"Seven. What is seven? Seven children; seven ideas; seven times in a row; seventh grade; a lucky roll of the dice; seven yards of cotton; seven miles from here; seven acres of land; seven degrees of incline; seven degrees below zero; seven grams of gold; seven pounds per square inch; seven years old; finishing seventh; seven thousand dollars of debt; seven percent of alcohol; The Magnificent Seven. How can an idea with one name be used in so many different ways, denoting such various senses of quantity?"

(Kilpatrick 2001)

Even from the youngest age children should be encouraged to enjoy playing with numbers, exploring how they work in a variety of situations, and developing fluency and flexibility in their use. It is likely for instance that some difficulties with place value and subtraction may have their roots in lack of fluency with counting on and counting back and with number sequences and patterns. Many children when asked to perform a subtraction, such as, "I had six lollies and I gave 2 to my friends. How many lollies do I have left?" will solve the problem by counting on from 4 rather than counting back from 6. Most early subtraction is done by counting on and adding and for some students the understanding of take away is very difficult to carry out. If we relate this back to number and number sense it seems likely that more time needs to be spent exploring counting on and back from different starting numbers and in different amounts so that fluency and flexibility is achieved. Mental routines can target these areas effectively and with fun and understanding.

Estimation

Estimation is another neglected area of the development of number sense. When children first arrive at school they make many informal guesses and estimates of quantity. They do not seem too concerned about 'right answers'. This state of affairs does not persist though as all too often children begin to seek the exact answer and will often be seen rubbing out an incorrect estimate and replacing it with the actual answer arrived at after the estimate. In real life estimations are frequently used. When shopping for instance it is quite a common practice to round prices up and down as items are added to the shopping trolley. A precise total is not needed but keeping within a budget or being able to know that the prices have been entered correctly is. Sometimes when cooking, estimation is important too, for instance knowing how many potatoes to chop for French Fries or how many carrots to chop is not usually treated as a precise mathematical activity. The important thing is that everyone has sufficient food without too much waste. Estimation develops with practice and experience. Only after concrete experiences do the judgements about quantity, size or chance etc develop. These experiences can be built into the mental routines. If from an early age children expect to estimate a ball park figure before actually computing mentally, with paper and pencil or a calculator they will begin to expect to find realistic solutions to those computations and will spot any errors that occur and hopefully stop and look for reasons for the differences between the estimates and the computed answers.

Micklo (1999) explains that becoming efficient at estimating is crucial to becoming a good problem solver and suggests that children are introduced to estimation as early as possible so that they learn how to apply the skill effectively.

Fluency with Counting and Mathematical Operations

Fluency with counting on and back and with number sequences is central to the ability to do addition and subtraction. Just because a child counts and touch counts to 9 or beyond does not mean that they have sufficient fluency, familiarity and recall of number sequences to be able to develop automaticity with number facts. Children need to be able to state the number after, before, or between given numbers spontaneously if they are to be expected to use count on 1, 2 or 3 facts or count back, 1, 2, or 3 facts. If children are restricted to counting all strategy to work out simple number facts then they are likely to become dependent on finger counting or nods of the heads to work out unknown number facts. Our recent experiences with upper primary and even with students doing the harder maths subjects at years 11 and 12 have shown how dependent on fingers and calculators even the mathematically able can be. Surely fluency and confidence with number strategies should be firmly developed as the foundation for all later computation. There are many routines in this book designed to enhance counting fluency and flexibility and to build from them into the development of a full range of strategies, listed below that will lead to automaticity with basic number fact recall as well as to the ability to extend those strategies beyond ten and to other situations such as early multiplication and sharing activities.

The following list of strategies include the very earliest ones used by children and shows a sequence of increasing complexity which seems to correspond with the natural progression of the children's interest in numbers. The list, however, is not meant to be a teaching sequence as such because children do not necessarily progress tidily through them and may use one strategy effectively one day and revert to a different one on another occasion or in a different situation. It is also more than likely that you will have children at different stages in their development. Planning for three levels across this range is important and so too is the immersion of children in the strategies and their meta-language so that when they have had sufficient time, experience and exposure to them they will confidently try them out and make them a part of their repertoire.

Subitizing (being able to recognise how many in a small group based on their appearance and with no need to count them at all)

Counting All (even after counting say 3 objects and then counting 2 more when asked how many altogether the whole set is counted from the beginning again)

Counting On by 1, 2, or 3 (when the number in one group is known the total of the number in two groups is found by counting on from the first number)

Doubles (children love to learn their doubles to ten and beyond and some will learn them faster than they learn the count on 2s and 3s)

Skip Counting (counting by 2s, 3s and 5sis the beginning of unitisation.

Turnarounds (when presented with say 2+5 the children will automatically count on 2 from the larger number rather than count on 5 from the smaller number)

Near Doubles (when the children are confidently using doubles they can be introduced to near doubles which are double plus or minus 1. So for 4 + 5 a child can work it out as double 4 plus 1 or by doing double 5 take away 1)

Rainbow Facts or Friends of 10 (all the number pairs that add to ten as shown below)

Bridge to 10 where a number is broken into two parts that enable 10 to be used as a bridge (7 + 6 = 7 + 3 + 3 = 10 + 3 = 13)

Build on Tens (applying the rule for counting on from ten)

Extended Number Facts (applying any of the above strategies to larger numbers, for instance using knowledge of 3+4 work out 30 + 40 or 13 + 4)

Place Value

Fluency with counting on and back to 100, looking for patterns in the 100 square and a solid foundation in understanding the composition of single digit numbers are the prerequisites for later understanding of place value. Constance Kamii explains in great detail how early concepts of quantity associated with single digit numbers in a sense has to be unlearned for place value to be understood. A great deal of time goes into children developing the understanding of number as a description of quantity. The three-ness of three for instance shows that the child now realises that three means the number of objects in a group not the name (label) of the last object counted. The child now has a view of three as the whole group. When working with tens and ones however children have difficulty switching from the whole to the parts, thus understanding that a ten can also be ten ones can take some time to develop. For quite some time the children will be able to see the ten either as a ten, as in a bundle of straws, or as ten ones as in a loose collection of straws but not as either or simultaneously. The process of grouping in twos, fives, tens etc is called unitising:

"To construct an understanding of unitizing, children almost have to negate their earlier idea of number. They have just learned that one object needs one word-that one means one object, that ten means ten objects. Now ten objects are one. How can something be simultaneously ten and one?"

Fosnot and Dolk (2001)

Early Money Concepts

The mental routines in the money section are designed to familiarise children with the characteristics of coins and notes and their identification of them and also to introduce equivalence of coins and notes. Simple addition and combining of coins are also included. In the early stages of working with coins children expect that larger coins will have greater value. They also find it difficult to understand why two or more coins of a lesser value can be exchanged for only one coin of an equivalent value to the other coins. As with the place value adaptive reasoning needs to be at a stage where a two 5 cent coins can be seen as one lot of ten cents simultaneously as seeing it as two lots of 5.

Bibliography

 Kilpatrick, J., Swafford, J., and Bradford, F., (2001) Adding it Up: Helping children Learn Mathematics, National Academy Press.

Estimation: It's More Than a Guess (1999) Childhood Education; v75 n3 p142-145 Spring 1999.

Fosnot, C.,T. and Dolk, M., (2001) Young Mathematicians at Work: Constructing Number Sense, Addition, and Subtraction, Heinemann, Portsmouth, NH.