Teaching Fractions through Situations: A Fundamental Experiment PDF
Preview Teaching Fractions through Situations: A Fundamental Experiment
Guy 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 © Springer Science+Business Media B.V. 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifi cally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfi lms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifi cally for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specifi c statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) 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
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