The Effect Of External Representation And Working Memory On Primary School Mathematical Problem Solving: An Empirical Study

Problem representation is a means of representation that individuals present andbehave the problem in the mind. It is the first stage of problem solving in the basiccognitive activity, essentially it is a recording form of the information of problemsolving that input to people’s mind, and it is a kind of mental representation. At themacro level it is generally considered that a representation of subject matter can bedivided into the external representation and the internal representation. Externalrepresentation refers to the subject matter content and structure of itself and thequestion of the situation, where the individual perceptual system can detect, analyzeand process the information; the internal representation referred to individual’sinternal understanding of the initial condition problem, constraint conditions and theways to solve the problem, which is also the cognitive structure related to theproblems built in the individual’s mind. To solve the mathematic word problems, firstwe understand the problem and then collect integrate and process the informationincluding the initial conditions, relationships among the numbers and questions withinthe problems and finally achieve an in-depth understanding of the problem. Thus, it ismathematic problem representation. In this process, first of all students will readproblems; second, they understand the problem (external representation) according tothe three elements of the mathematic word problems (the variable name, key wordsand numbers), and then they build problems related to the frame with the aid ofrelated knowledge and experience to further understand the problem in their mind(internal representation), finally students can understand the problem thoroughly andaccurately, which helps them get the problem answered successfully.Working memory is a platform of information processing, and it is one of theadvanced cognitive activities for human. The researchers do further study on working memory theory, cognitive mechanism, measurement methods and other aspects ofworking memory, of which there are some research results relevant to mathematicalproblem solving with working memory. However, in the primary school, the study onthe relationship between mathematic word problems solving and working memory,the researchers have not reached consensus while the combination of the two factorsabove has gradually become a hot issue for researchers. On the basis of the previousresearches, the author in this study self-makes a set of mathematic word problemmaterial according to literature and teaching actual situation. It examines the influenceof external representation, problem representation and working memory on primaryschool mathematics word problem solving.This research consists of three experiments. the first experiment compared howthree kinds of external representation forms (decorative pictures, informationalpictures and verbal description representation) respectively take effects onmathematical problem solving in primary school; in the second experiment, it furtherdiscussed the effect of two ways of problems representation on external representation;the third experiment respectively studied the effect of the two subsystems of workingmemory (phonological loop and the visual-spatial sketchpad) on mathematics wordproblem solving in primary school. Conclusions are drawn as follows:1three external representations, namely decorative pictures, and verbaldescription representation influence students’ word problem solving performance todifferent extents, among which the performance with the informational picturesrepresentation is better than that with decorative pictures, and verbal descriptionrepresentation.2the form of problem presentation has no effect on mathematical word problemsolving. Under Either dynamic conditions, or static condition, their representationresults are not significant difference.3Phonological loop has effect on primary school mathematical problem solvingwhich has different representation types，the results show that the representationperformance of high verbal working memory span is better than low verbal workingmemory span significantly. 4visual-spatial sketchpad has effect on difference representation types ofprimary school mathematical problem solving, manifested as the higher visual andspatial working memory span is, the better mathematical problem representationperformance is; the lower span is, the worse performance is.The study offers some guidelines for primary school mathematics teaching.educators in the actual teaching process can employ a flexible, diverse ways to guidethe students to master the knowledge of mathematics, and help them to adopt effectivemethods to understand the application problem. At the same time they can make useof students’ working memory characteristics to teach content organization, which willhelp to impart knowledge to students and cultivate students’ abilities of solvingproblems.