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Teaching Fractions through Situations: A Fundamental Experiment

Guy Brousseau, Nadine Brousseau, and Virginia Warfield
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Annie Selden
, on

As the title indicates, this book, intended for teachers, mathematics education researchers, and mathematicians, is about developing and teaching a curriculum for teaching fractions. According to the very short first chapter, the book describes three “adventures” — the adventure of the researchers while discovering and testing a curriculum for rational number learning, developed using the theory of didactical situations (Chapter 4), the adventure of the teachers implementing that curriculum (Chapter 3), and the adventures of the students experiencing that curriculum (Chapter 2).

First, a bit about the authors. Guy Brousseau is a French mathematics education researcher who is perhaps best known for his Theory of Didactical Situations. He was awarded the Felix Klein Medal by the International Commission on Mathematical Instruction (ICMI) for his mathematics education research conducted over four decades. His wife, Nadine Brousseau, was an elementary school teacher who was among the first to implement the experimental rational number curriculum. Virginia Warfield is a mathematician interested in mathematics education who previously helped translate Brousseau’s Theory of Didactical Situations in Mathematics into English and also collaborated with him on a series of four articles, covering parts of the Rational and Decimal Number curriculum, published in the Journal of Mathematical Behavior. This book is the result of putting together and expanding those articles.

Next, a brief explanation about the Theory of Didactical Situations. By a didactical situation is meant a mathematical situation, usually in the form of an activity that engages students, or mathematicians, in the construction of some mathematical concept by investigating that situation. However, these are not just any activities — they need to be carefully constructed and tested.

Perhaps the best way to understand the idea of didactical situations is via a description of Lesson1, Module 1 (Chapter 2). The idea is to have students invent a way to measure something so thin that their previously learned measurement methods do not work. On the table at the front of the classroom are five stacks of paper containing 200 sheets each. All the paper is the same color and the same size, but each stack has paper of a different thickness. Each team of five students has two slide calipers — a device for measuring thickness, standard in French elementary school classrooms. The students quickly decide they cannot measure just a single sheet of paper, so they try small stacks of 5 or 10 sheets of paper. When most teams have found a measuring system, the teacher introduces a communication game, in which each team splits in two groups — one with two transmitters and one with three receivers — separated by a curtain. The transmitters choose one type of paper, send a message to the receivers to enable them to pick the appropriate paper from another set of identical (but scrambled) 5 stacks of paper.

The lesson continues in this way until all of the students know: how to measure the thickness of a certain number of sheets of paper; how to write corresponding ordered pairs of (number of sheets of paper, thickness in millimeters); how to use an ordered pair to accept, or reject, any type of paper as belonging, or not belonging, to the original 5 stacks of paper; and how to find equivalent ordered pairs. Most of the students are also able to analyze a chart of measurements and point out inconsistencies, thereby making implicit use of proportionality.

Altogether Chapter 2 has 15 such modules with 44 lessons in all. These cover products of rational numbers, division of rational numbers, the rational number line, decimals, linear mappings and their composition, various representations of rational numbers, and multiplication and division of decimals. Chapter 2 is by far the longest — 118 pages of this 210-page book.

Now for a brief flavor of the of teachers’ implementations of this curriculum (Chapter 3). First, the implementations took place in a regular public school in a “blue collar district on the edge of Bordeaux,” but equipped with an observation classroom containing video cameras, and an extra teacher at each level. This “research classroom” was used for two and a half decades. The teachers had a lot of help from the researchers who were developing the curriculum. At first the teachers had to learn how to teach the lessons and respond to the children, but later on, there was a danger of having the lessons become “rituals” and this had to be fixed.

Chapter 4 is a retelling and reinterpretation of developments in mathematics education in France from about the 1960s to the more-or-less present, focusing on Brousseau’s contributions. In the 1960s a group of international mathematics education researchers saw the need for coordinated, collaborative research into the teaching and learning of mathematics. In France, this resulted in a number of IREM’s — Research Institutes for Mathematics Teaching. Brousseau was an early supporter of these and was instrumental in bring an early IREM to the University of Bordeaux, where he was on the faculty. There he created COREM, the Center for Observation and Research on Mathematics Teaching, which took the form of the school described above.

It was decided to focus on fractions and decimals, often a challenge for teachers, and to develop an experimental curriculum. Brousseau looked into the history of how these topics had been taught in different cultures and times. He concluded that although rational numbers can be taught in several ways — measurement, proportion, and operation — these are generally taught as if they were all equivalent, with the result being that students have to accept many things on the teachers say-so. “The curriculum was not made to be used in other classes. The sole purpose of the reproducibility was to consolidate the scientific observations that we needed in order to test certain hypotheses.” Brousseau’s wife Nadine and another teacher helped implement the various versions of the didactical situations that were developed and implemented over the course of 25 years. Data now include video recordings of 65 lessons —these are available at the ViSA site (Vidéos de Situations d’enseignement et d’Apprentissage

There is a lot of think about in this book. Perhaps most interesting for teachers and mathematicians are the lessons and the reactions of the students to the lessons.

Annie Selden is Adjunct Professor of Mathematics at New Mexico State University and Professor Emerita of Mathematics from Tennessee Technological University. She regularly teaches graduate courses in mathematics and mathematics education. In 2002, she was recipient of the Association for Women in Mathematics 12th Annual Louise Hay Award for Contributions to Mathematics Education. In 2003, she was elected a Fellow of the American Association for the Advancement of Science. She remains active in mathematics education research and curriculum development. 

1. The Adventure of the Students.- 2. Viewing the Adventure from the Perspective of Teachers and Researchers.- 3. Some Key Concepts and Terms from the Theory of Situations.- 4. The Setting for the Adventure.- 5. Description of the Center for Observation for Research in Mathematics Education.- 6. Conclusions and future directions.