Once again, I did the performance task that (I don't think) any of the other teachers did. I followed the regular task with the extension activity. I was anxious because I could tell this would be hard and I wasn't sure I'd be able to get my students to the point of knowing the distance formula in a workable way in time to take a test before winter break. I still wonder if a class that is not advanced would require A LOT more walking through. To my kids, this was just another week in math class and nothing groundbreaking. It took a lot of time, 3 - 4 class days, more than the amount of time we'd normally teach this. However, the difference in the process was sobering. My kids made connections to the Pythagorean Theorem almost immediately and didn't give what is normally a very difficult skill a whole lot of thought.
This may not seem outwardly obvious to people looking at the tests or quizzes these kids took over the distance formula, however, I know that they are thinking about it in an entirely different way. 99% of my students are using the distance formula "inside out." In their mind, they are imagining a right triangle every single time, first deciding what the horizontal distance, then the vertical distance between points is. Then, they use the Pythagorean Theorem to find the hypotenuse length between the two points. Except for a very few students, none of them are thinking about it as a formula. Ironically, it's the more advanced kids that just realize they can plug and chug to get the distance. I'm pretty sure this is all a very good thing, but am not underestimating the differences in what is happening. I'm a bit concerned that in a math program that is almost entirely very traditional, these kids might end up struggling. On the other hand, the deeper understanding of WHY the algorithms work instead of HOW to perform the algorithms will, undoubtedly better serve them.
It feels like the investment of time in a few key concepts is truly helping our kids be problem solvers and thinkers, not just algorithm followers. I know this is a very, very good thing. I know this is very, very different, even if some pass it off as not really different from what we've always done. I wish I could find a way to communicate this with other parents, teachers, educators, without sounding like a braggart or know-it-all. I wish I could know that this type of learning could continue through Algebra, Geometry and heck, even in next year's pre-Algebra classes.