![]() ![]() One can add players, add or change the directions of the arrows, and ask other kinds of questions - for example, "If there were 7 players, how many games would there be if everyone played everyone else exactly once?" Variants and extensions: Several variants come to mind immediately. Also, note that question 3, "Which of those games did Lee win?" has been translated as "❼uál o cuáles de esos partidos ganó Lee?" Using only "cuál" or "cuáles" by itself would prejudge the situation by telling the student whether there is one game or more than one. Some students (for example, those from Costa Rica) will not be used to this form. This translation, like the one of the Taxman task, has been done in the informal form. The Spanish translation of the task is included to highlight the need for assessment designers and teachers to be sensitive to the nuances of language. Because there was nothing to be gained by the ambiguity, it is now "How many more games need to be played to finish the tournament?" Games are needed?" Children in a pilot test interpreted the question in both ways. In fact, the games were deliberately arranged so that one trio of players illustrates this: José beat Alex and Alex beat Lee, but José did not beat Lee.Īn earlier version of question 6 was phrased as "How many games need to be played to finish the tournament?" This was found to be somewhat ambiguous it could mean either "How many games in all are needed?" or "How many more Task design considerations: Note that the relation "X beats Y in a game of checkers" is not transitive that is, if X beats Y and Y beats Z, then it is not necessarily the case that X beats Z. (The student must see the one-to-one correspondence between the arrows and the games, and devise a way to count the arrows properly, keeping track of which arrows have been counted and which have not, or simply realize that each game has a winner and loser, and divide 18 by 2.) It is even more challenging to count the games that have not yet been played. When counting the games that have been played, the student may realize that each of the six players has played three games, because there are three arrows associated with each dot nonetheless, it is not true that 18 games have been played. Students are thus asked to explore a very basic notion - counting, in this instance - in a new context. One interesting feature of this task is that it requires children to count sets of objects (the games that have been played and the games that have yet to be played) when it is not immediately clear how these objects are represented in the given picture. In addition, the task shows that mathematics can be non-computational. Students must use analytical skills and demonstrate reasoning to answer the questions, two fundamental attributes of mathematical power. The last two questions, in particular, allow a variety of strategies to be applied. Yet the task is linked to more traditional material as students are asked to convert the graphical representation to a familiar ranked table, thus illustrating the connections between such different representations. Such mathematics broadens the curriculum usually thought appropriate for the fourth grade. The content involved in this task is the elementary use of a network in graph theory to present a familiar situation. Rationale for the mathematics education community (Clearly it would be inappropriate to use this task to compare classes to whom this notation is familiar with classes to whom it is not.) The task, however, does not assume familiarity with the notation. In this case, the dots represent people they are connected with arrows indicating a certain mathematical relation: an arrow from A to B means ''A won the game that A and B played." Some groups of children, for example ones that use the Comprehensive School Mathematics Program (McREL, 1992), will be familiar with a similar notation, and hence will need little teacher introduction. Translate information from one form to anotherĪssumed background: This task presents information in the form of a directed graph, which is a way of showing relationships among objects. Broaden the view of mathematics appropriate for the 4th grade
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