L'ús dels jocs d'estrategia a la classe de matemàtiques

THE USE OF STRATEGY GAMES FAVORS LEARNING OF FUNCTIONAL DEPENDENCIES FOR DE-MOTIVATED STUDENTS Lluís Mora Cañellas IES Llavaneres (Spain) lmora1@xtec.cat

ABSTRACT This article refers the way a strategy game, DRAGO, can help students from ESO (Obligatory Secondary Teaching) to develop strategies aimed at solving problems, and, in our case, to develop strategies aimed at solving problems where functional dependencies are involved. More specifically, we want to see how the use of games helps enhance mathematical activity in high capacity students that do not manifest it due to an excess of routine, that provokes in them de-motivation towards discipline. We will show the results of this strategy on two students as an example of what we say. But it is not only for these reasons. Bishop (1998) says: “Game has a close relationship with mathematical reasoning, and we can consider true the statement that says that it is the base of hypothetical reasoning.”

BACKGROUND Many teachers and investigators have dealt with games as an important element for motivating math works at the classroom. We know that all cultures have adopted games as part of their pastime. Thus, Asher (1991) shows how American Indians (the Cayuga) used a wood bowl and 6 disks as a game. The game consisted in throwing the 6 disks in the air at the same time and count how many fell inside the bowl. According to the number of disks that effectively fell inside the bowl, they scored more or less points.

This kind of game would be equivalent to dice-throwing or coin-throwing with a high degree of chance and probability.

According to Bishop (1991), there are up to six important mathematical activities that all cultures practice: to count, locate, measure, draw, play and explain. We agree with Bishop (1998) that playing is a universal activity and that mathematics is also a universal area of knowledge.

Another important aspect of the use of strategy games is the specific kind of situations it develops, because it allows for important social interactions between players, for instance, the group or class, if the game is conducted in a classroom.

Strategy games, as Corbalán (1998) points out, can be considered as a particular class of problems, and thus can be treated using the channel defined for problemresolution by Polya: PROBLEM 1.- Understand the problem.

2.- Elaborate a plan.

3.- Execute the plan.

4.- Examine the results.

GAME 1.- Understand the game rules.

2.- Elaborate a game strategy.

3.- Apply the strategy while playing.

4.- Review the results of the game.

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In this sense, we asked ourselves if students would be motivated to work in a strategy game. Would they develop strategies that could drive them to establish relationships between the different elements involved in the game in order to deduct the best way to play? In other words, would they be able to deduct how to win most of the games?

DEFINING THE EXPERIMENT We decided to conduct research with two groups of third grade of ESO students (aged 14-15) at our Public Institute (secondary school) “IES de Sant Andreu de Llavaneres,” situated in an upper-medium-class residential town, 30 km north of Barcelona. Though conditions may seem advantageous for students, because of this social profile, and in many cases this was true, the index of school failure is quite high, about 25% for third ESO graders.

We chose for this research two different groups: 3rd –A and 3rd – C out of 6 that comprise the whole of third grade at our institute. Initially, all groups consisted of 16 students and were relatively homogeneous in terms of composition, because one of our school’s strategies is to try to distribute students in equilibrated groups as far as their capacities are concerned. (1) Group A included, among others, two grade repeaters, a student with special teaching needs, another student who reached third grade but did not pass any subjects of second grade, and, finally another student with digestive dysfunctions. When the research work ended, only 11 students out of the original 16 remained in the group. (2) Group C consisted of two grade repeaters as well, one student that needed special attention during the previous two years, and four students that finished the second grade without fulfilling mathematics requirement. By the end of our study, the group was reduced to 13 students.

During the first two years of ESO, mathematics teaching can be considered as “text-book:” explanation by teachers at the blackboard, practical exercises of subject that was explained, and, finally, problems with brief explanations. It is a very typical teaching system. We may say that problem-resolution processes are not dealt with, or, if they are, it is more in a very self-learning way, as many authors point out.

We developed the experiment along five 50-minute sessions. The first session was dedicated to the introduction of goals that students would have to achieve when they finish their work, and also to the introduction of the game, the materials needed and its rules. In this session we also began to work with the game with the simplest possible situations. Students had to work in pairs randomly organized, and they hade to complete activities included in the dossier given to them at the beginning of the session.

At the beginning of the second session, we presented the results of the previous one and discussed them. At the beginning of every session, teachers’ role was extremely important; because they had to make sure that the majority of students participated in the discussion. We ended the second session working with the same pairs formed in the previous one, but with more sophisticated situations. At the beginning of the third session, we again presented the results, tried to involve all the participants, and continued working in pairs in order to deepen the understanding of the game’s mechanisms. At the beginning of the fourth session we presented the results achieved by each of the two selected groups. The fifth session was dedicated to testing practical understanding by means of a competition among all the students.

Data used for an analysis of the experiment results included: (a) a summary of the work done by the students during the five sessions; and b) a video recording of the two sessions where both groups participated.

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In this article we want to present the results of 2 out of 32 students involved.

These students, a boy from 3rd-A and a girl from 3rd-C, were chosen because of their mathematical capabilities, according to their teachers, and also because they presented clear de-motivation trends regarding studying.

The girl, Laura, comes from a well-off family that owns a family business. She is the youngest daughter. During the first six months, she exhibited an increasing demotivation towards studying, together with incipient absenteeism, especially during firsthours classes, due to staying-up-late sleeping habits. Even with her background, the results were not that bad.

On the other side, Fran is the youngest son of a well-off family. He is an absentee with parental permission; that is, when he does not want to go to school, he is allowed to. Although in first and second grades he succeeded, he is a firm candidate to fail in third grade. He loves to be in the center of other’s attention. During the first six months of school, they both demonstrated an outstanding capacity for mathematics.

The game chosen for this experiment is called DRAGO, and also, PLANÇÓ. It is a game that fulfills all requirements to be considered a strategy game, according to Gómez Chacón’s definition (1992): (1) it can be played by more than one player; (2) it has a set of fixed rules; (3) the rules establish the goals to be achieved by each of the players; (4) the players have to choose their specific paths in order to achieve their goals; (5) the rules clearly establish when one of the players is the winner.

Many other games fulfill these conditions, but we found that this game is very suitable for third-ESO-graders for introducing the first degree functional dependencies.

The player that starts has to link two points with a continuous line, or one point with itself. Once the player does this, they have to draw a new point in the arc just drawn.

The rules to be followed are: 1) From one point, no more than three lines can be drawn. (fig 1) 2) Line cannot intersect.(fig 2) 3) The winner is the last player to do a valid movement.

new point

Fig.1

Fig. 2 Forbidden movement

This point is not usable for new movements

Figure 1.

We will now introduce the game.

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GAME DESCRIPTION DRAGO is a strategy game for two players created by John Conway in 1966.

The only things needed to play it are paper and pencil. The game begins by deciding, between both players, a certain number of points to be drawn in a paper. It is advisable that the number of points should be between 10 and 15. If the number is smaller than 10, there is no game, and if it is higher than 15, the game may take too long. Let us suppose we chose number 5. The points are drawn in the paper.

The mathematical interest of the game is based on the two following aspects: 1) The relationship between the initial number of points and the maximum or minimum number of movements that can be done, as also with the final number of points drawn. A detailed mathematical analysis of the game can be found in Mora (1991) who shows this dependency to be a linear one of the type y= ax+b.

2) To find, and be able to explain, which is the best strategy to try to win the highest possible number of games.

RESULTS AND STRATEGIES OF STUDENTS DURING THE SESSIONS As mentioned above, the first sessions consisted in explanation how to work during the next sessions, and in introduction of the game and its rules. Once this was done, students had to begin to play. They were divided in pairs (or groups of 3) and began playing the first games in order to see if they understood the rules and mechanisms of the game, with the simplest cases; 1 and 2 initial points.

One of the first things to remark was the trend we observed in students aged 1415 who did not register what they observed, so they just played, as if there was no other goal. Teachers’ task at this stage was to make sure that players registered the movements they made. At the beginning of the second session, there was a brainstorming to discuss what was done and to try to deduce some results. Below are the graphics drawn by students at this stage:

1 point

fig 1

fig 2

2 points

fig 3

Figure 2.

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Obtaining Fig. 1 does not pose a major problem to students. When drawing it on the blackboard, teachers should ask them to consider which elements are especially interesting. Only one student pays attention to the resulting form. As it is that figures 1 and 2 are obtained naturally and immediately, figure 3 requires some more work. To produce this form, one needs to have a winner’s strategy.

This is Laura’s description of different movements that can be made with 2 points. It is remarkable in two ways: 1) It introduces the idea of unblocked points, basic in a winner strategy.

2) It intuitively establishes a relationship between initial and final number of points. She’s already relating these two variables.

Once this stage was completed, we had to go on with more initial points. At the beginning of session 3, we did some brainstorming to analyze the results of working with 3 or more points. At this point, the task of the teacher was very important in order to organize the ideas that were introduced so far. The way students had to represent results was not the best one to find relationships, so teachers’ guidance became basic.

In the following drawing (Figure 3), we can see how they represent their results.

Laura drew some of the possible situations that can be obtained in a 3-initial-points game. The work is quite exhaustive, for she established the final number of movements, and the winner. She did the same with 4- and 5-point games.

Harmonization of results became essential for teachers to be capable to orientate the presentation of results in the best possible way.

After harmonization, we obtained the following table (see Figure 4):

Figure 3.

With results put in a table, the relationships between different elements (initial points, final points, number of movements, non-blocked points and winner) were more easily established.

Figure 4.

One of the advantages of having a table was that from now on we could go on completing it without the need to play. We could deduct how games would end just by looking at this table.

That was what Laura did, and she completed a table with all possible movements with 4, 5 and 6 points. And not only this, she was also able to formulate part of a winner strategy. It is to be said, though, that a complete formulation of the winner strategy is quite complicated and none of the students was able to complete it.

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In the following drawing (Figure 5) we can see the final part of the table drawn

by Laura, based on the previous table shown before. Laura followed this reasoning: With 5 points there are 5 possible endings. To obtain the number of movements we should subtract the initial number of points from the final number of points, and based in this number we can deduct if the winner will be the first to play or the second.

From this, she could establish a small part of a winner strategy: If the game starts with an even number of points, both players have the same chances to win. If it is not even, the second player has higher probabilities of winning.�

Figure 5.

But Fran went further. With the same data he was able to obtain an algebraic expression that related initial and final number of points. And knowing this we achieved a complete mathematical analysis. In the following drawing (Figure 6) we can see the formula proposed by Fran, and the reasoning he followed.

Fran found that beginning with n points the game ends with a maximum of 4n-1 points, and it takes 3n-1 movements.

The same as Laura, he was capable of guessing and writing down, a small part of the winner strategy: If starting number of points is uneven, the first player has a certain advantage. If not, both players have the same chances.

Figure 6.

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Both lack relating this fact to the number of unblocked points remaining at the end of the game. It is possible, though, that they both were conscious, although did not put it in writing that this relationship exists.

In the fourth session we explained to everybody the results that were found.

These results included the table with all the data that was written down and the formulas worked out by Fran. In order to establish a first-degree functional dependency, it was needed that the teacher completed the information with a graphical representation. In this way, we used the game to establish the three main elements of a functional dependency: table of values, formula and graphical representation.

The 5th session was devoted to a DRAGO championship between all the students of the class.

CONCLUSIONS Does the use of strategy games favor the learning of functional dependencies in de-motivated students? I think that strategy games are a very important motivating element that can be applied in a very positive way to help learning functional dependencies. Suitable games could be found to match different subject contents for students in secondary obligatory teaching. In this same sense, we could generalize the use of strategy games to the teaching of mathematics in general.

Due to the parallelism between problem resolution processes and strategy game analysis processes, the latter can be used to establish a set of activities that can be used in classes with high cultural diversity. Games have a universal range, and those games we have qualified as strategy games, are known for not establishing social or cultural preferences.

Concerning students with remarkable capacity but low motivation, according to results, we may say that strategy games allow them to fully develop all their abilities, as opposed to traditional teaching that drives them to school failure due to the abuse of routine.

It is important to also state that it is basic to try not to loose these students with demonstrated capacities, and that this task should be a joint task of all areas involved in teaching.

In our case, in order to emphasize our last assessment, we want to publicize that Fran will have to repeat third grade, because he has not complied with the criteria established by the Education Law in Spain.

It is also very important to guarantee that all students achieve their goals, and that teachers develop the ability to carry out different kind of works in class.

On the one hand, there must be aspects that have to be worked out individually, or in very small groups. On the other hand, it is essential that brainstorming sessions can be held in order to make results available to all of participants, and also to establish order in them. In these sessions, teachers should try to promote participation of everyone involved, and, if necessary, focus on proposed goals and achievements.

REFERENCES Asher, M. (1991). Ethnomatematics: A Multi-Cultural View of Mathematical Ideas.

Pacific Grove, California: Brooks/Cole.

Bishop, A. (1998). “El papel de los juegos en la educación matemàtica.” UNO. 18, 9-19.

Corbalán, F. (1998). “Juegos de estrategia en la enseñanza secundaria.” UNO. 18, 59-71.

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Gómez Chacón, I.M. (1992). Los juegos de estrategia en el curriculum de matemáticas, Madrid: NARCEA. Mason, J., Burton, L., Stacey,K. (1988). Pensar matemáticamente. Barcelona: MECLabor.

Mora, L. (1991). “El DRAGO: del juego a las funciones” [DRAGO: from game to functions] revista SUMA. 7, 47-52.

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