It is very clear that effective mathematics instruction begins with effective teaching. But over the course of history, effective mathematics teaching has been defined in many ways. In the early half of the 20th century, proficiency was defined by facility with computation, while in the later half of the century, the standards-based movement emphasized problem solving and reasoning. Such debate has often been acrimonious and has led to many false beliefs about successful mathematics teaching. At the turn of the 21st century, however, the National Research Council published Adding It Up: Helping Children Learn Mathematics (NAP, 2001) in which it defined mathematical proficiency as having five interwoven components.
1. CONCEPTUAL UNDERSTANDING
Conceptual understanding “reflects a student’s ability to reason in settings involving the careful application of concept definitions, relations, or representations of either.” With conceptual understanding, students are able to transfer their knowledge to new situations and contexts in order to solve the problem presented. It is this transfer of knowledge that is so vital for success not only in mathematics but in all disciplines and in the workplace. The authors of Principles and Standards for School Mathematics (NCTM, 2000)summarize it best2: “Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge.”
2. PROCEDURAL FLUENCY
In a position page on procedural fluency, the National Council of Teachers of Mathematics (NCTM) defines procedural fluency3 as “the ability to apply procedures accurately, efficiently, and flexibly; to transfer procedures to different problems and contexts; to build or modify procedures from other procedures; and to recognize when one strategy or procedure is more appropriate to apply than another”.
It should be noted that procedural fluency is more than memorizing procedures and facts. Procedural fluency builds on the foundation of conceptual understanding, so knowledge of procedures is no guarantee of conceptual understanding. For example, many secondary students learn to use the “FOIL” routine for the multiplication of binomials, without realizing that multiplying two binomials is a function of the distributive property.
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3. STRATEGIC COMPETENCE
Strategic competence is the ability to “formulate mathematical problems, represent them, and solve them.” While some may see this strand as similar to what has been called problem solving and problem formulation in mathematics education, it is important to point out that strategic competence involves authentic problem solving—problems for which students must formulate a mathematical model to represent the problem context and then determine the operations necessary to come up with a viable solution. Learning to solve these authentic problems is the essence of mathematics and developing such ability should be the primary goal of mathematics teaching. Many would argue that a primary goal of mathematics teaching and learning is to develop the ability to solve a wide variety of complex mathematics problems. Thus, mathematics instruction should be designed so that students experience mathematics as problem solving.
4. ADAPTIVE REASONING
Adaptive reasoning is “the capacity to think logically about the relationships among concepts and situations.”Adaptive reasoning is the “glue that holds everything together, the lodestar that guides learning.” The importance of adaptive reasoning cannot be understated. Students with adaptive reasoning can think logically about the math and they can explain and justify what they are doing.
5. PRODUCTIVE DISPOSITION
Productive disposition is “the tendency to see sense in mathematics, to perceive it as both useful and worthwhile, to believe that steady effort in learning mathematics pays off, and to see oneself as an effective learner and doer of mathematics….Developing a productive disposition requires frequent opportunities to make sense of mathematics, to recognize the benefits of perseverance, and to experience the rewards of sense making in mathematics.” 7 This balance of all five components is crucial to successful and effective mathematics teaching and ultimately, to teaching for student understanding. An effective mathematics program must focus on building students’ mathematical proficiency by helping them develop these five critical components.
1 NAEP (2003). What Does the NAEP Mathematics Assessment Measure? Online at nces.ed.gov/nationsreportcard/mathematics/abilities.asp.