Does Practice and Knowledge Equal Knowledge and Practice

Does Practice and Knowledge Equal Knowledge and Practice? Miriam Rosenberg-Lee (mrosenbe@andrew.cmu.edu) Department of Psychology Carnegie Mellon University Pittsburgh, PA 15213 USA Marsha C. Lovett (lovett@cmu.edu) Department of Psychology Carnegie Mellon University Pittsburgh, PA 15213 USA Abstract Debate surrounds the value of procedural practice in learning conceptual material in mathematical domains (Schoenfeld, 2004). We investigated whether purely procedural practice could lead to conceptual gains and explored cognitive load theory as a mechanism for those gains (Sweller, 1988). In a laboratory experiment, 93 undergraduates practiced a procedure by solving 30 problems of an algebra analog and were given conceptual tests before, during and after practice. The conceptual tests tapped students’ understanding of the underlying structure of the domain. Participants’ conceptual knowledge increased with procedural practice, particularly between no practice and some practice. Consistent with cognitive load theory performance on the conceptual test after practice was significantly related to procedural performance at the end of practice. However, this relationship between procedural learning and conceptual learning only held if participants had been alerted earlier in practice to the conceptual nature of the task. These results are consistent with the proposal by Rittle-Johnson, Siegler, & Alibali, (2001) that there is an iterative relationship between conceptual understanding and procedural skill. Introduction The role of practice in mathematics is a matter of intense controversy in the current curriculum reform debates. The National Council of Teachers of Mathematics’ 1989 Curriculum and Evaluation Standards for School Mathematics launched the present debate by advocating for, among other reforms, the de-emphasis of rote practice and rote memorization of rules and algorithms (Schoenfeld, 2004). Consistent with this approach were the opinions of education researchers such as Robert Davis who wrote in If “mathematics” is seen as conformity to memorized rituals, if it is taught without meaning … if meaninglessness compels a slow pace and a vast investment in repetition, and if routine calculation is the main goal, very little mathematics will be included in the curriculum (pp. 272-273). However, this shift of emphasis did not seem warranted to all parents and an anti-reform movement developed, advocating for back to basics. The ‘traditionalists’ actively sought mathematicians to support their side, having them send letters to state decision makers (Schoenfeld, 2004). David Ross, a mathematician at Kodak Research Labs states: “The best way to advance students’ conceptual thinking about mathematics is to have them master the traditional algorithms” (2001, ¶13). The traditionalists contend that practice is a necessary prerequisite without which deep understanding of mathematical concepts is impossible. Conversely, reformers fear that emphasizing practice can only lead to superficial understanding of mathematical concepts. Clearly with such opposing perspectives, consensus may be difficult to reach. Even before the debate reached its current level, Schoenfeld (1994) suggested that these issues must be settled through empirical research. He argued that a particular question worth investigation is: “how much mastery of some basics is required for competent, flexible performance on more demanding tasks. ” (¶ 19) That phrasing of the question highlights the underlying goal of both sides of this debate, namely, competent flexible performance on ‘demanding’ tasks. Competent and flexible performance requires two different kinds of proficiency: competent performance implies that calculations are fast and effortless; whereas, flexible performance suggests not just solving familiar problems but also being able to tackle novel problems built on the same principles. These two proficiencies can be likened to the distinction between procedural and conceptual knowledge. According to Rittle-Johnson, Siegler, & Alibali, (2001), procedural knowledge is “the ability to execute action sequences to solve problems” (p. 346) and conceptual knowledge is “an implicit or explicit understanding of the principles that govern a domain and of the interrelations between units of knowledge in a domain” (p. 346-7). For example, in learning multiplication, a student could understand the concept that multiplication is equivalent to multiple additions, and this knowledge could be independent of knowing the procedure “add number A to itself B times. ” Thus, the debate can be reformulated as: does procedural practice result only in procedural improvements, or can it also lead to conceptual learning?