Understanding mathematics for young children: a guide for foundation stage and lower primary teachers
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Los Angeles [u.a.]
Sage
2008
|
| Edition: | [3] fully rev. and expand. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | IX, 260 S. |
| ISBN: | 9781412947251 9781412947268 |
Staff View
MARC
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| 245 | 1 | 0 | |a Understanding mathematics for young children |b a guide for foundation stage and lower primary teachers |c Derek Haylock and Anne D. Cockburn |
| 250 | |a [3] fully rev. and expand. ed. | ||
| 264 | 1 | |a Los Angeles [u.a.] |b Sage |c 2008 | |
| 300 | |a IX, 260 S. | ||
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Record in the Search Index
| _version_ | 1804137927513473024 |
|---|---|
| adam_text | CONTENTS
Acknowledgements x
Introduction
1
The mathematics curriculum
1
The aims of the book
2
Input from teachers
2
Classroom activities
3
A note on terminology
4
1
Understanding mathematics
5
Learning and teaching mathematics with understanding
6
Learning with understanding. Teaching with understanding.
Concrete materials, symbols, language and pictures
7
An example in a nursery class.
Understanding as making connections
9
Connections between the four key components. An illustration: 7-year-olds
and the concept of division.
Understanding place-value notation
12
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
15
Mathematical symbols are not just abbreviations. A symbol represents
a network of connections.
Transformation and equivalence
17
What is different? What is the same? What stays the same when
things change?
VI UNDERSTANDING MATHEMATICS FOR YOUNG CHILDREN
The equals sign
19
The equals sign representing an equivalence. The equals
sign representing a transformation. One symbol, two meanings.
Some activities to use with children
22
Summary of key ideas
28
Suggestions for further reading
28
2
Understanding number and counting
30
What is three?
31
Numbers and numerals. Sets of three and one-to-one matching.
Adjective or noun? Nominal, cardinal and ordinal.
Understanding number
34
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
39
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
45
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
52
Summary of key ideas
56
Suggestions for further reading
58
3
Understanding addition and subtraction
59
What is addition?
60
The aggregation structure: union of two sets. The augmentation
structure: counting on and increasing. The relationship between
the ttvo addition structures.
What is subtraction?
62
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.
CONTENTS
Verbal miscues
74
Some activities to use with children
75
Summary of key ideas
82
Suggestions for further reading
82
4
Understanding multiplication and division
83
Understanding multiplication
84
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
92
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
100
Some activities to use with children
100
Summary of key ideas
104
Suggestions for further reading
105
5
Understanding the principles of arithmetic
106
Cornimi
tativity
107
The principle of complements
108
Subtraction. Division.
Compensation
110
Associativity
110
Associativity of addition. Associativity of multiplication.
Identities
113
Zero and one. Multiplication and division with zero.
Inverses
114
Some activities to use with children
115
Summary of key ideas
117
Suggestions for further reading
118
6
Understanding patterns in calculations
119
Pattern in number
120
Odd and even. Spatial patterns and visual images related to
numerical patterns. Patterns of dots for numbers.
Pattern in complements
122
Ten-complements. Making a number up to
10.
Pairs of complements
for all numbers up to
20.
Hundred-complements.
VIII
UNDERSTANDING MATHEMATICS FOR YOUNG CHILDREN
Patterns of fives and doubles
126
Fives. Doubles.
Pattern in multiplication tables
128
Tens, fives and twos. Reducing the workload.
The hundred square
129
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
136
Overemphasis on vertical layout. Some typical errors in vertical layout.
The empty number line
138
Developing the number line. Additions and subtractions on the
empty number line.
Some activities to use with children
139
Summary of key ideas
145
Suggestions for further reading
146
7
Understanding measurement
147
What do we measure?
148
Length and distance. Volume and capacity. Time. Mass and weight.
Measurement in general
154
Comparison. Ordering and transitivity. Conservation. Units. SI base
units and other metric units. Approximation and accuracy.
Types of measurement scales
164
Ratio scales. Interval scales. Ordinal scales.
Some activities to use with children
169
Summary of key ideas I74
Suggestions for further reading
175
8
Understanding shape and space I76
Number and shape: two branches of mathematics
177
Guess my rule. Equivalence and transformation again.
Three-dimensional or two-dimensional shapes
180
A mathematical analysis of shape and space
180
Translation. Rotation. Reflection. Similarity. Family likeness.
Perspectivity. Topological transformation.
Some activities to use with children I96
Summary of key ideas
202
Suggestions for further reading
202
9
Understanding data-handling
203
Meaningful, purposeful and cross-curricular
204
Meaningfulness. Purposefulness. Cross-curricular. An example.
CONTENTS ¡x
Pictorial representation
205
Making connections. Differences between numerical and pictorial
representations.
Ways of representing data
207
Representing discrete sets. Intersecting sets. Databases and
spreadsheets. The language of logic. Other ways of representing
classifications.
Representing frequency
213
Frequency tables. Moving towards a bar chart.
Different kinds of variable
216
Unordered and ordered discrete data. Grouped discrete data.
Continuous data.
Some activities to use with children
219
Summary of key ideas
223
Suggestions for further reading
224
10
Using and applying mathematics
225
The nature of mathematics
226
Content and cognitive processes. Using and applying.
Two dimensions in using and applying mathematics
227
Abstract or real life. Closed or open. The range of activities for children.
Solving problems
229
What is a problem?
Givens,
goal and gap.
Representing
231
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
236
Reasoning
237
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
244
Communicating results and findings. Explaining and proving.
Some activities to use with children
246
Summary of key ideas
250
Suggestions for further reading
252
References
253
Index
256
|
| adam_txt |
CONTENTS
Acknowledgements x
Introduction
1
The mathematics curriculum
1
The aims of the book
2
Input from teachers
2
Classroom activities
3
A note on terminology
4
1
Understanding mathematics
5
Learning and teaching mathematics with understanding
6
Learning with understanding. Teaching with understanding.
Concrete materials, symbols, language and pictures
7
An example in a nursery class.
Understanding as making connections
9
Connections between the four key components. An illustration: 7-year-olds
and the concept of division.
Understanding place-value notation
12
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
15
Mathematical symbols are not just abbreviations. A symbol represents
a network of connections.
Transformation and equivalence
17
What is different? What is the same? What stays the same when
things change?
VI UNDERSTANDING MATHEMATICS FOR YOUNG CHILDREN
The equals sign
19
The equals sign representing an equivalence. The equals
sign representing a transformation. One symbol, two meanings.
Some activities to use with children
22
Summary of key ideas
28
Suggestions for further reading
28
2
Understanding number and counting
30
What is three?
31
Numbers and numerals. Sets of three and one-to-one matching.
Adjective or noun? Nominal, cardinal and ordinal.
Understanding number
34
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
39
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
45
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
52
Summary of key ideas
56
Suggestions for further reading
58
3
Understanding addition and subtraction
59
What is addition?
60
The aggregation structure: union of two sets. The augmentation
structure: counting on and increasing. The relationship between
the ttvo addition structures.
What is subtraction?
62
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.
CONTENTS
Verbal miscues
74
Some activities to use with children
75
Summary of key ideas
82
Suggestions for further reading
82
4
Understanding multiplication and division
83
Understanding multiplication
84
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
92
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
100
Some activities to use with children
100
Summary of key ideas
104
Suggestions for further reading
105
5
Understanding the principles of arithmetic
106
Cornimi
tativity
107
The principle of complements
108
Subtraction. Division.
Compensation
110
Associativity
110
Associativity of addition. Associativity of multiplication.
Identities
113
Zero and one. Multiplication and division with zero.
Inverses
114
Some activities to use with children
115
Summary of key ideas
117
Suggestions for further reading
118
6
Understanding patterns in calculations
119
Pattern in number
120
Odd and even. Spatial patterns and visual images related to
numerical patterns. Patterns of dots for numbers.
Pattern in complements
122
Ten-complements. Making a number up to
10.
Pairs of complements
for all numbers up to
20.
Hundred-complements.
VIII
UNDERSTANDING MATHEMATICS FOR YOUNG CHILDREN
Patterns of fives and doubles
126
Fives. Doubles.
Pattern in multiplication tables
128
Tens, fives and twos. Reducing the workload.
The hundred square
129
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
136
Overemphasis on vertical layout. Some typical errors in vertical layout.
The empty number line
138
Developing the number line. Additions and subtractions on the
empty number line.
Some activities to use with children
139
Summary of key ideas
145
Suggestions for further reading
146
7
Understanding measurement
147
What do we measure?
148
Length and distance. Volume and capacity. Time. Mass and weight.
Measurement in general
154
Comparison. Ordering and transitivity. Conservation. Units. SI base
units and other metric units. Approximation and accuracy.
Types of measurement scales
164
Ratio scales. Interval scales. Ordinal scales.
Some activities to use with children
169
Summary of key ideas I74
Suggestions for further reading
175
8
Understanding shape and space I76
Number and shape: two branches of mathematics
177
Guess my rule. Equivalence and transformation again.
Three-dimensional or two-dimensional shapes
180
A mathematical analysis of shape and space
180
Translation. Rotation. Reflection. Similarity. Family likeness.
Perspectivity. Topological transformation.
Some activities to use with children I96
Summary of key ideas
202
Suggestions for further reading
202
9
Understanding data-handling
203
Meaningful, purposeful and cross-curricular
204
Meaningfulness. Purposefulness. Cross-curricular. An example.
CONTENTS ¡x
Pictorial representation
205
Making connections. Differences between numerical and pictorial
representations.
Ways of representing data
207
Representing discrete sets. Intersecting sets. Databases and
spreadsheets. The language of logic. Other ways of representing
classifications.
Representing frequency
213
Frequency tables. Moving towards a bar chart.
Different kinds of variable
216
Unordered and ordered discrete data. Grouped discrete data.
Continuous data.
Some activities to use with children
219
Summary of key ideas
223
Suggestions for further reading
224
10
Using and applying mathematics
225
The nature of mathematics
226
Content and cognitive processes. Using and applying.
Two dimensions in using and applying mathematics
227
Abstract or real life. Closed or open. The range of activities for children.
Solving problems
229
What is a problem?
Givens,
goal and gap.
Representing
231
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
236
Reasoning
237
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
244
Communicating results and findings. Explaining and proving.
Some activities to use with children
246
Summary of key ideas
250
Suggestions for further reading
252
References
253
Index
256 |
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| id | DE-604.BV035007985 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T21:42:55Z |
| indexdate | 2024-07-09T21:20:04Z |
| institution | BVB |
| isbn | 9781412947251 9781412947268 |
| language | English |
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| spelling | Haylock, Derek Verfasser aut Understanding mathematics for young children a guide for foundation stage and lower primary teachers Derek Haylock and Anne D. Cockburn [3] fully rev. and expand. ed. Los Angeles [u.a.] Sage 2008 IX, 260 S. txt rdacontent n rdamedia nc rdacarrier Mathematik Mathematics Study and teaching (Elementary) Grundschulkind (DE-588)4022350-4 gnd rswk-swf Mathematikunterricht (DE-588)4037949-8 gnd rswk-swf Grundschulkind (DE-588)4022350-4 s Mathematikunterricht (DE-588)4037949-8 s b DE-604 Cockburn, Anne D. Verfasser (DE-588)104406997X aut Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016677279&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Haylock, Derek Cockburn, Anne D. Understanding mathematics for young children a guide for foundation stage and lower primary teachers Mathematik Mathematics Study and teaching (Elementary) Grundschulkind (DE-588)4022350-4 gnd Mathematikunterricht (DE-588)4037949-8 gnd |
| subject_GND | (DE-588)4022350-4 (DE-588)4037949-8 |
| title | Understanding mathematics for young children a guide for foundation stage and lower primary teachers |
| title_auth | Understanding mathematics for young children a guide for foundation stage and lower primary teachers |
| title_exact_search | Understanding mathematics for young children a guide for foundation stage and lower primary teachers |
| title_exact_search_txtP | Understanding mathematics for young children a guide for foundation stage and lower primary teachers |
| title_full | Understanding mathematics for young children a guide for foundation stage and lower primary teachers Derek Haylock and Anne D. Cockburn |
| title_fullStr | Understanding mathematics for young children a guide for foundation stage and lower primary teachers Derek Haylock and Anne D. Cockburn |
| title_full_unstemmed | Understanding mathematics for young children a guide for foundation stage and lower primary teachers Derek Haylock and Anne D. Cockburn |
| title_short | Understanding mathematics for young children |
| title_sort | understanding mathematics for young children a guide for foundation stage and lower primary teachers |
| title_sub | a guide for foundation stage and lower primary teachers |
| topic | Mathematik Mathematics Study and teaching (Elementary) Grundschulkind (DE-588)4022350-4 gnd Mathematikunterricht (DE-588)4037949-8 gnd |
| topic_facet | Mathematik Mathematics Study and teaching (Elementary) Grundschulkind Mathematikunterricht |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016677279&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT haylockderek understandingmathematicsforyoungchildrenaguideforfoundationstageandlowerprimaryteachers AT cockburnanned understandingmathematicsforyoungchildrenaguideforfoundationstageandlowerprimaryteachers |