Chess and Primary School Mathematics

Transcription

1 Chess and Primary School Mathematics

2 SOME FUNDAMENTAL QUESTIONS

3 1) Why is chess a good game?

4 1) Why is chess a good game? 2) What are the benefits of chess in education?

5 1) Why is chess a good game? 2) What are the benefits of chess in education? 3) How does mathematics relate to chess?

6 1) Why is chess a good game? 2) What are the benefits of chess in education? 3) How does mathematics relate to chess? 4) What are the benefits of chess in mathematics education?

7 1) Why is chess a good game?

8 Chess is a combinatorial game Sequencial game of no chance and no hidden information

9 In this part of the talk, «good board game» means «nice game to play».

10 In this part of the talk, «good board game» means «nice game to play». Enjoyable game

11 In this part of the talk, «good board game» means «nice game to play». Enjoyable game Challenging game

12 In this part of the talk, «good board game» means «nice game to play». Enjoyable game Challenging game Addictive game

13 Interesting properties of a «good combinatorial board game»

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15 Depth Depth means that human beings are capable of playing at many different levels of expertise. A player may continue to learn how to improve his play for a long time. Depth can be measured.

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17 Clarity Clarity is the player's ability to mentally visualize a number of future moves. If a game is opaque, a player has no instincts. Clarity helps «eureka» moments.

18 Chess is a clear game.

19 Lines of Action is a opaque game.

20 We have a tension Depth vs. Clarity. However Depth and Clarity are not incompatible. A good game needs to be simultaneously deep and clear. Chess is deep and clear.

21 Drama A good game should have Drama: it should be possible for a player to recover from a weaker position and still win the game. Game's drama might be measured roughly by matching a strong player against a weak player, and having them switch sides.

22 Decisiveness In addition to drama, a game must also have Decisiveness: it should be possible ultimately for one player to achieve an advantage from which the other player cannot recover.

23 In Hex there are many positions in which it is possible through general principles to realize that an advantage is decisive.

24 Abalone has been criticized as lacking decisiveness: a player may choose to defend (clumping his pieces together and never extending them, even to attack).

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26 We have a tension Drama vs. Decisiveness. A game position can not be simultaneously dramatic and decisive. A good game should originate good dramatic problems and nice decisive puzzles. Chess has a good balance Drama/Decisiveness.

27 Game Perception Game Perception is the player's ability to understand what he is doing. A game with terrible game perception can be clear. A game can provide partial goals. For instance, a player may have fun playing Go just looking at the local fights.

28 Game Perception Also, a game with general principles (as Chess) usually has nice game perception.

29 Nim is a clear game (we can visualize a good number of future moves). Nim has no game perception. Without mathematics it is very hard to understand the game.

30 Distinct Phases A game should have Distinct Phases. Distinct phases provide different problems and puzzles. Distinct phases provide different types of games (and different styles of play).

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32 2) What are the benefits of chess in education?

33 Focus

34 Focus A person chooses to pay attention so intently to one thing that everything else seems to disappear.

35 Focus A person chooses to pay attention so intently to one thing that everything else seems to disappear.

36 Focus A person chooses to pay attention so intently to one thing that everything else seems to disappear.

37 Focus A person chooses to pay attention so intently to one thing that everything else seems to disappear.

38 Visualization of future situations

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44 Also, «abstract visualizations»

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46 Trees for decision making

47 Think first, act later!

48 Abstract thinking

49 Smothered Mate

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51 It works!

52 Different pieces: similar situation.

53 Different places on the board: similar situation.

54 It does not work: the queen can be captured with the king.

55 It does not work: the knight can be captured.

56 Smothered Mate (abstract observations):

57 Smothered Mate (abstract observations): Often, it needs the combined action of a heavy piece and a knight;

58 Smothered Mate (abstract observations): Often, it needs the combined action of a heavy piece and a knight; Different pieces or different places on the board may result in the same type of configuration;

59 Smothered Mate (abstract observations): The sacrificed heavy piece should not be captured with the king;

60 Smothered Mate (abstract observations): The sacrificed heavy piece should not be captured with the king; The knight should not be captured.

61 General and abstract observations:

62 General and abstract observations: do not relate to a specific game.

63 3) How does mathematics relate to chess?

64 Symmetry

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66 A pattern: «the same thing».

67 Central symmetry

68 Mirror symmetry Reti, 1928

69 Mirror symmetry Reti, 1928

70 Mirror symmetry If black chooses a side Reti, 1928

71 Mirror symmetry then white chooses the other side of the mirror. Reti, 1928

72 Mirror symmetry Reti, 1928

73 Mirror symmetry Reti, 1928

74 Mirror symmetry Let us look back. Reti, 1928

75 Mirror symmetry If white chooses a side Reti, 1928

76 Mirror symmetry then black chooses the same side of the mirror. Reti, 1928

77 If we understand one side, we also understand the other by symmetry. We «feel» the pattern and, so, the symmetry.

78 Lines and intersections

79 Intersections Kling e Horwitz, 1873

80 Intersections Kling e Horwitz, 1873

81 Intersections Kling e Horwitz, 1873

82 Intersections Kling e Horwitz, 1873

83 Intersections Kling e Horwitz, 1873

84 The black rook cannot leave the intersection point without opening one of the lines. Kling e Horwitz, 1873

85 Intersections Rinck, 1929

86 Intersections Rinck, 1929

87 Intersections Rinck, 1929

88 The rook is occupying the intersection point. Because of that it is no longer able to do its task. Rinck, 1929

89 Intersections Rinck, 1929

90 Intersections Rinck, 1929

91 Intersections Rinck, 1929

92 Distances and regions

93 Pawn Square Rule

94 Pawn Square Rule

95 Pawn Square Rule White king

96 Pawn Square Rule

97 Pawn Square Rule Grigoriev, 1930

98 Pawn Square Rule Grigoriev, 1930

99 Pawn Square Rule Grigoriev, 1930

100 Pawn Square Rule Grigoriev, 1930

101 Pawn Square Rule Grigoriev, 1930

102 Pawn Square Rule Grigoriev, 1930

103 Pawn Square Rule Grigoriev, 1930

104 Pawn Square Rule Grigoriev, 1930

105 Pawn Square Rule Grigoriev, 1930

106 Pawn Square Rule Grigoriev, 1930

107 Pawn Square Rule Grigoriev, 1930

108 Pawn Square Rule Grigoriev, 1930

109 Pawn Square Rule Grigoriev, 1930

110 Pawn Square Rule Grigoriev, 1930

111 Pawn Square Rule Grigoriev, 1930

112 Pawn Square Rule Grigoriev, 1930

113 Pawn Square Rule Grigoriev, 1930

114 Pawn Square Rule Grigoriev, 1930

115 Pawn Square Rule Grigoriev, 1930

116 Pawn Square Rule Grigoriev, 1930

117 Pawn Square Rule Grigoriev, 1930

118 Pawn Square Rule Grigoriev, 1930

119 Pawn Square Rule Grigoriev, 1930

120 Pawn Square Rule Grigoriev, 1930

121 Pawn Square Rule Grigoriev, 1930

122 Pawn Square Rule Grigoriev, 1930

123 Pawn Square Rule Grigoriev, 1930

124 Pawn Square Rule Grigoriev, 1930

125 Pawn Square Rule Grigoriev, 1930

126 Pawn Square Rule Grigoriev, 1930

127 Pawn Square Rule Grigoriev, 1930

128 Pawn Square Rule Grigoriev, 1930

129 Pawn Square Rule Grigoriev, 1930

130 Borders

131 Pawn Square Rule

132 Chess distance Reti, 1921

133 Chess distance Reti, 1921

134 Chess distance Reti, 1921 There is more than one shorter path (3 moves).

135 Chess distance Reti, 1921

136 Chess distance Reti, 1921

137 Chess distance Reti, 1921

138 Chess distance Reti, 1921

139 Chess distance Reti, 1921

140 Chess distance Reti, 1921

141 Parity

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144 Black 1

145 Black 1

146 Black White 1 2

147 Black White 1 2

148 Black White Black 1 2 3

149 Black White Black White

150 Black White Black White Black

151 After an odd number of moves, the knight is on a black square. After an even number of moves, the knight is on a white square. Black White Black White Black

152 This behavior does not depend on the knight's path. It is an invariant. Black White Black White Black

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155 Quantification

156 Blitz game

157 1.Nc7

158 1.Nc7 Nc7

159 Line 2.Kc7 stalemate

160 1.Nc7 Nc7

161 2.Ra8!

162 Line 2.Ra8! Ka8

163 3.Kc7

164 3.Kc7 Ka7

165 4.Kc6

166 4.Kc6 Ka8

167 5.Kb6

168 5.Kb6 Kb8

169 6.Ka6

170 6.Ka6 Ka8

171 7.b6

172 7.b6 Kb8

173 8.b7

174 8.b7 Kc7

175 9.Ka7

176 2.Ra8!

177 Line 2.Ra8! Kb7

178 Line 3.Ra7!

179 2.Ra8!

180 2.Ra8! Na8

181 3.Kc8!

182 Zugzwang German for «compulsion to move» Zugzwang for full point

183

184 { } = 0

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186

187

188 }

189 {0 }

190 {0 } = 1

191

192 1}

193 1}

194 1}

195 1}

196 {0 1}

197

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199 Which advantage is bigger?

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203 How bigger?

204

205 {0 1}= 1 2

206 = 0 2

207 = 0 2

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209 4) What are the benefits of chess in mathematics education?

210 1) Indirect reasons: focus, visualization of future situations, trees for decision making, think first act later!, abstract thinking

211 1) Indirect reasons: focus, visualization of future situations, trees for decision making, think first act later!, abstract thinking 2) Direct reasons: symmetry, lines and intersections, parity, distances and regions, quantification

212 Implementation: How to do? Some opinions and suggestions

213 How to use chess to teach decimals?

214 How to use chess to teach decimals? Usually, these questions do not make sense

215 First idea: The «good» first question should be Is it natural?

216 First idea: The «good» first question should be Is it natural? Second idea: Simplification; small boards, removal of pieces,.

217 First idea: The «good» first question should be «Is it natural?». Second idea: Simplification; small boards, removal of pieces,. S1: Use final positions of some chess studies to build problem solving sessions.

218 First idea: The «good» first question should be «Is it natural?». Second idea: Simplification; small boards, removal of pieces,. S1: Use final positions of some chess studies to build problem solving sessions. S2: Ocasionally, put marks on the board to show geometric configurations.

219 First idea: The «good» first question should be «Is it natural?». Second idea: Simplification; small boards, removal of pieces,. S1: Use final positions of some chess studies to build problem solving sessions. S2: Ocasionally, put marks on the board to show geometric configurations. S3: Use other games. Why not?

220 Portuguese Championship of Mathematical Games (since 2004)

221 Average Duration (average number of turns)

222 Average Duration There are good games that converge quickly to the end. This property is very useful when we want to organize tournaments for kids.

223 Average Duration Interaction (how the pieces of both armies influence each other)

224 Average Duration Interaction A game without interaction is basically a race (tendentially a boring game).

225 Average Duration Interaction Pleasant Moves, Pleasant Rules, Pleasant Boards

226 Average Duration Interaction Pleasant Moves, Pleasant Rules, Pleasant Boards Sometimes the players want to take back one move (or more). If the moves are very complicated this important procedure may be difficult...

227 Average Duration Interaction Pleasant Moves, Pleasant Rules, Pleasant Boards When we think about nice games to use in schools, it is great to have simple paragraphs explaining the rules!

228 Average Duration Interaction Pleasant Moves, Pleasant Rules, Pleasant Boards When we think about good games to use in schools, it is quite nice to obtain nice boards and pieces using a printer.

229 Chess and Primary School Mathematics