## Wednesday, September 07, 2011

### Collaboration Time

Since I began employment in this school district about a decade ago, our school days have been lengthened a few minutes so that students can leave 75 minutes early on Thursdays. Students have "short Thursday" so that during the final 75 minutes of the work day, teachers can meet and "collaborate".

It's not a horrible idea, and it's produced some benefits in our math department. I wonder, though, if we're now reaching the point of diminishing returns, and have collaboration time for its own sake.

I don't need a specific time to collaborate with my fellow teachers. I do it all the time. So do they.

Today I made up a problem in my pre-calculus class--the domain of a composite function. It turned out to be a little more difficult than I at first thought, but I worked it out. A student asked a question that now, in hindsight, is easy to answer, but at the moment it caused me to stumble. I was pretty sure I determined the correct answer, but when I used software to graph the curve, my answer appeared incorrect. I doubted my work.

Rather than hemming and hawing, I went straight to another teacher, and right after class he came over. I walked him through my work and we found the source of the confusion--I typed "sqr" instead of "sqrt" for "square root" into the graphing program I used. As soon as I fixed that, the graph showed exactly the domain I had calculated. He had noticed right away that the graph didn't "look" correct, and that's where he focused his attention.

Less than five minutes and everything was cleared up. That is collaboration. And it happens all the time; it doesn't need time set aside for it to occur.

If you'd like to try the problem out yourself, here it is:
f(x) = 1/x
g(x) = squareroot (x+4)
What is the domain of the composite function g(f(x))?

Joshua Sasmor said...

G(f(x))=sqrt(4+1/x) = sqrt((4x+1)/x). So in order for the square root to be defined, x and 4x+1 have to have the same signs; I.e., (x<0 and 4x+1<0) or (x>0 and 4x+1>0) so your domain is all x<-1/4 and x>0. I would have made a sign chart for the students using (4x+1)/x, but that's the way I saw it...

I agree that having a quick chat with a colleague can be more beneficial than a structured meeting time, but those meetings can also be useful. However my department is really small (2 of us), so YMMV.

Darren said...

That's yet another way of looking at it, one that's specific to this problem. Teaching a method "generally" applicable to different types of problems, my algebra got a little more convoluted. I teach that they not try to determine the composite function itself, since that isn't always as convenient as it is in this case, but to determine the domain anyway. You're right, though, that a sign chart would have helped. I found an error in the answer book once by doing a sign chart, and confirmed my work by graphing the function.

It was that darned silly question that threw me off!

mrelliott said...

I have the same concerns over so much teacher inservice time. The state I teach in requires 60 hours of inservice for teachers every year. The majority of this concentrates on good teaching techniques. After a couple of years, it ends up being a re-hashing of what's already been presented, ad nausea. And, some of these so-called teaching techniques are trendy and have no research backing that they even work. So teacher's productive work time is high-jacked over and over again.

I recently changed from teaching public high school to private college level. I now know how unproductive so much of my work time was because of these state mandates. The problem is someone, somewhere in the state education department, thought up this inservice idea, mandated it, then promptly forgot about it.

PeggyU said...

[-4,0) U (0, inf) (?)

Peggy U said...

Oops ... I realized I forgot to do the composition! [-1/4,0) U (0, inf) ...

PeggyU said...

Aack, less than or equal to -1/4 ... think first, type later!

(- inf, -1/4] U (0, inf)