Number concept
From an early age children need to be taught how to apply their mathematical knowledge in a range of settings and contexts. Children learn to handle numbers before they enter school. Numbers are a child’s world. They are heard in conversation (I have 2 biscuits, I am 3years old today) and stories, rhymes and jingles (1 2 3 4 5 once I caught… one step, two steps like a teddy bear). Various stages can be recognised in the development of number concepts such as in:
- Repeating numbers: one, two, seven, four
- Matching numbers words to objects: there are 3 cars, 2 lorries
- The correct order of numbers: 1,2 3,4
Learning the meaning of ‘first’, ‘second’ and ‘third’
- Comparing numbers: more than, less than
- Understanding that the number of things is constant regardless of size or position: i.e. 6 things are still six regardless of how they are placed
- Learning to recognise and write numbers
- Manipulating with numbers by doing various calculations
My investigation has led me to think about lots of ideas that can be used with children in Early Years classroom as well as with KS1. Numbers displayed on telephone sets and remote controls are probably the first form of numbers in an order that young children come across with. Primarily, number lines 0 – 10 & 0 – 20, as a part of the 100 square can be used where children familiarise themselves with numbers and recognise the correct order of numbers. Number lines are very versatile. They can be of any size, for individual or whole- class use. They can start on any number. Blank number lines are infinitely adaptable; they can be used for counting calculations using all four number operations. They are a good way to practice and overlearn the number bonds to 20 that children need to be able to remember fluently. Visual counting pattern on number lines can help children to understand relative numbers and number sequences. Moving a step forward from the correct order of numbers, the children can identify and colour all the odd and even numbers and establish rules for recognition.
When children are confident with bigger numbers, a large 100 square is ideal to work with the whole class to learn timetables, addition & subtraction of larger numbers by counting numbers above it or below it. Children can also identify multiples of 2,3, 5, 10 and others by highlighting numbers in different colours and demonstrate sequential patterns. They can reverse the two digit numbers, read them and make new numbers. Problem solving activities such as pick a number between 0-10 or 0-20, double it and add 1, is very exciting as children manipulate with numbers at their own pace.
Lots of different games can be introduced. Snakes & Ladders, number dominions, dice games, dot to dot, finding the difference between two dices; place value cards are few examples. By playing and replaying a selection of games, children can practice to grapple ideas, number facts and concepts in a way that they can enjoy and strengthen skills at the same time. Active involvement aids their learning and enhances their attitude towards the subject. But it is important to focus on the particular learning target that the game is reinforcing.
Hundred squares can be cut into several parts to form a jigsaw and children can be challenged to put the pieces together again. (See some activity ideas in the appendix)
To calculate numbers for the investigation, it required me use a calculator. In KS1, children can be introduced to calculators to find number patterns, predict and discuss results. ‘By the end of KS2, pupils should have the knowledge and competence to use a calculator to work out, say, (56+97)
* (133-85) and round the answers to one decimal place’ (NNS 1999, p. 8).
During university sessions we were given lots of ideas and hands on practical activities which include: working with number lines, 100 squares, blank number lines, beads, number games, number songs &rhymes, cubes, picture stories, books, small investigations and lots more. I planned a series of maths lessons during my school practices and used different activities with confidence where children enjoyed and explored numbers in many ways. ‘Millie’s Maths’ was popular software with children in the nursery to develop early maths skills and number recognition.
Mathematics Educational Theories
The dictionary definition of Numeracy is ‘the ability to use numbers especially arithmetical operation’ (Collins English Dictionary 1991). The system of numeration utilised currently was derived from an ancient Hindu system. However there have been many other systems developed by various cultures through the centuries.
In recent years there has been a considerable discussion as to how children develop number concepts but it is generally agreed that they need experiences with counting, matching, grouping and comparing before reaching an understanding of number. There is now extensive evidence to show that the child’s ability to master Piaget’s tests of one-to-one correspondence does lay down some of the necessary foundations for counting, adding and subtracting. Constructivist theorist such as Vygotsky and Bruner encouraged the move towards viewing children as ‘active’ learners where they construct new ideas through social experience. Mental functioning is not merely absorbed or transmitted from teacher to student but actively constructed by the individuals as the result of social experience. To help children to understand concepts better it is necessary to scaffold their learning so that the next step can be taught. This demands both pedagogic and subject knowledge and recognition of the child’s actual and potential levels of achievement
The Cockcroft Report had an impact on the teaching of mathematics. It stated: ‘we do not believe that mathematics in the primary years should be seen solely as a preparation for the next stage of education. The primary years ought also to seen as worthwhile in themselves- a time during which doors are opened on to a wide range of experience’. (1982c,para.287)
The report reinstated the need for ‘mental arithmetic’ in the curriculum. It was recognised that there was a general lack of ability amongst adults in applying mathematical knowledge or using it to communicate effectively and therefore more practical work, problem solving, investigations and discussions were recommended and mental arithmetic was given substantial importance.
Children need to learn mathematics in order to solve everyday practical problems. During KS1 pupils develop their knowledge and understanding of mathematics through practical activities and use mathematics language to gain satisfaction and enjoyment from exploration and discussion. They also develop a range of mental calculation skills and use them in different settings and other areas of learning. They talk about their methods and explain their reasoning when solving problems. At KS2 pupils use numbers systems more confidently and discuss and present their methods and reasoning by using a wider range of mathematical language and diagrams.
The National Numeracy Strategy (1999) has laid emphasis on more interactive direct teaching of mathematics in primary schools. It states that: ‘Numeracy is a proficiency which involves confidence and competence with numbers and measures’. (NNS, p. 4)
The teaching approach recommended by the National Numeracy Strategy is based on four principles:
- dedicated maths lesson everyday;
- direct teaching and interactive oral work with the whole class and groups;
- an emphasis on mental calculations;
- controlled differentiation, with all pupils engaged in mathematics relating to a common theme.
(NNS, 1999, p. 11)
Number cards and number fans help to ensure that all the children respond to the questions and enable teachers to quickly see which children have understood a concept or know certain facts. They can be tailored to be larger or smaller and on different thickness of laminated card for pupils with poor fine motor skills. These numbers can be textured to become tactile to help in developing number writing skills and also for visually impaired children.
Number squares can be stuck into the back /front of an exercise book so it can be used whenever the pupil is working with numbers. For right-handed pupils it is better to attach number lines/squares inside the front cover and inside the back cover for left handed pupils so the writing arm doesn’t cover the number line/square.
Children have different learning styles in maths as well as in literacy. Sharma refers to quantitative and qualitative learners but Chinn and Ashcroft (1998) call them ‘inch worms and grasshoppers’. The qualitative learners are the ‘inch worms’ who prefer a step by step practical approach whereas qualitative learners or ‘grasshoppers’ are better at seeing patterns and relationships but they are not good at following procedures correctly. With whole class teaching the teacher needs to be aware of the needs of pupils by giving lots of examples, looking for and generating the rule or principle and then practising more examples.
Mathematical Investigations
Investigation plays a vital part in any mathematics curriculum (Cockcroft report 1982). Investigations provide children with opportunity to engage in discussion and language related to mathematical activity. In an investigative approach, pupils are encouraged to think of alternate strategies to consider related to the problem (Fisher, R & Vince, A 1990). Children require different skills and strategies to carry out investigations. They need to bring certain knowledge to the investigation. From my own experience of the above investigation and of teaching practice I believe that children can extend their mathematical knowledge by working with other children and sharing skills and experiences. Opportunities for discussions and collaborative tasks will allow children to see many mathematical possibilities, where individuals are dependent on each other to achieve a common goal. Children also develop personal qualities through investigational work, for example being persistent and systematic.
ICT and Mathematics
ICT can offer a range of opportunities and contexts for developing and extending mathematical understanding at any stage of educational development. It is a cross-curriculum subject. There are many possibilities to foster and develop these skills.
In maths ICT can enhance teaching and learning by enabling pupils to:
- explore, describe and explore number patterns
- practice and consolidate their number skills
- explore and explain patterns in data
- estimate and compare measures of lengths, distance, angle, time etc.
- develop their mathematical vocabulary, logical thinking and problem solving skills (Draft NNS, Nov. 98, p.31)
We had hands on activity using Roamers during one of ICT sessions at the university. Roamers can be used with younger or older children. By using directional language such as forward, back, left and right teachers can introduce concept of distance, number of steps, and angles of turn to move from one part of the room to the other. The children can predict the route; programme the Roamers and test out their predictions. Inexperienced children will need to put one or two instructions at a time, and then adjust the next accordingly, but later they can be challenged to write all the instructions in one go. They can also hold simple and complex sequences to solve problems that may require investigation, calculation and recording with groups of younger or older children accordingly
Having collected data such as “Favourite colour” or “Favourite food” the children can enter the data into simple handling package (e.g. Graphers). Using ‘click and drag’ they can sort the data on screen and also produce pictogram/graphs/charts etc to support their conclusions. Using a drawing package the children can create geometrical shapes. By copying, pasting, reflecting and translating the shapes the children can produce images.
LOGO is a computer programme based on a cursor in the shape of a turtle, moving around the screen. The children enter instructions to turn the turtle and move the turtle FD, BK, LT, RT to draw desired shapes and create patterns.
The constructivist approach to using computers has shown that computers can support children’s understanding particularly in problem solving. The implications are that teachers need to have a clear view of the aims and purposes of using ICT. When planning tasks teachers need to ensure that all activities will be accessible to all pupils, including pupils of different gender, ethnicity and ability (including children with special needs).
BIBLIOGRAPHY
Chin, S and Ashcroft, M. (1998) ‘Mathematics for Dyslexics.’ A Teaching Handbook, 2nd edn. London: Whurr.
Davis. J et al (1990) ‘Supporting Primary Mathematics-Algebra’, Milton Keynes: The Open University Press
DfEE (1999) The National Numeracy Strategy. A framework for Teaching Mathematics.
Fisher, R & Vince, A (1990) ‘Investigating Maths Book 2’, Oxford: Basil Blackwell Ltd.
Sharma, M. C. (1990) ‘Concept of Number’. Math Notebook 8 (1, 2). Mass, USA. Centre for Teaching/Learning Mathematics.
The Cockcroft Report, (1982) ‘Mathematics Counts’, London: HMS
INDICTIVE REFERENCES
ILEA Learning Resources Branch (1986) ‘Working With Number Lines’
Ian Thompson (2001) ‘Issues in teaching Numeracy in primary schools’: Open University Press, Buckingham
Division of Education: ‘Mathematics Selected readings’, PGCE 2002-2003
David Wood (2002) ‘How Children Think and Learn’: Blackwell Publishing Ltd.
Draft National Numeracy Strategy (Nov 1998)
Investigation
An investigation was chosen to find a formula that would calculate the difference of products of four numbers in the opposite corners of any size of rectangles/squares when multiplied, that an be drawn on a ‘hundred square’. For example for square of 4x4,
4x37 =148
34x7=238
Difference of Products = 238 -148=90
Hence it was found that the following formula solved this:
(m-1) (n-1) 10
(4-1) (4-1) 10
(3x3) 10
9x10=90
