- Browse
- » Understanding mathematics for young children: a guide for foundation stage and lower primary teachers
Understanding mathematics for young children: a guide for foundation stage and lower primary teachers
Author
Publisher
SAGE
Publication Date
2008
Edition
Rev. and expanded ed.
Language
English
Description
Loading Description...
Table of Contents
From the Book - Rev. and expanded ed.
Acknowledgements
Introduction
The mathematics curriculum
The aims of the book
Input from teachers
Classroom activities
A note on terminology
1. Understanding mathematics
Learning and teaching mathematics with understanding
Learning with understanding
Teaching with understanding
Concrete materials, symbols, language and pictures
An example in a nursery class
Understanding as making connections
Connections between the four key components
An illustration: 7-year-olds and the concept of division
Understanding place-value notation
The principle of place value
The principle of exchange
Connecting materials, symbols and arrow cards
Connecting the number names with the symbols
Zero as a place holder
Connecting symbols for numbers with the number line
The function of a mathematical symbol
Mathematical symbols are not just abbreviations
A symbol represents a network of connections
Transformation and equivalence
What is different?
What is the same?
What stays the same when things change?
The equals sign
The equals sign representing an equivalence
The equals sign representing a transformation
One symbol, two meanings
Some activities to use with children
Summary of key ideas
Suggestions for further reading
2. Understanding number and counting
What is three?
Numbers and numerals
Sets of three and one-to-one matching
Adjective or noun?
Nominal, cardinal and ordinal
Understanding number
Connecting symbols with the number line
A network of connections
Laying the foundations for later experiences of number
Overemphasis on the cardinal aspect
Assimilation, restructuring and accommodation
Understanding zero
Understanding counting
Pre-counting experiences
The order of numbers is invariant
One-to-one matching of number names to objects
Connecting cardinal and ordinal aspects
Counting as an abstraction
The order of the objects is irrelevant
The arrangement of the objects is irrelevant
Matching the names to the numerals
Connecting 'one more' and the 'next number'
The pattern in counting
Numbers go on for ever
Not all numbers are counting numbers
A mathematical analysis of number
Unstable truths, changing properties and new possibilities
The natural numbers
Integers
Rational numbers
Real numbers
The significance of this mathematical analysis
Some activities to use with children
Summary of key ideas
Suggestions for further reading
3. Understanding addition and subtraction
What is addition?
The aggregation structure: union of two sets
The augmentation structure: counting on and increasing
The relationship between the two addition structures
What is subtraction?
The partitioning structure: take away
Subtraction is not just partitioning
The comparison structure
The language of comparison
The complement of a set structure
The reduction structure: counting back
The inverse of addition structure
Overemphasis on 'take away'
The network of connections for understanding subtraction
Verbal miscues
Some activities to use with children
Summary of key ideas
Suggestions for further reading
4. Understanding multiplication and division
Understanding multiplication
Children's difficulties in understanding multiplication
The repeated addition structure for multiplication
The commutative principle (multiplication and addition)
Rectangular arrays
A network of connections for multiplication
Contexts for repeated addition
The scaling structure
Understanding division
Two structures for division: equal sharing and inverse of multiplication
Overemphasis on sharing
Division in money and measuring contexts
Rectangular arrays and division
Division as repeated subtraction
Division as ratio
Experiences of sharing that do not correspond to division
The language of division
Conclusion
Some activities to use with children
Summary of key ideas
Suggestions for further reading
5. Understanding the principles of arithmetic
Commutativity
The principle of complements
Subtraction
Division
Compensation
Associativity
Associativity of addition
Associativity of multiplication
Identities
Zero and one
Multiplication and division with zero
Inverses
Some activities to use with children
Summary of key ideas
Suggestions for further reading
6. Understanding patterns in calculations
Pattern in number
Odd and even
Spatial patterns and visual images related to numerical patterns
Patterns of dots for numbers
Pattern in complements
Ten-complements
Making a number up to 10
Pairs of complements for all numbers up to 20
Hundred-complements
Patterns of fives and doubles
Fives
Doubles
Pattern in multiplication tables
Tens, fives and twos
Reducing the workload
The hundred square
Pattern in the hundred square
Adding ones and adding tens
Subtracting ones and subtracting tens
Patterns for adding nines and eights
Additions with two-digit numbers on the hundred square
Subtractions with two-digit numbers on the hundred square
The two-hundred grid
Children's errors and vertical layout
Overemphasis on vertical layout
Some typical errors in vertical layout
The empty number line
Developing the number line
Additions and subtractions on the empty number line
Some activities to use with children
Summary of key ideas
Suggestions for further reading
7. Understanding measurement
What do we measure?
Length and distance
Volume and capacity
Time
Mass and weight
Measurement in general
Comparison
Ordering and transitivity
Conservation
Units
SI base units and other metric units
Approximation and accuracy
Types of measurement scales
Ratio scales
Interval scales
Ordinal scales
Some activities to use with children
Summary of key ideas
Suggestions for further reading
8. Understanding shape and space
Number and shape: two branches of mathematics
Guess my rule
Equivalence and transformation again
Three-dimensional or two-dimensional shapes
A mathematical analysis of shape and space
Translation
Rotation
Reflection
Similarity
Family likeness
Perspectivity
Topological transformation
Some activities to use with children
Summary of key ideas
Suggestions for further reading
9. Understanding data-handling
Meaningful, purposeful and cross-curricular
Meaningfulness
Purposefulness
Cross-curricular
An example
Pictorial representation
Making connections
Differences between numerical and pictorial representations
Ways of representing data
Representing discrete sets
Intersecting sets
Databases and spreadsheets
The language of logic
Other ways of representing classifications
Representing frequency
Frequency tables
Moving towards a bar chart
Different kinds of variable
Unordered and ordered discrete data
Grouped discrete data
Continuous data
Some activities to use with children
Summary of key ideas
Suggestions for further reading
10. Using and applying mathematics
The nature of mathematics
Content and cognitive processes
Using and applying
Two dimensions in using and applying mathematics
Abstract or real life
Closed or open
The range of activities for children
Solving problems
What is a problem?
Givens, goal and gap
Representing
A real-life problem about transport
Mathematical modelling
The mathematical solution
Interpreting the mathematical solution
Taking into account the constraints of the real world
Finding a route through a problem
Round the modelling cycle again
Enquiry
Reasoning
An investigation with newspapers
Articulating patterns
The language of generalizations
Sequential generalizations and conjectures
Global generalizations
Comparing sequential and global generalizations
Counterexamples and special cases
Hypotheses and higher-order generalizations
Communicating
Communicating results and findings
Explaining and proving
Some activities to use with children
Summary of key ideas
Suggestions for further reading
References
Index
Author Notes
Loading Author Notes...
More Details
Contributors
ISBN
9781412947251
141294726
9781412947268
141294726
9781412947268
Staff View
Loading Staff View.
