Deborah Schifter, Virginia Bastable, and Susan Jo Russell
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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
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
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.