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Teaching and Learning Middle Grades Mathematics

Rheta Rubenstein, Charlene Beckmann, and Denisse Thompson
Key College Publishing
Publication Date: 
Number of Pages: 
[Reviewed by
George Ashline
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At the outset of Teaching and Learning Middle Grades Mathematics (abbreviated TLMGM below), Rheta Rubenstein, Charlene Beckmann, and Denisse Thompson acknowledge the complexity of teaching middle grades mathematics, and emphasize the practical need for teachers to effectively integrate knowledge of teaching (methods) and knowledge of mathematics (content). One of their goals in writing this text is "to provide research, reading, foundations, experiences, and strategies to support teachers in reaching" the ambitious vision for school mathematics described in the NCTM's Principles and Standards for School Mathematics (2000). They have created a middle grades mathematics content and methods text which challenges both pre-service and in-service teachers to become "reflective practitioners and career-long learners". In this book review, I will offer some thoughts concerning the fundamental concepts and issues it addresses, as well as the abundant professional development resources it offers. As a member of an instructional team delivering a proportional reasoning course this summer to in-service middle grades teachers in Vermont through the Vermont Mathematics Partnership (one of the NSF-sponsored math and science partnership programs), I am keenly aware of the importance of appropriate mathematics teacher preparation, and I feel that this text makes a significant contribution toward that critical goal.

To begin, TLMGM contains five units, with the first four focusing on mathematics embedded in content strands recommended by the NCTM, including reasoning with patterns and algebra; rational numbers; geometry, measurement, and transformations; and data analysis and probability. In addition to these mathematical areas, the five units address important elements of effective mathematics teaching:

  • creation of a supportive environment for student learning and use of cooperative learning strategies

  • integration of research about adolescent learning into mathematics instruction

  • combination of the first two elements in planning instruction, creating lessons, and implementing instructional strategies

  • assessment providing feedback to all other aspects of teaching

  • collaboration with various educational partners in areas such as curricular integration with other disciplines and curricular material evaluation

Throughout TLMGM, lessons are often implemented using a three-pronged "Launch", then "Explore", then "Share and Summarize" approach adapted from the Connected Mathematics Project approach (Lappan et. al). This allows mathematical and pedagogical problems and issues to be systematically introduced, investigated, and synthesized.  Also, TLMGM contains many valuable "student pages", designed to actively engage middle grades students. For implementation and personalization purposes, many of these pages, along with other material and manipulatives, are helpfully provided in an electronic format in the CD-ROM accompanying the text.

At the end of each section, the authors provide various questions to "Deepen Mathematical Understanding", "Develop Mathematical Pedagogy", and "Reflect on Professional Reading". Some questions are marked with special portfolio or fieldwork icons, suggesting possible material for portfolios and tasks for middle grades students and relevant fieldwork, respectively. Some of the suggested readings are pertinent sections of the NCTM's Principles and Standards for School Mathematics, an important resource for any teacher faced with the educational expectations in the United States in light of the recent NCLB legislation. I found the reading reflection questions particularly helpful to analyze accessible (often via the companion CD-ROM), enriching, thought-provoking, and timely articles and manuscripts. Furthermore, some questions include sample student work and misconceptions, which focus teacher consideration on classroom dilemmas and suggest possible ways to respond to such challenges. Throughout TLMGM, the authors emphasize the need to carefully reflect upon mathematics content and instructional strategies. Rather than advocating a particular teaching method or style, the authors call for an analysis of different pedagogical approaches and a discussion of the pros and cons of the materials included in the text. This analysis is often outlined in end-of-section questions, as well as section and unit summaries, and is important when realistically deciding the best approach to meet a teacher's needs.

To illustrate the makeup of a TLMGM unit, let's consider the third unit on planning and instruction, with its geometry and measurement content focus. Like other units, this one began with an overview of the pertinent pedagogical issues and a brief historical survey of the roots of geometry in the time of the ancient Babylonians, Egyptians, and Greeks. After detailing an investigation of area formula evolution for various planar figures, the authors suggest several possible interview topics for middle grades students related to area and perimeter, with investigation and interview results expanded upon later in the unit. Then, in the unit sections, the authors consider a rich collection of important middle grades mathematics topics, including measurement formulas, proportions and similar figures, quadrilateral classifications, tessellations, and transformations, and the Pythagorean Theorem. Throughout the unit, lesson suggestions and engaging activities illustrate ways in which such mathematics can come alive in a middle grades classroom. For example, in one section, the authors model a lesson on similarity with "Gavin and the Giant's Button".  In another section, they detail the "Quadrilateral Scavenger Hunt" and "Quadrilateral Sort Game", for which one of the appendices (and the CD-ROM) contains sample quadrilaterals in various "real life" contexts. In addition to these mathematical topics, the authors concurrently describe a number of considerations and methods to effectively deliver this content. In this unit, important pedagogical tools are detailed to identify worthwhile mathematical tasks, to plan a middle grades lesson, to develop effective classroom discourse, and to integrate guided discovery, open-ended explorations, and direct instruction into lesson planning.

Next, I would like to highlight some of the materials included in TLMGM which I found particularly attractive:

  • the opening survey on "Mathematics Education: Where Do I Stand?", offering teachers a means to assess their feelings about a number of significant teaching issues and to re-visit these later in the text to see if their opinions have changed.

  • the rich investigations presented in the first four units — including explorations of historical area formulas (mentioned above), figurate numbers and their patterns, equivalent expressions for integers (created with "four fours"), and Pascal's Triangle and its historical significance.

  • a variety of engaging activities and lesson plan suggestions, such as:

    • Graphing Distance versus Time
    • Solving Equations with Cups and Beans, and Two Color Chips for Integers
    • Function contexts, such as the Sports Complex Manager and the Biologist
    • Candy Bar Fractions and Fraction Strip Comparisons
    • Pattern Block and Tessellation Investigations
    • Mean vs. Median: Which Measure is Better?
    • Games: Four in a Row, Fraction Card, Quadrilateral Sort and Quadrilateral Scavenger Hunt, and Diagonal Duel
  • the analysis of student responses and possible misconceptions provided at the end of some sections — for example, for rational number operations, some common student errors in communicating mathematically are given, such as the infamous NAEP (National Assessment of Educational Progress) problem in which eighth graders commonly estimated (7/8) + (12/13) as 19 or 21, rather than 2.

  • several illustrations of integrating technology into lessons — for instance, spreadsheets in unit one yielding function outputs and first and second differences, a calculator-based motion detector activity in unit two investigating graphical representations of distance versus time, a Geometer's Sketchpad student page in unit three exploring various rotations and translations, and a Fathom Dynamic Statistics handout in unit four investigating numerical and graphical ways of describing data and measures of central tendency.

  • consideration of pertinent mathematics education approaches and issues, such as:

    • implementing cooperative learning ground rules and techniques, including think-pair-share, roundtable, coop, random reporting, pairs check, and jigsaw
    • problem solving effectively, with reference to some of G. Pólya's work
    • developing rational number understanding and arithmetic operation meaning, including sample stories and analyses adapted from M. Bell & Z. Usiskin
    • identifying worthwhile mathematical tasks and considering the level of cognitive demand of each, as suggested by M. S. Smith & M. K. Stein
    • asking guiding questions when planning a lesson, such as those relating to a sample middle grades lesson on proportionality
    • creating effective classroom discourse, and incorporating the framework for advancing children's thinking by J. Fraivillig
    • analyzing various formative and summative assessment contexts questions, such as those relating to a given 7th grade probability unit
    • evaluating venues for educational collaboration, with colleagues, administrators, counselors, and parents

I do have a couple of suggestions for possible improvements in TLMGM. As illustrated in the above Unit 3 details, the authors focus on important content areas and offer ways in which these can engage students in a middle grades classroom. The mathematics terminology underlying each topic arises periodically over the course of each section, and is addressed in the well constructed "Descriptive Glossary" in the first appendix. In my opinion, it would have been helpful to include at the beginning of each section or unit a list of the new terminology to be introduced, so that the reader could systematically find the new mathematical topics in the text. Also, to complement the questions provided at the end of each section, especially those on the content-side but even the more open-ended pedagogical ones, it would have been useful to offer in another appendix a collection of answers and/or possible directions for response for the questions.

Teaching and Learning Middle Grades Mathematics provides an in-depth analysis of fundamental issues faced by middle grades mathematics teachers. Offering an important alternative to texts which primarily focus on elementary or secondary grades, TLMGM is well-written, offers guidance to both pre-service and in-service middle grades teachers to effectively integrate mathematical content and pedagogy, and is overflowing with materials designed to meet that goal. Ultimately, I expect that teachers will find its plethora of professional development resources invaluable now and for future reference.

George Ashline ( is associate professor of mathematics at St. Michael's College in Colchester, VT. He is a member of Project NeXT, a professional development program for new or recent Ph.D.s in the mathematical sciences who are interested in improving the teaching and learning of undergraduate mathematics. He is also actively involved in professional development programs in mathematics for elementary and middle school teachers.
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