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))?