In 1999, I was working as a special educator in an inclusive elementary school. My ideas about mathematics instruction were old-fashioned.  A few of my fifth-grade students had particular difficulties in multiplication, and it was hard to see what they could do—I was so focused on what they couldn’t do.   That summer I attended the Summer Institute at Mathematics in the City at the City College of New York.  One (of many) ideas that transformed my work was the idea of the landscape of learning.  Cathy Fosnot and Martin Dolk (Fosnot & Dolk, 2001) developed this idea from the concept of developmental trajectories (Simon, 1995), which detailed the complex paths through which children  learned concepts like multiplication.  This helped tremendously in my work as a special educator in mathematics.  Instead of seeing a child who had difficulty multiplying two digit numbers as a child who couldn’t multiply, I was able to see the complex set of strategies and big ideas that the child could use.  Perhaps that same child can easily double numbers using a ratio table.  Could I design a problem that would help the child move beyond doubling to multiplying by tens? The landscape of learning helped me see where a child was, and where my next moves were as a teacher.

Last summer I taught a section on multiplication and division at the Summer Institute.  The image that began this post was collectively built during that workshop as the participants explored various big ideas, strategies, and models for multiplication and division.  The landscape of learning frames all children’s mathematical development as dynamic and unique. And it proves to be a critical frame for differentiating learning.