Underlying Cognitive Components and Conceptual Knowledge in Arithmetic
Edwards, William Tomos
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Within the field of mathematical cognition there is extensive research on conceptual knowledge of arithmetic operations. There is also extensive research on the link between mathematical ability and spatial ability. This research study seeks to build on both areas of research and identify ways in which they are interrelated. Conceptual knowledge of arithmetic operations is the subject of ongoing research. When solving a three-term problem of the form a x b ÷ b, those who understand the inversion concept do not need to perform any calculations because they know that the multiplication and division operations cancel each-other out. When solving a three-term problem of the form a x b ÷ c, those who understand the associativity concept know that they can do b ÷ c first, or a x b first. Research indicates that there is a complex relationship between spatial ability and mathematical ability. In some studies spatial ability is shown to have an especially strong relationship with certain measures of mathematical performance while in other studies this is not the case. Theories have already been put forth that visual-spatial abilities are initially cardinal to learning mathematics in children, but verbal and general intelligence become more important to mathematical performance later on, after these mathematical skills and forms of knowledge have been well learned. In this study it is theorized that spatial abilities are more important than other cognitive abilities for acquiring new mathematical knowledge across the lifespan, and not just in childhood. Conversely, general intelligence is more important to mathematical performance after the relevant mathematical knowledge has been well learned. This theory is supported by past research as well as the results of this study. This study also provides important clues about the development of conceptual knowledge of arithmetic by showing that knowledge of the inversion concept and the associativity concept are both strongly related to spatial ability. Verbal reasoning ability doesn’t relate to knowledge of these concepts but it is related to performance with mathematical skills that are more basic.