# Contexts for Learning Mathematics

I’m Cathy Fosnot, I was a professor for many years at City College of New York, I founded the Mathematics in the City, a National Centre of in-service for Math education and the author of various books and articles on mathematics education including the Contexts for Learning Series from Heinemann, and I am the Senior Content Advisor with the academic team in DreamBox learning.

When children are constructing multiplication they often start by needing to build a whole system of ones. So, if the context is “What’s 3 x 6?”, kids will count out 6, count out another 6, count out another 6, make sure they have 3 groups and to get the answer they go all the way back and count from 1 again. Eventually they begin to skip count and use repeated addition, and repeated addition usually builds into kids beginning to realize they can regroup groups. So, something like 12 fifteens, kids will regroup the fifteens putting two together and turn 12 fifteens into 6 thirties. So, this is a developmental pathway to really support children to begin to move from just a system of ones to skip counting to a system of that’s based on regrouping of groups that gets the kids the doubling and halving, and that can build into the use of 10 times, which eventually build into partial products. So, in Dreambox we built digital tools that would ensure that children would move along these developmental pathways. For example, we built a DreamBox tool that would be a Digital Array, we built a Digital Ratio Table, we built a Digital Number Line where we could show 6 + 6 + 6 + 6 as being equivalent to 2 twelve’s. And really building this whole foundation, one of the things that was really important to us in DreamBox was to ensure that we weren’t just focusing on procedures, that we were really building the underline foundation for Algebra.

One of the things we are trying to do in Dreambox is build multiple tools that allow multiple ways into content. And some children choose one over another, like some better than others, do better with some than they do with others. And we also build repertoires that way of multiple ways to represent concepts. In many programs other than Dreambox, they may call themselves adaptive, but usually what they mean is, you do this activity first, the sequence of activities or this unit, and when you’ve finished it, you’ve mastered it, you go to the next one. And if that next one is too hard for you, it puts you back in the one prior again and you repeat it or repeat a variation on it – they call that adaptive. And that’s not intelligent adaption. That’s not providing children with any kind of personalised program. That’s just a feedback loop, that is a linear pathway. What we try to do in Dreambox is something quite different. We’ve tried to develop multiple pathways with multiple tools to build multiple representations. This allows children to actually think they’re choosing where they want to go in the environment and in many ways they are, but we’re also choosing which openings to provide. So, a child maybe has three or four different games that are available to him or her and these are now open, and depending on how a child does with one, the child may then choose to do another, or new doors open. And there’s always multiple games for children to play and multiple places to go in the environment.

Cathy Fosnot, Ed.D.
Professor Emeritus of Education at the City College of New York.
Founding Director of Mathematics in the City, CCNY
Acclaimed author of the “Contexts for Learning Mathematics Series (K-6)”