**1. Make a Bigger Puddle, Make a Smaller Worm (1972), by Marion Walter.**

**1.1. Review by: Geraldine Green.**

*The Arithmetic Teacher*

**20**(1) (1973), 73.

The very young child discovers the magic of mirrors in this fascinating and unusual library book. By moving a mirror across the delightful illustrations, the young child sees shapes and patterns change. As he experiments, he also creates new pictures. This book will have great appeal, for it invites the young mathematician to explore, to discover, and to learn by doing as he uses the safe metal mirror that is provided with the book to answer the stimulating questions or to follow the simple directions. The reader has fun doing, seeing, thinking, and using his imagination. While having much fun, the child will grasp simple mathematical concepts. An added bonus is the opportunity provided by this book for language development.

**2. Look at Annette (1972), by Marion Walter.**

**2.1. Review by: Geraldine Green.**

*The Arithmetic Teacher*

**20**(1) (1973), 73.

In this library book, the reader meets Annette and helps her do many interesting things by moving a mirror across the appealing pictures that are in the book. Wholes are made out of halves, more things appear, and some things disappear - all by using the magic mirror. The young reader will have fun seeing, doing, thinking, and imagining as he uses the safe, metal mirror, which comes with the book, to follow the simple suggestions and to answer the intriguing questions. This delightful book encourages the young mathematician to explore, to discover, and to learn by doing. Simple mathematical concepts will be developed while the child is having fun. This book also stimulates language development.

**3. The Art of Problem Posing (1983), by Stephen I Brown and Marion I Walter.**

**3.1. Review by: James R Hersberger.**

*The Mathematics Teacher*

**77**(6) (1984), 482.

With the current emphasis on development of mathematics curricula that foster problem-solving ability, 'The Art of Problem Posing' is a welcome addition to the literature. Nearly every curriculum model presented describes a developmental process in which students progress from learning to solve problems that are presented to them to seeking out and posing the problems themselves. The stated intent of the authors is "to try to understand: 1. What problem posing consists of and why it is important. 2. What strategies exist for engaging in and improving problem posing. 3. How problem posing relates to problem solving." They have done just what they intended and in a clear and timely manner. It has been my experience that the problem-posing stage is the most difficult for students and teachers alike. The techniques often appear obtuse and certainly do not develop naturally in the majority of students. Brown and Walter have presented a systematic, lucid treatment of the topic, with several excellent examples discussed in great detail. I especially recommend the chapters in which the " What-If-Not " approach is described in theory and in practice and also the chapter in which problem posing is related to problem solving. In conclusion, I heartily recommend this book for the target audience, which includes mathematics teachers in secondary schools and higher levels of education. I would further recommend it to any elementary or middle school teachers who are interested in techniques that will be invaluable if the curriculum is ever to be effectively centred around problem solving.

**3.2. Review of 3rd edition (2005) by: Keith R Leatham.**

*The Mathematics Teacher* **99** (3) (2005), 223.

The premise of this book is twofold: that the art of problem posing is an integral part of mathematical thinking, and that involving teachers and students in this art can be a powerful way for them to engage in mathematical exploration and learning. The authors describe various levels of problem-posing strategies, pro vide opportunities to see such strategies in action, and then encourage readers to experiment with the strategies them selves. The text is easy to read and actively involves the reader. Teacher educators will easily see ways they can use this book with pre-service or in-service teachers. Although they include no delineated problem sets, the authors invite readers to engage in a number of problem-posing activities. This third edition also includes a new chapter devoted to describing ways in which the content and principles of this book have been used successfully in classroom settings. Teacher educators, in-service teachers, and pre-service teachers will find many examples of problem posing activities they can use with their students. 'The Art of Problem Posing' has the potential to influence significantly the way teachers think about the nature of mathematics and what it means to "do" mathematics. The authors envision that this will, in turn, influence mathematics students of all ages as they engage in solving problems they themselves have posed.

**3.3. Review of 3rd edition (2005) by: Robert Buyea.**

*Teaching Children Mathematics* **12** (8) (2006), 428.

This book is best suited for teachers of mathematics in middle school, secondary school, and higher levels of education. However, as a teacher of elementary students, I did find some of the book helpful and applicable for me as well. The book is well organized, beginning with an introduction to the topic. It presents what problem posing consists of along with its importance, then discusses strategies for engaging in problem posing. Finally, the book highlights the relation- ship between problem solving and problem posing. The authors also include a chapter about having students assume the various roles of journal writing throughout their course to engage heavily in both problem solving and problem posing, an interesting idea for teachers of higher-level mathematics.

**4. The Mirror Puzzle Book (1985), by Marion Walter.**

**4.1. Review by: A Dean Hendrickson.**

*The Arithmetic Teacher*

**34**(1) (1986), 54-55.

This book is a series of problems involving the use of a mirror on one drawing to match other related drawings. The one drawing is the "Mirror Master," and those drawings to be matched by arranging the mirror on the Mirror Master are called puzzles. A person familiar with the ESS unit Mirror Cards will see that these are similar to, but generally more difficult than, those. However, earlier experience with Mirror Cards will make this little problem-solving book much more enjoyable. The drawings to be matched cannot always be made with the mirror on the Mirror Master, and students must classify the puzzles as either "do-able" or impossible. The purpose of these puzzles is to give students experience with finding line symmetry in a shape, determining whether more than one line of symmetry exists for the shape, and so on. This focus helps in the development of spatial relations and recognition of certain properties of geometric figures, as these might be treated more formally at a later time. Two mirrors are provided so that children can work together or so that games can be developed for two to play with each other. Teachers should find the book a valuable addition to their store of nonverbal problem-solving activities.

**4.2. Review by: A Dean Hendrickson.**

*The Arithmetic Teacher* **34** (4) (1986), 40.

In a review of 'The Mirror Puzzle Book' in the September 1986 issue ("New Books for Teachers," pp. 54-55), I failed to mention Marion Walter as the originator of the ESS unit Mirror Cards I referred to. This oversight was regrettable and I apologize. Walter should take pride in the fact that the reviewed book won an honourable mention in the 1986 Children's Book Award Program conducted by the New York Academy of Sciences, a fact unknown to me at the time the review was written.

**4.3. Review by: Julie Ragan Madison.**

*The Arithmetic Teacher* **36** (7) (1989), 58.

Fun challenging, possible, impossible, symmetry are all words that describe Marion Walter's The Mirror Puzzle Book. The book presents twelve mirror puzzle pages. Students attempt to reproduce twelve figures with a mirror from each individual mirror master. The puzzles move from simple to more challenging, along with some impossible puzzles. If students approach these puzzles in a systematic fashion, it is possible for them to develop a concept of symmetry from using this book. The book can be placed without supplementary materials in a mathematics centre. It even includes two mirrors and an extra set of mirror masters so that two students can attempt the puzzles at the same time. Some puzzles are appropriate for primary grades; they increase in difficulty to a level that adults may find challenging. The Mirror Puzzle Book offers students many hours of entertaining activity. Activities involving symmetry and perception are compiled in a colourful, inviting format.

**4.5. Review by: A 4th grade teacher.**

*Amazon Reviews* (2000).

This book provides a fun way to help students recognize patterns, relationships and how to organize information that is necessary for building their math and critical thinking skills. It is a great boost to the child that is spatially oriented and will help enhance this orientation in those who need to improve this skill. I have used this in my 4th grade class for many years and it has been enjoyed by all. Some students went so far as to create their own patterns. What a great brain teaser!

**4.6. Review by: Heather Scott.**

*Amazon Reviews* (2013).

Marion Walter has written a very clever book. The puzzles are timeless - use the mirror to make the images in the book. Some puzzles are not possible - and that is one of the brilliant things about her writing - she covers all the angles to provide you with some challenging ideas that gets your brain working mathematically. This is a must for every one who is interested in supporting young learners in learning mathematics.

**5. Problem Posing: Reflections and Applications (1993), by Stephen I Brown and Marion I Walter.**

**5.1. Review by: Jerry Johnson.**

*The Mathematics Teacher*

**87**(1) (1994), 55.

This thought-provoking reference fulfils a promise made in the author's second edition of 'The Art of Problem Posing' (Hillsdale, N.J.: Lawrence Erlbaum Associates, 1990). It is a collection of readings that represent how other colleagues have used and extended the pedagogical scheme of problem posing. The majority of the articles are edited reprints, with two original articles. Three broad sections form the book. The first section reflects on issues associated with problem posing by touching on pedagogical, psychological, and philosophical concerns. The two subsequent sections illustrate the potential of using problem posing to enrich students' learning and understanding of arithmetic, algebra, and geometry. Two specific opportunities give readers a chance to practice posing problems, although most readers will find themselves posing problems throughout the book. Problem posing is an infectious idea; both teachers and students need to be exposed to it. The authors' first work introduced the idea, and many converts were made. This reference demonstrates how some converts implemented and extended the idea. Although this collection can be used on its own and is highly recommended, readers will benefit most if they have first read 'The Art of Problem Posing'.

**6. Mirror Explorations Set, 4 activity books for teaching with and about mirrors, gr. K-8: "M" Is for Mirror: Find the Hidden Pictures, The Mirror Puzzle Book, The Magic Cylinder Book: Hidden Pictures to Color and Discover, Mirror Explorations (1998), by Duncan Birmingham, Marion Walter, Ivan Moscovich and Peggy McLean.**

**6.1. Review by: Mark S Levy.**

*Teaching Children Mathematics*

**4**(7) (1998), 428.

The Mirror Exploration Set contains four books that allow students to discover the principles of image, reflection, and symmetry. '"M" Is for Mirror' is a colourful picture book that gives children the opportunity to find symmetrical patterns after reading such clues as "A is for Astronomer, who studies the moon." The Mirror Puzzle Book presents fourteen puzzles that ask the reader to use the enclosed mirrors to explore lines of symmetry. As the level of difficulty increases, critical-thinking skills need to be applied. 'The Magic Cylinder Book' contains pictures that are examined through a cylindrical mirror. Students are asked to predict what they will see through the mirror. The book gives instructions for creating pictures on square and polar grids that enable learners to apply their graphing skills. 'Mirror Explorations' is an activity book that suggests way of discovering symmetry through the use of manipulative materials and mirrors. The activities cover the following topics: geometric shapes, reflection, addition, subtraction, multiplication, division, angles, and real-world applications.