Session Information
24 SES 12, Mathematics and Beliefs. The Affective Dimension.
Paper Session
Contribution
The project presented here is part of a four-year intervention project where we collaborate with two primary school teachers to create a successful learning culture in early learning of mathematics, with special emphasis on language development. Two of the main objectives of the project is to improve pupils’ proficiency in expressing mathematical concepts and ideas using a variety of representations, and to improve their proficiency in mathematical reasoning, arguing and justification. Recent research shows that mathematical reasoning is important for children’s later achievement in mathematics (Nunes, Bryant, Sylva & Barros, 2009). There is thus a need to conduct research on how young pupils learn mathematics, especially through language use and reasoning, and how they understand and use mathematical concepts and symbols in these processes.
Young children can benefit from early introduction to the concept of fractions, before they develop an ingrained theory of numbers that is restricted to counting numbers (Siegal & Smith, 1997, p. 18). While others have analyzed how young children make notations for fractions to gain insight in children’s learning of fractions (Brizuela, 2006), we have investigated how children reason while trying to make sense of fractions and mixed numbers using a variety of representations. The research question we focus on here is:
How do young pupils give meaning to semiotic representations of mixed numbers?
In mathematics, the use of semiotic representations is crucial. We need semiotic representations both to think about and explore mathematical concepts, and to communicate mathematical ideas. All mathematical objects are abstract in their nature, and we only gain access to them through semiotic representations (Duval, 2006). For learners of mathematics, this gives rise to a potential cognitive conflict; how can the mathematical object be distinguished from the representations at use, if the use of semiotic representations is the only way to get access to the mathematical object (Duval, 2006, p.107)? The ability to interpret semiotic representations and to change between different representations is thus a critical threshold for the development of mathematical understanding.
As a theoretical framework we use semiotic theory as described by Steinbring (2005, 2006). Steinbring views mathematical signs as “instruments for coding and describing mathematical knowledge, for communicating mathematical knowledge as well as for operating with mathematical knowledge and generalizing it”. He emphasizes that all mathematical signs have both an epistemological and a semiotic function (Steinbring 2006, pp.133-134). Thus, a sign can be thought of as representing an abstract mathematical concept as well as a concrete object or reference context (Rønning, 2013, p.162). Learning in this framework is seen as meaning making through mediation between the sign/symbol and the object/reference context. This process is captured in Steinbring’s epistemological triangle (where the third corner is the concept) (Steinbring, 2006, p.135). One can see the sign/symbol corner as containing the signs that the learner tries to give meaning to, and the object/reference context corner as containing the actual situation that is represented by the signs, i.e. the context that the learner uses to give meaning to the mathematical sign. The mathematical concept mediates between the sign/symbol and the object/reference context. If there is no mediation through a concept, there is just an associative connection between the sign/symbol and the object/reference context (the sign’s semiotic function). It is worth noticing that the epistemological triangle is not a static system that is independent of the learner. The connections between the three corners have to be actively constructed, often in interaction with others (Steinbring, 2006). When the learner’s mathematical knowledge develops the reference context will tend to become less concrete (Steinbring, 2005, p. 30).
Method
Expected Outcomes
References
Brizuela, B. M. (2006). Young children's notations for fractions. Educational Studies in Mathematics, 62(3), 281-305. Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1-2), 103-131. Klette, K. (2009). Challenges in strategies for complexity reduction in video studies. Experiences from the PISA+ study: A video study of teaching and learning in Norway. In T. Janik & T. Seidel (Eds.), The power of video studies in investigating teaching and learning in the classroom. (pp. 61-82). New York, NY: Waxmann. Marshall, H. (1994). Discourse analysis in an occupational context. In C. Cassell & J. Symon (Eds.), Qualitative methods in organizational research: A practical guide (pp. 91-106). London: Sage. Munthe, E. (2006). Visuell analyse: et mangfold av muligheter. In M. Brekke (Ed.), Å begripe teksten. Om grep og begrep i tekstanalyse (pp. 175-198). Kristiansand: Høyskoleforlaget. Nunes, T., Bryant, P., Sylva, K., & Barros, R. (2009). Development of maths capabilities and confidence in primary school. London: Department for Education. Retrieved from https://www.gov.uk/government/publications/development-of-maths-capabilities-andconfidence-in-primary-school Rønning, F. (2013). Making sense of fractions in different contexts. Research in Mathematics Education, 15(2), 201-202. Siegal, M., & Smith, J. A. (1997). Toward making representation count in children's conceptions of fractions. Contemporary Educational Psychology, 22(1), 1-22. Steinbring, H. (2005). The construction of new mathematical knowledge in classroom interaction: An epistemological perspective (Vol. 38). Springer Science & Business Media. Steinbring, H. (2006). What makes a sign a mathematical sign?–An epistemological perspective on mathematical interaction. Educational Studies in Mathematics, 61(1-2), 133-162.
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