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# Dedicated to Teachers

Learning arithmetic facts well takes years. Adults use retrieval in only a fraction of situations. But even such retrieval isn't simply accessing memorized information, because many people use cover strategies, quickly and unconsciously calculating an answer based on related facts or other reasoning strategies.

Meaningful and effective learning of number facts involves three phases: (1) building foundational concepts of number and arithmetic and learning to figure out simple facts with counting and visually based strategies, (2) learning reasoning strategies to determine facts more efficiently, and (3) achieving full fact fluency.

Let’s consider each in turn:

## Building Foundations and Initial Strategies

Students with greater conceptual knowledge are more likely to use sophisticated strategies and retrieve facts accurately. With more positive beliefs, early childhood teachers can do much to build strong foundations and initial arithmetic strategies. They can ensure that children can count with understanding and apply counting strategies to solve problems.

## Learning Reasoning Strategies

Effective teachers attend explicitly and directly to the important conceptual issues students are more likely to encounter. They help students develop important conceptual understandings. Research is clear that effective strategies include counting strategies, conceptual subitizing, and break-apart-to-make-ten.

## Achieving Full Fact Fluency

To achieve full fact fluency, children must practice their arithmetic strategies. This is not drill but repeated encounters combining different ideas, and experiences for learning and internalizing them. Children might engage in whole-group, individual-within-whole-group, and independent practice.

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Linda Ruiz Davenport is the Director of K–12 Mathematics for Boston Public Schools and supports mathematics teaching and learning district-wide.

Connie S. Henry is an Assistant Director of K–12 Mathematics for Boston Public Schools. She has taught and coached math for many years.

Douglas H. Clements is the Kennedy Endowed Chair in Early Childhood Learning and Distinguished University Professor at the University of Denver.