According to Jean Piaget, children must go through certain developmental stages. No stages are skipped by anyone. Those stages in math are “concrete” (hands-on objects being manipulated), “symbolic” (pictures or tally marks representing concrete objects), and “abstract” (numerals that bear no visual relationship to concrete objects). Piaget stated that no one was ready for abstract thinking until at least the age of eleven. However, most math programs expect children to use numerals fluently at a much earlier age. For example, most of our students could tell you that 2 + 2 = 4. They have memorized it by rote. But, they may not understand that if I have 2 concrete objects in one pile and combine them with 2 objects in another pile, I will have one pile with 4 objects.

Project: In the fall of 2001, I had a class of struggling 5th grade math students. I decided I needed to go back and help my students through the levels of math development from the beginning. So, we started teaching every concept using hands-on manipulatives with simpler problems before we worked our way through tally marks and finally numerals. With a concrete foundation laid, I started seeing the lights come on.

Results for 21 Fifth Grade Students: (There were actually more students, but only 21 were there all year.)

NWEA Level Math Test*

• Average gain in RIT points* = +13.9 RIT points (5 points was the expected growth.)

STAR Math Test (Advantage Learning Systems)

• Average growth in Grade Equivalent* = +1.51 grades
  

• Average growth in percentile ranking* = +20 percentage points

* See Glossary