Arithmetic and Cognitive Contributions to Algebra
Author: Cirino, Paul T.
Date: 2013-00-00
Type: Reports - Research
Subjects: Arithmetic,Algebra,Mathematics Skills,Cognitive Processes,Predictor Variables,Mathematics Achievement,Secondary School Mathematics,Middle School Students,Mathematics Instruction,Quasiexperimental Design,Mathematics Tests,Statistical Analysis,Mathematical Concepts,Vocabulary,Short Term Memory

Algebra is a prerequisite for access to STEM careers and occupational success (NMAP, 2008a), yet algebra is difficult for students through high school (US DOE, 2008). Growth in children's conceptual and procedural arithmetical knowledge is reciprocal, although conceptual knowledge has more impact on procedural knowledge than the reverse (Rittle-Johnson &amp; Alibali, 1999; Rittle-Johnson et al., 2001). However, research produce a complete picture of whether and how algebra procedural skill and conceptual knowledge are distinct versus related. Arithmetic skills are likely strongly related to algebra, and cognitive processes influence algebra directly or indirectly through arithmetic. Relations between cognition and arithmetic are more established relative to what is known about how either influence algebra. The broad purpose is to identify skills important for algebra with the ultimate goal of improving algebra preparedness. This presentation focuses on 3 inter-related components: (a) predictors of procedural and conceptual algebra performance (School Algebra); (b) predictors of algebraic precursor skills such as arithmetic concepts, fractions, and proportional reasoning (Skills); and (c) experimental manipulation of procedural and conceptual algebraic skill (Experimental). The studies took place in diverse intermediate/middle schools in Texas and Tennessee. This group of studies sought to identify cognitive concomitants of arithmetic, as well as cognitive and arithmetic predictors of algebra, and to examine the manipulation of procedural and conceptual instruction. Cognitive predictors of arithmetic in Grade 6 include language (vocabulary), line estimation, magnitude estimation (symbolic and nonsymbolic), and working memory. The dominant predictors were clearly line estimation and symbolic magnitude estimation. Both involve identification and mapping of symbolic information, quickly, which is consistent with the nature of written arithmetic, as well as the timed nature of many of these measures. For the Algebra study, vocabulary was a significant cognitive contributor to algebra skills, but remained so even when arithmetic predictors were included. Arithmetic concepts and fractions were dominant arithmetic predictors. Even when pretest performance was included, the same pattern of predictive variables remained relevant. For the Experimental study, results suggest that procedural and conceptual knowledge are partially separable. Students may acquire procedural knowledge without conceptual knowledge, but may not necessarily acquire conceptual knowledge without also acquiring some procedural knowledge. However, the evidence also suggests that the order in which students receive procedural and conceptual instruction (i.e., all procedural followed by all conceptual, all conceptual followed by all procedural, or procedural and conceptual together) does not affect the overall procedural or conceptual learning. Tables and figures are appended.