Chess and Primary School Mathematics Chess and Primary School Mathematics . SOME FUNDAMENTAL QUESTIONS

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  • Chess and Primary School Mathematics

  • SOME FUNDAMENTAL QUESTIONS

  • 1) Why is chess a good game?

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

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

  • 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?

  • 1) Why is chess a good game?

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

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

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

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

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

  • Interesting properties of a «good combinatorial board game»

  • www.thegamesjournal.com/articles/DefiningtheAbstract.shtml

  • 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

    Depth can be measured.

  • 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

    Clarity helps «eureka» moments.

  • Chess is a clear game.

  • Lines of Action is a opaque game.

  • We have a tension Depth vs. Clarity.

    However Depth and Clarity are not incompatible.

    Chess is deep and clear.

    A good game needs to be simultaneously deep and clear.

  • 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.

    Drama

  • 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.

    Decisiveness

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

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

  • 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 .

  • Game Perception is the player's ability to understand what he is doing.

    A game with terrible game perception can be clear.

    Game Perception

    A game can provide partial goals. For instance, a player may have fun playing Go just looking at the local fights.

  • Game Perception

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

  • 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.

  • A game should have Distinct Phases.

    Distinct phases provide different problems and puzzles.

    Distinct Phases

    Distinct phases provide different types of games (and different styles of play).

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

  • Focus

  • Focus

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

  • Focus

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

  • Focus

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

  • Focus

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

  • Visualization of future situations

  • Also, «abstract visualizations»

  • Trees for decision making

  • Think first, act later!

  • Abstract thinking

  • Smothered Mate

  • It works!

  • Different pieces: similar situation.

  • Different places on the board: similar situation.

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

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

  • Smothered Mate (abstract observations):

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

  • 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;

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

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

  • General and abstract observations:

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

  • 3) How does mathematics relate to chess?

  • Symmetry

  • A pattern: «the same thing».

  • Central symmetry

  • Mirror symmetry

    Reti, 1928

  • Mirror symmetry

    Reti, 1928

  • Mirror symmetry

    Reti, 1928

    If black chooses a side…

  • Mirror symmetry

    Reti, 1928

    then white chooses the

    other side of the mirror.

  • Mirror symmetry

    Reti, 1928

  • Mirror symmetry

    Reti, 1928

  • Mirror symmetry

    Reti, 1928

    Let us look back.

  • Mirror symmetry

    Reti, 1928

    If white chooses a side…

  • Mirror symmetry

    Reti, 1928

    then black chooses the

    same side of the mirror.

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

  • Lines and intersections

  • Kling e Horwitz, 1873

    Intersections

  • Kling e Horwitz, 1873

    Intersections

  • Kling e Horwitz, 1873

    Intersections

  • Kling e Horwitz, 1873

    Intersections

  • Kling e Horwitz, 1873

    Intersections

  • The black rook cannot leave the intersection point without

    opening one of the lines.

    Kling e Horwitz, 1873

  • Rinck, 1929

    Intersections

  • Rinck, 1929

    Intersections

  • Rinck, 1929

    Intersections

  • Rinck, 1929

    The rook is occupying the intersection point. Because of that

    it is no longer able to do its task.

  • Rinck, 1929

    Intersections

  • Rinck, 1929

    Intersections

  • Rinck, 1929

    Intersections

  • Distances and regions

  • Pawn Square Rule

  • Pawn Square Rule

  • White king

    Pawn Square Rule

  • Pawn Square Rule

  • Grigoriev, 1930

    Pawn Square Rule

  • Pawn Square Rule

    Grigoriev, 1930

  • Pawn Square Rule

    Grigoriev, 1930

  • Pawn Square Rule

    Grigoriev, 1930

  • Pawn Square Rule

    Grigoriev, 1930

  • Pawn Square Rule

    Grigoriev, 1930

  • Pawn Square Rule

    Grigoriev, 1930

  • Paw