PME-NA Part II: Tale of Three Great Tasks

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!