Rationale
for Use
This activity provides
mathematics teachers with an opportunity to reflect on practice by analyzing
a multiapproach problem suitable for students in the middle grades. The intent
of the activity is to analyze students’ algebraic thinking by examining
their solution to a rich problem.
Suggestions
for Use
This activity may
be used with groups of middle school teachers or by individuals who are interested
in a careful examination of middle school students’ algebraic thinking.
Procedures
The following is a suggested approach
to using the article in professional development sessions with middle school
teachers.
- Before
participants read the article, have them solve the Crossing the River problem
(Figure 1 on p. 284 of the article) individually. They are to show all their
work.
- Have
participants discuss their problem-solving approaches in small groups and then
post their solutions on easel paper.
- In
a whole-group discussion, have small groups share their approaches to solving
the problem that they posted on easel paper. Facilitate a discussion on the various
approaches used to solve the problem. A gallery walk may replace the whole-group
discussion. For this, participants should be instructed to study the various
approaches to solve the problem that they observe.
- Give
participants Figures 2 and 3 (pp. 284–285 of the article) and ask them,
in small groups, to analyze the two students’ approaches to solving the
Crossing the River problem.
- Part-way
through the small-group discussion, ask them the overarching question of the
article, “What do these students know about building rules to represent
functions?” They should record their comments and be prepared to share
them in a whole-group discussion.
- Ask
each group to share their responses to the overarching question. Record their
comments on easel paper.
- Have
participants individually read the entire article.
- Ask
groups to compare their responses with those of the
authors. They should be prepared to share their conclusions in a whole-group
discussion.
- Ask
participants to use the Crossing the River problem with their own students.
- If there is
another scheduled session with the participants, ask them to bring in samples
of their students’ work displaying a variety of approaches.
- Ask
teachers, in pairs, to analyze the work of those students and report on what
they learned about the algebraic thinking of the students. They should report
on their analyses in the large group.
- Hold a large-group
discussion about how these results would affect participants’ instructional
practices.
Extensions
If participants
are familiar with developing rubrics, have them work in small groups to develop
a rubric for scoring the Crossing the River problem. This provides a different
activity. The previous activity had participants analyzing students’ algebraic
thinking. The extension has participants evaluating students’ work.
Connections
to Other NCTM Publications
- Moses,
Barbara, ed. Algebraic
Thinking, Grades K–12, Readings from NCTM’s
School-Based Journals and Other Publications. Reston, Va.: NCTM, 1999.
- Friel,
Susan, Sid Rachlin, and Dot Doyle. Navigating
through Algebra in Grades 6–8. Reston,
Va.: NCTM, 2001.
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