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8MATHEMATICAL GAMES IN TEACHING PROCESS (MATEMATHHHI irPH y HABHAHHI)
A.Braverman,
Ben-Gurion University of Negev, Beer-Sheva, ISRAEL,
E.Kizner,
Ben-Gurion University of Negev, Beer-Sheva, ISRAEL,
P.Samovol,
Ben-Gurion University of Negev, Beer-Sheva, ISRAEL, Kaye Academic College of Education, Beer-Sheva, ISRAEL,
M.Applebaum,
Kaye Academic College of Education, Beer-Sheva, ISRAEL
Розглядаються математичш ггри як оптимальш навчальш засоби для розвитку креативност1. Запропоновано досв1д автор1в за проблемою досл1дження.
The science of pedagogy has long established the importance of game in teaching process. However, the quest of the game optimal content remains a teaching challenge, an issue of discussion for many researchers, even today. The absence of corresponding methodological manuals for teachers should also be stressed, Below we present the authors' experience in the problem investigation (see examples).
I. Basic notions.
Mathematical games and brain-twisters certainly differ from sportive games, and one of the differences is that in mathematical games one can always predict the game results. Their common feature is the game excitement, ambition to win, anticipation of victory and victory itself. Let us refine a number of basic notions:
• The "mathematical game" notion is a game of two rivals, possessing the following quality: in each playing moment the game is characterized by the position that changes only as a function of players' moves. Certain positions are winning for one of the players. The goal of each player is to achieve his winning position. An assumption is made that in a mathematical game the players do not err, i.e. they play in the best possible manner (using game terminology, "they play correctly").
Sometimes, a game can end in a draw, which means that neither player can achieve a
winning position, or that some positions are tie-scored.
• The notion of "mathematical game winning strategy" comprises a set of rules (i.e. an instruction or an algorithm) that result in obligatory victory of one or other player who follows them (independently of his rival's game manner). The notion of "tie-score strategy" assumes such strategy that definitely results either in a victory or in a draw, when a player follows its instructions.
The experience shows that the beginners' common stumbling block is to comprehend "what do they want from us in general", what reasoning is valid for the problem solution and what - not.
It is generally assumed that two players make their moves in turn. One cannot pass his move! The task question is: "Who will win in a regular game?" The students' standard error is misunderstanding of "a regular game" phrase.
In 2004, the authors conducted the Mathematical Game Methodological Problems study among the students of Kaye College of Education (Beer Sheva). The study was conducted within the framework of the Studies in Mathematics B course. During the study it was found that the majority of students, while solving the tasks of mathematical games, replace the winning strategy notion by various specific notions. In
other words, they view only one game version out of many feasible ones.
Example 1. The "Foma-Yerema Game. (It was devised on the basis of Problem No. 5 of the 2005 International Spring Tournament of the Cities).
Foma and Yerema divide a heap of N coins, having value of 1,2,3,4,5,... altyns. (Note. - Altyn is the Old Russian coin value), where the altyn sum of all coins is an odd number. By each move one of them chooses a coin out of the heap, whereas the other says who should have it. Foma is the first to choose, the next move is made by him who has more altyns, if the number of altyns is equal, then the player who made the previous move is playing. Who of the players has a winning strategy? Find this strategy for N = 5 .
Discussion.
Thus, it follows from the conditions that the sum of all coins 1 + 2 + 3 +... + N = 2k +1 is an odd number.
Let us formalize the game conditions.
Let us assume at a certain game stage that Foma already has f altyns, while Yerema has e altyns, and they play in the M = {a1,a2,...,ak}set. Foma has the first move, if f > e . Otherwise, the first move is Yerema's. In case of equal sum, the move is made by the same person as before. In such reformulation of game conditions the assemblies are(f, e, M). It is evident that each such set is either winning or losing for Foma, while for Yerema it is either losing or winning, respectively, because such item-by-item examination is finite. The no-score is impossible, following from the condition of odd sum of altyns for all coins. All winning assemblies for Foma comprise the F set, while all winning assemblies for Yerema comprise the E set. These sets do not intersect (according to the winning strategy definition).
Let us take a look at the very beginning of the game and use the proof by contradiction.
Let us assume that Foma has a winning strategy. It means that there exists such /coin that both (f1,0, M )and (0, f1, M) positions are winning for Foma.
Let us further examine the (f1,0, M) position. According to the problem condition,
Foma continues the game and wins, basing on the assumption of his winning strategy.
But then Yerema must "make his move" in the (0, f1, M) position, and making use of Foma's winning strategy for the (f1,0, M) position, he wins. So we have a contradiction. Consequently, Yerema has a winning strategy.
Actually, the game is a brilliant example of the situation when the winning strategy in general does exist but is beyond the reach and use of students. The game itslf is like chess. And already for case of N = 5 coins we have a huge number of possible versions.
The authors have also written game computer version for the case of 5 coins (the compilation of such software can be given to the students, because actually we have here a mere item-by-item examination of versions, though rather cumbersome).
II. The mathematical game psychological aspects.
The most significant psychological factors of a mathematical game are in our opinion the following:
1) Encouragement of search for the problem solution;
2) Game excitation during participation in the game;
3) Personality self-assertion based on the increasing self-importance sensation.
III. Mathematical games and brain-teasers as an optimal teaching means for the development of individual creativity.
Application of mathematical games and brain-teasers in teaching process is very extensive. However, many teachers regretfully ignore this powerful educational tool. One of the reasons is the underdeveloped technology of this operation. Along with it, the instructing effect of, for instance, brain-teasers is extremely powerful. The brain-teaser apparent simplicity urges a student to test more and more new ideas and their solutions. In this process, intellectual endurance is formed in the students. In fact, bright ideas are manifested only within negligible time intervals whereas most of the time is occupied by hard work (Maslow, 1972). In our case, the task
intrinsic elegance multiplied by ardor can command child's attention to the problem for a sufficiently long period.
Let us take the examples of the most elegant and efficient brain-teasers used in our study:
Example 2
Using three 0 figures and one mathematical sign several times, make an expression equal to 6. (Answer: (0!+0!+0!)!= 6)
Example 3
Daniel bought a calculator, where he can only add, subtract, and multiply numbers, calculate integral part of a figure (i.e. [x]), calculate square root, factorial and double factorial of numbers:
(2n -1)!= 1 • 3 • 5 •... • (2n -1); (2n)!=
= 2 • 4 • 6 •... • (2n)
Can Daniel obtain number 101 of two unities, using his calculator?
Answer: Yes. [VTTll]= 101
Example 4
Using figures 1,9,9,8 taken exactly in this given sequence, and various mathematical signs, obtain 59. The figures cannot be united into a number.
Answer
1 -[(V9) !] ! !+V9 + 8 = .. = 1 -[3!]! !+V9 + 8 =
= 1 • []!!+V9 + 8 == 2 • 4 • 6 + 3 + 8 = 59
The results of our study show that mathematical games display all major parameters of creativity by Guilford u Torrance (1967, 1968):
• Ability of problem detecting and setting;
• Ability of generating numerous
ideas;
• Flexibility - ability of producing diversified ideas;
• Ingeniousness - ability of nonstandard response to the stimuli;
• Ability of improving the object by adding various details;
• Ability of problem solving, i.e. analytical and synthetic ability.
We find it expedient to apply mathematical brain-teasers regularly by giving them the appropriate organizational
format, for instance, a week brain-teaser, a brain-teaser to celebrate......(holiday), etc.
IV. Didactic mathematical games.
Didactic games are a special kind of mathematical games. They can be graded as the teaching games, the controlling games, and the generalizing games.
The game will be teaching if the participating students acquire new knowledge, skills and habits, or are compelled to acquire them during their preparation to the game. In doing this, the more pronounced is the cognitive activity motif (not only in the game but in the very content of mathematical material), the better are the results of knowledge apprehension.
The game will be controlling, if its didactic aim is repetition, reinforcement, checking of previously acquired knowledge. To participate in such a game, a student needs certain mathematical training.
The generalizing games demand integration of knowledge. They promote the establishment of inter-subject relations, are aimed at the acquisition of skills in various training situations.
The best didactic games are based on self-teaching principle, i.e. in such a manner that they direct the students to self-acquisition of knowledge and skills. As a rule, instruction includes two components: collection of necessary information and making correct decision. These two components provide the students' didactic experience. However, experience gain requires much time. It is important to enhance "experience gain" by the students, to teach them training these skills independently. The Game Discussion stage is a crucial factor in a didactic game.
A teacher conducts discussion, where he reviews and characterizes the game events and their perception by the participants. In this process special attention is paid to the establishment of bonds between the game content and the program curriculum or topic.
V . Organizational issues of didactic games.
When a teacher offers his students a didactic game, he must have answers to the following questions:
• The game objective. What mathematical skills and habits will the students acquire
during the game? What game moment should be paid particular attention?
• What didactic materials and aids are required for the game?
• What conclusions should be told to the students after the game?
• How can the students be involved in the process of devising new didactic games?
Game is always creation. During a mathematical game a habit of concentration, of independent thinking, is imparted to the students, their attentiveness, striving to knowledge are developing. "The consequence of any game impressing is the formation of new cognitive habit, the quality of new
perception of a certain problematic domain, etc." (Druzhinin, 1999).
It is a must for any practical teacher to have optimal set of curriculum-encompassing didactic games at his disposal. Below are given the examples of such games on selected topics. In our opinion, the learning material reinforcement is the best field fot such kind of games. Didactic games can evidently be devised for any curriculum topic.
Example 5. «Oral Count». «Who will be the first to tell number N = 100 ?».
Two persons are playing. The first one tells any integer from the 1 < m < 9 interval. The second one adds any other integer out of the same 1 < m < 9 interval and names the sum obtained. The First player again adds any integer out of the 1 < m < 9 interval to the sum and so one. The winner is he who first tells number N = 100.
Game analysis.
Let us imagine the Winner's final move. To win his last move, the would-be Winner must start from 91 < x < 99 number. Therefore, he must force the First player start his last move from n = 90 . Analogously, coming from "the end to the beginning", the game "losing numbers" will be 0,10,20,...,90 ... Actually, the First player cannot tell number 10 immediately. If the First player tells any number of 1 < m < 9 = K0, the Second one adds
(10 - m) number to the result obtained. Following this winning strategy, the First player, at that or other moment, forces the Second one to start from n = 90, and thus wins the game.
Remark.
The game rules can be generalized. For instance:
«Who will be the first to tell number N = 100?».
Two persons are playing. The first one tells any integer from the 1 < m < K0 interval. The second one adds any other integer out of the same 1 < m < K0 interval and names the sum obtained. The First player again adds any integer out of the 1 < m < K0 interval to the sum and so one. The winner is he who first tells the number N = 100.
а) If N can be divided by (1 + K0), then the Second player always wins.
Let us formulate the Second player's winning strategy. Let the First player tell any number out of the 1 < m < K0 interval. The
Second player must tell the (1 + K0 - m) < K0 number. Then the First player will have to again start from any (1 + K0)-divisible number.
б)If N is not divided by (1 + K0),
then, acting rightly, the First player is always the winner. Let us assume that when N is divided by (1 + K 0), the residual is 0 < r < K0. Then, the First player must first tell the ml = K0 +1 - m number. We obtain the sum, which being divided by (1 + K0), gives the previous residual of 0 < r < K0 . Note that by virtue of 1 < m < K0 constraints, the Second player, following his move, cannot have the number which divided by (1 + K0), gives the same 0 < r < K0 residual. Therefore, in reply to the Second player's move, the First player tells the Ni = r [mod. (K0 +1)] sum,
and wins in a couple of moves.
Example 6. «Quadratic trinomial».
The #*x2 + #*x + # = 0 quadratic equation, where # sign denotes any quadratic trinomial factor, is written on the blackboard. A first player gives any three numbers, whereas the second places them at his choice, replacing the # signs. Can the first player arrive at the situation when the equation obtained has different rational roots, or would the second player always interfere with his actions?
The answer is: he can. We define only the mathematical idea here. The first player
will win if he tells in pairs such various A, B, C integers whose sum is equal to zero (for instance, 1, -3, 2). Then A*x2 + B*x + C = 0
quadratic equation has x1 = 1 ^ x2
C
roots.
Example 7. «Decimal notation of an integer»
Two players play "guesses". The first player thinks of three two-digit numbers a, b, c . The second player gives three X ,Y, Z numbers to the first one, and in response receives the S = a • X + b • Y + c • Z expression. Can the second player guess all three numbers of the first player acting in this manner?
Answer: yes, he can {1,100,10000}.
(The students we are testing are of opinion that this game has high impressingpotential).
Example 8 'Divisibility and prime numbers"
First, the 2004! (1 • 2 • 3 •... • 2003 • 2004) number is written on the blackboard.
Two players make their moves in turn. A player subtracts from the number on the blackboard any integer divisible by no more than 20 different prime numbers (so that the remainder is non-negative), writes this remainder on the blackboard, and erases the previous number. He who reaches 0 is the winner. What is his game strategy?
Solution.
Let us denote all prime numbers in ascending order as
P = 2, P2 = 3, P, = 5, ..., P20 = 71, P21 = 73. Denote N = P • P2 •... • P21. It is evident that N is a minimal number divisible by at least 21 different prime numbers; hence for m , 1 < m < N , we have the number of different prime divisors of m less than 21. It is also evident that M = 2004!MN
Let us describe the Second player's
winning strategy.
Let the First player in a certain move subtract the number that being divided by N, gives the residual m ; by the prerequisite virtue he cannot subtract any N -divisible number. By the next move the Second player subtracts N - m, and since1 < N - m < N, N - m also has no more than 21 different prime divisors.
By such a strategy, the first player always begins from the N -divisible number,
but by virtue of the problem condition he cannot nullify N -divisible number at a single move. In a finite number of steps the Second player will receive a less than N positive number from the First player, and will end the game.
Remark:
1. Parameters: the amount of prime divisors (20 in our case), and M = 2004! can be varied so that the First player can win, depending on M, for instance when M = 2004!-1.
2. Note that the powers of prime multipliers are not involved in the condition.
Example 9. Given that N = 60. At each move the given number can be diminished by any of its divisors. The loser is he who reaches zero.
Solution:
The winner of the game is he who can obtain a unity. The First player is the winner, but in order to win, he must leave his rival only the odd numbers.
An example from "The properties of the point of intersection of the triangle medians" topic.
Example 10. (Minimax).
A cake is shaped like an arbitrary triangle. Two sweet-teeth divide it as follows. The first one indicates a point on the cake, whereas the second one makes a right-angled cut through this point and takes larger share. What is the largest share of the cake the first sweet-tooth can take? The cake is thought to have equal thickness everywhere.
Answer: 4/9. Let us demonstrate that if the first sweet-tooth chooses the point of intersection of its medians (point M), he can thus ensure 4\9 of the cake for himself. If the second sweet-tooth cuts the cake along the line passing through point M and parallel to a triangle rib, the first sweet-tooth will have 4\9 of the cake. If the second sweet-tooth makes a cut that is not parallel to the triangle ribs, then the first sweet-tooth's share even increases. Let us show that if the first sweet-tooth chooses any other M1 point inside the triangle, the second sweet-tooth can make a cut in such a way that his (the first sweet-tooth) share will be less. In fact, three triangles that are cut off the given triangle by straight lines, which are parallel to the triangle ribs and pass through
point M, cover the entire triangle. Therefore, point M1 is inside a triangle that we denote as
T. The second sweet-tooth makes a cut parallel
to the triangle base. The triangle obtained lies inside T; thus, he is less than 4\9.
Example 11 «Linear and nonlinear programming»
The robbers Grabber and Eyer share 100 coins. Grabber grabs a handful of coins, and Eyer looks at it and decides who will take it. So it goes on until one of then takes 9 handfuls - then the other one takes the remaining coins (their distribution may end in such a way that the coins are divided between them before anyone has 9 handfuls). Grabber can grab any number of coins. What largest amount of coins can he grab independently of Eyer's actions.
Solution.
Answer: 46 coins. Eyer's optimal strategy: if a handful contains 6 or more coins, he should take it; if it contains 5 or less coins, he should give it to Grabber. In this case, if Eyer collects 9 handfuls, he has 54 or more coins; if Grabber collects 9 handfuls, they will contain no more than 45 coins, and Grabber will have 55 or more coins. Using this strategy, Eyer ensures 54 coins for himself. It can be easily verified that any alternative Eyer's strategies are less beneficial. Grabber, respectively, can have 46 coins at best. He can achieve the result by grabbing 6-coin handfuls. Eyer will take 9 such handfuls - 54 coins - and Grabber will have the remaining 46 coins.
Conclusion.
Mathematical game carries labor, entertainment, and learning functions. Only superficially such a game seems carefree and easy. In fact, it demands from the player to contribute maximal energy, intelligence, knowledge, restraint, and independence.
Mathematical game promotes dramatic spur of student's creative and search activity, triggers his creativity. The skills acquired can later be easily implemented in any alternative practical activity domain.
And finally, as it was already noted, mathematical game is the ideal ground for search and study of positive impressing mechanism in students.
(Braverman, Samovol, Applebaum, 2004).
«It is positive or negative impressing that will encourage a child or teenager to reveal interest or to ignore knowledge in mathematics, literature, or arts, that will impart ethical or antisocial aims". (Efroimson, 1997, 1998).
1. Applebaum, M. V., 2000, Using Mathematics Games in Teacheng Mathematics, Mathematics in School (ALEH) No. 25, 4 pp., Hebrew University, Jerusalem, Israel,(in Hebrew).
2. Braverman, A., Samovol, P., Applebaum, M. (2004). Positive impressing of a school research problem. In Didactics of Mathematics: Problems and investigations (International collection of scientific works) n 22, Donetsk, TEAN, 116-120.
3. Druzhinin, V. N. (1999). Psyhologiya Obschih Sposobnostey (Psychology of General Faculties), Sankt-Petersburg., "Piterkom", p. 190. (in Russian).
4. Efroimson, V. P. (1997, 1998). Predposylki Genialnosti (Prerequisites of Genius), "Chelovek", 2-6:1997; 1:1998 (In Russian).
5. Guilford J.P. (1967). The nature of human intelligence, McGraw-Hill, Inc.
6. Maslow, A. H. (1972). A holistic approach to creativity. In C. W. Taylor (Eds.) Climate for creativity, Pergamon Press, Inc.
7. Torrance, E. P. (1968). Education and the Creative Potential, The University ofMinnesotapress, Minneapolis.
Резюме. Braverman A., Kizner E., Samovol P., Applebaum M. МАТЕМАТИЧЕСКИЕ ИГРЫ В ОБУЧЕНИИ. Рассматриваются математические игры как оптимальные учебные средства для развития креативности. Предложен авторский опыт по проблеме исследования.
Summary. Braverman A., Kizner E., Samovol P., Applebaum M. MATHEMATICAL GAMES IN TEACHING PROCESS. It is examined mathematical games as optimum educational facilities ^ for development of creative. Offered experience of authors after the problem of research.
Надшшла до редакцп 13.11.2005р.
