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Symbolic and situational mathematics are the two major representations of mathematical knowledge. Although previous literature has studied the relationship between the two from the perspective of teaching practice, learning effectiveness and behavioural performance, there is still a lack of empirical psychological research on cognitive mechanisms to explore the psychological processes of the two. The current study investigated the relationship between symbolic and situational mathematics by determining the differences in cognitive correlates between the two in fourth-grade children. Their symbolic and situational mathematics abilities were assessed using symbolic and situational enumeration tests under the same conditions. Several types of general cognitive abilities, language processing and academic achievements were also examined. Results showed that both situational and symbolic mathematics are crucial for mathematical achievement. Arithmetic computation is closely correlated with symbolic mathematics, whereas spatial processing and inductive reasoning ability are uniquely correlated with situational mathematics. The results suggest that situational and symbolic mathematics have separate cognitive correlates, which means the two are distinct in terms of psychological processing.

Background: When mathematical knowledge is expressed in general language, it is called verbalized mathematics. Previous studies on verbalized mathematics typically paid attention to mathematical vocabulary or educational practice. However, these studies did not exclude the role of symbolic mathematics ability, and almost no research has focused on verbalized mathematical principles. Aims: This study is aimed to investigate whether verbalized mathematics ability independently predicts mathematics achievement. The current study hypothesized that verbalized mathematics ability supports mathematics achievement independent of general language, related cognitive abilities and even symbolic mathematical ability. Sample: A sample of 241 undergraduates (136 males, 105 females, mean age = 21.95, SD = 2.38) in Beijing, China. Methods: A total of 12 tests were used, including a verbalized arithmetic principle test, a mathematics achievement test, and tests on general language (sentence completion test), symbolic mathematical ability (including symbolic arithmetic principles test, simple arithmetic computation and complex arithmetic computation), approximate number sense ability (numerosity comparison test) and several related cognitive covariates (including the non-verbal matrix reasoning, the syllogism reasoning, mental rotation, figure matching and choice reaction time). Results: Results showed that the processing of verbalized arithmetic principles displayed a significant role in mathematics achievement after controlling for general language, related cognitive abilities, approximate number sense ability and symbolic mathematics ability. Conclusions: The results suggest that verbalized mathematics ability was an independent predictor and provided empirical evidence supporting the verbalized mathematics role on achievement as an independent component in three-component mathematics model.