Thank you to all those who were able to join us for the maths information night this past Tuesday. The following is a rationale for the experiences that were presented on the evening.
A context for the mathematics information night:
It is a central belief of mathematics education that learning occurs through the construction of ideas, processes and understandings in a social setting rather than by the transfer of pre-formed knowledge from teacher to student. Consequently two factors are important in the development of mathematical understanding
- The use of materials that assist children in the construction of these understandings
- The use of a consistent language that is appropriate to the capabilities and needs of each child.
Sequences of development that build on well understood prior knowledge are necessary. But even well designed games and activities are not enough on their own. Learning situations need to have a context that encourages discussion in order to elicit emerging understandings. Talking about what is happening, and reflecting on the concepts being constructed enables concepts to take shape in the learner’s mind.
Children often have beliefs and methods which can appear very different from accepted mathematical practice. Ill-formed ideas and inappropriate generalisations need to be challenged, using activities that require children to revise former ways of thinking. The challenge is to lead children to understand and accept the new way of thinking as their own rather than to get them to learn another person’s method by rote. Evidence from children who have experienced difficulty in learning mathematics has shown that those who simply acquire teacher-taught techniques by rote are often unable to apply this knowledge outside of the situation it is taught in. By contrast children who participate actively in their own learning are more able to apply knowledge and understanding and to maintain future use and adaption.
At Princes Hill Primary the following four ideas underpin all mathematical learning and these exist at all year levels. It is the complexity and the sophistication of these ideas that develop as children progress in their mathematical learning. Not all children progress at the same pace which is why teachers differentiate the learning.
‘Really big ideas’:
- Representation – numbers can be modelled and represented in many different ways (e.g., materials, diagrams, number charts, partial/open number lines)
- Enumeration – whole numbers are used to count collections, counts can be achieved in multiple ways, and different units can be used to say how many or how much
- Equivalence – numbers can be renamed in many different but equivalent ways, renaming is a special type of representation
- Relationships – numbers can be used to compare and order; relationships between numbers lead to different number sets (e.g. fractions, ratios, per cents, etc)
This evening we are focusing on introducing the concept of place value. Place value is an essential concept to learn because it underpins computation processes. It involves much more than recognising place value parts. Place value is a system of assigning values to digits based on their position (a base 10 system of numeration, positions represent successive powers of 10)
Big Ideas for Place Value
- Whole numbers can be recognised as cardinal numbers as well as composite units, that is, as numbers that tell how many in a set (e.g. 6 ones) or as units in their own right (e.g. 1 six)
- A sense of numbers beyond 10 as ‘a ten and some more’ is necessary to appreciate the two-digit place-value pattern.
- Two patterns underpin place-value understanding at this level of schooling: ’10 of these is 1 of those’ and ‘1000 of these is 1 of those’.
- Place value knowledge is developed by making (representing) numbers in terms of their place value parts, naming and recording
- Place value knowledge is consolidated by comparing, ordering, counting forwards and backwards in place value parts, and renaming
Before they are ready to meet the ‘big ideas’ of place value, children need to be able to:
- Count fluently by ones using the number naming sequence to 20 and beyond
- Model, read and write numbers to 10 using materials, diagrams, words and symbols
- Recognise collections to 10 without counting
- Trust the count for each of the numbers to 10 without having to model or count by ones
- Demonstrate a sense of numbers beyond 10 in terms of 1 ten and some more
- Count larger collections by two’s, fives, and tens
Cardinal number – a specific number name for how many in a given collection of objects
Composite unit – a unit made up of other units ( when children understand 6 as one 6 rather than a collection of 6 ones.
Conceptual understanding – understanding that is made between new and existing ideas ( eg: a conceptual understanding of area allows students to apply this knowledge to an unfamiliar problem such as determining the dimensions of a garden given the length of the fencing around it)
Context – the situation or circumstances that require the application of numeracy skills
Renaming – writing a number in an equivalent form, usually in terms of its place value parts (eg 365 is 3 hundreds 6 tens and 5 ones but it can be renamed as 36 tens and 5 ones or 3 hundreds and 65 ones and renaming when adding/multiplying or subtracting/dividing (eg 5 tens and 8 tens is 13 tens, it is regrouped for recording purposes as 1 hundred and 3 tens but when subtracting 28 from 45 the 8 ones can only be taken if 1 of the 4 tens is renamed as 10 ones)