Influence of Problem Representation and Working Memory Span on Pupils’ Mathematical Problem Solving
2011, 43 (11):
Since 1980’s systematic researches have been made on mental processes, influencing factors and psychological models of problem solving. The researches found that problem context, problem representation types, mental preparation, working memory, cognitive styles, and self-evaluation are the main factors for problem solving (Lu haidong, 2004; You xuqun, 2006; Chen yinghe, 2004; Beckmann et al., 2007). Some researchers pointed out that problem representation and problem solving plans are the basic inner processes (Kintch & Greeno, 1985; Greeno, 1986). Along with deeper studies, direct translation strategy and problem-model strategy (Mayer, 1996) were considered as the main representation types, and working memory as the major factor for problem representation type (Baddeley, 1992; Cornoldi, 1999; Mclean, 1999; Wang enguo, 2007; Wanyan, 2007). However, the past studies paid much attention to how verbal working memory affected problem representation, while there were few studies on how visual-spacial memory span influenced problem representation in mathematics. As a hypothesis, we thought that it was the individual differences in working memory that resulted in different problem representation in mathematics, which led to different performances in mathematical assignments. Those with a high spacial working memory span, based on an overall comprehension of the problem, could construct a good problem representation, and would show superiority in high speed and accuracy in mathematical problem solving. Those with a high verbal working memory span, might display an advantage in comprehension of mathematical problems. However, compared with the high visual-spacial memory span group, they might have different problem representation. This study was especially concerned about whether there were differences in problem representation and problem solving between the two groups with different working memory.
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This study included four separate experiments. 104 Grade 6 pupils from primary school (male, 61, female, 43, Mage=11.26±0.56) participated in the study. Four tests were carried out, concerning mathematical problems, problem representation, verbal working memory span and visual-spacial working memory span respectively. All subjects were tested by mathematical works and cognitive behavior.
By using one-way analysis of variance and χ2 test, this study found:
1. Verbal working memory span showed a main effect in high level difficult word problem solving (F (1,102)=5.78, p<0.05), and had no main effect in low and middle level difficult word problem solving (F (1,102)=0.21, p>0.05; F (1,102)=0.58, p>0.05).
2. Visual-spacial working memory span showed a main effect in the three level difficult word problem solving (F (1,102)=8.13, p<0.01; F (1,102)=4.59, p<0.05; F (1,102)=11.01, p<0.01).
3. The mathematical scores were correlated with problem representation. When solving the three level difficult problems, the high score group had a tendency to choose problem-model strategy (82.0%; 81.9%; 55.8%). There were significant differences in problem representation in the three level difficult problems between the high and low score group.
4. There were no relations between verbal working memory span and problem representation types (low difficult problem: χ2(2)=2.31, p>0.05; middle difficult problem: χ2(2)=2.21, p>0.05; high difficult problem: χ2(2)=2.95, p>0.05). When solving the three level difficult problems, both the high and low verbal working memory span groups tended to choose problem-model strategy (high group: 65.7%; 63.9%; 46.6%; low group: 71.3%; 66.3%; 42.9%), and there were no significant differences between the high and low verbal working memory span group in selecting problem representation types.
5. There were correlations between visual-spacial working memory span and problem representation types (low difficult problem: χ2(2)=16.44, p<0.001; middle difficult problem: χ2(2)=18.16, p<0.001; high difficult problem: χ2(2)=8.64, p<0.01). In solving the three level difficult problems, there were significant differences between the high and low visual–spacial working memory span group in selecting problem representation types, and the high visual-spacial working memory span group more likely chose problem-model strategy (75.4%; 72.4%; 49.9%).