# 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.

### 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

### 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

### 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

### A Book of Abstract Algebra by Charles C Pinter Book Summary:

Accessible but rigorous, this outstanding text encompasses all of the topics covered by a typical course in elementary abstract algebra. Its easy-to-read treatment offers an intuitive approach, featuring informal discussions followed by thematically arranged exercises. This second edition features additional exercises to improve student familiarity with applications. 1990 edition.

### Mathematical Mindsets by Jo Boaler Book Summary:

Banish math anxiety and give students of all ages a clear roadmap to success Mathematical Mindsets provides practical strategies and activities to help teachers and parents show all children, even those who are convinced that they are bad at math, that they can enjoy and succeed in math. Jo Boaler—Stanford researcher, professor of math education, and expert on math learning—has studied why students don't like math and often fail in math classes. She's followed thousands of students through middle and high schools to study how they learn and to find the most effective ways to unleash the math potential in all students. There is a clear gap between what research has shown to work in teaching math and what happens in schools and at home. This book bridges that gap by turning research findings into practical activities and advice. Boaler translates Carol Dweck's concept of 'mindset' into math teaching and parenting strategies, showing how students can go from self-doubt to strong self-confidence, which is so important to math learning. Boaler reveals the steps that must be taken by schools and parents to improve math education for all. Mathematical Mindsets: Explains how the brain processes mathematics learning Reveals how to turn mistakes and struggles into valuable learning experiences Provides examples of rich mathematical activities to replace rote learning Explains ways to give students a positive math mindset Gives examples of how assessment and grading policies need to change to support real understanding Scores of students hate and fear math, so they end up leaving school without an understanding of basic mathematical concepts. Their evasion and departure hinders math-related pathways and STEM career opportunities. Research has shown very clear methods to change this phenomena, but the information has been confined to research journals—until now. Mathematical Mindsets provides a proven, practical roadmap to mathematics success for any student at any age.

### Mathematics & Mathematics Education: Searching for Common Ground by Michael N. Fried,Tommy Dreyfus Book Summary:

This book is the fruit of a symposium in honor of Ted Eisenberg concerning the growing divide between the mathematics community and the mathematics education community, a divide that is clearly unhealthy for both. The work confronts this disturbing gap by considering the nature of the relationship between mathematics education and mathematics, and by examining areas of commonality as well as disagreement. It seeks to provide insight into the mutual benefit both stand to gain by building bridges based on the natural bonds between them.

### 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.

### Teaching Students to Communicate Mathematically by Laney Sammons Book Summary:

Students learning math are expected to do more than just solve problems; they must also be able to demonstrate their thinking and share their ideas, both orally and in writing. As many classroom teachers have discovered, these can be challenging tasks for students. The good news is, mathematical communication can be taught and mastered. In Teaching Students to Communicate Mathematically, Laney Sammons provides practical assistance for K–8 classroom teachers. Drawing on her vast knowledge and experience as a classroom teacher, she covers the basics of effective mathematical communication and offers specific strategies for teaching students how to speak and write about math. Sammons also presents useful suggestions for helping students incorporate correct vocabulary and appropriate representations when presenting their mathematical ideas. This must-have resource will help you help your students improve their understanding of and their skill and confidence in mathematical communication.

### 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.

### The Knot Book by Colin Conrad Adams Book Summary:

Knots are familiar objects. We use them to moor our boats, to wrap our packages, to tie our shoes. Yet the mathematical theory of knots quickly leads to deep results in topology and geometry. The Knot Book is an introduction to this rich theory, starting from our familiar understanding of knots and a bit of college algebra and finishing with exciting topics of current research. The Knot Book is also about the excitement of doing mathematics. Colin Adams engages the reader with fascinating examples, superb figures, and thought-provoking ideas. He also presents the remarkable applications of knot theory to modern chemistry, biology, and physics. This is a compelling book that will comfortably escort you into the marvelous world of knot theory. Whether you are a mathematics student, someone working in a related field, or an amateur mathematician, you will find much of interest in The Knot Book.

### Contexts for Learning Mathematics by Catherine Twomey Fosnot,Pearson Education,Fosnot Book Summary:

Contexts for Learning consists of: Investigations and Resource Guides - workshop structure involves students in inquiring, investigating, discussing, and constructing mathematical solutions and strategies - investigations encourage emergent learning and highlight the developmental landmarks in mathematical thinking - strings of related problems develop students' deep number sense and expand their strategies for mental arithmetic Read-Aloud Books and Posters - create rich, imaginable contexts--realistic and fictional--for mathematics investigations - are carefully crafted to support the development of the big ideas, strategies, and models - encourage children to explore and generate patterns, generalize, and develop the ability to mathematize their worlds Resources for Contexts for Learning CD-ROM - author videos describe the series' philosophy and organization - video overviews show classroom footage of a math workshop, including minilessons, investigations, and a math congress - print resources include research base, posters, and templates

### The Princeton Companion to Mathematics by Timothy Gowers,June Barrow-Green,Imre Leader Book Summary:

This is a one-of-a-kind reference for anyone with a serious interest in mathematics. Edited by Timothy Gowers, a recipient of the Fields Medal, it presents nearly two hundred entries, written especially for this book by some of the world's leading mathematicians, that introduce basic mathematical tools and vocabulary; trace the development of modern mathematics; explain essential terms and concepts; examine core ideas in major areas of mathematics; describe the achievements of scores of famous mathematicians; explore the impact of mathematics on other disciplines such as biology, finance, and music--and much, much more. Unparalleled in its depth of coverage, The Princeton Companion to Mathematics surveys the most active and exciting branches of pure mathematics. Accessible in style, this is an indispensable resource for undergraduate and graduate students in mathematics as well as for researchers and scholars seeking to understand areas outside their specialties. Features nearly 200 entries, organized thematically and written by an international team of distinguished contributors Presents major ideas and branches of pure mathematics in a clear, accessible style Defines and explains important mathematical concepts, methods, theorems, and open problems Introduces the language of mathematics and the goals of mathematical research Covers number theory, algebra, analysis, geometry, logic, probability, and more Traces the history and development of modern mathematics Profiles more than ninety-five mathematicians who influenced those working today Explores the influence of mathematics on other disciplines Includes bibliographies, cross-references, and a comprehensive index Contributors incude: Graham Allan, Noga Alon, George Andrews, Tom Archibald, Sir Michael Atiyah, David Aubin, Joan Bagaria, Keith Ball, June Barrow-Green, Alan Beardon, David D. Ben-Zvi, Vitaly Bergelson, Nicholas Bingham, Béla Bollobás, Henk Bos, Bodil Branner, Martin R. Bridson, John P. Burgess, Kevin Buzzard, Peter J. Cameron, Jean-Luc Chabert, Eugenia Cheng, Clifford C. Cocks, Alain Connes, Leo Corry, Wolfgang Coy, Tony Crilly, Serafina Cuomo, Mihalis Dafermos, Partha Dasgupta, Ingrid Daubechies, Joseph W. Dauben, John W. Dawson Jr., Francois de Gandt, Persi Diaconis, Jordan S. Ellenberg, Lawrence C. Evans, Florence Fasanelli, Anita Burdman Feferman, Solomon Feferman, Charles Fefferman, Della Fenster, José Ferreirós, David Fisher, Terry Gannon, A. Gardiner, Charles C. Gillispie, Oded Goldreich, Catherine Goldstein, Fernando Q. Gouvêa, Timothy Gowers, Andrew Granville, Ivor Grattan-Guinness, Jeremy Gray, Ben Green, Ian Grojnowski, Niccolò Guicciardini, Michael Harris, Ulf Hashagen, Nigel Higson, Andrew Hodges, F. E. A. Johnson, Mark Joshi, Kiran S. Kedlaya, Frank Kelly, Sergiu Klainerman, Jon Kleinberg, Israel Kleiner, Jacek Klinowski, Eberhard Knobloch, János Kollár, T. W. Körner, Michael Krivelevich, Peter D. Lax, Imre Leader, Jean-François Le Gall, W. B. R. Lickorish, Martin W. Liebeck, Jesper Lützen, Des MacHale, Alan L. Mackay, Shahn Majid, Lech Maligranda, David Marker, Jean Mawhin, Barry Mazur, Dusa McDuff, Colin McLarty, Bojan Mohar, Peter M. Neumann, Catherine Nolan, James Norris, Brian Osserman, Richard S. Palais, Marco Panza, Karen Hunger Parshall, Gabriel P. Paternain, Jeanne Peiffer, Carl Pomerance, Helmut Pulte, Bruce Reed, Michael C. Reed, Adrian Rice, Eleanor Robson, Igor Rodnianski, John Roe, Mark Ronan, Edward Sandifer, Tilman Sauer, Norbert Schappacher, Andrzej Schinzel, Erhard Scholz, Reinhard Siegmund-Schultze, Gordon Slade, David J. Spiegelhalter, Jacqueline Stedall, Arild Stubhaug, Madhu Sudan, Terence Tao, Jamie Tappenden, C. H. Taubes, Rüdiger Thiele, Burt Totaro, Lloyd N. Trefethen, Dirk van Dalen, Richard Weber, Dominic Welsh, Avi Wigderson, Herbert Wilf, David Wilkins, B. Yandell, Eric Zaslow, Doron Zeilberger

### Integrating Prosocial Learning with Education Standards by Kristie Fink,Jonathan Cohen,Sean Slade Book Summary:

Integrating Prosocial Learning with Education Standards demonstrates how to meet educational standards that privilege cognitive aspects of learning while also advancing prosocial or Whole Child efforts (e.g., social emotional learning, character education, and mental health promotion). The book utilizes a growing body of research to reveal effective ways to implement a curriculum that integrates social, emotional, ethical, and civic aspects of learning with required state standards, and a wide range of "real world" examples describe how any school, anywhere, can lay a foundation for all young people to succeed.

### Academic Language in Diverse Classrooms: Mathematics, Grades 35 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

### Equity in Mathematics Education by Constantinos Xenofontos Book Summary:

Following in the steps of the socio-political turn of the discipline, Equity in Mathematics Education: Addressing a Changing World emerged as a response of the editor and the chapter authors to the enormous changes that have in the last years occurred at a global level (for example, the ongoing war in Syria, the political [in]actions of powerful nations to fight climate change, the rise of far-right parties in many countries around the world, and so on). In recent years, massive migration waves from the Middle East have caused significant demographic changes to many European countries, Canada and the US, that are reflected in schools and classrooms. These observations have led this book’s contributors to reconsider the concept and/or practice of equity, and its related concept, social justice, and the role of mathematics education research in addressing and promoting a fairer world. Contrary to other, perhaps highly specialized books concerned with similar topics, this book aims to provide a smooth, yet deep introduction to those who are new to this research area. Equity in Mathematics Education: Addressing a Changing World contributes to the understanding of equity and its complex relations to mathematics education. It is anticipated that it will support individuals in teaching, educational research, policy making and planning, and teacher education, in becoming more aware of the interplay between school mathematics and socio-political issues that, ultimately, impacts the lives of learners and their communities, teachers as practitioners and as citizens, the wider society, and the world as a whole. Even though each chapter can be read independently of others, an engagement with all chapters in this volume will provide readers with a solid holistic understanding of the research territory of equity and mathematics education.

### Making Mathematics Practical by Tin Lam Toh,Khiok Seng Quek,Yew Hoong Leong,Jaguthsing Dindyal,Eng Guan Tay Book Summary:

This book is the first of its kind, as it includes both mathematics content and pedagogy. It is a professional instructional manual on how mathematical problem solving curriculum can be implemented in the classrooms. The book develops from the theoretical work of Polya and Schoenfeld, and explicates how these can be translated to the actual implementation in schools. It represents the work of a group of researchers from the Singapore National Institute of Education, after experimenting with it in the Singapore school classrooms. This book includes a set of scheme of work, lesson plans and a choice of mathematics problems that teachers can actually use in teaching problem solving. Certain pedagogical considerations are developed and suggested in this book. In addition, the book includes an assessment framework on how mathematical problem solving can be assessed.

### College Algebra by Jay P. Abramson,Valeree Falduto,Rachael Gross (Mathematics teacher),David Lippman,Rick Norwood,Melonie Rasmussen,Nicholas Belloit,Jean-Marie Magnier,Harold Whipple,Christina Fernandez Book Summary:

"The text is suitable for a typical introductory algebra course, and was developed to be used flexibly. While the breadth of topics may go beyond what an instructor would cover, the modular approach and the richness of content ensures that the book meets the needs of a variety of programs."--Page 1.

### Young Tableaux in Combinatorics, Invariant Theory, and Algebra by Joseph P.S. Kung Book Summary:

Young Tableaux in Combinatorics, Invariant Theory, and Algebra: An Anthology of Recent Work is an anthology of papers on Young tableaux and their applications in combinatorics, invariant theory, and algebra. Topics covered include reverse plane partitions and tableau hook numbers; some partitions associated with a partially ordered set; frames and Baxter sequences; and Young diagrams and ideals of Pfaffians. Comprised of 16 chapters, this book begins by describing a probabilistic proof of a formula for the number f? of standard Young tableaux of a given shape f?. The reader is then introduced to the generating function of R. P. Stanley for reverse plane partitions on a tableau shape; an analog of Schensted's algorithm relating permutations and triples consisting of two shifted Young tableaux and a set; and a variational problem for random Young tableaux. Subsequent chapters deal with certain aspects of Schensted's construction and the derivation of the Littlewood-Richardson rule for the multiplication of Schur functions using purely combinatorial methods; monotonicity and unimodality of the pattern inventory; and skew-symmetric invariant theory. This volume will be helpful to students and practitioners of algebra.

### Adding It Up by National Research Council,Division of Behavioral and Social Sciences and Education,Center for Education,Mathematics Learning Study Committee Book Summary:

Adding It Up explores how students in pre-K through 8th grade learn mathematics and recommends how teaching, curricula, and teacher education should change to improve mathematics learning during these critical years. The committee identifies five interdependent components of mathematical proficiency and describes how students develop this proficiency. With examples and illustrations, the book presents a portrait of mathematics learning: Research findings on what children know about numbers by the time they arrive in pre-K and the implications for mathematics instruction. Details on the processes by which students acquire mathematical proficiency with whole numbers, rational numbers, and integers, as well as beginning algebra, geometry, measurement, and probability and statistics. The committee discusses what is known from research about teaching for mathematics proficiency, focusing on the interactions between teachers and students around educational materials and how teachers develop proficiency in teaching mathematics.

### 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

### Mathematics Learning in Early Childhood by National Research Council,Division of Behavioral and Social Sciences and Education,Center for Education,Committee on Early Childhood Mathematics Book Summary:

Early childhood mathematics is vitally important for young children's present and future educational success. Research demonstrates that virtually all young children have the capability to learn and become competent in mathematics. Furthermore, young children enjoy their early informal experiences with mathematics. Unfortunately, many children's potential in mathematics is not fully realized, especially those children who are economically disadvantaged. This is due, in part, to a lack of opportunities to learn mathematics in early childhood settings or through everyday experiences in the home and in their communities. Improvements in early childhood mathematics education can provide young children with the foundation for school success. Relying on a comprehensive review of the research, Mathematics Learning in Early Childhood lays out the critical areas that should be the focus of young children's early mathematics education, explores the extent to which they are currently being incorporated in early childhood settings, and identifies the changes needed to improve the quality of mathematics experiences for young children. This book serves as a call to action to improve the state of early childhood mathematics. It will be especially useful for policy makers and practitioners-those who work directly with children and their families in shaping the policies that affect the education of young children.

### Programming for Computations - MATLAB/Octave by Svein Linge,Hans Petter Langtangen Book Summary:

This book presents computer programming as a key method for solving mathematical problems. There are two versions of the book, one for MATLAB and one for Python. The book was inspired by the Springer book TCSE 6: A Primer on Scientific Programming with Python (by Langtangen), but the style is more accessible and concise, in keeping with the needs of engineering students. The book outlines the shortest possible path from no previous experience with programming to a set of skills that allows the students to write simple programs for solving common mathematical problems with numerical methods in engineering and science courses. The emphasis is on generic algorithms, clean design of programs, use of functions, and automatic tests for verification.

### Building Powerful Numeracy for Middle and High School Students by Pamela Weber Harris Book Summary:

"I continue to be amazed at the power we can harness in our secondary students by teaching ourselves and our students real numeracy." --Pamela Harris As secondary math teachers, we're often frustrated by the lack of true number sense in our students. Solid research at the elementary level shows how to help all students become mathematically proficient by redefining what it means to compute with number sense. Pam Harris has spent the past ten years scrutinizing the research and using the resulting reform materials with teachers and students, seeing what works and what doesn't work, always with an eye to success in higher math. This book brings these insights to the secondary world, with an emphasis on one powerful goal: building numeracy. Developing numeracy in today's middle and high school students is reflective of the Common Core State Standards mission to build "the skills that our young people need for success in college and careers." (CCSS 2010) Numeracy is more than the ability to do basic arithmetic. At its heart, numeracy is the ability to use mathematical relationships to reason with numbers and numerical concepts, to think through the math logically, to have a repertoire of strategies to solve problems, and to be able to apply the logic outside of classrooms. How can we build powerful numeracy in middle and secondary students? Harris's approach emphasizes two big ideas: Teach the importance of representation. The representation of student strategies on models such as the open number line, the open array, and the ratio table promote discussion on relationships rather than procedures Teach with problem strings. Introduced by Catherine Twomey Fosnot and her colleagues in the Young Mathematicians at Work series, problem strings are purposefully designed sequences of related problems that help students construct numerical relationships. They encourage students to look to the numbers first before choosing a strategy, nudging them toward efficient, sophisticated strategies for computation. Understanding numerical relationships gives students the freedom to choose a strategy, rather than being stuck with only one way to solve a problem. Using the strings and activities in this book can empower your students to reason through problems and seek to find clever solutions. They'll become more naturally inclined to use the strategies that make sense to them. Students become engaged, willing to think, and more confident in their justifications. When we give secondary students this numerical power, we also help them learn higher mathematics with more confidence and more success.

### Thirty-three Miniatures by Jiří Matoušek Book Summary:

Contains a collection of clever mathematical applications of linear algebra, mainly in combinatorics, geometry, and algorithms. Each chapter covers a single main result with motivation and full proof in at most ten pages and can be read independently of all other chapters (with minor exceptions), assuming only a modest background in linear algebra. --from publisher description

### Increasing Student Learning Through Multimedia Projects by Michael Simkins Book Summary:

Addressed to K-12 teachers, discusses enhancing student achievement through project-based learning with multimedia and offers principles and guidelines to insure that multimedia projects address curriculum standards.

### Euclid's Elements by Euclid,Dana Densmore,Thomas L. Heath Book Summary:

The classic Heath translation, in a completely new layout with plenty of space and generous margins. An affordable but sturdy student and teacher sewn softcover edition in one volume, with minimal notes and a new index/glossary.

### The Differentiated Classroom by Carol Ann Tomlinson Book Summary:

Although much has changed in schools in recent years, the power of differentiated instruction remains the same—and the need for it has only increased. Today's classroom is more diverse, more inclusive, and more plugged into technology than ever before. And it's led by teachers under enormous pressure to help decidedly unstandardized students meet an expanding set of rigorous, standardized learning targets. In this updated second edition of her best-selling classic work, Carol Ann Tomlinson offers these teachers a powerful and practical way to meet a challenge that is both very modern and completely timeless: how to divide their time, resources, and efforts to effectively instruct so many students of various backgrounds, readiness and skill levels, and interests. With a perspective informed by advances in research and deepened by more than 15 years of implementation feedback in all types of schools, Tomlinson explains the theoretical basis of differentiated instruction, explores the variables of curriculum and learning environment, shares dozens of instructional strategies, and then goes inside elementary and secondary classrooms in nearly all subject areas to illustrate how real teachers are applying differentiation principles and strategies to respond to the needs of all learners. This book's insightful guidance on what to differentiate, how to differentiate, and why lays the groundwork for bringing differentiated instruction into your own classroom or refining the work you already do to help each of your wonderfully unique learners move toward greater knowledge, more advanced skills, and expanded understanding. Today more than ever, The Differentiated Classroom is a must-have staple for every teacher's shelf and every school's professional development collection.

### Creating Cultures of Thinking by Ron Ritchhart Book Summary:

Discover why and how schools must become places where thinking is valued, visible, and actively promoted As educators, parents, and citizens, we must settle for nothing less than environments that bring out the best in people, take learning to the next level, allow for great discoveries, and propel both the individual and the group forward into a lifetime of learning. This is something all teachers want and all students deserve. In Creating Cultures of Thinking: The 8 Forces We Must Master to Truly Transform Our Schools, Ron Ritchhart, author of Making Thinking Visible, explains how creating a culture of thinking is more important to learning than any particular curriculum and he outlines how any school or teacher can accomplish this by leveraging 8 cultural forces: expectations, language, time, modeling, opportunities, routines, interactions, and environment. With the techniques and rich classroom vignettes throughout this book, Ritchhart shows that creating a culture of thinking is not about just adhering to a particular set of practices or a general expectation that people should be involved in thinking. A culture of thinking produces the feelings, energy, and even joy that can propel learning forward and motivate us to do what at times can be hard and challenging mental work.

### Valuation Theory in Interaction by Antonio Campillo,Franz-Viktor Kuhlmann,Bernard Teissier Book Summary:

For more than a century, valuation theory has had its classical roots in algebraic number theory, algebraic geometry and the theory of ordered fields and groups. In recent decades it has seen an amazing expansion into many other areas. Moreover, having been dormant for a while in algebraic geometry, it has now been reintroduced as a tool to attack the open problem of resolution of singularities in positive characteristic and to analyze the structure of singularities. Driven by this topic, and by its many new applications in other areas, the research in valuation theory itself has also been intensified, with a particular emphasis on the deep open problems in positive characteristic. The multifaceted development of valuation theory has been monitored by two International Conferences and Workshops: the first in 1999 in Saskatoon, Canada, and the second in 2011 in Segovia and El Escorial in Spain. This book grew out of the second conference and presents high quality papers on recent research together with survey papers that illustrate the state of the art in several areas and applications of valuation theory. This book is addressed to researchers and graduate students who work in valuation theory or the areas where it is applied, as well as a general mathematical audience interested in the expansion and usefulness of the valuation theoretical approach, which has been called the ``most analytic'' form of algebraic reasoning. For young mathematicians who want to enter these areas of research, it provides a valuable source of up-to-date information.

### Category Theory in Context by Emily Riehl Book Summary:

Category theory has provided the foundations for many of the twentieth century's greatest advances in pure mathematics. This concise, original text for a one-semester course on the subject is derived from courses that author Emily Riehl taught at Harvard and Johns Hopkins Universities. The treatment introduces the essential concepts of category theory: categories, functors, natural transformations, the Yoneda lemma, limits and colimits, adjunctions, monads, and other topics. Suitable for advanced undergraduates and graduate students in mathematics, the text provides tools for understanding and attacking difficult problems in algebra, number theory, algebraic geometry, and algebraic topology. Drawing upon a broad range of mathematical examples from the categorical perspective, the author illustrates how the concepts and constructions of category theory arise from and illuminate more basic mathematical ideas. Prerequisites are limited to familiarity with some basic set theory and logic.

### Essential Questions by Jay McTighe,Grant Wiggins Book Summary:

What are "essential questions," and how do they differ from other kinds of questions? What's so great about them? Why should you design and use essential questions in your classroom? Essential questions (EQs) help target standards as you organize curriculum content into coherent units that yield focused and thoughtful learning. In the classroom, EQs are used to stimulate students' discussions and promote a deeper understanding of the content. Whether you are an Understanding by Design (UbD) devotee or are searching for ways to address standards—local or Common Core State Standards—in an engaging way, Jay McTighe and Grant Wiggins provide practical guidance on how to design, initiate, and embed inquiry-based teaching and learning in your classroom. Offering dozens of examples, the authors explore the usefulness of EQs in all K-12 content areas, including skill-based areas such as math, PE, language instruction, and arts education. As an important element of their backward design approach to designing curriculum, instruction, and assessment, the authors *Give a comprehensive explanation of why EQs are so important; *Explore seven defining characteristics of EQs; *Distinguish between topical and overarching questions and their uses; *Outline the rationale for using EQs as the focal point in creating units of study; and *Show how to create effective EQs, working from sources including standards, desired understandings, and student misconceptions. Using essential questions can be challenging—for both teachers and students—and this book provides guidance through practical and proven processes, as well as suggested "response strategies" to encourage student engagement. Finally, you will learn how to create a culture of inquiry so that all members of the educational community—students, teachers, and administrators—benefit from the increased rigor and deepened understanding that emerge when essential questions become a guiding force for learners of all ages.

### The Rabbit Problem by N.A Book Summary:

In Fibonacci's Field, Lonely and Chalk Rabbit meet, snuggle together and then spend a year trying to cope with their ever-increasing brood and the seasonal changes that bring a new challenge each month. Presented in calendar format with one pop-up illustration and other special features.