The Cognitive Complexity of Learning and Doing Arithmetic

Abstract

A theory of the cognitive processes involved in doing and learning place value arithmetic is proposed. The theory is embodied in a computer model that simulates the learning of multicolumn subtraction under one-on-one tutoring. The model is used to measure the relative difficulty of two different methods for subtraction, with either a conceptual or a procedural representation. The model predicts that regrouping is more difficult to learn than an alternative method, particularly in a conceptual representation, a result that contradicts current practice in U.S. schools. The notion of a problem or a topic being difficult to solve or to learn is a key organizing concept in the design of mathematics instruction. Arguments concerning the sequencing of the curriculum frequently turn on the notion of difficulty. Novel ways of teaching particular mathematical topics are typically promoted with claims that they make those topics less difficult to learn. A series of lessons on a mathematical topic is likely to confront students with practice problems of increasing difficulty. The difficulty of a topic is not the only factor that determines a successful instructional design, but it is an important one. Difficulty is a quantitative concept. The entire point of the concept is to be able to say that something (a concept, a problem, a topic) is more or less difficult than something else. However, until recently we had no way of measuring or quantifying cognitive difficulty. If two researchers disagreed whether one practice problem was more difficult than another, they could only appeal to intuition. There was no objective method to settle such disputes. The development of measures in other sciences suggests that a precise, quantifiable concept of difficulty must be based on a theory of the underlying cognitive processes. For example, the measurement of heat had to await the modem concept of energy and the atomic theory of matter. Similarly, a measure of pressure requires the distinction between mass and weight, and measures of current and voltage are impossible without a theory of electricity. The purpose of this article is to demonstrate that our current theory of cognitive processes is powerful enough to support a precise definition of difficulty as well as a technology for measuring this quantity. The past 30 years have seen the emergence of a theory of cognitive processes based on information processing concepts (Gardner, 1985). Briefly put, this theory claims that complicated processes such as those involved in learning and thinking