*Teaching Mathematics Vocabulary in Context: Windows, Doors, and Secret Passageways* looks at "how vocabulary acquisition impacts the learning of mathematics" (p. 1). The book is full of good ideas that Miki has found to be effective in teaching her middle school classes and that could be used by anyone who teaches mathematics. Miki's students, with their ideas and understandings taking center stage, learn how to communicate mathematically.

Chapter one sets the stage to establish the role of mathematics vocabulary in Miki's classroom. On the first day of class she communicates her expectations to her students who in turn pass these expectations on to their caregivers. Students are then immersed in mathematics vocabulary through "warm-ups, problems, investigations, explorations, assessments, recording, reporting, writing, speaking, and reading" (p. 22). Miki begins their vocabulary studies by investigating what the word *mathematics* means. If you teach calculus just ask your students what the word *derivative* means and you will appreciate what Miki is doing with her students. By developing her students' mathematics vocabulary it makes it possible for her students to meaningfully discuss their multiple representations and/or solutions of a problem, and their mathematical conjectures with Miki.

In chapter two her students share their interpretations of a problem, work collaboratively to solve the problem and finally present their solutions. Throughout, the three phases of students' thinking, pairing up and sharing, Miki circulates listening to hear how students are applying their mathematics vocabulary. Recognizing that quality classroom conversations just don't happen she talks with her students about improving their listening skills and together they develop a rubric that describes the characteristics of a quality mathematics conversation.

While attending a workshop Miki learned that in order for a word to become part of one's vocabulary it must be used meaningfully for about 30 times. As a result she encourages the use of mathematics vocabulary by her students, in as many formats as possible, throughout the three phases of students' thinking, pairing up and sharing.

Of interest in chapter three is a classroom game Miki uses as a warm-up that focuses on students' understanding and use of mathematics vocabulary. As well at the end of the first two trimesters she holds portfolio conferences with her students and their caregivers. Miki describes how her students place notes from her mini lessons in their portfolios to demonstrate how well their note taking has improved. The students also include self-made concept maps in their portfolios to help them demonstrate at their portfolio conferences what mathematics they have learned during the trimester.

In chapter four Miki explains how students at least once a month, using a format discussed in class, summarize how they solved a problem. A general rubric developed by Miki and her former students is used as a guide for the students' writing process and for their partners to give them feedback on their writing. The problem-solving write-ups are then submitted to Miki. I am always skeptical of students making comments and suggestions on other students' work; yet, Miki's students believe that this process is very important in helping them learn mathematics.

Chapter five describes how students use 'mathematical reflections' when they complete an 'investigation.' The reflections become a vehicle for students to use their mathematics vocabulary and "provide insight into individual student concept development and evaluative reasoning" (p.103). The reflections are then subject to peer review [which again sends up a red flag with me] utilizing a teacher-student developed rubric.

Chapter six looks to expand students' mathematics vocabulary using their journals as a 'personal textbook.' Examples from former students are used to illustrate how her students should use their journals. In order to minimize the amount of work that goes into checking and giving students feedback on their journals Miki asks her students to periodically put specific journal entries into what she calls a 'journal folder,' which she evaluates later. The students in a homework log keep a record of what has been assigned, when it was assigned, and when it is due. A page from the students' homework log is included in their portfolios along with an assignment they liked and one they found most difficult. Students must be prepared to defend their choices. Homework is also shared with tablemates and any disagreements are submitted to the class for further consideration. Finally the work is handed in to Miki for feedback.

Chapter seven shares information with the reader on biweekly self-evaluations where students through prompts focus on reviewing their work, evaluating their understanding, checking their homework and vocabulary, summarizing what they have learned and asking questions, and reflecting on their effort and behavior. As mentioned before, at the end of the trimester the students will provide feedback to both Miki and their caregivers regarding what they have learned and accomplished mathematically. Miki states, "The variety of individual responses lets me appreciate the different gifts that students bring to learning and teaching" (p. 141).

Miki stresses that studying mathematics with an emphasis on mathematics vocabulary "is contingent on *not* relying on dictionary definitions until *after* the concepts represented are well developed and understood" (p. 135). Consider the implications that this alone would have for students learning mathematics.

Chapter eight describes how Miki's students use a collection of terms provided by her to generate essays, stories, letters or poems to improve their use of mathematics vocabulary. Chapter nine contains more ideas about writing mathematics poetry. At this point the reader could quite easily question if there is enough time in the day/term to write math poetry but Miki insists "Now, I don't think I could teach mathematics without incorporating the reading and writing of related poetry" (p. 166).

In chapter ten Miki looks back at the activities/strategies in her book and their importance for the mathematical success of her students. The development of students' mathematics vocabulary and resulting mathematical culture serves to make explicit students' understanding of mathematical concepts but it also has the potential to initiate students' progression through the Van Hiele levels of geometric thought from visualization to formal deduction.

This book is valuable resource for mathematics teachers because of the many ideas and accompanying student examples it contains for teaching mathematics. It is a book that demonstrates many ways the mathematics teacher can make students' thoughts and ideas overt.

Rick Seaman (

[email protected]) is Associate Professor of Mathematics Education at the University of Regina in Regina, SK, Canada.