Session Information
24 SES 10, Arithmetics and Number Sense. Proportions, Fractions and Multiplication
Paper Session
Contribution
The recently revised primary mathematics curriculum of England (Department for Education, 2013) imposes higher expectations on younger children, particularly in relation to children’s multiplicative thinking. Whereas previously children were only expected to “recall multiplication facts to 10 x 10” by the end of Key Stage 2 (11 years old) (Department for Education and Employment, 1999, p. 69), the new expectation is that children should master the 12 times table as early as the end of Year 4 (9 years old). Arguably, such expectation puts a great deal of pressure on primary teachers to deliver, which could result in an overreliance on rote memorization, at the expense of teaching for conceptual understanding.
Being able to represent a mathematical concept in different ways is crucial in developing one’s conceptual understanding of that concept (Kilpatrick, Swafford & Findell, 2001). Such view of conceptual understanding is shared by Hiebert and Carpenter (1992) who posit “The mathematics is understood if its mental representation is part of a network of representations. The degree of understanding is determined by the number and strength of its connections” (p. 67). More recently, Barmby, Harries, Higgins and Suggate (2007) propose that mathematical understanding be understood as “the resulting network of representations associated with that mathematical concept” (p. 42).
Representations can manifest in two ways: internally and externally. While the former is taken to refer to “abstractions of mathematical ideas or cognitive schemata that are developed by a learner through experience” (Pape & Tchoshanov, 2001, p. 119), the latter is concerned with mathematical representations that can “act as stimuli on the senses” (Janvier, Girardon & Morand, 1993, p. 81). Simply put, while internal representations are private to the learner, external representations can be externalised and shared with or experienced by others. In the context of this study, the focus will be on exploring mathematics learners’ external representations as they can be examined externally.
There is a range of external representations available for mathematics learners to make use of. Bruner (1966), for example, argues that students can represent their (mathematical) thinking in three different ways: enactively (e.g. using manipulatives or concrete objects); iconically (e.g. representing mathematical concepts or processes visually); and symbolically (e.g. through the use of mathematical notations). To an extent, the usefulness of his framework to help develop and assess children’s mathematical understanding is arguably limited by its lack of emphasis on the role of meaningful context in which learning needs to be embedded in. Yoong (1999) refers to this missing component as the ‘story’ component in his multi-model strategy framework. By getting children to pose word problems, Yoong (1999) argues that it can bridge the gap between textbook mathematics and real-world applications. This raises the importance of the ability of teachers to formatively assess the quality of these word problems.
While there are existing frameworks (e.g. Fischbein et al., 1985; Greer, 1992; Kouba, 1989) showing different types of multiplication word problems, they do so in a way that capture only exemplary word problem types (e.g. repeated addition, multiplying factors, scale or one-to-many correspondence, Cartesian product) without taking into account of those that are less straightforward to assess as well as those which demonstrate children’s misconceptions. Arguably, these two latter types of word problems posed by children are equally important in helping to develop teachers’ ability to assess the quality of their children’s multiplication word problems. Drawing from this research gap, this study thus sets out to develop a framework that incorporates a wider range of types of multiplication word problems as posed by the children.
Method
Expected Outcomes
References
Barmby, P., Harries, T., Higgins, S., & Suggate, J. (2007, July). How can we assess mathematical understanding? In J.-H. Woo, H.-C. Lew, K.-S. Park & D.-Y. Seo (Eds.), Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education (pp. 41-48). Seoul, South Korea: PME. Bruner, J. (1966). Towards a theory of instruction. Cambridge, MA: Belkapp Press. Department for Education (2013). Mathematics programmes of study: Key Stages 1 and 2 National Curriculum in England. London: Department for Education. Department for Education and Employment (1999). The National Curriculum: Handbook for primary teachers in England. London, UK: Department for Education and Employment. Fischbein, E., Deri, M., Nello, M. S., & Marino, M.S. (1985). The role of implicit models in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education, 16(1), 3-17. Greer, B. (1992). Multiplication and division as models of situations. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 276-298). New York, NY: Macnillan. Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 65-97). New York, NY: Macmillan. Janvier, C., Girardon, C., & Morand, J. (1993). Mathematical symbols and representations. In P. S. Wilson (Ed.), Research ideas for the classroom: High school mathematics (pp. 79-102). Reston, VA: National Council of Teachers of Mathematics. Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press. Kouba, V. L. (1989). Children’s solution strategies for equivalent set multiplication and division word problems. Journal for Research in Mathematics Education, 20(2), 147-158. Ofsted (2014). [Anonymised] Primary school report. Manchester, UK: Ofsted. Pape, S. J., & Tchoshanov, M. A. (2001). The role of representation(s) in developing mathematical understanding. Theory into Practice, 40(2), 118-127. Yoong, W. W. (1999). Multi-model approach of teaching mathematics in a technological age. Paper presented at the 8th South East Asian Conference on Mathematics Education, Manila, the Philippines, 30th May – 4th June 1999.
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