The ESCOT “Fish” problem uses different representations of the ratio of male to female fish in each of three ponds. All the representations are linked to each other, and to the ratio of fish in the ponds, which can be manipulated directly by the student. Students are asked to distribute the fish with specific ratios to each of the ponds.

The Rationale Behind the Feature (Specific Design Principle):

Doing this gives students different views of ratios, allowing them to check their work, to give the opportunity for students to resonate with one or more representations, and to help them understand the concept of ratios more deeply.

Context of Use:

Students can see both a numeric and a pie chart representation of a ratio of male to female fish in the ponds, which changes as students move fish from pond to pond. The manipulation of the fish (a third iconic representation) is dynamically linked to the numeric and graphical representation and gives students a direct way to interact with the concept of ratio and to visualize changes in a ratio by means of concrete, numeric, and graphic representations.

Field-based Evidence:

In the Math Forum Problem of the Week environment, where this software was used, students are asked to explain how they solve the problem. Many students used the pie charts and/or the numeric ratios that are shown in the software to justify their reasoning. However, there was some confusion with the pond that had a 1:2 ratio – some students wrote it as ˝, and went on to explain that the pie chart would be split in two. If they were actively looking at the representations in the software, they would see that this is not the case, and would question their own reasoning leading to a teachable moment. This leads us to believe that while the idea of having multiple representations is powerful, the layout of the representations could be improved. That is, students spend much of their time in the ponds, as it were, and their eyes would have to roam to the right side of the screen in order to see the representations. A better design would be to have the representations closer to where their eyes are, to somehow be more salient.

References:

Ainsworth, S. E, Bibby, P. A., & Wood, D. J. (1997). Information technology and multiple representations: New opportunities – new problems. Journal of Information Technology for Teacher Education 6(1), 93-105. [Special Issue: Multimedia and Multiple Representations]

Drier, H. S. (2000a). The Probability Explorer: A research-based microworld to enhance childrens intuitive understandings of chance and data. Focus on Learning Problems in Mathematics 22(3-4), 165-178.

Drier, H. S. (2000b). Childrens meaning-making activity with dynamic multiple representations in a probability microworld. In M. Fernandez (Ed.), Proceedings of the twenty-second annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, (vol 2, pp.691-696). Tuscon, AZ. Can be viewed online at http://www.west.asu.edu/cmw/pme/resrepweb/PME-rr-Drier.htm

Edwards, L. D. (1998). Embodying mathematics and science: Microworlds as representations. Journal of Mathematical Behaviour 17(1), 53-78.