The Expanding column on the continua addresses children’s development during the ages of 7, 8 and 9. We know that children process and assimilate information at different rates; thus, it is typical to see the skills listed in this column demonstrated by children in this age range.

Students who are currently operating in this column are deeply involved with learning the operations for subtraction, multiplication, division and fractions. They use manipulatives with varied levels of complexity to demonstrate their thinking and prove their hypotheses. They are also learning the traditional algorithms and the way in which the manipulatives illustrate each algorithm learned. They draw detailed models of their work that include labels and write reflections about their process and the overall task. Children working at this level are comfortable assigning representational variables to each manipulative and explaining those assumption to others. These students are able to listen to their peers and offer feedback to them based on what they hear and see.

Socially, students in this column have moved out of concrete learning. They thrive in talking about what others think and feel. They can follow another persons thinking and respond reflectively. Math discussions at this level are highly interactive