# Young Mathematicians At Work Constructing Algebra

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### Learning to Support Young Mathematicians at Work by Catherine Twomey Fosnot,Maarten Dolk,Kara Louise Imm,William Jacob,Despina Stylianou Book Summary:

"Our digital-lab environment provides an active, more meaningful professional development experience, which empowers teachers to integrate theory and practice. We offer teachers the opportunity to embark upon their own landscape of learning journey." --Catherine Twomey Fosnot The bestselling Young Mathematicians at Work series has helped tens of thousands of teachers inspire deep mathematical understanding in students with its signature workshop approach. The Contexts for Learning Mathematics series has helped even more teachers bring that approach into their classrooms to align their math program to the Standards of Practice in the Common Core. Now Cathy Fosnot and her colleagues have developed an invaluable resource that gives teachers ownership of the core ideas and essential understandings of early algebra. Learning to Support Young Mathematicians at Work offers Professional Development providers interactive, meaningful tools to help teachers deepen their algebraic thinking within a unique digital environment. Two DVDs feature classroom sessions that show students exploring the sequence of investigations and activities found in the popular Contexts for Learning Mathematics algebra unit books Trades, Jumps, and Stops and The California Frog-Jumping Contest. Extensive classroom video footage allows participants to study children over time, and examine and analyze their development as well as the teacher's pedagogy. The user can respond to prompts, create video clips and presentations, and explore related materials all from the DVD, either alone at home or in a workshop setting. The PD facilitator's guide features a flexible menu of workshops, ranging from two-hour sessions that focus on a particular topic to comprehensive 5-day institutes. As participants engage in the investigations, they build their own conceptual and pedagogical understanding of algebra and proof, questioning and conferring, observing children at work in mathematics, and using the powerful tools of context and representations. These learning experiences foster teachers' algebraic thinking and set the stage for robust and active classroom practice that promotes students' deep understanding. The DVDs are also available individually without the facilitator guide for teachers who may prefer to study the material outside of a workshop setting. Trades, Jumps and Stops DVD California Frog Jumping Contest DVD NOTE: The DVD-ROMs are compatible with Windows XP, Vista, Windows 7, and Mac OS X up to 10.6. They are not compatible with Mac OS X 10.7 (Lion) or above.

### La lamentation d'un mathématicien by Paul Lockhart Book Summary:

Voilà donc une illustration de l’art de faire des mathématiques : jouer avec des motifs, remarquer des tendances, imaginer des conjectures, chercher des exemples et des contre-exemples, être inspiré pour inventer et explorer, fabriquer des arguments et les analyser, et soulever de nouvelles questions. C’est tout cela à la fois. Je ne dis pas que c’est vital, car ce ne l’est pas. Je ne dis pas que cela va guérir le cancer, car ce n’est pas le cas. Je dis juste que c’est amusant et que cela m’apporte du plaisir. De plus, c’est complètement inoffensif. Quelles sont les activités humaines dont on pourrait en dire autant ? Dans cet essai, Paul Lockhart se désespère de la manière dont les mathématiques sont enseignées dans nos écoles et vues par le grand public. Les mathématiques sont un art, un acte créatif, et nul ne devrait être privé de leur beauté. Livre illustré de 136 pages - Illustrations de Jérôme Poloczek.

### Working with the Ratio Table, Grades 5-8 by Antonia Cameron,Catherine Twomey Fosnot,Sherrin B. Hersch Book Summary:

In their series of professional books for teachers, Young Mathematicians at Work, Catherine Twomey Fosnot and Maarten Dolk described Mathematics in the City, an innovative project where teachers helped young children construct a deep understanding of number and operation in a math-workshop environment. Now they and two colleagues from the project have developed a flexible, video-based, digital context for inquiry into the teaching and learning of mathematics that will change how professional development is conducted. Designed for you, the workshop leader or college instructor, the Working with the Ratio Table Resource Package enables your in- or preservice teachers to not only watch but interact with video that depicts classroom teachers as they listen to, question, and interpret students' thinking; develop connections between mathematical ideas and strategies; and, ultimately, develop vibrant mathematical communities in their classrooms. The Resource Package includes three valuable components: A completely interactive CD-ROM, where your workshop participants can explore-independently or under your guidance-videos of instruction and assessment; sample children's work over time to analyze development; take and save notes on what they see; capture specific frames or footage; and then email their captured video clips and notes to other members of your professional development workshop. The context of the classroom will be at the fingertips of your participants for exploration. A Professional Development Overview Manual that provides general advice on how you can use the CD-ROM for staff development. A Facilitator's Guide whose field-tested content is specific to the CD-ROM and includes helpful suggestions for using video clips and student examples on the CD to design rich professional development experiences; sample dialogue to help you anticipate what your participants might say; tips for facilitating discussions among teachers; and descriptions of the mathematical ideas being explored. In Working with the Ratio Table, your workshop participants will observe sixth graders as they construct some of the big ideas related to fractions, making connections between ratios and equivalence and uncovering landmark division strategies like comparison through common denominators. By studying the use of carefully crafted problems designed both to generate a range of solution strategies and to highlight the power of ratio tables and other models for division, teachers will discover what a valuable tool real-life contexts are for building a solid foundation in mathematics. System Requirements for CD-ROM Windows/PC Pentium II Processor 266MHz (or higher) Windows 98 (or higher) 64 MB RAM (more recommended) SVGA Color Display (or better) 4x CD-ROM Drive (or faster) Sound Card 16-bit Flash(TM) Player and Acrobat Reader(R) Quicktime 6.0 (or higher) Mac PowerPC Processor G3/233MHz (or higher) System 9.2 or 10.2 (or higher) 64 MB RAM (more recommended) SVGA Color Display (or better) 4x CD-ROM Drive (or faster) Sound Card 16-bit Flash(TM) Player and Acrobat Reader(R) Quicktime 6.0 (or higher) *Please note CD-ROM is not compatible with Mac OS X 10.7

### The California Frog-Jumping Contest by Fosnot,Bill Jacob,Catherine Twomey Fosnot Book Summary:

The California Frog-Jumping Contest: Algebra is one of five units in the Contexts for Learning Mathematics' Investigating Fractions, Decimals, and Percents (4 - 6) This unit uses the context of the famous short story by Mark Twain - The Celebrated Jumping Frog of Calaveras County - to develop equivalence and its use in solving algebraic problems. The context of a frog jumping along a track is used to foster number line representations in which students solve for an unknown amount, which is usually the length of a frog jump. Equivalent sequences of jumps are represented naturally on a double number line by having them start and end at the same location, with one expression shown on top of the line and the other shown underneath the line. The representation can then be used as a tool for solving the problem. The unit begins with a problem in which students find the length of a bullfrog's jump, knowing the full length of a sequence of his jumps and steps. This context leads to using the number line as a tool for solving problems with unknowns. Next, students must find various approaches for lining up six- or eight-foot benches for two jumping tracks of lengths 28 and 42 feet. Students utilize the equivalence 6 + 6 + 6 + 6 = 8 + 8 + 8 to change one possible solution into a second possible solution and use the number line to represent this equivalence. A similar problem about fences is used to develop a combination chart, which is a useful representation for determining net gain (or loss) after an exchange. The second half of the unit includes more frog-jumping problems as the frogs plan for their Olympic Games. Now students further explore the use of variables to represent more complex situations and solve for unknown amounts. Here, students use the number line to represent jumps in the problems and can separate off equal amounts of unknown lengths to determine the lengths of unknown amounts. As the unit progresses, the questions require that students investigate equivalent lengths of different-sized jumps and work with these equivalences flexibly to solve problems. The complexity of learning to symbolize has been the subject of extensive research. One study, summarized in Adding It Up (National Research Council 2001, 264), illustrates typical difficulties students may have. Known as the reversal error, it is illustrated by work on the following problem: At a certain university, there are six times as many students as professors. Using S for the number of students and P for the number of professors, write an equation that gives the relation between the number of students and the number of professors. A majority of students, ranging from first-year algebra students to college freshmen, wrote the equation 6S=P. Apparently they used 6 as an adjective and S as a noun, following the natural language in the problem. However, they needed to multiply the number of professors by 6 to find the number of students. The correct response is 6P=S. Because learning to write algebraic expressions is so difficult, we don't push symbolizing early in this unit. The representation of the number line is used to fix students' attention on the distinction between the lengths of jumps and the number of jumps. Once this is set, students can begin symbolizing in problems like this in a meaningful way. The unit ends with the students constructing more formal algebraic notation as they develop methods to simplify their earlier representations. To learn more visit http://www.contextsforlearning.com

### Tep Vol 26-N4 by Teacher Education and Practice Book Summary:

Teacher Education and Practice, a peer-refereed journal, is dedicated to the encouragement and the dissemination of research and scholarship related to professional education. The journal is concerned, in the broadest sense, with teacher preparation, practice and policy issues related to the teaching profession, as well as being concerned with learning in the school setting. The journal also serves as a forum for the exchange of diverse ideas and points of view within these purposes. As a forum, the journal offers a public space in which to critically examine current discourse and practice as well as engage in generative dialogue. Alternative forms of inquiry and representation are invited, and authors from a variety of backgrounds and diverse perspectives are encouraged to contribute. Teacher Education & Practice is published by Rowman & Littlefield.

### Groceries, Stamps & Measuring Strip Grade 2 by Fosnot,Harcourt School Publishers Book Summary:

The California Frog-Jumping Contest: Algebra is one of five units in the Contexts for Learning Mathematics' Investigating Fractions, Decimals, and Percents (4 - 6) This unit uses the context of the famous short story by Mark Twain - The Celebrated Jumping Frog of Calaveras County - to develop equivalence and its use in solving algebraic problems. The context of a frog jumping along a track is used to foster number line representations in which students solve for an unknown amount, which is usually the length of a frog jump. Equivalent sequences of jumps are represented naturally on a double number line by having them start and end at the same location, with one expression shown on top of the line and the other shown underneath the line. The representation can then be used as a tool for solving the problem. The unit begins with a problem in which students find the length of a bullfrog's jump, knowing the full length of a sequence of his jumps and steps. This context leads to using the number line as a tool for solving problems with unknowns. Next, students must find various approaches for lining up six- or eight-foot benches for two jumping tracks of lengths 28 and 42 feet. Students utilize the equivalence 6 + 6 + 6 + 6 = 8 + 8 + 8 to change one possible solution into a second possible solution and use the number line to represent this equivalence. A similar problem about fences is used to develop a combination chart, which is a useful representation for determining net gain (or loss) after an exchange. The second half of the unit includes more frog-jumping problems as the frogs plan for their Olympic Games. Now students further explore the use of variables to represent more complex situations and solve for unknown amounts. Here, students use the number line to represent jumps in the problems and can separate off equal amounts of unknown lengths to determine the lengths of unknown amounts. As the unit progresses, the questions require that students investigate equivalent lengths of different-sized jumps and work with these equivalences flexibly to solve problems. The complexity of learning to symbolize has been the subject of extensive research. One study, summarized in Adding It Up (National Research Council 2001, 264), illustrates typical difficulties students may have. Known as the reversal error, it is illustrated by work on the following problem: At a certain university, there are six times as many students as professors. Using S for the number of students and P for the number of professors, write an equation that gives the relation between the number of students and the number of professors. A majority of students, ranging from first-year algebra students to college freshmen, wrote the equation 6S=P. Apparently they used 6 as an adjective and S as a noun, following the natural language in the problem. However, they needed to multiply the number of professors by 6 to find the number of students. The correct response is 6P=S. Because learning to write algebraic expressions is so difficult, we don't push symbolizing early in this unit. The representation of the number line is used to fix students' attention on the distinction between the lengths of jumps and the number of jumps. Once this is set, students can begin symbolizing in problems like this in a meaningful way. The unit ends with the students constructing more formal algebraic notation as they develop methods to simplify their earlier representations. To learn more visit http: //www.contextsforlearning.com

### Un mathématicien aux prises avec le siècle by Laurent Schwartz Book Summary:

" Je suis mathématicien. Les mathématiques ont rempli ma vie : une passion pour la recherche et l'enseignement tour à tour comme professeur à l'Université et à l'École polytechnique. J'ai en même temps réfléchi aux rôles des mathématiques, de la recherche et de l'enseignement, dans ma vie et dans celle des autres, aux processus mentaux de la recherche, et je me suis consacré pendant des décennies aux réformes bien nécessaires de l'Université et des grandes écoles.[ ... ] Mais j'ai eu bien d'autres activités, parfois au point de démolir ma recherche. J'ai consacré une grande partie de mon temps à lutter pour les opprimés, pour les droits de l'homme et les droits des peuples, d'abord comme trotskiste, puis en dehors de tout parti. Il était normal de ma part de vouloir décrire ces activités, comme témoignage pour l'avenir. "

### Best Buys, Ratios, and Rates by Bill Jacob,Catherine Twomey Fosnot Book Summary:

Best Buys, Ratios, and Rates: Addition and Subtraction of Fractions is one of five units in the Contexts for Learning Mathematics' Investigating Fractions, Decimals, and Percents (4 - 6) The focus of this unit is the development of equivalence of fractions, proportional reasoning, and rates. It begins with a comparison of the cost of cat food at two stores: Bob's Best Buys where it is on sale, $15 for 12 cans, and Maria's Pet Emporium where it is on sale, $23 for 20 cans. Several important ideas and representations develop as students explore this problem, among them finding ways to determine the cost of a common numbers of cans for comparison and the use of the ratio table to represent their proportional reasoning about the context. The development of the ratio table is further supported in the next investigation as students work to determine the cost of several different amounts of bird seed sold by weight. As the unit progresses, proportional reasoning is once again the focus as students develop recipes for a variety of containers, using the recipe of Maria's gourmet puppy snack mix. In the second week the double number is introduced for computation as students investigate the readings on a farm truck's gas tank over the course of trips to several neighboring farms to pick up produce. A trip across the Pennsylvania Turnpike is also explored. This unit also includes several minilessons for addition and subtraction of fractions. Strings of related problems are used initially using money and clock models. Double number lines are introduced later in the unit to enable students to develop generalizable, strategies for addition and subtraction. This model supports students to choose a common multiple (or factor) to work with as well as further opportunities to explore equivalent fractions. Note: The context for this unit assumes that your students have had prior experience with fractions and their relationship to division with whole numbers. If this is not the case, you might find it helpful to first use the units Field Trips and Fund-Raisers. To learn more visit http://www.contextsforlearning.com

### Calcul mathématique avec Sage by Paul Zimmermann,Laurent Fousse,François Maltey Book Summary:

Sage est un logiciel libre de calcul mathématique s'appuyant sur le langage de programmation Python. Ses auteurs, une communauté internationale de centaines d'enseignants et de chercheurs, se sont donné pour mission de fournir une alternative viable aux logiciels Magma, Maple, Mathematica et Matlab. Sage fait appel pour cela à de multiples logiciels libres existants, comme GAP, Maxima, PARI et diverses bibliothèques scientifiques pour Python, auxquels il ajoute des milliers de nouvelles fonctions. Il est disponible gratuitement et fonctionne sur les systèmes d'exploitation usuels.Pour les lycéens, Sage est une formidable calculatrice scientifique et graphique. Il assiste efficacement l'étudiant de premier cycle universitaire dans ses calculs en analyse, en algèbre linéaire, etc. Pour la suite du parcours universitaire, ainsi que pour les chercheurs et les ingénieurs, Sage propose les algorithmes les plus récents dans diverses branches des mathématiques. De ce fait, de nombreuses universités enseignent Sage dès le premier cycle pour les travaux pratiques et les projets.Ce livre est le premier ouvrage généraliste sur Sage, toutes langues confondues. Coécrit par des enseignants et chercheurs intervenant à tous les niveaux (IUT, classes préparatoires, licence, master, doctorat), il met l'accentsur les mathématiques sous-jacentes à une bonne compréhension du logiciel. En cela, il correspond plus à un cours de mathématiques effectives illustré par des exemples avec Sage qu'à un mode d'emploi ou un manuel de référence.La première partie est accessible aux élèves de licence. Le contenu des parties suivantes s'inspire du programme de l'épreuve de modélisation de l'agrégation de mathématiques.Ce livre est diffusé sous licence libre Creative Commons. Il peut être téléchargé gratuitement depuis http://sagebook.gforge.inria.fr/.

### The Mystery of the Meter by Fosnot,Bill Jacob,John Michael Siegfried,Catherine Twomey Fosnot Book Summary:

The Mystery of the Meters: Decimals is one of five units in the Contexts for Learning Mathematics' Investigating Fractions, Decimals, and Percents (4 - 6) This unit begins with the story of Zig - who discovers five mysterious dials on the side of his house. The dials are part of the electric meter for his house. At first Zig does not know this and he sets out to investigate how the dials work. As he collects data (readings every ten minutes), he notices that the hands on the dials turn in relation to each other (since each dial represents a different power of ten). Using Zig's data, students investigate how the dials are related. As the unit progresses, students use readings from the meter to measure energy to the thousandth of a kilowatt-hour to calculate the amount of energy used during a specific time period, and to determine readings on missing or obscured dials, working with place value equivalents. The unit focus is on decimals, and since the electric meter in this unit represents kilowatt-hours to the thousandths, it can be used as a model to represent decimals. Because students can see how the numbers expressed as decimals increase with time, the meter is a powerful tool for students to use to determine equivalents and to examine how decimals increase and are ordered. The electric meter consists of five circular dials numbered zero through nine that are lined up in place value order. As the hand on each dial makes a complete revolution, the number indicated on the dial to its left increases by one (one-tenth of a revolution.) This model was chosen because the position of the dials supports understanding of place value with decimals in tenths, hundredths, and thousandths. In examining how the hands on the dials move as the values increase, students may have opportunities to confront basic cognitive obstacles in making sense of decimal representations. These dials advance in the same way that the mechanical odometers of old cars worked, so exploring this mechanism provides an experience that students don't get these days due to the use of electronic digital odometers. The electric meter used in this unit measures thousandths of a kilowatt-hour (or watt-hours), while the meters in most homes measure kilowatt-hours. This change was made to enable students to interpret the action of the meter in operating everyday electric devices, such as a refrigerator, light bulb, or television. When students go home and look at their meters, they may discover this as well as other similarities and differences. You can explain that in the story the meter is different, and you can invite students to think about the relationship of the problems in this unit to the meter they have at home. To learn more visit http://www.contextsforlearning.com

### Teaching and Learning Algebraic Thinking with 5- to 12-Year-Olds by Carolyn Kieran Book Summary:

This book highlights new developments in the teaching and learning of algebraic thinking with 5- to 12-year-olds. Based on empirical findings gathered in several countries on five continents, it provides a wealth of best practices for teaching early algebra. Building on the work of the ICME-13 (International Congress on Mathematical Education) Topic Study Group 10 on Early Algebra, well-known authors such as Luis Radford, John Mason, Maria Blanton, Deborah Schifter, and Max Stephens, as well as younger scholars from Asia, Europe, South Africa, the Americas, Australia and New Zealand, present novel theoretical perspectives and their latest findings. The book is divided into three parts that focus on (i) epistemological/mathematical aspects of algebraic thinking, (ii) learning, and (iii) teaching and teacher development. Some of the main threads running through the book are the various ways in which structures can express themselves in children’s developing algebraic thinking, the roles of generalization and natural language, and the emergence of symbolism. Presenting vital new data from international contexts, the book provides additional support for the position that essential ways of thinking algebraically need to be intentionally fostered in instruction from the earliest grades.

### Mathematics Education in Singapore by Tin Lam Toh,Berinderjeet Kaur,Eng Guan Tay Book Summary:

This book provides a one-stop resource for mathematics educators, policy makers and all who are interested in learning more about the why, what and how of mathematics education in Singapore. The content is organized according to three significant and closely interrelated components: the Singapore mathematics curriculum, mathematics teacher education and professional development, and learners in Singapore mathematics classrooms. Written by leading researchers with an intimate understanding of Singapore mathematics education, this up-to-date book reports the latest trends in Singapore mathematics classrooms, including mathematical modelling and problem solving in the real-world context.

### Constructivism Reconsidered in the Age of Social Media by Chris Stabile,Jeff Ershler Book Summary:

No longer relegated to just the classroom, learning has become universal through the use of social media. Social media embodies constructivism itself as the users engage in the development of their own meaning. And, constructivism is relevant to education, and learning theory and technological advance can be better understood in the light of one another. This volume explores: particular areas influenced by constructivist thinking and social media, such as student learning, faculty development, and pedagogical practices, practical and useful ways to engage in social media, and dialogue and discussions regarding the nature of learning in relation to the technology that has changed how both faculty and students experience their educational landscape. This is the 144th volume of this Jossey-Bass higher education series. It offers a comprehensive range of ideas and techniques for improving college teaching based on the experience of seasoned instructors and the latest findings of educational and psychological researchers.

### Academic Language in Diverse Classrooms: Mathematics, Grades 3–5 by Margo Gottlieb,Gisela Ernst-Slavit Book Summary:

Make every student fluent in the language of learning. The Common Core and ELD standards provide pathways to academic success through academic language. Using an integrated Curricular Framework, districts, schools and professional learning communities can: Design and implement thematic units for learning Draw from content and language standards to set targets for all students Examine standards-centered materials for academic language Collaborate in planning instruction and assessment within and across lessons Consider linguistic and cultural resources of the students Create differentiated content and language objectives Delve deeply into instructional strategies involving academic language Reflect on teaching and learning

### La Symétrie, ou les maths au clair de lune by Marcus Du Sautoy Book Summary:

Il était une fois les mathématiques Au fil de douze chapitres, un pour chaque mois de l'année, Marcus du Sautoy explore la nature de la symétrie, et nous offre un aperçu sans égal de la vie d'un mathématicien. Il navigue des pyramides au football, des atomes à la peinture, des insectes à l'architecture, de l'informatique à la psychologie, pour rendre compte d'un des concepts les plus significatifs des mathématiques. Il nous présente les mathématiciens farfelus, ceux du passé comme ceux du présent, qui se sont battus pour en comprendre les qualités insaisissables. Dans Symétrie, ou les maths au clair de lune, Marcus du Sautoy montre aux novices avec humour et pédagogie ce que c'est d'être aux prises avec une idée, parmi les plus complexes que l'esprit humain peut appréhender.