Thursday, September 30, 2010

A Conversation Worth Having

I have a student who is very creative with her math.  She just tries stuff, learns from mistakes and constructs her own understanding.  It's pretty cool to watch.  Today, she was describing to the class how she came up with the standard form of a linear equation we were playing with yesterday by using the slope of the line. 

M.M. pipes up, "I don't get how she did that." 

What he meant was, "Yeah, I see how that equation works, but I have no idea how she thought of doing that." 

She follows her intuition and sees where it takes her.

I told M.M., "I'm not so sure that you really need to know how she came up with it any more than you need to know how an artist paints or how an author writes.  What you need to appreciate is the fact that she created some understanding on her own."

I launched into a quick discussion on Duplication, Application and Creation and how many "advanced" kids actually live in the duplication stage, but they just do it faster than everyone else.  I can teach duplication and provide opportunity for application, but creation is all them.

Here's what went down as a result.

I.N. : Mr. Cox, your class is the only one I do my homework in.


I.N. : Because you don't make us do it.  Like at home, if my mom makes me go out for a sport, I don't really try because I feel like I'm forced to do it, but I'll go play sports on my own with friends or even with random people.  I want to do my homework in here. 

J.T.:  Actually, Mr. Cox, you don't really force anything on us.

Try not to, anyway.

The conversation went to grades and grading in general and I said I don't really care for grades and wouldn't give them if I didn't have to.

S.N. : Then you need to start your own school, Mr. Cox.

Yeah, that'd be nice but I don't have that kind of money.

L.F.:  I'll go work at McDonald's and give you my paycheck. 

One kid said, "you're gonna change the system, Mr. Cox" and I kinda let the comment go.  But after a few minutes I came back around to it.

You guys really think I'm gonna change the system?


"Nah, we're gonna change the system."

Tuesday, September 28, 2010

Equations Three Ways

I've never really been satisfied with how I teach students to solve equations.  No matter what, it ends up being one big algorithm and kids have no idea why one side of the equation is equal to the other.  Here's what I'm doing to try to fix that.


Strength:  It's not math.  It's a puzzle. 
Weakness:  Dealing with negatives is a real pain in the butt.

Guess and Check

I actually really like this method.  Guess and check is probably my most under-used problem solving strategy, but using it to solve equations has been really helpful.  I've noticed a greater understanding of rate(s) of change, using information from wrong answers to help find right ones and checking answers--something most kids don't want to do--is embedded in the process. 

We've gotten to the point where we can nail the answer on the third guess by using the information gained in the first two--even for equations with non-integer solutions. 

Strength: Students understand that simplified expressions on each side of the equal sign end up looking the same every time (ax + b = cx + d).  Rate of change is very useful.  Being wrong helps you to become right.  Did I mention they are checking their answers?
Weakness:  Leave 'em in the comments. 

From Construction to Deconstruction

We spend a lot of time teaching kids how to break things down whether it's reducing fractions, simplifying radicals or solving equations but we rarely (read: I rarely) have taught them how to construct things that may eventually need to be deconstructed.

Constructing a more complicated equation from a simple equation has helped my students understand that, no kidding, the two expressions on opposite sides of the equal sign are equivalent.   I've used the just-unwrap-the-present illustration many times, but we really need to teach the students to wrap one up first.  Having them list their steps for construction makes the process for actually solving the equation seem much more natural.  When I say, "just use the inverse order of operations" --or whatever completely abstract thing I've been known to throw out there in order to make myself feel better when they keep screwing it up--it makes no sense to them.  This helps. 

Strength: Kids get a grasp of which operation to tackle first while solving for x.

Weakness: Very complicated equations with variables on both sides don't seem so natural when you begin with x = 2.

I've heard rumors that there are some teachers who actually teach solving equations by graphing.  Never seen it in the wild, though. 

Thursday, September 16, 2010


Teaching can be stressful.  So, I find it important to loosen things up a little.

Like, maybe, accidentally leaving the guidance office with things that happen to belong on the secretaries' desks only to return them in the form of  gifts on Secretary Appreciation Day in the Spring.   We all get a good laugh out of it--except that one time a fight almost broke out because Mrs. P thought Ms. A had taken her tape dispenser. 

Or, rearranging the furniture in a teacher's room while they're out on school business.  (Note: don't do that one again.)

Or dressing up like a colleague on Nerd Day.  (Well, I thought it was funny.)

Most of these antics were the result of me being part of a cohort of younger teachers who were now teaching at the same high school we had attended. 

Things changed when I moved to the new middle school.  Most of the "new" staff had come from the same school, so I had to take some time adjusting to the culture that many had brought with them.  Seemed like the right thing to do.  As a result, I quickly became  good friends with Mr. G, who had come to us by way of a neighboring district.  We'd usually end up in one of our rooms by the end of the lunch period.  But one day, I walked into his room and he had a visitor.  I felt like I had kind of intruded so the next time I walked up there, I knocked as I entered to let him know I was coming.  He made the mistake of letting me know I was now welcome to bang on his door every time I enter had scared the crap out of him.  Slowly, more staff have put themselves in my crosshairs by thinking it'd be funny to get me first decided to join in the fun. I have one rule: Administrators are off limits...except Mrs. H.  But, she started it.  She got me good. 

Problem is, I can't scare the woman.  She has this mother-grandma-teacher-administrator-eyes-in-the-back-of-my-head-I-feel-a-disturbance-in-the-force kinda Jedi thing going.  I can't sneak up on her.  This morning, I spotted her about 100' away. Her back was turned. Kids were walking around everywhere.  Conditions were perfect.  There was no chance she saw me.  None.  I get 10' away and she turns and says, "Good morning, David." 

"Dangit!  How do you do that?" (I think I actually jumped up and down while I said that.)

I'm not giving up.  I'll get her someday.  But in the meantime, she better keep a close eye on her stapler.

Wednesday, September 15, 2010

If Only It Was Always Like This

Instead of working through the problems in the geometry book, we decided it would be fun to try to prove all the theorems as we come to them.  A couple of weeks ago he proved the midsegment theorem.  Now we are on to trying to prove that the centroid is 2/3 of the way down the median.  So we sit around the white board easel and discuss how we might go about this.  At this point, I don't even know how we're gonna prove this thing.  We learned from the midsegment theorem that defining the vertices using (x1, y1),  (x2, y2), (x3, y3) helped out greatly.  So, what the heck, let's try it again. 

We know we can define the midpoints generally as well, so that's what we do.  Then it hits! We can define the vector from the vertex to the midtpoint of the opposite side, use a scalar of 2/3 to determine the vector from the vertex to what's supposed to be the centroid and then translate the vertex to said point. 


 Turns out that the centroid is 1/3(x1+ x2+x3, y1+ y2+y3). 

I didn't know that.  If this is what it's like to become a co-learner with your students, then sign me the heck up.

Tuesday, September 14, 2010

WCYDWT: Space Shuttle Discovery

I've tipped my hand here a bit because simply showing a shuttle launch would be way too open ended. I have the original lesson plan as designed by NASA and I'm pretty sure how I'd go about presenting this lesson, but I'm curious as to how y'all'd do it.

STS 121 Launch (Initial/Final values) from David Cox on Vimeo.

Thursday, September 2, 2010

How Much is TOO Much?

What else can I ask him to do?  
He's 13.

I mean, c'mon. I can ask for a more rigorous proof, but the kid can manipulate these expressions and then say things like, "I think this looks really cool with all x's and y's all over the place. "