# Use of money as a decimal representation

- Cambridge Mathematics
- Research
- Issues around the Framework
- Use of money as a decimal representation

### 01 July 2016

## Executive summary: Use of money as a decimal representation

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.

### Summary of advantages and disadvantages of money as a decimal representation

**Advantages**

- In theory, counting money could help students with decomposition of numbers to the value of their digits
- Can help students leverage informal knowledge
- Fewer conceptual errors were made by students using perceptually rich money
- Might be assets in activities placing a greater emphasis on problem-solving than accuracy

**Disadvantages**

- Two noted misconceptions associated with money as a symbolic representation (longer-is-larger or shorter-is-larger)
- Money as a concrete representation is associated with one of these misconceptions
- Money as a perceptually rich concrete representation can adopt the disadvantages of perceptually rich manipulatives (see below)
- Thinking and talking about money using everyday language, whether in conjunction with manipulatives or symbols, can also reinforce one of the above misconceptions

### Money, symbolisation, and language

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:

- Money thinking: students are able to order decimals correctly up to two decimal places but then fail to correctly order numbers which only differ beyond two decimal places corresponding to cents (Steinle, 2004).
- String-length thinking: students “interpret the decimal as two whole numbers separated by a marker (Steinle, 2004, p. 2). This misconception has been similarly described in relation to money by other researchers (Babcock LDP, accessed 2016; NCTM, 2003).

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 and manipulatives

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.

### Conclusions: recommendations from research

- Perceptually rich money might be helpful in activities with a greater focus on problem-solving, but might be counterproductive in activities depending on greater accuracy, like summative tests (McNeil et al., 2009).
- A process of gradual decontextualisation might help bring students closer to realising the essential ideas behind the particular problems they’re solving (Gravemeijer, 2002, in McNeil et al., 2009).
- NCTM emphasizes vocal reading to reinforce the meaning of each place and the distinction and connection between the right and left sides of the decimal point – but specifically mentions the vocal reading of currency as an exception, since this could instead reinforce misunderstandings (NCTM, 2003 - unpublished working group circular) like those discussed above. This problem could potentially be avoided by reinforcing standard decimal vocal reading of currency notation (e.g. "two point five four pounds" rather than "two pounds fifty-four p."). Ball, (1993) emphasizes that it is important to carefully develop the context for classroom discourse in order to make money work as a representation.

### References

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

Ball, D. L. (1993). Halves, pieces, and twoths: Constructing and using representational contexts in teaching fractions. Rational Numbers: An Integration of Research, 157–195.

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

McNeil, N. M., Uttal, D. H., Jarvin, L., & Sternberg, R. J. (2009). Should you show me the money? Concrete objects both hurt and help performance on mathematics problems. Learning and Instruction, 19(2), 171–184. http://doi.org/10.1016/j.learninstruc.2008.03.005NCTM. (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.