.  plans change

5. Go do the plan. (ie. get your data)
See #4

6. Make sense out of the data.
This was the sweet spot.  How can math be used to turn data into an answer?  Kids are getting the hang of this and it's fun to watch.

7. Answer your question. 
Cross check the answer with the guess.

8. Tell someone about it.  
I use the word "presentation" very loosely here.  This was anything from a write-up to a group presentation to an informal interview after an activity.


None of this is new.  But, for some reason, it seemed new.  The first few activities we did would focus on a particular piece (I'll blog about these--this year. Promise.).  The challenging part was to keep from over-planning.  Not because I'm that kind of teacher, but because the more I planned, the less students had to.  And, well, #4.  Oh, and time.  It takes a lot more time to have students make the plan and we have bells.

CCSS 8: Unit Building

At the end of February, representatives from grade levels K-8 spent two days unpacking the CCSS and clustering them into units of study. My previous experience with unpacking standards became a process of identifying "essential" standards which assumed the existence of non-essential standards.  Those standards that didn't make the cut were ultimately ignored in favor of those that were most heavily tested essential.
We have the same provider leading these new sessions, so I was a little worried we would end up looking for content to cut rather than incorporate.  So far, that hasn't been the case.  Obviously, there are certain topics that will require more focus (eg. linear relationships as opposed to exponents) but the goal has been to see how and where these supporting standards fit with the focus standards.
Our team has come up with the following units of study.  We haven't reached the point where I can discuss specific activities/tasks, but I'd like some feedback on the pedagogy that motivated the clustering and sequencing.

Transformational Geometry

Use the coordinate plane to discuss transformations, congruence and similarity.  Use dilations as an application of the ratios/proportions work done in grades 6 and 7.  Use a graph as a tool to describe proportional relationships.

Data Analysis

Use bivariate data to create scatter plots which can then be the jumping off point for informal line of best fit (where the line may have an initial value other than zero) and an introduction for a future defining of function/non-function.

Linear Relationships

Graphing, graphing and more graphing.  Take the informal line-of-best-fit and formalize the definition.  Allow math to be it's own context.  Graph systems of equations and look for common point (read: solution to system).

Equations

Use the work done in graphing systems to motivate more abstract symbolic manipulation required for solving linear equations.

Exponents

This was a tough one.  Expressions with integer exponents and scientific notation seemed like an island unto themselves.  We are still working on finding a place that fits nicely for these ideas.

Functions

Use the informal introduction to functions and formally define a function.  Look at linear and non-linear functions.  Compare functions using different representations (ie. graph vs. table vs. equation vs. verbal).

Pythagorean Theorem

I think this one speaks for itself.

3D Geometry

Problem solving involving the volume of cones, cylinders and spheres.
We are trying to move from the concrete/informal to the abstract/formal while allowing students to explore these ideas while creating their own formal definitions.  I'm particularly interested in the sequence that runs from Data Analysis to Functions (note: Exponents look to be a unit that can be dropped in and our school calendar lends itself to having that unit kick off the second semester) as it may receive the most push-back from our high school colleagues.  Traditional textbooks usually go the route of
Functions-->Equations-->Graphing-->Applications so we're going to have to have solid rationale.
No one pushes back better than you all.  I'm counting on that.

Do you see what I see?

Who knew that Volkswagen could nail the state of Math Ed in :32? 

More Fraction Multiplication

My 7th grade students are in the middle of exploring fractions.  They are currently researching the question:

What are fractions and how can I add, subtract, multiply and divide them?

I'm not sure how much instruction I want to give here. Is this intuitive? What can I do to make this more usable?

 Applet with more instruction is here.

Update
John Golden:

@dcox21 @danielpearcy worth letting students adjust estimate afterfirst fraction is shown?
— John Golden (@mathhombre) November 8, 2012

Good idea.  Updated version.

Please Wu, Don't Hurt 'Em

Yes, this is a mission that Wu's on—taking out the weak.

Hung-Hsi Wu

[Textbook School Math] gives students (and teachers) a gimmick; the CCSMS require that students actually learn mathematics.

You can't touch this.

Integrated Teaching

For the past six years, our middle school science and social studies departments have kinda been given the short end of the stick.  The minutes given to math and ELA doubled at the expense of these two departments.  I don't know what the teachers were worried about, though, because only the 8th graders are tested and the test only covered, I don't know, three years worth of standards.  (And besides, Jason says teaching science in CA is easy.)
This year we decided to give some relief by reducing the minutes to math and ELA and adding a semester of integrated math/science and a semester of integrated ELA/social studies to each student's schedule.
My assignment this year is to teach one section of 7th grade math and four sections of the integrated class.  It's been tough because we are really creating this class as we go.
So far, this is what I've learned:

1. Students will do exactly what you tell them to do.  


2. Students have trouble breaking large (essential) questions down into smaller (guiding questions) questions. 


3. Students think "try harder" is a plan. 


4. 
   Polya and the guys who invented this were life coaches.   
   
   


5. Establishing protocols is essential. 


6. Scaffolding doesn't just refer to content. 


7. Changing "Why?" into "Tell me more about that" is magic. 


8. "Common knowledge" isn't so common.  


9. We need to do a better job of helping students distinguish between opinion and argument. 


10. After 16 years of teaching, I still get excited when students realize that thinking is better than memorizing. 


11. Any list worth reading stops at 10.  

The Best Thing

I'm overwhelmed.  I'm tired.  I have pockets of excellence in a sea of mediocrity. My wife deserves more than I'm giving. My two-year-old is challenging every thing I ever thought I knew about parenting--we have put child locks on the freaking upper cupboards, for crying out loud. My other boys need more of me.  Four of my five periods are spent teaching a class we are inventing as we go.  I'm convinced that everything revolves around the scientific method or some variation of it.  Questions still are way more satisfying than statements.  My planning is always weaker than the adjustments I make in the middle of the period.  A good problem is more engaging than engagement strategies, but I don't know enough good problems.  Expo markers dry out way too fast.  Some of my students have bigger problems than I'll ever encounter and somehow I have to make what we do matter. 

And yet, my students keep showing up, my boys still greet me with "I love you, Daddy" and my wife still loves me.  

And tomorrow, I get to be better.