One of the most profound academic challenges students may face is the transition from arithmetic to algebra, which typically occurs in middle school. This shift in focus from arithmetic to algebra is beyond the simple incorporation of new facts or techniques; rather, it entails bigger and broader conceptual changes, necessitating a deepening understanding of abstract ideas and the need to engage the enigmatic unknown. For many learners, this reorientation from the concrete to the more abstract is disquieting, and as such, the role of the teacher in purposefully and thoughtfully scaffolding the new way of thinking and processing mathematics becomes even more central, particularly as standards-based assessments carry greater importance. Fortunately, a new text, Getting from Arithmetic to Algebra: Balanced Assessments for the Transition, is now available, and it can serve as a rich resource for educators and families committed to supporting their learners.
Organized with both the Common Core State Standards (CCSS) (2012) and the National Council of Teachers of Mathematics (NCTM) Standards in mind, Getting from Arithmetic to Algebra draws heavily from the work of the Balanced Assessment Group at the Harvard Graduate School of Education. Authors Judah Schwartz and Joan Kenney, both members of the original Balanced Assessment Group, arranged this book with an eye towards making it relevant and accessible to educators familiar with either the CCSS or the NCTM Standards, or for those transitioning from one set of standards to the other. As such, the text includes several useful tables that compare key components of the CCSS and NCTM Standards, with the additional inclusion of alignment to the language and conceptual frameworks of the Balanced Assessment Group.
The mathematics tasks in the book are divided into 5 discrete sections, each with multiple tasks included. Echoing the language used by the Balanced Assessment group, the text is organized into chapters, with the mathematics content chapters titled Number and Quantity, Pattern and Function, Shape and Space, Chance and Data, and Arrangement, with the most tasks concentrated in the Number and Quantity section. Although the student versions of the tasks are included in the book as copy-ready reproducibles, they are also available for free download at www.tcpress.com, which is indicated in the text at the beginning of each chapter.
Although much of the text is devoted to the mathematics tasks themselves, Getting from Arithmetic to Algebra: Balanced Assessments for the Transition isnt just a ready-to-use compendium of mathematics exercises intended to be reproduced for implementation with students. What makes this text unique and particularly useful for educators is the fact that each of the dozens of tasks includes a teachers guide, possible solutions, and a scoring rubric. The level of detailed information for each task is exceptionally useful to busy educators keen to support their learners.
The teachers guide that precedes each task fills an entire page, and is rich with information intended to support the work of the educator. In addition to a short description of each problem, the guide also includes an indication of which mathematics object is emphasized (which maps onto content standards from CCSS and NCTM). There are also indications of which actions are involved in the task, which correspond to the NCTM Process Standards and the CCSS Standards for Mathematical Practice. Each of the actions is given a weight (from 0-4) to indicate the level of prominence of that skill in the task, which may be very useful for educators planning instruction.
Another useful component of the teachers guide for each task is the explanation of the assumed mathematical background. For example, for the task titled, Smart Money, teachers may assume that, Students should have experience calculating with percentages and decimals, and organizing large amounts of data (p. 48). This may be useful in deciding which tasks are appropriate for each student. Additionally, the teachers guide for each task includes core elements of performance, which consists of 2 or 3 statements about the expected outcomes for the tasks. There are also general instructions for using each task (which include suggestions on pacing, materials the teacher may wish to have available, or concepts to review before engaging in the task).
Each task is followed with a detailed solution and rubric for evaluating student performance. The rubrics, echoing the language of the math actions indicated in the teachers guides, provide criteria for students working at the partial competency level and at the full competency level. Evaluating student work with these rubrics may provide educators with clear ideas about which processes may benefit from review.
Throughout the text, Schwartz and Kenney make clear that although the tasks may serve as rich and informative assessments of student learning, there are many other possible uses for them as well. The tasks strongly resemble open-ended items in standardized assessments, and as such, may be useful in test preparation. The tasks also align neatly to the mathematics standards used in most schools, and could easily function as supplemental assignments or activities. Further, the tasks and rubrics may provide rich fodder for the professional development of mathematics teachers, particularly those involved in establishing interrater reliability in evaluating student work using rubrics.
Although the tasks, teachers guides and rubrics make this text a worthwhile addition to middle school mathematics teachers libraries, what makes this book exceptional is the section at the end of each chapter titled, Making the transition to algebra. These sections elevate the academic content of the book by highlighting the ways in which each task provides an access point for students to engage in more rigorous, conceptual mathematical thinking. These sections move the book beyond a simple compendium of fun tasks; in these sections, Schwartz and Kenney connect the dots for teachers, underscoring the deep conceptual shifts each problem is intended to evoke or elicit. As such, this text may be a useful guide in helping craft responses to the persistent question, Why are we doing this? Additionally, the thinking outlined in these sections may be useful in sharing with families the ways in which their students are learning mathematics.
Through their artful weaving of standards-based mathematics, worthwhile learning activities, accessibility and equity, Schwartz and Kenney have created a text that may serve as a valuable resource not just for middle school mathematics educators, but for families of middle school students, as well. The engaging and even fun mathematics tasks, along with their clear and accessible solutions, may provide opportunities for rich mathematics experiences in a range of contexts.
Common Core State Standards Initiative. (2012). Mathematics. Retrieved on August 6, 2012, from http://www.corestandards.org/Math.
National Council of Teachers of Mathematics. (2012). Principles and Standards for School Mathematics. Retrieved on August 6, 2012, from http://www.nctm.org/standards/content.aspx?id=16909