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Mathematics

Mathematical Development in Young Children: Exploring Notations by Barbara M. Brizuela (Teachers College Press) (Hardcover) Using data from interviews and in depth conversations with children from five to nine years of age, Brizuela's study examines how children understand and learn mathematical notations in their development as mathematics learners. Each chapter focuses on a different notational system--written numbers, commas and periods, fractions, data tables, number lines, graphs, and student-invented systems drawing from established conventions. The chapters are organized chronologically in terms of the ages of the children described in each chapter, so the resulting order is by both increasing age of the children and increasing complexity of the mathematical content.

Excerpt: This volume is organized chronologically in terms of the ages of the chil­dren described in each of the chapters. This leads to two overlapping se­quences: an order in terms of increasing age of the children (5 through 9 years of age), and an order in terms of the increasing complexity of the mathematical content dealt with in the notations. I have used pseudonyms to protect the privacy of the children I have studied. In Chapters 3, 5, 6, and 7, I use children's real first names, at children's and parents' request or with their permission.

In Chapter 2, I focus on George, a 5-year-old kindergarten student who will be presented trying to create a systematic way of writing numbers, in his own idiosyncratic way, and trying to understand how the written number system works. Specifically, I will focus on George's way of writing "teens," and how this might relate to his developing understandings about the written number system and about place value.

In Chapter 3, my focus is Paula, another 5-year-old kindergarten stu­dent who, in the process of developing a system for naming and writing two-digit numbers beyond 12, invents the idea of "capital numbers." The interaction between inventions and conventions, one of the main issues presented in this book, will be highlighted in Paula's chapter.

Thomas is the focus of Chapter 4. Thomas is a very sophisticated 6-year-old kindergarten student who, during a series of extended clinical in­terviews, deals with the role and function of commas and periods in num­bers. His case highlights the connections between children's ideas and the history of mathematical notations, as well as the gradually evolving con­ceptual understandings about the written number system and the notations used to represent those evolving understandings.

The data in Chapters 5, 6, and 7 come from the Early Algebra project, directed by Analúcia Schliemann (Tufts University) and David Carraher (TERC). Sara is the central topic in Chapter 5. Sara is a third grader who will be shown developing notations for problems dealing with fractions. The focus of the chapter will be not only her evolving understanding of the problems, fractions, and their notations, but also her use of the notations to "help her think," as she explains it herself. The events that will be analyzed in Chapter 5 come from classroom events in which she discusses the most appropriate type of notation with her friends, and from a follow-up in a subsequent clinical interview.

In Chapter 6, we move into even more complex mathematics, while remaining in the K–3 focus of the book. This chapter contains descriptions of the work with data tables of a group of second and third graders, with an emphasis on Jennifer, one of these children. Through describing Jenni­fer's work with data tables, I will highlight the evolution of her thinking about mathematical concepts such as additive relations, her ideas about data tables in general, and the ways in which children bring into notations, such as data tables, their own idiosyncratic methods of representation as well as the conventions that they gradually appropriate.

In Chapter 7, I explore the interrelations of different types of notations. This chapter focuses on an extended clinical interview that I carried out with three third graders—Jennifer, Nathan, and Jeffrey—in which we explored a problem dealing with functions and sought different ways to represent the problem. During the interview, the children examined differ­ent types of notations: data tables, number lines, graphs, and natural lan­guage. This chapter will bring together many of the points that were raised in previous chapters: the ways in which the children used the notations to help them think through the problem; the ways in which they combined their own spontaneous ways of representing and the conventions that they had already appropriated; and the ways in which the notations reflect and embody the conceptual understandings about functions that the children held.

In Chapter 8, I gather together some of the main ideas presented throughout the book, highlighting the learning process behind mathematical notations and the importance of Piaget's theory in our thinking about this field.

Mind-Bending Math and Science Activities for Gifted Students (For Grades K-12) by Callard-Szulgit Rosemary (Rowman & Littlefield Education) Here is a reference and guide for teachers and parents that covers many aspects of gifted thinking in relation to math and science. It features competitions and curricula that can be easily adapted to students' lifestyles outside of the classroom and the materials are accessible to adults with limited scientific backgrounds. Advice, vignettes, and cartoons are included. Intended for grades K-12.

Excerpt: What does it take for a student to be brilliant? Certainly it entails a healthy dose of intelligence, but I think there also has to be an aspect of illumination mixed with it. A student needs to take his or tier "book smarts" and electrify that trait with creativity and imagination. Memorizing the Periodic Table of chemical elements is a neat parlor trick that requires quite a bit of brain power, but it's the brilliant student who will think of a new way to organize time elements in that table. Playing a piece of music flawlessly requires talent and discipline, but the performance will not be brilliant until the student makes the music speak in a new voice via his or her own interpretation. Intelligence is impressive, but brilliance is inspiring.

The material in this book is not meant to replace existing curricula—nor is it a panacea for bored or underachieving students. Instead, the literary suggestions, exercises, competitions, and so forth are meant to help students unlock creative potentials and combine them with intellectual rigor. Parents, teachers, and students should use this book as a guide toward finding opportunities that they may not have known about previously; allowing them to "take off" in whatever direction inspires them the most. Each student in question will "click- with different chapters, and if each is significantly inspired by just one idea within, then we will rest easily knowing that we have made a positive contribution to education.

We have included a few exercises that we have written in a student-friendly fashion. They can be copied and given to students, who should be able to follow the instructions with minimal guidance. In that sense, the assignments will grant teachers more freedom with their time, because they can meet with the students at convenient checkpoints. This does not, however, abdicate teachers from their responsibility to spend time with those students. As bright as they' may' be, they still benefit from a teacher's guidance in helping them reach higher levels of thinking. Keep in touch with them and let them inspire you.

Just before completing my part of the writing for this book, I had the pleasure of spending a morning at the North Carolina School of Science and Mathematics (NCSSM), a wonderful public, residential, coeducational high school for juniors and seniors with high intellectual ability and commitment to scholarship. I started my visit with an introduction to Dr. Steve Warshaw, senior vice-president for academic affairs, who would meet me later for an interview and graciously answer my multitude of questions. I then went on a tour with Jeanne Chen, a junior at the school. We never stopped talking! I was so excited to be on a campus where respect for learning was apparent and professional courtesies abounded.

I started by asking Jeanne what the major difference was between her former high school and NCSSM. She immediately replied, "Ini my home high school, I already knew most of the work. Here, I am challenged every day! We each strive for our own personal excellence."

Jeanne's reply pretty much summed up my entire experience at NCSSM that morning. Everywhere we went, I saw students dedicated to learning—mature, happy young adults, socially interactive with each other and with any other adults in their proximity.

In the music conservatory, I met Mr. Laird, who was just getting a rehearsal started for the school's spring concert of Haydn's Mass in D Minor. I also visited the recording studio, where students are taught about and experience professional recordings. Time art studio and classrooms were next, with the beautiful stained-glass windows designed and made by former students.

It was obvious that the faculty members were talented, well trained, and committed to their fields of expertise and to the students at NCSSM. Counseling and support services were plentiful. It would be hard for kids to fail here. There is an overriding commitment to excellence and success for everyone.

Dr. Warshaw proudly pointed out the school's Distance Learning Program, an information highway for cyber campuses. Coupled with a trimester calendar, the school's philosophy is to ensure that each student obtains a thorough grounding in the sciences, mathematics, language arts, history, and a foreign language; that each becomes proficient in the use of computers, laboratory facilities, and other means by which knowledge is acquired and processed in the sciences and mathematics; and that each adopts a sequence of study that includes investigating areas of academic interest in depth through high-level course offerings and such activities as mentorship, research courses, independent studies, seminars, directed group studies and mini-term.

I applaud the North Carolina School of Science and Mathematics as one more opportunity open to all of North Carolina's bright students with specific interests and talents to study, accelerate, learn, laugh, honor, and grow as young adults. Contact them at NCSSM, 1219 Broad Street, Durham, NC 27705, www.ncssm.edu.

Math Refresher for Scientists and Engineers, 3rd edition by John R. Fanchi (Wiley-IEEE Press) Expanded coverage of essential math, including integral equations, calculus of variations, tensor analysis, and special integrals

Math Refresher for Scientists and Engineers, Third Edition is specifically designed as a self-study guide to help busy professionals and students in science and engineering quickly refresh and improve the math skills needed to perform their jobs and advance their careers. The book focuses on practical applications and exercises that readers are likely to face in their professional environments. All the basic math skills needed to manage contemporary technology problems are addressed and presented in a clear, lucid style that readers familiar with previous editions have come to appreciate and value.

The book begins with basic concepts in college algebra and trigonometry, and then moves on to explore more advanced concepts in calculus, linear algebra (including matrices), differential equations, probability, and statistics. This Third Edition has been greatly expanded to reflect the needs of today's professionals. New material includes:

• A chapter on integral equations

• A chapter on calculus of variations

• A chapter on tensor analysis

• A section on time series

• A section on partial fractions

• Many new exercises and solutions

Collectively, the chapters teach most of the basic math skills needed by scientists and engineers. The wide range of topics covered in one title is unique. All chapters provide a review of important principles and methods. Examples, exercises, and applications are used liberally throughout to engage the readers and assist them in applying their new math skills to actual problems. Solutions to exercises are provided in an appendix.

Whether to brush up on professional skills or prepare for exams, readers will find this self-study guide enables them to quickly master the math they need. It can additionally be used as a textbook for advanced-level undergraduates in physics and engineering.

This book is, as the title says, a refresher. It's intended for the reader who has already complete course work in the material, but wants to review it, or use it possibly as a reference. Fanchi's book covers most branches of mathematics that the typical undergraduate is likely to encounter, with a few notable exceptions. There is no discussion about tensors and nothing on things like topology. The book's chapters follow what is probably the traditional route many of us took in our introduction to mathematics. The first chapter is a review of algebra, followed by geometry, trigonometry, and hyperbolic functions. These are followed by analytic geometry, linear algebra, and calculus (broken into chapters dealing with differential calculus, partial derivatives, integrals, and ordinary differential equations).

Since this is a refresher book, the author assumes that the reader has already been exposed, either through experience or course work, to the subjects being covered. So, the material is not introduced the way it typically is found in traditional texts. There is, for example, relatively little developmental material leading up to key concepts. This is both expected and appropriate.

The book excels in several areas, including the use of examples, which are found liberally throughout the text. I highly recommend readers work through these exercises, especially since many of them have solutions that are, themselves, important and useful material. Each exercise comes complete with worked-out solutions in the back of the book, so if you want, you can use this as a very effective book for self-education, almost as if you were going back to school to take a refresher course.

As with any book, reliability and accuracy is critical. Especially for a reference-type book that you are likely to use when you don't have a lot of time to question, or check, the accuracy of the material. Fanchi does an excellent job in this regard; I found almost no errors (and none that were serious). Another thing I look for in a reference book is a good index. I know that the next time I pick it up, it'll be because I remember reading something in it, and I know the answer I'm looking for is in there somewhere, but I cannot remember the page. The first place I'll go is the index, and with a good one (like the one in Fanchi's book) I should be able to track down the solution without too much wasted time.

There were a couple of pleasant surprises in this book. One was the three chapters on probability and statistics. The other was the section (in the chapter on ordinary differential equations) dealing with nonlinear dynamics and chaos. Frankly, I was surprised to see the subject of chaos discussed in this book at all (especially since tensors are not). My only complaint is that there is not an entire chapter devoted to the subject.

The book is easy to read, and well organized. I found it very enjoyable. It's the sort of book that you can easily read for an hour, cover 20 or 30 pages, and not feel too exhausted - if, that is, you are reading it as a review. However, as I said, I'd not recommend this as an introductory text for any of these subjects (and I think the author, by reason of his choice of title, would agree).

Teaching Mathematics in Primary Schools by Robyn Zevenbergen, Shelley Dole, Robert J. Wright (Allen & Unwin) A systematic, research-based introduction to the principles and practice of teaching mathematics at the primary school level, this inquiry moves beyond traditional lockstep approaches to teaching mathematics to emphasize how students can learn to think mathematically in terms of globalization and new technologies.

This book has been designed to present preservice and practising teachers with the most current methods of teaching mathematics to primary school students, drawing on significant bodies of contemporary research as the foundation for a book that aims to move teaching mathematics into the new millennium. While not always making it explicit, the authors have also drawn on the common wisdom of the field. Much of this wisdom is no longer published, having become part of the everyday practice of teaching mathematics. It is not our intention to endorse a particular approach; rather it is to alert readers to the different ways in which good teaching of mathematics can occur.

The introductory chapters are designed to provide an overview of the key issues that impact on the teaching of mathematics in primary schools, and are followed by several chapters related to planning and assessment. The remainder of the hook focuses on the teaching of particular content areas. Underpinning these chapters is the intention to draw on commonly used techniques and approaches coupled with more recent developments, particularly those arising from the use of technology.

The wider social movements today impacting on educational reforms have been incorporated into our approach to provide preservice teachers with a sense of the current debates within the field of mathematics education; we have counterbalanced them with the common wisdom of the field. This is particularly evident in two areas.

We have used the phrase `New Times' throughout the text to signify the growing impact of knowledge integration and the use of technology on education. Simultaneously, there is a growing recognition that many of the old methods of teaching are inappropriate for these New Times, and that for some students of mathematics the outcome of the old methods have always been poor. We have attempted to unite the best elements of the old wisdom with the best of the new approaches to teaching mathematics to give readers a broader appreciation of the field. Secondly, there is a significant groundswell in the teaching of early years that challenges the collective wisdom of old approaches. The stand-alone chapter 8, on the teaching of early number, embraces this new approach. It is our hope that readers will use this chapter to challenge many of the older approaches to teaching number, although we recognize that many students will enter systems where older approaches still dominate.

 

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