From Concrete to Abstract: Making Sense of Math in Lower School

Mathematics isn’t simply a set of methods; it is a set of connected ideas that need to be understood. When students understand the big ideas in mathematics, the methods and rules fall into place. -Jo Boaler, Professor of Education at Stanford University

Math is an important component of the Lower School experience. Beginning in Preschool, students explore mathematical ideas daily. Lower school students become mathematical thinkers through carefully scaffolded experiences and a rich vertical curriculum that includes regular opportunities for problem solving, making connections, and communicating ideas. Teachers strive to create classroom cultures where learners are persistent, open to challenges, and curious about how and why math works the way it does. Mistakes are accepted as part of the process, and students are encouraged to take risks with mathematical concepts and strategies.

Lower School teachers regularly introduce new mathematical ideas using the Concrete, Pictorial, Abstract (CPA) approach, based on research by American psychologist Jerome Bruner. Bruner determined that mathematical content is best learned when it follows a progression from concrete, hands-on experiences, to pictorial representations linked to the concrete learning, before finally reaching the abstract, or symbolic, stage. Progression from one stage to the next is purposefully linked to ensure deep and enduring understanding of concepts.

An example of the CPA approach is the development of number sense, a major focus in early math. Number sense is the ability to recognize numbers, identify their relative values, and understand how to use them in various ways. Simply put, it is fluidity and flexibility with numbers. Children are introduced to number sense in preschool through a variety of experiences counting concrete objects that they can hold or touch. They then progress to counting and drawing picture representations of concrete objects. Finally, they are introduced to the abstract number symbols that represent amounts of corresponding objects or pictures. Although CPA is described as three distinct stages, teachers naturally go back and forth between each stage to reinforce concepts.

Lower School teachers understand the importance of providing enriching, engaging mathematics for all students throughout their years in Lower School. Traits such as perseverance and curiosity are intentionally developed alongside a carefully designed, vertically connected curriculum from preschool through Class V.