Counting and understanding number

Aims: To understand how children learn to count and how visual images can support understanding of the number system and place value To be confident in subject knowledge regarding fractions, ratio and proportion

‘Of all the subjects, mathematics is perhaps the most demanding in terms of its need for in-depth subject knowledge, even at primary level. Confidence and flexibility in the classroom are essential prerequisites for the successful teacher of mathematics, and children are perhaps the most acutely sensitive barometer of any uncertainty on their part’ Williams Review (2008) ‘It is a combination of deep subject knowledge and a range of appropriate teaching and learning techniques which make for the most powerful interactions between teachers and pupils. Enhancing subject specialism therefore needs to be seen not as an end in itself, but as a way of bringing about excellence in teaching and learning to improve standards in our schools’ DfES (2003)

Mathematics is like cabbage, you either love it or hate it, but it is all dependent on how it was served up to you as a child!

Learning the words of counting sequences 1.Acquisition o Learning the sequence connected in a stream, of rhythmic sound. o Beginning to separate the individual words, maintaining their order. 2.Elaboration o Confirming the order of occurrence. o Knowing the order backwards o Knowing the sequence onward or backward, from a given point. o Confirming the connection of individual words to a related quantity.

Easy Counting Mistakes  Fail to correspond their pointing to individual objects.  Fail to correspond the sound with the pointing action.  Miss an object.  Itemise an object more than once.  Missing a number name.  Applying the same name twice.  Confusing the order of names.  Lose track of what has been counted and what remains to be counted.  Don't stop the verbal sequence at the last object, keeping on because of the rhythm.  Don't realise the last number is cardinal.  Miss some objects because they don't think they should be included in the count because of their colour, shape, position etc.

The Counting Principles How to count 1. The One to One Principle 2. The Stable Order Principle 3. The Cardinal Principle Applying counting 4. The Abstraction Principle 5. The Order Irrelevance Principle

Counting skills Knowing the number names in order Synchronising saying words and pointing or moving objects Keeping track of objects counted Recognising that the number associated with the last object touched is the total number of objects Recognising small numbers of objects without counting them Counting things you cannot move or touch or see, or objects that move around Counting objects of very different sizes Recognising that if a group of objects already counted is re-arranged then the number of objects stays the same Recognising that if objects are added or removed the number of objects changes

Slavonic Abacus For ‘spatial thinkers’ thought is composed primarily of mental images Mathematics is a subject in which one feels that spatial thinkers ought to do well but often they seem unable to ‘remember’ anything from one day to the next They may do well in areas like shape and space but simply cannot remember number bonds or multiplication tables Offer an image – concrete at first, then internalisation of the mental picture/model for the children to ‘see’ when they need it Particularly supportive for visual and kinaesthetic learners The colours enable children to ‘see’ (not count) numbers such as 6, 7, 8… One row can show complements to 10 Once the model is internalised, numbers which add to 10 can be seen in the mind’s eye as patterns of beads/cubes. These are far more meaningful, and therefore memorable, than the endless lists of number bonds to 10 which the spatial thinker can never recall In the same way, children can learn to ‘see’ two-digit numbers and their complements to 100 The abacus can also be used to ‘see’ a visual representation of multiplication (up to 10 x 10)

Fractions An opportunity to work collaboratively on activities related to fractions

Fractions is one of those concepts that many pupils find difficult. A reason for their difficulty is the relative nature of fractions: that the same fraction can refer to different quantities (e.g. ½ of 8 and ½ of 12 are different) and different fractions can be equivalent because they refer to the same quantity (1/3 and 3/9 for example)! Maths4life Fractions Booklet

What is a fraction? 1. A number in its own right (e.g. on a number line, the number is the result of dividing the top number by the bottom number) 2. A proportion of a whole (e.g. 2/3 of the class walk to school, 2 out of 3 children walk to school) 3. Relates to sharing objects (e.g. sharing two pizzas between three people)

Fractions can be introduced by asking learners to think about some fractions they have encountered in an everyday context. Before dealing with the written symbolic form of any fractions we can name some of them and talk about what they mean. Some interesting questions to raise: - When do people use fractions? - Do fractions matter if they are only small parts? - Is a half always the same size? - Can a fraction be bigger than one whole unit? - Is it possible to have three halves? - Are fractions anything to do with division?

Imagine a square 10 ? ? ? ? 12 ?