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. It was also seen that all other things being equal, the player going second ( White ) has a built-in advantage, thereby confirming a widely held belief among the checkers community.Broaden the view of mathematics appropriate for the 4th grade Though the problem definition makes it inherently hard to objectively quantify the results, the objectives were successfully achieved for the most part. Soar was chosen as the platform for building this agent because it came built-in with features that facilitate creating rules-based expert systems, has been proven to be fairly reliable in developing such systems (including flight simulators) for over twenty years and makes it relatively straight-forward to have multiple agents play each other, and to add or modify features or strategies to the agents. The objective behind Soar Checkers is to investigate if it is possible to create an agent-based game playing system that would beat novices with ease and at least challenge advanced novice to intermediate-level players by designing a rules-based expert system whose knowledge base consists of nothing more than the rules of checkers, and a set of guidelines for game-play based on good strategy. The classic board game of checkers is ideally suited for research in AI game-playing programs or agents.
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