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E-grāmata: Developing Fractions Knowledge

  • Formāts: 224 pages
  • Sērija : Math Recovery
  • Izdošanas datums: 21-Jul-2016
  • Izdevniecība: SAGE Publications Inc
  • Valoda: eng
  • ISBN-13: 9781473986855
  • Formāts - PDF+DRM
  • Cena: 36,87 €*
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  • Formāts: 224 pages
  • Sērija : Math Recovery
  • Izdošanas datums: 21-Jul-2016
  • Izdevniecība: SAGE Publications Inc
  • Valoda: eng
  • ISBN-13: 9781473986855

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Supporting and understanding your students’ fractional knowledge is crucial to their overall grasp of numbers and mathematics. By centralising around three key stages of development, this effective guide will help you to assess your students’ understanding of fractions and modify your teaching accordingly.

These key stages are identified as:

·         Stage 1a: Fair Sharing

·         Stage 1b: Part-Whole

·         Stage 2a: Disembedding and Iterating

·         Stage 2b: Measuring with Unit Fractions

·         Stage 2c: Reversing Fractions

·         Stage 3a: Fractions as Numbers

·         Stage 3b: Operating with Fractions

As the newest addition to the bestselling Maths Recovery series, this book will be a useful guide for all primary classroom teachers and assistants, including experienced Mathematics Recovery instructors.



From the bestselling Maths Recovery series comes another exciting resource for teachers. Focusing around the three stages of development that connect to whole number knowledge this toolkit will support effective and engaging mathematics teaching. 

Recenzijas

With its beginnings in whole-number arithmetic understanding, Developing Fractions Knowledge is the most comprehensive look at fractions learning to date. It journeys with the practitioner and learner to view fractions as part-whole concepts, measures, and ultimately as numbers with meaning. The assessment and instructional tasks and practical connections to research and Math Recovery principles make it a must-have resource for educators to use again and again! -- Carolyn Olijnek Developing Fractions Knowledge skilfully combines cutting-edge research with practical knowledge of the classroom to provide an essential resource for everyone interested in developing students understanding of fractions. I cant wait to try some of the tasks and activities with students! -- Dr Janette Bobis, Professor in Mathematics Education, All in, it was an enjoyable book (I have to admit thinking a book on fractions is not one I would choose) and I think would be a worthwhile investment for a department. -- N. G. Macleod The book includes many example assessment tasks and instructional activities to use with pupils who are learning about fractions for the first time, as well as pupils who are struggling and working to relearn concepts.

This book provides a comprehensive look at fraction knowledge and understanding. It is a great book for anyone teaching primary maths at any age, but would also be appropriate for a longer-term, on-going departmental professional development. -- Matthew Reames

List of Figures xv
List of Tables xix
About the Authors xxi
Acknowledgments xxiii
Series xxv
Introduction 1(2)
The Purpose of the Book
1(1)
The Structure of the Book
1(2)
1 Professional Learning and Three Grand Organizers for Arithmetic Instruction 3(12)
Scenario 1: School-Based Mathematics Leader
3(1)
Scenario 2: District-Wide Math Recovery® Leader
4(1)
Scenario 3: Mathematics Coach
4(1)
Three Grand Organizers for Arithmetic Instruction
5(10)
Guiding Principles for Instruction
5(3)
Domains of Arithmetic Knowledge
8(3)
Dimensions of Mathematizing
11(4)
2 From Whole Numbers to Fractions 15(12)
The Reorganization Hypothesis
15(1)
Unitizing
16(1)
Fragmenting and Partitioning
17(1)
Iterating
18(1)
Disembedding
19(1)
Distributing
20(2)
Units Coordination
22(2)
Introduction to Content
Chapters
24(1)
Stage 1
24(1)
Stage 2
25(1)
Stage 3
25(2)
3 Assessing Stages of Units Coordination 27(14)
Tasks for Assessing Units Coordination
27(6)
Rubric for Assessing Units Coordination
33(2)
Sample Responses
35(6)
4 Teaching Students at Stage 1: Fragmenting 41(16)
Domain Overview
41(1)
Fragmenting and Partitioning
42(2)
Testing Equality of Shares
44(1)
Working with Students at the First Three Levels of Fragmenting
45(5)
Fragmenting Problem Type 1: Equal Sharing
46(1)
Fragmenting Problem Type 2: Drawing Unit Fractions
46(3)
Fragmenting Problem Type 3: Making Wholes from Parts
49(1)
Assessment Task Groups
50(3)
A4.1 Equal Sharing of Single Items
50(1)
A4.2 Making One Equal Share
50(1)
A4.3 Drawing a Unit Fraction of One Bar and Justifying the Amount
51(1)
A4.4 Making Connected Numbers
51(1)
A4.5 Drawing a Unit Fraction of a Segmented Bar and Justifying the Amount
52(1)
Instructional Activities
53(4)
IA4.1 Share an Energy Bar Fairly
53(1)
IA4.2 Make Only Your Own Share
54(1)
IA4.3 Slicing Cakes in a Bakery
55(1)
IA4.4 Working with Connected Numbers
55(2)
5 Transitioning to Stage 2: Part-Whole Reasoning 57(14)
Domain Overview
57(1)
Fractions as Parts Within Wholes
58(1)
Fractions as Parts Out of Wholes
59(1)
Working with Students with Parts Within Wholes Fraction Schemes
60(4)
Assessment Task Groups
64(2)
A5.1 Parts Within the Whole
64(1)
A5.2 Parts Out of the Whole
64(1)
A5.3 Finding an Equal Share
65(1)
A5.4 Iterating a Part
65(1)
Instructional Activities
66(5)
1A5.1 Is It Fair?
66(1)
IA5.2 Tiny Creature Journeys
67(1)
IA5.3 Filling an Order
68(1)
IA5.4 Comparing Unit Fractions
68(1)
IA5.5 Finding Unit Fractions of Chocolate Bars
69(2)
6 Teaching Students at Stage 2: Measuring with
Unit Fractions
71(1)
Domain Overview
71(1)
Unit Fractions and the Whole
72(1)
The Measurement Scheme for Unit Fractions
73(1)
The Role of the Whole
73(1)
Reciprocal Relationship Between the Unit Fraction and the Whole
74(1)
Measuring Non-Unit Fractions
75(1)
Comparing Fractions as Quantities
75(2)
Commensurate Fractions
77(1)
Assessing Measurement Conceptions of Unit Fractions
77(1)
Instruction on Fractions as Measures
78(1)
Assessment Task Groups
78(2)
A6.1 Naming Unit Fractions
78(1)
A6.2 Producing Unit Fractions from the Whole
79(1)
A6.3 Producing the Whole from Unit Fractions
79(1)
A6.4 Producing Non-Unit Fractions from Unit Fractions
80(1)
Instructional Activities
80(5)
IA6.1 Comparing Unit Fractions
80(1)
IA6.2 Guess My Fraction (Fractions as Measures)
81(1)
IA6.3 Left, Right, or Just Right (Placing Fractions on a Number Line)
82(1)
1A6.4 Different but the Same (Commensurate Fractions)
82(3)
7 Teaching Students at Stage 2: Reversible Reasoning 85(12)
Domain Overview
85(1)
The Splitting Operation
86(1)
Splitting Task Responses
87(1)
Reversible Reasoning with Proper Fractions
87(1)
Coordinating Three Levels of Units in Activity
88(1)
Assessing Reversible Reasoning
88(1)
Instruction on Reversible Reasoning
89(1)
Assessment Task Groups
89(3)
A7.1 Splitting
90(1)
A7.2 Reversing Proper Fractions with Rods
90(1)
A7.3 Reversing Proper Fractions with Strips
91(1)
A7.4 Reversible Multiplicative Reasoning
91(1)
Instructional Activities
92(5)
IA7.1 Two Hidden Units
92(1)
IA7.2 My Number, Your Number
93(1)
IA7.3 My Fraction, Your Fraction
93(1)
IA7.4 Splitting a Part
94(1)
IA7.5 Reversing Multiplicative Reasoning
95(2)
8 Teaching Students at Stage 3: Fractions as Numbers 97(14)
Domain Overview
97(1)
The Iterative Fraction Scheme: Fractions as Numbers
98(1)
Assessing an Iterative Fraction Scheme
98(1)
Promoting an Iterative Fraction Scheme: Magic Candy Bars
99(2)
Promoting an Iterative Fraction Scheme: Fraction Comparisons and Drawing Longer Bars
101(1)
Tasks to Solidify Conceiving of Fractions as Numbers in Their Own Right
102(1)
Assessment Task Groups
103(3)
A8.1 Drawing 'Large' Fractions
103(1)
A8.2 Drawing the Whole Given a Fraction Larger than the Whole
104(1)
A8.3 Drawing a Mixed Number
104(1)
A8.4 Magic Candy Bars
105(1)
A8.5 Drawing a Longer Fruit Strip
105(1)
A8.6 Which Is Bigger?
106(1)
Instructional Activities
106(5)
IA8.1 Magic Candy Bars
106(1)
IA8.2 Fraction Comparison Game
107(1)
IA8.3 Drawing a Longer Fruit Strip
108(1)
IA8.4 Making a Fraction Number Sequence
108(3)
9 Teaching Students at Stages 2 and 3: Equal Sharing of Multiple Items 111(14)
Domain Overview
111(1)
Equally Sharing Multiple Items and the. Fifth Level of Fragmenting (Stage 3 Students)
112(2)
Equally Sharing Multiple Items: Using Repeated Halving
114(1)
Equally Sharing Multiple Items: Sharing Whole Bars First, and Then Partitioning a Single Bar or Using Repeated Halving
115(1)
Equally Sharing Multiple Items: Bricolage Solutions for Harder Problems
116(2)
Equally Sharing Multiple Items: Solutions Involving Multiplication and Division
118(1)
Assessment Task Groups
118(5)
A9.1 Equally Sharing 3 Cookie Dough Logs among 4 Bakers
119(1)
A9.2 Equally Sharing 6 Fruit Bars among 5 Friends
120(1)
A9.3 Equally Sharing 5 Sub Sandwiches among 7 People
121(1)
A9.4 Equally Sharing 3 Different Cakes among 5 People
122(1)
Instructional Activities
123(2)
IA9.1 Equally Sharing Multiple Items of the Same Size
123(1)
IA9.2 Equally Sharing Multiple Items of Different Sizes
124(1)
10 Teaching Students at Stages 2 and 3: Multiplying Fractions 125(18)
Domain Overview
125(1)
Taking Unit Fractions of Unit Fractions
125(3)
Taking Non-Unit Fractions of Unit Fractions
128(1)
Taking Unit Fractions of Non-Unit Fractions
129(1)
Taking Fractions of Fractions, Generally
130(3)
Assessment Task Groups
133(3)
A10.1 Sharing Shares
133(1)
A10.2 Taking Fractions of Unit Fractions
134(1)
A10.3 Taking Unit Fractions of Fractions
134(2)
A10.4 Taking Fractions of Fractions
136(1)
Instructional Activities
136(7)
IA10.1 Sharing Shares
136(1)
IA10.2 Taking Fractions of Fractions I
137(1)
IA10.3 Making 1/4 of a Recipe
138(1)
IA10.4 Taking Fractions of Fractions II
139(1)
IA10.5 Notating and Articulating Patterns in Reasoning
140(3)
11 Teaching Students at Stages 2 and 3: Adding and Subtracting Fractions 143(18)
Domain Overview
143(2)
Transforming Bars to Show Multiple Fractions
145(1)
Pulling Two Fractions Out of the Same Bar with a Co-Measurement
146(2)
Commensurate Fractions More Generally
148(3)
Assessment Task Groups
151(4)
A11.1 Transforming Bars
152(1)
A11.2 Pulling Tim Fractions from the Same Bar
153(1)
A11.3 Adding Fractions
154(1)
A11.4: Naming Fractions the Same Size
154(1)
Instructional Activities
155(6)
IA11.1 Transforming Bars I
155(1)
IA11.2 nansforming Bars II
156(1)
IA11.3 Add and/or Subtract
157(1)
IA11.4 Making Same-Size Fractions with Smaller Parts
158(1)
IA11.5 Making Same-Size Fractions with Larger Parts
159(2)
12 Teaching Students at Stages 2 and 3: Dividing Fractions 161(14)
Domain Overview
161(1)
Fraction Division Problems with a Measurement Meaning
161(2)
A Multiplicative Solution to Fraction Division Problems with a Measurement Meaning
163(2)
A Quantitative Meaning for a Reciprocal
165(1)
Problems to Elicit Multiplicative Solutions of Fraction Measurement Division Problems
166(1)
Assessment Task Groups
167(3)
A12.1 Drawing Pictures of Whole Number Division
167(1)
A12.2 Meanings of Division
167(1)
A12.3 Writing Division Stories
168(1)
A12.4 Fitting into one
169(1)
Instructional Activities
170(5)
IA12.1 Meanings of Division
170(1)
IA12.2 Writing Division Stories
171(1)
IA12.3 Solving Measurement Division Problems I
171(1)
IA12.4 Reciprocals
172(1)
IA12.5 Solving Measurement Division Problems II
173(2)
13 From Fractions to Algebra 175(10)
Fractions and Algebra: Similar Ways of Thinking
175(1)
Quantitative Unknowns
176(1)
Multiplicative Reasoning with Quantitative Unknowns
177(4)
The 'Other' Fraction Division
181(1)
Working on the Algebraic Meaning of Fraction Division
182(3)
Glossary 185(4)
Appendix 189(2)
References 191(4)
Index 195
Dr. Amy J. Hackenberg taught mathematics to middle and high school students for 9 years in L.A. and Chicago, prior to earning a Ph.D. in mathematics education from the University of Georgia. Amy is currently an associate professor of mathematics education at Indiana University-Bloomington. She conducts research on how middle school students construct fractions knowledge and algebraic reasoning and on how teachers can learn to develop productive student-teacher relationships. In her current project she is investigating how to differentiate instruction for diverse middle school students (see https://idream.sitehost.iu.edu/), studying her own teaching as well as working with practicing teachers. She is the proud co-author of the Math Recovery series book, Developing Fractions Knowledge.

Dr. Anderson Norton is a professor in the Department of Mathematics at Virginia Tech. His research focuses on building models of students mathematical development. This work has generated interdisciplinary collaborations with psychologists and neuroscientists. Prior to this volume, Norton served as chair of the steering committee for the North American Chapter of the International Group for the Psychology of Mathematics Education, co-editor on a pair of publications bridging psychology and mathematics education, and co-author of the Math Recovery series book, Developing Fractions Knowledge.



Dr Robert J. (Bob) Wright holds Bachelors and Masters degrees in mathematics from the University of Queensland (Australia) and a doctoral degree in mathematics education from the University of Georgia. He is an adjunct professor in mathematics education at Southern Cross University in New South Wales. Bob is an inter­nationally recognized leader in assessment and instruction relating to childrens early arithmetical knowledge and strategies, publishing six books, and many articles and papers in this field. His work over the last 25 years has included the development of the Mathematics Recovery Program, which focuses on providing specialist training for teachers to advance the numeracy levels of young children assessed as low-attainers. In Australia and New Zealand, Ireland, the UK, the USA, Canada, Mexico, South Africa and elsewhere, this programme has been implemented widely, and applied extensively to classroom teaching and to average and able learners as well as low-attainers. Bob has conducted several research projects funded by the Australian Research Council including the most recent project focusing on assessment and intervention in the early arithmetical learning of low-attaining 8- to 10-year-olds.