The Integration of internal and External Information in Numerical Tasks

Numerical tasks with Arabic numerals involve the integration of internal and external information and the interaction between perception and cognition. 2-digit number comparison task was selected to study these integration and interaction processes. To compare the magnitudes of two 2digit Arabic numerals, we can (1) compare them digitbydigit sequentially, (2) compare corresponding digits in parallel, or (3) encode them as an integrated representation and compare the whole numerical values. Previous studies showed that 2digit comparison was holistic when target numerals were compared with a standard held in memory. In our experiment target numerals and standards were presented on the same external display at the same time. Instead of a holistic comparison, we found that 2-digit comparison was a combination of sequential and parallel comparisons. The implications of this discrepancy were discussed in terms of the interplay between perception and cognition. Although different types of numerals (e.g., Arabic, Roman, Greek, etc.) all represent the same abstract quantities— numbers, they can produce dramatically different cognitive behaviors in numerical tasks (e.g., Nickerson, 1988; Zhang & Norman, in press). This representational effect, caused by different representations of a common structure, can be easily observed by comparing the difficulties and processes of two multiplication tasks: 735 × 278 (Arabic numerals) and DCCXXXV × CCLXXVIII (Roman numerals, equivalent to 735 × 278). In addition, to perform numerical tasks, people usually need to process information distributed across the internal mind and the external environment in an interactive manner (Zhang & Norman, 1994, in press). For example, to do 735 × 278 with paper and pencil, we need to process not just the information in internal representations (e.g., the value of each individual symbol, the addition and multiplication tables, arithmetic procedures, etc.) but also the information in external representations (e.g., the visual and spatial properties of the symbols, the spatial relations of the partial products, etc.). In this paper, we use a simple numerical task, number comparison, to examine how internal and external information is processed and integrated in numerical tasks and what effects such interactions and integrations have on behavior. Number Comparison The time to compare the magnitudes (larger or smaller) of two 1-digit Arabic numerals decreases with the numerical distance between them (Moyer & Landauer, 1967). For example, it is faster to compare 1 and 9 than 8 and 9. This distance effect resembles that found for physical stimuli such as dot patterns and line lengths, implying that Arabic digits might have an internal representation analogous to a physical continuum. Multidigit Arabic numerals have two dimensions: a base dimension represented by the shapes of the ten digits (0, 1, ..., 9) and a power dimension represented by positions of the digits (Zhang & Norman, in press). The base dimension is an internal representation because the numerical value of each digit has to be memorized, whereas the power dimension is an external representation because the position of a digit can be perceptually inspected. Unlike 1-digit comparison, which is solely internal, multidigit comparison is based on both internal information (the values of individual digits) and external information (positions of digits). Thus, multidigit comparison is a good task for the study of the integration of internal and external information and the interaction between perceptual and cognitive processes. To compare the magnitudes of multidigit Arabic numerals, we can (1) compare them digit by digit sequentially, (2) compare corresponding digits in parallel, or (3) encode them as an integrated representation and compare the whole numerical values. In the first case (sequential comparison), only the highest digits should affect the comparison unless they are not sufficient for making decisions. In the second case (p a r a l l e l comparison), lower digits may facilitate or interfere with the comparison of higher digits. In the third case (holistic comparison), only the absolute numerical distances should matter. Empirical studies have revealed a discrepancy between 2-digit comparison and higher multidigit (3 or more) comparison: 2-digit comparison is holistic (Dehaene, Dupoux, & Mehler, 1990; Hinrichs, Yurko, & Hu, 1981) whereas higher multidigit comparison is sequential (Hinrichs, Berie, & Mosell, 1982; Poltrock & Schwartz, 1984). The holistic comparison of 2-digit Arabic numerals was often cited as evidence of a holistic analog internal representation of Arabic numerals. This paper addresses two specific issues: (1) whether 2digit comparison is indeed holistic and (2) how internal and external information is processed and integrated in 2-digit comparison tasks.