Table Of ContentGuy Brousseau · Nadine Brousseau
Virginia Warfi eld
Teaching
Fractions through
Situations: A
Fundamental
Experiment
Teaching Fractions through Situations:
A Fundamental Experiment
Guy Brousseau (cid:129) Nadine Brousseau
Virginia Warfi eld
Teaching Fractions through
Situations: A Fundamental
Experiment
Guy Brousseau Nadine Brousseau
Université de Bordeaux Bordeaux , France
Bordeaux , France
Virginia Warfi eld
University of Washington
Seattle , WA , USA
ISBN 978-94-007-2714-4 ISBN 978-94-007-2715-1 (eBook)
DOI 10.1007/978-94-007-2715-1
Springer Dordrecht Heidelberg New York London
Library of Congress Control Number: 2013941883
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Preface
We would like to thank James King, Virginia Stimpson and Marion Walter for
their very helpful comments on this book and for the encouragement that came
with those comments.
We would also like to thank all of the teachers at the École Michelet in Talence
for their support, encouragement and very hard work without which the project we
are describing could never have happened.
Readers who would like to fi nd out more about the École Michelet or about
the fi eld of research whose foundation this book describes can do so at http://
faculty.washington.edu/warfi eld/guy-brousseau.com/ .
v
Contents
1 Why These Adventures? ........................................................................... 1
A Few Words by the Anglophone Author................................................... 3
First an Introduction to All Three Authors ............................................. 3
Next the Background of the Teaching Project Itself:
How and Why It Came to Exist .............................................................. 4
Introductory Remarks by Guy Brousseau ................................................... 7
2 The Adventure as Experienced by the Students .................................... 9
Module 1: Introducing Rational Numbers as Measurements ..................... 10
Lesson 1: Measurement of the Thicknesses of Sheets
of Paper by Commensuration .................................................................. 10
Lesson 2: Comparison of Thicknesses and Equivalent
Pairs (Summary of Lesson) ..................................................................... 15
Lesson 3: Equivalence Classes – Rational Numbers
(Summary of Lessons) ............................................................................ 16
Module 2: Operations on Rational Numbers as Thicknesses ..................... 18
Lesson 1: The Sum of Thicknesses (Summary of Lesson) ..................... 18
Lesson 2: Practicing the Sum of Thickness. What Should
We Know Now? ...................................................................................... 19
Lesson 3: The Difference of Two Thicknesses (As Measure) ................ 21
Lesson 4: The Thickness of a Piece of Cardboard Composed
of Many Identical Sheets: Product of a Rational Number
and a Whole Number .............................................................................. 22
Lesson 5: Calculation of the Thickness of One Sheet:
Division of a Rational Number by a Whole Number .............................. 24
Lesson 6: Assessment ............................................................................. 25
vii
viii Contents
Module 3: Measuring Other Quantities: Weight, Volume and Length ....... 26
Lesson 1: Making Measurements ........................................................... 26
Lesson 2: Construction of Fractional Lengths:
A New Method Appears ......................................................................... 28
Lesson 3: Comparison of Methods, and Demonstration
of Equivalence ......................................................................................... 30
Lesson 4: Fractions of Collections .......................................................... 31
Module 4: Groundwork for Introducing Decimal Numbers ....................... 33
Lesson 4: Whole Number Intervals Around a Fraction .......................... 33
Module 5: Construction of the Decimal Numbers ...................................... 36
Lesson 1: Bracketing a Rational Number with Rational Numbers:
Chopping up an Interval .......................................................................... 36
Lesson 2: Bracketing a Rational Number Between Rational Numbers,
Shrinking the Intervals, and Observing Decimal Filters ......................... 39
Lesson 3: Representation on the Rational Number Line ........................ 41
Lesson 4: From Writing Decimal Rational Numbers
as Fractions to Writing Them as Decimal Numerals .............................. 45
Module 6: Operations with Decimal Numbers (Summary) ........................ 48
Module 7: Brackets and Approximations (Summary) ................................ 49
Module 8: Similarity ................................................................................... 51
Lesson 1: Enlargement of a Puzzle ......................................................... 51
Lesson 2: The Image of a Whole Number .............................................. 53
Lesson 3: The Image of a Fraction ......................................................... 55
Lesson 4: The Image of a Decimal Number ........................................... 60
Lesson 5: Division of a Decimal Number
by 10, 100, 1,000, … (Summary) ........................................................... 62
Module 9: Linear Mappings ........................................................................ 63
Lesson 1: Another Representation of the Optimist
(Lesson Summarized) ............................................................................. 63
Lesson 2: (Summary of Lesson) ............................................................. 65
Lesson 3: Lots of Representations of the Optimist
(Summary of Lesson) .............................................................................. 65
Lesson 4: Good Representations, Not So Good Representations ........... 68
Lesson 5: Change of Model .................................................................... 70
Lesson 6: Reciprocal Mappings .............................................................. 73
Module 10: Multiplication by a Rational Number ...................................... 74
Lesson 1: Multiplication by a Rational Number ..................................... 74
Lesson 2: Multiplying by a Decimal (Summary of Lessons) ................. 78
Lesson 3: Methods of Solving Linear Problems
(Summary of Lessons) ............................................................................ 78
Lesson 4: The Search for Linear Situations
(Summary of Lessons) ............................................................................ 79
Contents ix
Module 11: The Study of Linear Situations in “Everyday Life” ................ 81
Lesson 1: Fraction of a Magnitude ......................................................... 81
Summary of the Remaining Paragraphs and Sections of Module 11 ..... 87
Module 12: More on the Problem Statement Contest ................................. 90
Lesson 1 .................................................................................................. 90
Module 13: New Division Problems in the Rationals ................................. 93
Lesson 2: (Extract) Division as Reciprocal Mapping
of Multiplication (The Term Is Not Taught to the Students) .................. 94
Module 14: Composition of Linear Mappings ............................................ 101
Lesson 1: The Pantograph ....................................................................... 101
Lesson 2: Composition of Mappings: First Session ............................... 103
Lesson 3: Composition of Linear Mappings: Designation
of Composed Mappings .......................................................................... 107
Lesson 4: Different Ways of Writing the Same Mapping ....................... 110
Lesson 5: Rational Linear Mappings ...................................................... 114
Module 15: Decomposition of Rational Mappings. Identifi cation
of Rational Numbers and Rational Linear Mappings ................................. 118
Lesson 1: Decomposition of Rational Mappings .................................... 118
Lesson 2: The Meaning of “Division by a Fraction”
(Summary of Lessons) ............................................................................ 121
Lesson 3: Division of Decimals .............................................................. 124
3 The Adventure as Experienced by the Teachers .................................... 127
Background of the Project .......................................................................... 127
The Relationship with the Theory of Situations ......................................... 131
The Perspective of the Teacher ................................................................... 134
Observable Aspects of C onnaissances ................................................... 140
Manifestation of S avoirs ......................................................................... 140
What Then Are the Causes of Learning and the Reasons
for Knowing? .......................................................................................... 144
How Does the Teacher Use Assessment of and Within the Curriculum? ... 145
The Assessment of Students and Groups of Students ............................. 146
The Types of Situations That Appear in the Lessons.................................. 147
The Types of Didactical Situation and How They Are Conducted ......... 148
The Types of A-didactical Situation and How They Are Conducted ..... 150
Presentation of the Rules of the Game .................................................... 153
Evaluation in A-didactical Situations ..................................................... 154
Obsolescence ............................................................................................... 155
Isolated Evaluation of S avoirs and Constant Evaluation;
the Necessity of the Uncertain and the Implicit .......................................... 156
The Play of the Real and the Fictional ........................................................ 156
The Inexpressible, the Said and the Unsaid ................................................ 157
Further Aspects of the Teachers’ Adventures ............................................. 157
