## Sunday, November 20, 2011

### Teaching Points, Lines, and Planes

The time has finally come! I get to teach the lesson I spent weeks developing!  For a bit of background:  see my previous post about the PCMI project.  Also: I teach 3 sections of geometry.  My co-worker, Ms. A, teaches 2 additional sections.  All of our 9th graders are enrolled in geometry except for a few students who took it online last year.  There are a few sophomores, juniors, and seniors in the classes as well, typically students from other districts who still have Algebra in the 9th grade.   We both taught the project at the same time.

When Ms. A and I decided to teach this lesson we wanted it to be as an introduction to proofs.   As a result, we wanted to emphasize them arguing for whatever they felt was correct.  The photos, diagrams, and sentences would be their supporting statements.

I started the activity by going through the definitions worksheet (their givens).  I changed the third column to be "real-world approximations."  I have a number of students who would need the scaffolding of having a list of what points, lines, and planes could be represented with.  I was really glad we did that because when students were struggling I just sent them back to the definitions sheet.  I was originally going to have the students use geogebra to make the diagrams, but I didn't ever really teach them how.  I left it open as an option, but encourages them to do it by hand or use the shapes/lines tools in SMART notebook (which they all have installed on their computers).   I really wish I had spent more pre-activity time teaching them how to make the drawings and explaining why they were important to know how to do.  Ms. A spent more time on explaining the drawings, so we shall see if she gets better results.

When I was explaining the project I spelled it out really clearly that part of my goal was to introduce them to a new way of thinking about mathematics.  During this unit, and going forward, there might not be one right answer.  I don't really care whether they pick always, sometimes, or never, as long as their photos, diagrams, and sentences back up their argument.  I would not be telling students if they were right or wrong at any point.  In fact, I wouldn't really be offering much help at all.  They have teammates, classmates, a definitions sheet, a textbook, and everything online.

I explained that there were a few statements that were purposefully unclear.  They needed to choose an interpretation and run with it.  I talked about how my math teacher friend and I argued for hours at math camp and that Ms. A and I got into a fight about the answer to one of the statements the night before.  One smart alek asked if we threw punches or just swore at each other.   When I said, "neither.  It was pretty civilized,"  he turned to the class and said, "So it was both. They totally cussed each other out.  Ms A took a swing, but Ms. W knocked her out just after." At least he was loyal and picked me to be the winner.

Throughout the project they kept trying to get me to tell them, but I stayed strong. One student was having a hard time with "Two lines that don't intersect are always parallel."  He recognized that with the lines being on separate planes they could never intersect and not be parallel and wanted to know if that was right.  I kept evading his demands to tell him if he was right and he got quite upset.  Eventually he told me, "I've never had a math teacher be like this.  Math has an answer.  You know the answer.  JUST TELL ME! Isn't it your job to teach?  You're not doing anything"  I just reiterated that in mathematics, people don't know if what they are researching is correct or not, they just have to make assertions and proceed.  After he calmed down and said, "Well, we haven't really talked about two planes before, so I'm going to assume it means one plane and go with Always True."  At the time, I was a bit sad that he went that route, but if I'm saying it's about their thought process and argument, I have to be really pleased with that moment.   He got over his frustration, chose a path, came up with an argument, and worked it out, which is what I wanted.

In being more hands-off, I know that a lot of the groups have some major misconceptions.  But I'm ok with that.  Points, Lines, Planes, and those crazy drawings of them intersecting etc are always a confusing topic for my students.  I'd rather them learn that math can have that fuzzy I don't know if I'm right or not, but I'm going to go with it feeling too.  After we come back from break, I will use Kate Nowak's wrap-up activity of presenting the "best" 10 statements and have them correct 4 slides with misconceptions.

Class has been a bit messy, but I'm pleased with the progress and the project.  Of course, the day that the kids are laying on the floor (a line on a plane), rearranging photos on the wall, and wandering the high school with cameras, is the day that the principal wanders by, the vice-principal stops in for an observation, and the tour group of potential families parks itself in front of the math rooms rooms.  It was a controlled chaos.  Ms. A was pretty calm about it and reminded me that with projects, things look a bit messier, and that's part of the point.  Messy is ok.  Neither my vice principal or principal said a word to me about it.  Didn't ask why math class needed 10 digital cameras or why the students were all over the school.   So my worries were unfounded and I love that they were unfounded.