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

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

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

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

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

Pawn Square Rule

Grigoriev, 1930

Pawn Square Rule

Grigoriev, 1930

Borders

Pawn Square Rule

Reti, 1921

Chess distance

Chess distance

Reti, 1921

Chess distance

There is more than one shorter path (3 moves).

Reti, 1921

Chess distance

Reti, 1921

Chess distance

Reti, 1921

Chess distance

Reti, 1921

Chess distance

Reti, 1921

Chess distance

Reti, 1921

Chess distance

Reti, 1921

Parity

Black

1

Black

1

Black White

1 2

Black White

1 2

Black White Black

1 2 3

Black White Black White

1 2 3 4

Black White Black White Black

1 2 3 4 5

Black White Black White Black

1 2 3 4 5

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

1 2 3 4 5

This behavior does not depend on the knight's path.

It is an invariant.

Quantification

Blitz game

1.Nc7

1.Nc7 Nc7

2.Kc7 stalemate

Line

1.Nc7 Nc7

2.Ra8!

2.Ra8! Ka8

Line

3.Kc7

3.Kc7 Ka7

4.Kc6

4.Kc6 Ka8

5.Kb6

5.Kb6 Kb8

6.Ka6

6.Ka6 Ka8

7.b6

7.b6 Kb8

8.b7

8.b7 Kc7

9.Ka7

2.Ra8!

2.Ra8! Kb7

Line

3.Ra7!

Line

2.Ra8!

2.Ra8! Na8

3.Kc8!

Zugzwang – German for «compulsion to move»

Zugzwang for full point

{ | } = 0

| }

{0| }

{0| } = 1

|1}

|1}

|1}

|1}

{0|1}

Which advantage is bigger?

How bigger?

{0|1}= 1

2

+ 1

2

1

2 = - 1 0

+ 1

2

1

2 = - 1 0

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

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

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…

Implementation: How to do? Some opinions and suggestions

How to use chess to teach decimals?

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

First idea: The «good» first question should be

Is it natural?

First idea: The «good» first question should be

Is it natural?

Second idea: Simplification; small boards, removal of pieces,….

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.

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.

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?

Portuguese Championship of

Mathematical Games (since 2004)

Average Duration (average number of turns)

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.

Average Duration Interaction

(how the pieces of both armies influence each other)

Average Duration Interaction

A game without interaction is basically a race (tendentially a boring game).

Average Duration Interaction

Pleasant Moves, Pleasant Rules, Pleasant Boards

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...

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!

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.

Chess and Primary School Mathematics