Many years ago, back in the the mid-90s, I was in Colorado and visited my friend Anne’s school.  I was planning to do a Geometer’s Sketchpad workshop with the teachers after school, teaching them how to author “scripts” (the pre-cursor to Custom Tools in Sketchpad), with a focus on the centers of triangles.  Anne was gracious enough to let me teach her geometry classes for the day in the computer lab, and so the students had the neat experience of learning the same thing their teachers were going to learn after school.  (“You mean our teachers don’t even know how to do this??  Cool!”)

We started by constructing a centroid and creating the script.  The students were familiar with Sketchpad, so things went quickly.  We moved on to the circumcenter, and this is where things got interesting.  I gave the definition and they started constructing.  And I heard this exchange:

Student 1: “Hey, man, what’s wrong with your circumcenter?”

Student 2:  “Nothing’s wrong with it.  I followed the directions!  What’s wrong with yours??”

I wandered over, and I find that Student 1′s circumcenter was inside the triangle, just like the (visually boring) centroid was.  Student 2′s circumcenter was outside the circle.  Opportunity time!  I ran to the front of the room and wrote a question on the board:  “When is the circumcenter inside, outside, or on the triangle?”

Not one minute later, I had 28 high school sophomores looking at me like I had two heads, basically telling me, “That’s a stupid question!  Obviously, when the triangle is acute, it’s inside, when it’s obtuse it’s outside, and when it’s right it’s on the triangle.  Why are you even asking us something so simple??”  I was beside myself with excitement because, traditionally, it’s not at all obvious. This is because most students are just told that fact.  They don’t discover it.  It’s just one more thing that they have to memorize that makes geometry class onerous and quite possibly their least favorite math class.

But these kids figured it out for themselves, independently, saying things to their neighbors like, “Uh, isn’t it just whether the triangle is acute, obtuse, or right?  Or am I missing something?”  Because they had a dynamic tool.  They dragged.  It moved.  They controlled it.  They had constructed the situation, so they knew how the objects were related to each other.  While this power of discovery never comes as a surprise to me, it’s still so cool to see it in action.  These are moments I never forget (obviously, since I’m blogging about it some 16 years later!).

We continued on to do the orthocenter and incenter, making scripts and exploring some properties of each.  This is where the title of this post comes in.  I noticed one boy staring off into space at one point, not doing anything on his computer.  I walked over and said, “What’s up?”

He replied, very seriously, “Let me get this right.  You actually like math.”

I said, “Yea, I do.”

He nodded and then stared off into space for another 20 seconds, before he went back to work at his computer.  It’s like he had never met someone who actually liked math.  His teacher was a great teacher, very committed to math and education, and I’m pretty sure she actually liked math, but maybe he just saw her as someone who was doing her job.  I was just some random person off the street who actually likes math.  Who knows?  I’d like to think that he might have thought differently about math from that moment on!