NUMBER WAR GAMES I: Number Concept and Relationships

Teaching Mathematics Facts Using Card Games

Children, all over the world, love to play games. I have successfully used games for initial teaching and remedial mathematics instruction, particularly, for learning arithmetic facts (addition, subtraction, multiplication, and division), comparison of fractions, and comparing and combining integers. An ordinary deck of playing cards, a pair of Dice and Dominos are good tools for teaching arithmetic, particularly, number conceptualization and simple arithmetic facts.  However, using cards from an ordinary deck assume number concept, in thier use and Dominos and Dice only teach subitizing. Whereas, A set of Visual Cluster Cards helps children to learn all the components of Nubmer Concept: learning visual clustering (generalization of subitizing), decomposition/recomposition, comparison of numbers, and their relationships.

Playing cards are used for playing games all over the world. Every culture has developed playing cards and games related to them. The games and their complexity vary from simple to complex and from simple comparison to strategies.  The number and type of games played using an ordinary deck of playing cards abound. Games, using playing cards, are enjoyed by all—from children to adults.  

One of the most popular games children play is called the Game of War. This is a family of card games. These games, under various names, are played by children and adults all over the world. I have adopted many of these games for teaching mathematics concepts and reinforcing them.  I call these games: ‘The Number War Games.’ 

I have designed a special set of playing cards called: Visual Cluster CardsTM for playing these games.  Visual Cluster Cards are without numbers on them. When children use these cards, within few days, they learn the most important component of number: decomposition/ recomposition. Through decomposition/ recomposition, they acquire the 45 sight-facts (addition and then subtraction facts of numbers up to 10). Visual Cluster Cards are better suited for these games[2].  

Visual Cluster Cards are modified ordinary deck of cards, in their design and in number. They are of two types: With face cards and without face cards. Both types of Visual Cluster cards have several arrangements of clusters for numbers such as: 0 (one blank card), 3 (two clusters), 8 (two), 9 (four), wild card (two), and 10 (two). All other numerals (1, 2, 4, 5, 6, and 7 have one card in each suit (spade, club, diamond, and heart).  There are 60 cards (the deck without face cards) in this deck.  The other deck of Visual Cluster cards includes face cards, in addition to all the other cards. The blank card represents zero and the wild card as a variable–assuming the value the context and the player assigns. In this deck, all face cards represent 10 (a good option when working on numbers up to 10) or the jack represents 11, queen represents 12, and king represents 13. 

In both decks, the cluster of objects (pips, icons) represents the numeral and the color (black = positive, red = negative) of the pips represent the number: e.g., five of clubs represents the number +5. Whereas, 5 of hearts represents, the number, 5. Thus, in both decks, when working with integers, red cards represent negative numbers, and black cards represent positive numbers.  As one can see, numeral is a representation of quantity and number is a directed numeral (it has a direction and quantity). Up to fifth grade, we do not make a differentiation between numeral and number. However, once the children beginn to learn about integers, we need to differentiate between numeral and numeber.

Children learn the quantity (numeral), number (positive and negative numerals) represented by the cards by observation (by sight), ultimately without counting. Since, children derive and learn the relationship between numbers up to ten by sight, these facts are called sight factsThere are a total of 45 sight facts[3]. Sight facts are like sight words. A child should master these 45 sight facts by the end of Kindergatrten.

The ordinary Game of War is played by children all over the world. My game begins in the same way as the Game of War. It is played essentially the same way and is easy to learn. Before, they play the game, however, it is important that children become familiar with the deck of Visual Cluster cards, particularly, the patterns of visual clusters on each card.  

Visual Cluster cards have clusters of objects displayed on the card. For example, there are five diamonds displayed in the middle in a particular pattern–a pattern that encourages decomposition/recomposition (see below).

An arrangement of this type is called a visual cluster[4]. The particular arrangement above is the visual cluster for five. It will be called the numeral 5 up to fifth grade. Later, it will be called numeral 5 and number +5.

Because of the patterns of pips, on individual visual cluster cards, they can be recognized, without counting, visualized, and then committed to memory with ease. The special nature of the visual pattern of a cluster of pips, representing the quantity, on a card helps a child to form a vivid image of that quantity, therefore, the numeral/quantity represented by the card, in their minds. Each Visual Cluster card is organized according to a particular cluster. This helps players to recognize the size of collections (up to 10) without counting. This also helps children to integrate: (a) orthographic image (5) of the numeral (when it is formalized in writing), (b) the auditory form (f-i-v-e), and, (c) the quantity represented by the cluster. This integration is called “numberness.” In this particular case, this integration is called “fiveness.” Writing should begin when a child can recognize the cluster representing a numeral instantly.

Children who are not able to form and hold these clusters in their minds and are, therefore, unable to recognize the size of a collection of objects by observation, have not conceptualized number, yet. This lack of integration of these three elements is a symptom and the manifestation of dyscalculia. Research supports this observation and shows that, in such a case, children  have difficulty in learning number concept, number relationships, particularly addition and subtraction facts and other higher concepts, and later operations on numerals and numbers (i.e., integers, etc.). These children keep on counting on fingers or on number line to find the sums and differences of even two small numbers. They also have great difficulty in automatizing arithmetic facts.  

The following games not only help children to conceptualize number but also help them to master arithmetic facts.  These games are highly motivating to children. 

There are several games, in this series, that are variations of other popular card games, such as “Go Fish.”  If you use or are aware of any card games that relate to number and number relationships, I would love to hear about them (maheshsharma@me.com).

Game One:  Visual Clustering and Comparison of Numbers

(For children age 3 to those who are having difficulty mastering arithmetic facts)

Objective:  To teach number concept—numberness, decomposition/ recomposition, and sight facts.

The game can be played between two or three players.  However, it is most effective between two players.

Materials:  Take a deck of Visual Cluster cards including jokers (joker can assume any number value, according to context). In the case of Visual Cluster Cards with face cards, each card’s value is the number of objects displayed by the visual cluster on the card (e.g., Ace = 1 and the blank card = 0).  For example, the four of diamonds, clubs, spades or hearts will be known as number/numeral four. 

Each face card, jack, queen, and king is initially given the value of ten.  The ace represents number one.  The joker can assume any value and can be different each time it is used. When children know the teen’s numbers, then you can introduce: jack = 11, queen = 12, King = 13.  

How to Play: 

  1. The whole deck is divided into two equal piles of cards (if two players).  
  2. Each child gets one pile of cards.  One can also distribute the cards equally by counting out loud (This teaches children sequence of numbers and their location on the sequence of numbers. This increases number vocabulary–lexical enries for number) . Each person keeps the cards face down.  
  3. When the game begins, each person turns a card face up.  The bigger value card wins. For example, one has the three of hearts (value 3), and the other person has the seven of diamonds (value 7). The seven of diamonds wins. The winner collects all the displayed cards and puts them underneath his/her pile. (When playing this game with integers, three of hearts represents -3 and 7 of diamonds will represent -7).
  4. If both players have the same value cards (for example, one has the five of hearts, and the other has the five of spades), they declare war: “I declare war.” 
  5. Each player puts three cards face down on each sound of the word, in succession, saying I (for the first card) declare (for the second card), and war (for the third card). Then each player displays a fourth card face up.  The bigger valued fourth card wins. If they match again, the same process is repeated.
  6. The winner collects all cards and places them underneath his/her pile.  
  7. The first person with an empty hand loses. 

This game is appropriate for pre-K, Kindergarteners, and other children who have not mastered number concept. Number conceptualization is dependent on five interconnected skills: (i) Having a large number vocabulary, (ii) one-to-one correspondence with sequence, (iii) visual clustering (extension of subitizing)—recognizing a cluster of objects up to five by observation (without counting) is called subitizing and recognizing up to 10 objects is called visual clustering, (iv) decomposition/recomposition, and (v) ordering.  This game develops all of these prerequisite skills and many more.  Children with a lack of number concept have great difficulty in learning arithmetic facts and can derive them only by sequential counting. Which is a very inefficient strategy. Initially, for a short while, children can count the objects on the cards. However, fairly soon they begin to rely on visual clusters to recognize the value of cards. In a game, children have the opportunity of comparing almost five hundred pairs of numbers. 

Game Two: What Makes This Number

(For children age 3 to those who are having difficulty mastering arithmetic facts)

Objective: To master addition sight facts

Materials:  Same as above

How to Play: 

  1. The whole deck is divided into two equal piles of cards.  
  2. Each child gets a pile of cards.  The cards are kept face down. 
  3. Each person displays one card face up.  Each one finds two numbers whose sum is their card. For example, one has the three of hearts (value 3) and, therefore, gives two sight facts: 1 + 2 = 3, 2 + 1 = 3. The other has the seven of diamonds, the sight facts are: 1+ 6, 2+ 5, 3 +4, 4+3, 5+2, 6+1. The one with more sight facts wins. If the child, with the bigger number, cannot produce all the sight facts, the other player gets a chance and if he/she can give all the sight facts, he/she wins. In general, the person who is able to produce all the sight facts correctly and has the bigger number wins. The winner collects all the displayed cards and puts them underneath his/her pile. 
  4. If both players have the same number of sight facts, there is war.  For example, one has the five of hearts (value 5) and gives all the sight facts and the other has five of clubs (value 5) and gives all the sight facts. Or, one has five of diamonds and gives three sight facts only, and the other has nine of clubs (value 9) and gives three sight facts only, they declare war. 
  5. Each one puts three cards face down. Then each one displays another card face up. The bigger number of sight facts wins. 
  6. The winner collects all the cards and places them underneath his/her pile.  
  7. The first person with an empty hand loses. 

This game is appropriate for children who have not been introduced to sight facts or have not mastered/automatized simple addition facts. 

Initially, children will count the objects on the cards. However, fairly soon they begin to rely on visual clusters to recognize and find the sums. Within a few weeks, they can master all the 45 sight facts[5]. Initially, the game can be played with dominos or with a deck of cards of numbers up to five.

This series of posts will continue. In future editions, number games relating to other operations (inlcusding algebraic operations) will be included. Next few games will be on arithmetic operations.


[1]Copyright 2008 with Mahesh Sharma. 

[2]Visual Cluster Cards are available from Center for Teaching/Learning of Mathematics ($15 per deck plus $4.00 for shipping and handling).

[3]Number Conceptualization by Sharma (2008).  

[4]Same as above.

[5]The list of sight facts and how to teach them is included in How to Teach Number Concept Using Visual Cluster cards (Sharma, 2017).  Also see the post on Sight Words and Sight Facts on this Blog.

NUMBER WAR GAMES I: Number Concept and Relationships

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