Conceptual Understanding

Conceptual understanding is a phrase used extensively in educational literature, yet one that may not be completely understood by many K-12 teachers. A Google search of the term produces almost 15 million entries from a vast arena of subjects. Over the last twenty years, mathematics educators have often contrasted conceptual understanding with procedural knowledge. Problem solving has also been in the mix of these two. A good starting point for us to understand conceptual understanding is to review The Learning Principle from the NCTM Principles and Standards for School Mathematics (2000). As one of the six principles put forward, this principle states: Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge. For decades, the major emphasis in school mathematics was on procedural knowledge, or what is now referred to as procedural fluency. Rote learning was the norm, with little attention paid to understanding of mathematical concepts. Rote learning is not the answer in mathematics, especially when students do not understand the mathematics. In recent years, major efforts have been made to focus on what is necessary for students to learn mathematics, what it means for a student to be mathematically proficient. The National Research Council (2001) set forth in its document Adding It Up: Helping Children Learn Mathematics a list of five strands, which includes conceptual understanding. The strands are intertwined and include the notions suggested by NCTM in its Learning Principle. To be mathematically proficient, a student must have: • Conceptual understanding: comprehension of mathematical concepts, operations, and relations • Procedural fluency: skill in carrying out procedures flexibly, accurately, efficiently, and appropriately • Strategic competence: ability to formulate, represent, and solve mathematical problems • Adaptive reasoning: capacity for logical thought, reflection, explanation, and justification • Productive disposition: habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one's own efficacy. As we begin to more fully develop the idea of conceptual understanding and provide examples of its meaning, note that equilibrium must be sustained. All five strands are crucial for students to understand and use mathematics. Conceptual understanding allows a student to apply and possibly adapt some acquired mathematical ideas to new situations.