How did you get into my 3rd Grade classroom?? I was dealing with all of this just a few weeks ago.
My situation was slightly different, and because it informs my thoughts about your questions, I”ll share it here.
I decided to move to measurement out of frustration with my students’ ability to subtract. In particular, I was asking them to find the distance between two numbers on the number line, and it felt like we weren’t going anywhere.
Based on previous years memories of students who start measuring at the 1′ mark, I began class with a picture of a ruler and a pencil on the board. I placed the pencil so one end was at the 1′ mark, the other at the 6′ mark. How long is this pencil, I asked the class? The first view was, as you might expect, 6′, and I don’t remember if another kid or I was the one to disagree, but I drew the class’ attention to the alternate view, which was based on two things:
a) Counting the spaces, not the lines
b) Our ability to slide the pencil to anywhere that we want on the ruler (i.e. it wouldn’t make sense for the length to change)
Then I gave out rulers and asked them to measure that piece of paper, that same Investigations activity.
OK, and one last thing: when kids end up with different measurements, one thing I’ve learned to do is to turn it into a statistics problem. At the end of their measurements I tallied their measurements on the board in a list. I point out that variation in measurement is totally normal — there’s no way to make a perfect measurement, there is always error. So, I asked, looking at our measurements, what do we figure the TRUE measurements of the paper to be?
What does all this mean? My first guess is that a lot of number line strategies don’t make a ton of sense to kids yet, but that measuring helps. When we want kids to realize that 20 + 30 = 15 + 25 because you can slide both down the number line? That seems like a strategy best built on something like experiences with a ruler.
(It also reminds me of this research paper that we talked about, where teachers who focused more on measurement seemed to help struggling students more than those teacher who dug in on arithmetic.)
Second, an alternative to asking kids to remeasure until they get the correct measurement might be to record their various measurements and then to ask the statistical question about them: what do we think the true measurement is, given the spread of measurements we’ve been making? can we remeasure to get closer to that true measurement? Because, unlike addition or subtraction, there isn’t a way to measure the true length — every measurement is an approximation.
All this said, I’m not really sure why it makes sense to kids to start at 1 instead of 0. I like your idea that moving between tiles and lengths might get at the true relationship, but I find myself still puzzled about how kids see the ruler.