Flexible knowledge of rational numbers with multiple graphical representations in ITS (more)

Sequencing learning with multiple representations of rational numbers (more)

Affordances of the problem context when learning with multiple graphical representations of fractions

Affordances of the problem context when learning with multiple graphical representations of fractions

Status: Difficulty factors assessment

Fractions pose a very significant challenge for students in the elementary and middle grades (e.g., Moss, 2005). Yet, fractions are an important foundation for later learning in mathematics. Conceptual difficulties arise as many of the intuitive properties of whole numbers no longer hold for rational numbers. For example, larger numbers do no longer always mean larger quantities (e.g., 1/13 is not larger than 1/4). There is substantial amount of evidence demonstrating that the use of multiple external representations (MERs) can significantly enhance student learning in complex domains (e.g., Ainsworth & Loizou, 2003; Berthold, 2006; Schnotz & Bannert, 2003), also in the domain of fractions (Rau, 2008). Another line of research has demonstrated that providing abstract concepts within a familiar problem context can promote students' understanding (e.g., Mack, 1995; Thußbas, 2001). Consider the problem: "Which quantity is greater, 1/4 or 1/3?" As Mack's (1993, 1995) studies indicate, it is not unusual for students to say that 1/4 is larger, as 4 is larger than 3. However, this error is less likely when asking: "What is more, a third of a pizza or a fourth of a pizza?" When using both MERs and cover stories in the domain of fractions, it should be especially helpful if the graphical representation structurally maps into the cover story. However, we know of no experimental investigations regarding this hypothesis.
The proposed project will focus on the design of a tutoring system in the domain of fraction comparison. Students will be given a series of fraction comparison problems that will be provided along with a cover story and a graphical representation. Tutors built for this project will be used for an experimental study investigating the effects of presenting students with a cover story that matches the graphical representations used to support students' understanding of the numerical problem at hand. Students working with the tutor will thus be presented with problems that include either a cover story that is neutral with regard to the given representation (e.g., "During World War II, 1/3 of the national steel production was done in Homestead, PA.
Mahoning Valley, OH accounted for 1/10 of the national steel production. Which cite produced more steel - Homestead or Mahoning Valley?"), or with a problem that is particularly suited for a specific graphical representation (e.g., "You're going down the river in a canoe. Your friend is in another canoe, just next to you. You are passing a long ship, and you see that your canoe has about one quarter of the length of the big ship, while your friend's canoe has about one fifth of the length of the big ship. Who has the longer canoe - you or your friend?" - which, dealing with length, maps into a number line representation). The same cognitive model can be used for both types of problems, since they vary only with regard to the cover story included in the interface.

References:

Ainsworth, S., & Loizou, A. T. (2003). The effects of self-explaining when learning with text or diagrams. Cognitive Science: A Multidisciplinary Journal, 27(4), 669-681.

Berthold, K. (2006). Learning from worked-out examples: Multiple representations, an integration help, and self-explanation prompts all forster understanding., Albert-Ludwigs-Universität, Freiburg i. Br.

Mack, N. K. (1995). Confounding whole-number and fraction concepts when building on informal knowledge. Journal for Research in Mathematics Education, 26(5), 422-441.

Moss, J. (2005). Pipes, Tubes, and Beakers: New approaches to teaching the rational-number system. In J. Brantsford & S. Donovan (Eds.), How people learn: A targeted report for teachers (pp. 309-349): National Academy Press.

Rau, M. A. (2008). Flexible knowledge of fractions with multiple graphical representations in intelligent tutoring systems. Albert-Ludwigs-Universität, Freiburg im Breisgau.

Schnotz, W., & Bannert, M. (2003). Construction and interference in learning from multiple representation. Learning and Instruction, 13(2), 141-156.

Thußbas, C. (2001). Beeinflussen Cover Stories mathematischer Texaufgaben analogen Transfer in der Mathematik? Zeitschrift für Experimentelle Psychologie, 48(4), 272-289.