Deborah Schifter, Virginia Bastable, and Susan Jo Russell

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Reasoning Algebraically about Operations (RAO) is one of the seven modules in the Developing Mathematical Ideas Series (DMI), a professional development curriculum designed to help teachers think through the major concepts of K–grade 8 mathematics and examine how children develop those concepts. Under the guidance of the facilitator, participants investigate mathematics content, analyze cases in the casebook as well as recorded classroom lessons, and inquire into the understanding of their own students.

The module consists of a casebook (sold separately) and an online facilitator’s package that contains everything necessary to prepare for and lead the seminar, including access to the casebook content and classroom videos. Reasoning Algebraically about Operationsparticipants examine generalizations at the heart of the study of the four operations in the elementary and middle grades. They express these generalizations in common language and in algebraic notation, develop arguments based on representations of the operations, study what it means to prove a generalization, and extend their generalizations and arguments when the domain under consideration expands from whole numbers to integers.

The primary goal of Reasoning Algebraically about Operations is to help elementary and middle school teachers learn the mathematics content they are responsible for teaching in a profound way. To this end, the program asks participants to make sense of the content, recognize where and how the content of their grade is situated in the trajectory of learning from kindergarten through middle school, build connections among different concepts, and analyze student thinking from a mathematical perspective. Through this work, teachers learn how to orient their instruction to specific mathematical goals and to develop a mathematics pedagogy in which student understanding takes center stage.

The facilitator’s package consists of an Introduction to DMI, Preseminar Preparation for the facilitator, and content for eight sessions:

Session 1: Discovering Rules for Odds and Evens

Session 2: Finding Relationships in Addition and Subtraction

Session 3: Reordering Terms and Factors

Session 4: Expanding the Number System

Session 5: Doing and Undoing, Staying the Same

Session 6: Multiplying in Clumps

Session 7: Exploring Rules about Factors

Session 8: Wrapping Up

For each session, there is an “Overview,” summarizing the main mathematical themes of the session, a facilitator preparation checklist and mathematics background notes, a “Detailed Agenda,” and “Maxine’s Journal,” a narrative account of the session from the point of view of a facilitator.

The facilitator’s checklist for each session links to all the readings, including those from the casebook, and downloadable materials the facilitator will need to complete or prepare before leading that session. For those sessions that include a video, the checklist also contains a link to that video.

The “Detailed Agenda” describes each activity of a session and the recommended amount of time for that segment. There are three versions of the detailed agenda that the facilitator can access: (1) the “reading” form to prepare for giving the seminar, (2) an MS Word document that can be downloaded and annotated by the facilitator, and (3) the “In-Class Agenda” that not only can be scrolled through during a session but also has the video for that session embedded within, providing easy access to the video for displaying to the participants.

“Maxine’s Journal” was created to convey a sense of what a Reasoning Algebraically about Operations seminar might be like—the type of discussions that might take place, the type of lessons participants might draw from the sessions—and how it might feel to facilitate one. Maxine is a composite character as are the teachers in her seminar. Though she is fiction, Maxine’s journal describes events and individuals observed and recorded by the developers of RAO and those who piloted the its first programs.

To sample all that Reasoning about Operations has to offer, click here for a preview of Session 2.