Too little time. Too many conceptual gaps. Here we are three weeks into a summer program with students. The young people who appear each day exhibit a huge disparity of skill levels and yet most are woefully unprepared in predictably obvious areas: fractions, decimal operations, estimation, multi-digit multiplication and of course. . .long division. Most of the students have been taught multi-step procedures with few links to the underlying concepts. They demonstrate, individually and collectively, the research supporting an emphasis on numeracy education. They cannot visualize quantity, therefore they cannot apply that knowledge to a useful purpose.
They love games which reinforce patterns. They cut their own fraction manipulatives. They illustrate multi-digit multiplication with place value objects and various representational pictures. They learn a few selected number facts and patterns which aid word retrieval. They apply all of the above to algebra. They simultaneously work on multiplication, division, fractions and solving equations. They perform mental math with decimal fractions as they apply language which supports visualization. "How many pieces the size of one fourth can I make if I start with three and one half?"
We play with dice. We use pictures of pizza drawn on dry erase boards. They chant the perfect squares. They learn the many appearances and applications of the number one. They are a joy to teach.