One reason I enjoy going to PME-NA is seeing the work of my
colleagues on children’s thinking. Inevitably what arises out of these studies
are incredibly rich tasks that teachers can use in their teaching. I might add
that these tasks are also good for teachers and prospective teachers to explore
as they are learning about the ways children think.
Higinio Dominguez at Michigan State is an excellent interviewer of children-- :)Somewhere between Herb Ginsburg and Mister Rogers. He just gets those kids to
talk, and does it with very little interference so what the kids say is
remarkably close to what they really think. One thing he routinely does is
leave off the question that usually ends the typical work problem. So for
instance, when introducing this problem: “Joey has 14 cars. He lost some, so now he has 5
cars," and then asking, "How many cars did he lose?” Higinio says he simply asks, "So what so the
problem?" The kids say all sorts of things such as, “Joey should really keep
better track of his cars,” or “Probably his brother took them,” or “They are
probably just under the bed.” Higinio’s point is that is you want kids to bring
in their own personal connections (or resources--this is a much deeper construct) and this approach is one way to give them the
opportunity to do that. Eventually the child or children will ask, "How many
cars did he lose?" Or the teacher/interviewer can ask, "So how many are under
the bed?”
Steven Greenstein, from Montclair State University has been investigating young children’s
understanding of geometry, specifically the geometric reasoning involved in
topology. He has noted that young children can identify important properties of
sameness between 3-d objects. His research methodology included the use of a
microworld, Configure, he developed to support children’s exploration on 3-D shapes. It is
free and it looks like such a great way to get children to talk about their
ways of thinking in geometry. The website includes suggestions for teachers who wish to use the program with their students. Here is the link: PLAYWITH SHAPES.com
Last, this idea comes from some researchers at Virginia
Tech, Andrew Norton and Steven Boyce. The task is about unitizing, and children
use their thinking about multiplication, and units to solve these problems
mentally. I was familiar with the candy roll problem used to develop base ten
understanding,
(see an explanation here: http://www-rohan.sdsu.edu/faculty/jbowers/macpics/intro.htm)
but was unfamiliar with this packing scenario:
A chip is worth 2 cents. When they are packed there are 3
chips to a cup, and 5 cups to a box. How many cents is the box worth? OR If you
have 12 cents, how many more cents do you need to make a box? With this
scenario, you can change the numbers around, and also you can begin the problem
with actual chips, cups, and boxes. Eventually, kids begin to re-unitize and can
solve them mentally. The problems can also extend to boxes per crate. I can see
a relationship with this task to factoring and even prime numbers as well
as fractions and folding problems!
