The teacher asked about the list at the top right of the student's paper, and she responded that she had made the tallies to show how many, and there were 21. A student's representation of the number of animals that went off to see the world Representations help students recognize the common mathematical nature of different situations. The first problem is to determine the number of objects left after three objects have been taken from a collection of five. The third problem asks the number of balls that must be put into a box if it is to have five balls and there are already three in the box. Seeing similarities in the ways to represent different situations is an important step toward abstraction.
Students use representations to organize their thinking.
Representations can carry some of the burden of remembering by letting students record intermediate steps in a process. For example, a student trying to find the number of wheels in four bicycles and three tricycles drew the picture shown in figure 4.36. Thus the student was able to compute the sums separately and add them together. The representation served as a placeholder for thoughts that were not yet internalized. Understanding and using mathematical concepts and procedures is enhanced when students can translate between different representations of the same idea.