Friday, February 22, 2013

Full Court Press

Man oh man I haven't taught Algebra 2 for years and I could not believe how bewildered everyone looked yesterday as we gave them 6 combination/permutation problems to work through.  I say 'we' because my student teacher has been running the show in the class.  And she's doing a good job.

How mean am I BTW that I throw her into THIS unit to start with?  What a jerk!  Maybe I should have waited for something easier....wait, for these Algebra 2 kids nothing seems easy.  Like the normal distribution or rational functions would have been easier?  Not so much.

So we had these six problems.  Nobody was getting anything.  Everybody was asking questions.  They find it difficult to actually think through the situations.  They find it difficult to actually think.

I asked my student teacher how involved she wanted me to be - do you want me to bounce around and help?  Yeah, that sounds like a good idea since they have a lot of questions.

We put on the full court press.  All students were furiously trying to understand and all students were trying to think through the problems.  She and I bounced from pair to pair to pair (the kids are in pairs in my classroom) answering questions about all six problems.  We slowly moved kids from "I don't get it" to "I might get it" to "I think I get it."  All kids finished with some level of confidence with these problems.

And after doing this two hours in a row, my student teacher & I were exhausted.

She felt very bad about how the week has gone.  She felt that it was disastrous since so many kids had questions.  I stepped her back.

"You know, it seemed really chaotic, but if you really think about it, there was a TON of learning going on today."

And truly there was.  For whatever reason, not one student chose to opt out and quit.  Everyone was trying. Maybe it was because they saw us working so hard, or maybe it was because the end of the trimester is coming up, or maybe it was because the problems were interesting, but on that day, yesterday, the kids were engaged, trying, thinking, and learning.

Great lesson or not, we had them.  Put that one in the WIN column.

NOTE: While I was writing this, Matt Vaudrey throws this in my face.
I'll agree with the esteemed Mr. Vaudrey that one measure of success might be that students don't need me.  But I'll argue for a different definition today.

Thursday, February 14, 2013

Combinations

I have been thinking really hard lately about something awesome I'm doing in the classroom.  If nothing else, I want to put a little distance between me and my recent rant about how inept our administration seems to be.  But I have been hard pressed to really love anything I'm doing lately.

Enter my student teacher.  She's been here a couple of weeks and has begun to take over some of my classes.  In fact, she's taking over at the beginning of my favorite unit: counting principles & probability.  It makes me sad because I LOVE this stuff.  I LOVE the problems.  It makes me happy that I can look at a problem like this and immediately know what to do:


The truth is, however, that my love and comfort with these kinds of problems has at times made me less effective of a teacher.  What is obvious to me is not to the kids, and sometimes I have trouble remembering that.  The tables, trees, and pictures that float through my brain - sometimes I forget to let the kids in on all of the secrets...OR...I let them in on too many and confuse the crap out of them.

So I have been asking myself how I can be a better teacher of this stuff for a few years now.  And in focusing on the negative, I forgot how good I really am at teaching some of the most fundamental concepts.

Take combinations for example.  I don't even know where I learned this or maybe it's obvious to most math teachers, not sure.  But my student teacher was pretty impressed and it didn't seem obvious to her.  So it's probably worth blogging about.

"Here's how I'd teach it," I said.  "It all starts out with a bucket of blocks."


So let's say we have six different colored blocks and we distribute those to the six kids in the front row...five to five....four to four...FACTORIALS.  Awesome.

But what if we have six and we only distribute them to four kids, or three, or two...PERMUTATIONS.  Awesome.  If you do it right, you can even make sense out of this bizarre looking formula:


But this is the really cool part, and the part I thought was obvious to most people but I guess maybe it's not.  What if we just want to pick four from the six?  And not give them to anyone.  Order doesn't matter here.  This is what we call COMBINATIONS.  But how could we calculate THAT?

Well, consider again 6P4.  One way to deal with this would be to do what we did above.  But another way would be to pick the four and THEN arrange them.  If we pick the four and THEN put them in order, this would be represented by this expression: 6C4∙4!.  And the two strategies should yield the same answer, SO...

and


Generalized:


And this is what the stork did when he dropped combinations on the ancient world.