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Research addressing the strengths and weaknesses of using money as a representation of decimal concepts falls within the general categories of research on teaching rational numbers, fractions, and decimals. This research is linked to the literature examining strengths and weaknesses of real-world problem solving, perceptual richness in contextualized activities, misconceptions about decimals, and forms of representation of decimal concepts. Research that directly addresses money as a representation has been done in two connected areas: (1) money, symbolisation, and language, and (2) money and manipulatives. Theory and empirical findings in both areas suggest a few key strengths and weaknesses of money as a representation of decimal concepts which would be important to take into account for the framework.
Decontextualisation (Gravemeijer, 2002), progressive symbolisation (Enyedy, 2005), concreteness fading (McNeil & Fyfe, 2012), and mathematisation (Freudenthal, 1991, in van den Heuvel-Panhizen, 2003) are all terms referring, in part, to the development of students’ abilities to work with mathematical concepts to greater degrees of abstraction as they become more experienced. Freudenthal considered this a cyclical rather than a purely linear process, with students at any stage building further abstractions from prior abstractions and experiences as they progress (van den Heuvel-Panhuizen, 2003). Using money as a representation of decimals can be helpful, but it also carries particular associations with language that can complicate its use in the classroom.
Steinle (2004) identified ten categories of misconceptions within the three broad error-displaying categories. She linked one of these categories, money thinking, to money directly, while another, string-length thinking, was not linked to money in this study but closely resembles misconceptions around money and decimals noted by other researchers:
There is evidence that money may be a good way to bring students’ existing competencies into decimal learning (Carraher, Sowder, Sowder, & Analúcia Dias, 1988), although the above misconceptions can still be problematic. Existing evidence supports further exploration of the idea that using money as a decimal representation can help students to draw on their informal knowledge; however, few studies have directly compared money to other decimal representations, and these have been small (Martinie & Bay-Williams, 2003; NCTM, 2003).
Money can be used as a manipulative in either a perceptually rich (more detailed and contextualised) or a bland way (less detail may allow for more general application) (McNeil, Uttal, Jarvin, & Sternberg, 2009). Research reviewed by Carbonneau et al. (2013) suggests that perceptual richness in manipulatives carries risks that might interfere with development of a more generalised understanding of targeted concepts. However, both Carbonneau et al. (examining manipulatives in general) and McNeil et al. (focusing on money in particular) reported tradeoffs depending on the degree of perceptual richness. Interestingly, while Carbonneau et al. found that, counter to expectations, high perceptual richness significantly benefited transfer outcomes, but was marginally detrimental to problem-solving and was not as effective for retention as low perceptual richness, whereas McNeil et al. concluded that perceptually rich money manipulatives might interfere with transfer. Overall, more research in this area will be needed to inform the use of money as a manipulative in greater detail.
Babcock LDP. (2016, April 11). The Gaps and Misconceptions Tool - Why do fractions and decimals seem difficult to teach and learn? Retrieved 11 April 2016, from http://www.annery-kiln.eu/gaps-misconceptions/fractions/why-fractions-difficult.html
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Carraher, T. N., Sowder, J., Sowder, L., & Analúcia Dias, S. (1988). Using Money to Teach about the Decimal System. The Arithmetic Teacher, 36(4), 42–43.
Enyedy, N. (2005). Inventing mapping: Creating cultural forms to solve collective problems. Cognition and Instruction, 23(4), 427–466.
Freudenthal, H., 1905-. (1991). Revisiting mathematics education : China lectures. Mathematics Education Llibrary, xi, 200 .
Gravemeijer, K. (2002). Preamble: from models to modeling. In Symbolizing, modeling and tool use in mathematics education (pp. 7–22). Springer.
Martinie, S. L., & Bay-Williams, J. M. (2003). Investigating students’ conceptual understanding of decimal fractions using multiple representations. Mathematics Teaching in the Middle School, 8(5), 244.
McNeil, N. M., & Fyfe, E. R. (2012). ‘Concreteness fading’ promotes transfer of mathematical knowledge. Learning and Instruction, 22(6), 440–448. http://doi.org/10.1016/j.learninstruc.2012.05.001
NCTM. (2003). Misconceptions with the Key Objectives (Working Group Circular). Unpublished.
van den Heuvel-Panhuizen, M. (2003). The didactical use of models in realistic mathematics education: An example from a longitudinal trajectory on percentage. Educational Studies in Mathematics, 54(1), 9–35.
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