What We Know

Children need a conceptual understanding of operations and the effect of operations on numbers as well as computational proficiency. Traditionally, teaching of operations (addition, subtraction, multiplication and division and the effect of these operations) has been limited to a focus on drill and practice, procedures and algorithms. While drill and practice may enable students to compute routine problems, it does not provide a foundation for students encountering non-routine problems (Markovitz & Sowder, 1994). For this reason, developing a conceptual understanding is imperative.

Research has shown that developing conceptual understanding results in higher achievement in standardized tests and tests of conceptual understanding (Carpenter, Franke, Jacobs, Fennema, & Empson, 1998; Fuson, Smith, & Lo Cicero, 1997; Woods & Sellers, 1997). Strong gains are seen in conceptual understanding and use of calculation methods when students are actively involved in activities that make mathematics meaningful (Fuson, Smith, & Lo Cicero, 1997; Hiebert & Wearne, 1993).

Students move through a fairly well-defined sequence of solution methods when they learn to perform operations with single-digit numbers. Connections between addition and subtraction and between multiplication and division can make the learning of subtraction and division combinations easier (National Research Council, 2001).

Computational proficiency means being fluent with the procedures of adding, subtracting, multiplying and dividing mentally or on paper and knowing when and how to use these procedures appropriately. Fluency includes being proficient with operations across mathematical content areas.

Students need to compute basic number combinations quickly and accurately. They also need to be accurate and efficient with algorithms--step by step procedures for adding, subtracting, multiplying and dividing multi-digit whole numbers, fractions and decimals. For example, all students should have an algorithm for multiplying 64 X 37 that they understand, that is reasonably efficient, that is general enough to be used with other two-digit numbers and that can be extended to use with larger numbers (National Research Council, 2001).

With conceptual understanding, primary-aged children can learn to solve number problems in ways that make sense and solve problems far more difficult than those we have traditionally presented to them (Carpenter, Fennema, Franke, Levi, & Empson, 1999).