Published 2005 Revised 2014

The term "number sense" is a relatively new one in mathematics education. It is difficult to define precisely, but broadly speaking, it refers to "a well organised conceptual framework of number information that enables a person to understand numbers and number relationships and to solve mathematical problems that are not bound by traditional algorithms" (Bobis, 1996). The National Council of
Teachers (USA, 1989) identified five components that characterise number sense: number meaning, number relationships, number magnitude, operations involving numbers and referents for numbers and quantities. These skills are considered important because they contribute to general intuitions about numbers and lay the foundation for more advanced skills.

Researchers have linked good number sense with skills observed in students proficient in the following mathematical activities:

- mental calculation (Hope & Sherrill, 1987; Trafton, 1992);
- computational estimation (for example; Bobis, 1991; Case & Sowder, 1990);
- judging the relative magnitude of numbers (Sowder, 1988);
- recognising part-whole relationships and place value concepts (Fischer, 1990; Ross, 1989) and;
- problem solving (Cobb et.al., 1991).

An intuitive sense of number begins at a very early age. Children as young as two years of age can confidently identify one, two or three objects before they can actually count with understanding (Gelman & Gellistel, 1978). Piaget called this ability to instantaneously recognise the number of objects in a small group 'subitising'. As mental powers develop, usually by about the age of
four, groups of four can be recognised without counting. It is thought that the maximum number for subitising, even for most adults, is five. This skill appears to be based on the mind's ability to form stable mental images of patterns and associate them with a number. Therefore, it may be possible to recognise more than five objects if they are arranged in a particular way or practice and
memorisation takes place. A simple example of this is six dots arranged in two rows of three, as on dice or playing cards. Because this image is familiar, six can be instantly recognised when presented this way.

Usually, when presented with more than five objects, other mental strategies must be utilised. For example, we might see a group of six objects as two groups of three. Each group of three is instantly recognised, then very quickly (virtually unconsciously) combined to make six. In this strategy no actual counting of objects is involved, but rather a part-part-whole relationship and rapid mental addition is used. That is, there is an understanding that a number (in this case six) can be composed of smaller parts, together with the knowledge that 'three plus three makes six'. This type of mathematical thinking has already begun by the time children begin school and should be nurtured because it lays the foundation for understanding operations and in developing valuable mental calculation strategies.

Learning to count with understanding is a crucial number skill, but other skills, such as perceiving subgroups, need to develop alongside counting to provide a firm foundation for number sense. By simply presenting objects (such as stamps on a flashcard) in various arrangements, different mental strategies can be prompted. For example, showing six stamps in a cluster of four and a pair prompts
the combination of 'four and two makes six'. If the four is not subitised, it may be seen as 'two and two and two makes six'. This arrangement is obviously a little more complex than two groups of three. So different arrangements will prompt different strategies, and these strategies will vary from person to person.

If mental strategies such as these are to be encouraged (and just counting discouraged) then an element of speed is necessary. Seeing the objects for only a few seconds challenges the mind to find strategies other than counting. It is also important to have children reflect on and share their strategies (Presmeg, 1986; Mason, 1992). This is helpful in three ways:

- verbalising a strategy brings the strategy to a conscious level and allows the person to learn about their own thinking;
- it provides other children with the opportunity to pick up new strategies;
- the teacher can assess the type of thinking being used and adjust the type of arrangement, level of difficulty or speed of presentation accordingly.

To begin with, early number activities are best done with moveable objects such as counters, blocks and small toys. Most children will need the concrete experience of physically manipulating groups of objects into sub-groups and combining small groups to make a larger group. After these essential experiences more static materials such as 'dot cards' become very useful.

Dot cards are simply cards with dot stickers of a single colour stuck on one side. (However, any markings can be used. Self-inking stamps are fast when making a lot of cards). The important factors in the design of the cards are the number of dots and the arrangement of these dots. The various combinations of these factors determine the mathematical structure of each card, and hence the types of
number relations and mental strategies prompted by them.

Consider each of the following arrangements of dots before reading further. What mental strategies are likely to be prompted by each card? What order would you place them in according to level of difficulty?

Card A is the classic symmetrical dice and playing card arrangement of five and so is often instantly recognised without engaging other mental strategies. It is perhaps the easiest arrangement of five to deal with.

Card B presents clear sub-groups of two and three, each of which can be instantly recognised. With practice, the number fact of 'two and three makes five' can be recalled almost instantly.

Card C: A linear arrangement is the one most likely to prompt counting. However, many people will mentally separate the dots into groups of two and three, as in the previous card. Other strategies such as seeing two then counting '3, 4, 5' might also be used.

Card D could be called a random arrangement, though in reality it has been quite deliberately organised to prompt the mental activity of sub-grouping. There are a variety of ways to form the sub-groups, with no prompt in any particular direction, so this card could be considered to be the most difficult one in the set.

Card E shows another sub-group arrangement that encourages the use (or discovery) of the 'four and one makes five' number relation.

Obviously, using fewer than five dots would develop the most basic number sense skills, and using more than five dots would provide opportunities for more advanced strategies. However, it is probably not useful to use more than ten dots. (See the follow-on article focusing on developing a 'sense of ten' and 'place value readiness'). Cards such as these can be shown briefly to
children, then the children asked how many dots they saw. The children should be asked to explain how they perceived the arrangement, and hence what strategies they employed.

Games can be very useful for reinforcing and developing ideas and procedures previously introduced to children. Although a suggested age group is given for each of the following games, it is the children's level of experience that should determine the suitability of the game. Several demonstration games should be played, until the children become comfortable with the rules and procedures of
the games.

Deal and Copy (4-5 years) 3-4 players

Materials: 15 dot cards with a variety of dot patterns representing the numbers from one to five and a plentiful supply of counters or buttons.

Rules: One child deals out one card face up to each other player. Each child then uses the counters to replicate the arrangement of dots on his/her card and says the number aloud. The dealer checks each result, then deals out a new card to each player, placing it on top of the previous card. The children then rearrange their counters to match the new card.
This continues until all the cards have been used.

Variations/Extensions

- Each child can predict aloud whether the new card has more, less or the same number of dots as the previous card. The prediction is checked by the dealer, by observing whether counters need to be taken away or added.
- Increase the number of dots on the cards.

Memory Match (5-7 years) 2 players

Materials: 12 dot cards, consisting of six pairs of cards showing two different arrangements of a particular number of dots, from 1 to 6 dots. (For example, a pair for 5 might be Card A and Card B from the set above).

Rules: Spread all the cards out face down. The first player turns over any two cards. If they are a pair (i.e. have the same number of dots), the player removes the cards and scores a point. If they are not a pair, both cards are turned back down in their places. The second player then turns over two cards and so on. When all the cards have been
matched, the player with more pairs wins.

Variations/Extensions

- Increase the number of pairs of cards used.
- Use a greater number of dots on the cards.
- Pair a dot card with a numeral card.

What's the Difference? (7-8 years) 2-4 players

Materials: A pack of 20 to 30 dot cards (1 to 10 dots in dice and regular patterns), counters.

Rules: Spread out 10 cards face down and place the rest of the cards in a pile face down. The first player turns over the top pile card and places beside the pile. He/she then turns over one of the spread cards. The player works out the difference between the number of dots on each card, and takes that number of counters. (E.g. If one card showed 3
dots and the other 8, the player would take 5 counters.) The spread card is turned face down again in its place and the next player turns the top pile card and so on. Play continues until all the pile cards have been used. The winner is the player with the most counters; therefore the strategy is to remember the value of the spread cards so the one that gives the maximum difference can be
chosen.

Variations/Extensions

- Try to turn the spread cards that give the minimum difference, so the winner is the player with the fewest counters.
- Roll a die instead of using pile cards. Start with a set number of counters (say 20), so that when all the counters have been claimed the game ends.
- Use dot cards with random arrangements of dots.

The next article in this series is entitled A Sense of 'ten' and Place Value.

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Bobis, J. (1996). Visualisation and the development of number sense with kindergarten children. In Mulligan, J. & Mitchelmore, M. (Eds.) Children's Number Learning : A Research Monograph of the Mathematics Education Group of Australasia and the Australian Association of Mathematics Teachers. Adelaide: AAMT

Case, R. & Sowder, J. (1990). The development of computational estimation: A neo-Piagetian analysis. Cognition and Instruction , 7, 79-104.

Cobb, P., Wood, T., Yackel, E., Nicholls, J., Wheatley, G., Trigatti, B., & Perlwitz, M., (1991). Assessment of a problem-centred second-grade mathematics project. Journal for Research in Mathematics Education , 22, 3-29.

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Hope, J. & Sherril, J. (1987). Characteristics of unskilled and skilled mental calculators. Journal for Research in Mathematics Education , 18, 98-111.

Mason, J. (1992). Doing and construing mathematics in screen space, In Perry, B., Southwell, B., & Owens, K. (Eds.). Proceedings of the Thirteenth Annual Conference of the Mathematics Education Research Group of Australasia . Nepean, Sydney: MERGA.

Ross, S. (1989). Parts, wholes, and place value: A developmental view. Arithmetic Teacher , 36, 47-51.

Sowder, J. (1988). Mental computation and number comparison: Their roles in the development of number sense and computational estimation. In Heibert & Behr (Eds.). Research Agenda for Mathematics Education: Number Concepts and Operations in the Middle Grades (pp. 192-197). Hillsdale, NJ: Lawrence, Erlbaum &
Reston.

Presmeg, N. (1986). Visualisation in high school mathematics. For the Learning of Mathematics , 6 (3), 42-46.

Trafton, P. (1992). Using number sense to develop mental computation and computational estimation. In C. Irons (Ed.) Challenging Children to Think when they Compute . (pp. 78-92). Brisbane: Centre for Mathematics and Science Education, Queensland University of Technology.