# 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

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