A r c h i v e d  I n f o r m a t i o n

Toward a New Science of Instruction: Programmatic Investigations in Cognitive Science and Education--August 1993

Thinking Through Mathematics

Concerns of Research: Connections Between Learning and Understanding in Arithmetic

Mathematics as a discipline encompasses many forms of reasoning. Interpreting a word problem, judging the best way to express it in formal mathematical notation, performing the necessary steps in calculation, and then assessing the appropriateness of the answer to the situation in the problem--all these involve somewhat different problem-solving skills. The learning of mathematics, then, has great potential to foster the higher-order thinking skills that are increasingly demanded of instruction. But if mathematics is taught only as a series of mechanical procedures, and not as a rich field of inquiry, instruction will usually yield--at best--speed and accuracy of calculation but little grasp of underlying principles. Generations of children in the United States have begun their mathematics studies in strictly traditional classrooms that use just this approach. Today, many students are still taught that there is only one correct way to perform each basic arithmetic operation, and they are required to work quietly, by themselves, on repetitive drills designed to indelibly imprint correct procedures.

Although cognitive research has yielded much evidence that a different approach--one which allows children to experiment, discuss, and question--is more effective, some children in traditional settings have nevertheless done very well, even going on to excel as mathematics researchers or teachers. Far too many other children have done poorly, however. They have never understood the motivating principles that lie beneath their calculations, have never seen a connection between math and their lives outside the classroom, have never appreciated the patterns and symmetries of numerical relationships.

With the problem-solving and reasoning skills that math can instill becoming more important, and with test scores of U.S. students lagging far behind those of children in other countries, it has become a pressing concern of education research to discover why arithmetic is so difficult for so many children and what can be done about that difficulty. As one researcher at NRCSL has pointed out, early math is no harder than mastering many video games, yet too many students who excel with a joystick in their hands go blank when it is replaced by a pencil or a piece of chalk in math class.

It is important that the effort to replace the blankness with enthusiasm and comprehension be a balanced one. The fact that some children learn well in some traditional settings has led to strands of research at NRCSL that ask just what it is that works in these settings--as well as what can go wrong. Other research projects investigate promising, newer approaches to mathematics instruction, emphasizing full comprehension of both principles and procedures in arithmetic. But whether researchers focus on refining or on substantially replacing traditional methods of instruction, their projects all ask: "How should arithmetic be taught?" And in this sense they all challenge, inform, and stimulate each other. In addition, they all investigate, to varying degrees and from divergent perspectives, most of the major influences on knowledge acquisition and construction that have been identified by cognitive research. These influences include, as we have noted earlier, the nature and role of existing knowledge, the coherence of learners mental representations of knowledge, students and teachers metacognitive processes (which help them to diagnose and repair learning difficulties), and the many features of the settings in which learning takes place.
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[Introduction ] [Table of Contents] [What Makes Counting Easy, Subtraction Hard?]