When the teacher moves to a 6x6 grid...

students have an opportunity to apply their thinking to another situation (which is essential if the lessons learned are to stick), and to generalize to a grid of any size. Students thus enter the algebra territory with the benefit of having talked through a problem so thoroughly that they are able to make sense of algebraic relationships. And they have the opportunity to appreciate the beauty of mathematics: because an equation will work for a grid of x size, problem solving becomes much more efficient. 

This same approach of using students’ varied strategies to build flexible reasoning can be used in mathematics classrooms at all levels. By starting in the earliest grades, students’ appreciation of mathematics and identity with respect to mathematics competencies can be developed right from the beginning.