Current projects

Flexible knowledge of rational numbers with multiple graphical representations in ITS

Sequencing learning with multiple representations of rational numbers (more)

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

Flexible knowledge of rational numbers with multiple graphical representations in ITS

Status: data analysis

The understanding of rational numbers is foundational not only in mathematics, but also in many real-world situations. Yet, part-whole numbers constitute a major challenge for middle-school students. The National Council of Teachers of Mathematics (NCTM) emphasizes the importance of helping students to develop flexible understanding of rational numbers. But how can teachers achieve this goal?
Previous research studies lead to the hypothesis that having students solve problems with multiple visual representations promotes their learning in a variety of domains (e.g., Ainsworth & Loizou, 2003; Berthold, 2006; Butcher, 2006; Hegarthy & Just, 1993; Kaput, 1989; Moss & Case, 1999; Schnotz & Bannert, 2003). In the domain of rational numbers, several authors have argued, based on observational studies, that learning with multiple representations can lead to deeper conceptual understanding of fractions (Corwin et al., 1990; Cramer et al., 1997a, 1997b; Steiner & Stoeckling, 1997). However, we know of no experimental studies that have investigated the advantages of instructions with multiple (graphical) fraction representations over instruction that focuses on a single representation.
In this study, we therefore focus on the question whether and how multiple representations of fractions help students to understand fractions more deeply. We hypothesize that a learning environment that provides students with the opportunity to actively work on multiple visual representations, in addition to practice on the conventional symbolic computational representation, leads to better learning outcomes for both conceptual and procedural knowledge as compared to instruction incorporating only one visualization in addition to symbolic computation. We anticipate, however, that multiple representations make the students' task more challenging, since they must link the representations to, and distill from them, a common concept or principle. In the experiment presented in this thesis, we are investigating whether additional support is needed in order to help students connect multiple representations. Specially, we address this issue by integrating prompts for sense-making activities (i.e. self-explanation) as a factor in our experiment. This leads to a 2 x 2 experimental design with independent factors (1) whether the students work with a single representation or with multiple, and (2) whether or not students are prompted, by the computer tutor, to self-explain.


Fig. 1. Example of an addition tutor with the numberline.


Fig. 2. Representations additionally used in the multiple representations conditions.


Fig. 3. Self-explanation prompts as used in this study.

We conducted an in vivo experiment in which 132 6th-grade students used one of four versions of CTAT-built tutors (see figures 1-3). Students worked on fraction conversion and fraction addition problems provided by the tutor for 2.5 hours in the classroom.

The results shed light into the effect of learning with multiple graphical representations in the domain of fractions. In line with previous research in other domains, we provide evidence that instruction with multiple representations is only beneficial when students actively link the representations to one another, e.g. through self-explanation activities. Students in our study were not able to make these links themselves; rather they needed to be prompted to engage in sense-making activities. We attribute the effect of combining multiple representations and self-explanation prompts to students' benefit from the opportunity to test and re-think the understanding they gained from individual graphical representations on the constraints of other representations which resulted in deeper understanding of general fraction concepts. In line with this interpretation, the benefit of learning with multiple representations and self-explanation prompts becomes most apparent in conceptual knowledge (see figues 4 and 5). Both students classified as having low and high prior knowledge benefit from such an intervention.


Fig. 4. Reproduction of conceptual knowledge assessed by a delayed post-test.


Fig. 5. Transfer of conceptual knowledge assessed by a delayed post-test.

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.

NCTM. (2000). Principles and Standards for School Mathematics. Reston, VA: Na-tional Council of Teachers of Mathematics.

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

Butcher, K. R. (2006). Learning From Text With Diagrams: Promoting Mental Model Development and Inference Generation. Journal of Educational Psychology, 98(1), 182-197.

Hegarty, M., & Just, M. A. (1993). Constructing mental models of machines from text and diagrams. Journal of Memory and Language, 32(6), 717-742.

Kaput, J. J. (1989). Linking Representations in the Symbol Systems of Algebra. In S. Wagner (Ed.), Research issues in the learning and teaching of algebra (pp. 167-195). Reston, Va.: Erlbaum, National Council of Teachers of Mathematics.

Moss, J., & Case, R. (1999). Developing children's understanding of the rational num-bers: A new model and an experimental curriculum. Journal for Research in Mathematics Education, 30(2), 122-147.

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

Corwin, R. B., Russell, S. J., & Tierney, C. C. (1990). Seeing fractions: A unit for the upper elementary grades. Sacramento, CA: California Dept. of Education.

Cramer, K., Behr, M., Post, R., & Lesh, R. (1997a). Rational Number Project: Fraction Lessons for the Middle Grades: Level 1. Dubuque, IA: Kendall/Hunt Publishing.

Cramer, K., Behr, M., Post, R., & Lesh, R. (1997b). Rational Number Project: Fraction Lessons for the Middle Grades: Level 2. Dubuque, IA: Kendall/Hunt Publishing.

Steiner, G. F., & Stoecklin, M. (1997). Fraction calculation--a didactic approach to constructing mathematical networks. Learning and Instruction, 7(3), 211-233.