Mathematical Misconceptions: A Guide for Primary Teachers

177 Pages·2008·12.41 MB·English

by  Dr Anne Cockburn

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Mathematical
Misconceptions
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Mathematical
Misconceptions
A Guide for Primary Teachers
Edited by
Anne D. Cockburn and Graham Littler
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©AnneD.CockburnandGrahamLittler2008
Firstpublished2008
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Contents
List of figures vi
List of pictures ix
List of tables x
Introduction 1
0 Zero: understanding an apparently paradoxical number 7
Anne D. Cockburn and Paul Parslow-Williams
1 Equality: getting the right balance 23
Paul Parslow-Williams and Anne D. Cockburn
2 Beginning to unravel misconceptions 39
Sara Hershkovitz, Dina Tirosh and Pessia Tsamir
3 Insights into children’s intuitions of addition, subtraction, 54
multiplication and division
Dina Tirosh, Pessia Tsamir and Sara Hershkovitz
4 Right or wrong? Exploring misconceptions in division 71
Pessia Tsamir, Sarah Hershkovitz and Dina Tirosh
5 Developing an understanding of children’s acquisition of
number concepts 86
Anne D. Cockburn
6 Highlighting the learning processes 101
Graham Littler and Darina Jirotková
7 Everyday numbers under a mathematical magnifying glass 123
Carlo Marchini and Paola Vighi
Appendix 152
Index 159
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List of figures
Chapter 0 7
Figure 0.1 Penny’s ‘47’ 15
Figure 0.2 Using a number line as part of the ‘squeeze game’ 17
Chapter 1 23
Figure 1.1 Extract from RobertRecorde’s The Whetstone of
Witte, 1557 (Cajori, 1928: 165) 24
Figure 1.2 A series of equality problems based on addition 25
Figure 1.3 A series of equality problems based on subtraction 26
Figure 1.4 Examples of pupils’ recording strategies for solving
equality problems 28
Figure 1.5 Equality problems represented in a visual form 32
Figure 1.6 Visual representation of a ‘missing number’ equality
problem using a suspension balance 34
Figure 1.7 Picture cards used to represent objects in the equality
relationship 2 = 3 – 1 35
Figure 1.8 An example of a pictorial model that could be
used as a basis for discussion of number sentence
structures 35
Chapter 2 39
Figure 2.1 Arranging four digits to create a range of
calculations 41
Figure 2.2 Arranging four digits to create a calculation
involving addition 41
Figure 2.3 Addition tasks to explore children’s understanding of
place value 43
Figure 2.4 Addition task to deepen the pupils’ understanding 43
Figure 2.5 New task developed by teachers 44
Figure 2.6 Mathematical reasoning for the addition task 44
Figure 2.7 Arranging four digits to create a calculation
involving subtraction 44
Figure 2.8 Subtraction tasks to explore children’s understanding
of place value 47
Figure 2.9 Mathematical reasoning for the subtraction task 47
Figure 2.10 Arranging four digits to create a calculation
involving division 48
Figure 2.11 Arranging four digits to create a calculation
involving multiplication 49
Figure 2.12 Mathematical reasoning for the multiplication task 51
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LISTOFFIGURES vii
Figure 2.13 Division and multiplication tasks to explore children’s
understanding of place value 52
Figure 2.14 Arranging four digits to create a calculation involving
multiplication – the case of impossible solution 52
Chapter 3 54
Figure 3.1 Debby’s calculation task 55
Figure 3.2 Debby’s trueor false statements 56
Figure 3.3 A sample of Ben’s response 56
Figure 3.4 Children’s correct and incorrect responses to
‘addition makes bigger’ 57
Figure 3.5 Debby’s subtraction task 58
Figure 3.6 Correct and incorrect responses to ‘subtraction
makes smaller’ 59
Figure 3.7 The multiplication task the teachers were set 60
Figure 3.8 Debby’s division task for teachers 61
Figure 3.9 Common intuitive beliefs about division 62
Figure 3.10 Debby’s estimation task 63
Figure 3.11 Dividing decimals: algorithmic, formal and intuitive
mistakes 64
Figure 3.12 Teaching by analogy: Dan’s example 65
Figure 3.13 The three steps of the cognitive conflict teaching
method 66
Chapter 4 71
Figure 4.1 Right or wrong? 76
Figure 4.2 A division task with one divisor 80
Figure 4.3 A division task with several divisors 80
Figure 4.4 Tammy’s first subtraction task 83
Figure 4.5 Tammy’s second subtraction task 83
Figure 4.6 Tammy’ s final task 84
Chapter 5 86
Figure 5.1 Two buildings with the same number of rooms, illustrating
the commutativity of multiplication 90
Figure 5.2 A wood illustrating the commutativity of multiplication 90
Figure 5.3 A child’s system to assist in the counting out
100 one-penny coins 93
Figure 5.4 Exchanging ten ones for a ten rod 96
Figure 5.5 Making connections 97
Chapter 6 101
Figure 6.1 Cubes lying down 104
Figure 6.2 Cubes as a tower 105
Figure 6.3 Three piles of matchsticks 106
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viii MATHEMATICALMISCONCEPTIONS
Figure 6.4 Matchsticks in pattern 106
Figure 6.5 Predicted strategies for completing task B 115
Chapter 7 123
Figure 7.1 A first landscape of natural numbers 125
Figure 7.2 A landscape ofNwith addition 127
Figure 7.3 A landscape ofNwith addition and subtraction 128
Figure 7.4 A landscape ofNwith operations (addition, multiplication),
and subtraction 129
Figure 7.5 A landscape ofNwith operations (addition, multiplication),
and algorithms (subtraction, division) 132
Figure 7.6 A landscape ofNwith operations (addition,
multiplication), algorithms (subtraction, division), and
relations (equality, order) 135
Figure 7.7 The newspaper game 138
Figure 7.8 Possible solutions of the game (disregarding accompanying
text conditions) 138
Figure 7.9 The (newspaper) solution of the game 139
Figure 7.10 A diagram illustrating the building up of theNandz
number systems 140
Figure 7.11 Nandz, their propertiesand the connections between them 140
Figure 7.12 Carlo’s game 141
Figure 7.13 The number systemsN,zandq, including the absolute
rational numbers,Q 144
a
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List of pictures
Introduction 1
‘I want 3 as I’m 3’ 4
The honey collecting task 4
The 3 little pigs 5
Patsy’s bus stop task 5
‘You are only nothing’ 6
Chapter 0 7
0 in the bag 9
A fist of zero 10
18 number fan 18
400 and 30 and 9 18
439 19
Chapter 2 39
A primary mathematics lesson in Israel 39
Chapter 3 54
Debby’s group of teachers 55
‘4 ÷0.5 = 8’ or 8 half-apples 68