problem solvers: problem
J u l i e J a me s a n d A l i c e S t e i ml e
Jesse’s
train
complete this task. If this tool is not readily
available, directions and a template for creating
the manipulative are available; access them at
http://bit.ly/16caAud.
Has your class gone digital? If so, access
http://bit.ly/1aOdG66 for an inexpensive rod
applet.
WAVEBREAK MEDIA LTD/THINKSTOCK
Problem scenario
◗
Persevering in problem solving and
constructing and critiquing mathematical
arguments are some of the mathematical practices included in the Common Core
State Standards for Mathematics (CCSSI 2010).
To solve unfamiliar problems, students must
make sense of the situation and apply current
knowledge. Teachers can present such opportunities by engaging students in structured
group activities or collaborative problemsolving activities.
But encouraging students to work together
on a problem-solving task can be difficult.
Many teachers can probably identify with
attempting to have students work in groups
only to have one student take charge while
the others sit back and observe. One way of
helping students to learn to think and work
collaboratively is to present them with “groupworthy” problems (Horn 2005, p. 219). Such
problems meet the following criteria:
• Illustrate important mathematical concepts
• Include multiple tasks that draw effectively
on the collective resources of a group
• Allow for multiple representations
• Have several possible solution paths
The following problem is modeled after
those in United We Solve 116 Math Problems
for Groups (Erickson 1996). Provide each group
with a set of clues needed to solve the problem.
Every group member is responsible for one
clue. Students will need Cuisenaire™ Rods to
346
Use the clues and Cuisenaire Rods to build
Jesse's train.
See the activity sheet on page 349.
Classroom setup
Each group of students is presented with a set
of clue cards and a set of Cuisenaire Rods. If
students are unfamiliar with Cuisenaire Rods,
allow time for them to explore the manipulative
and make observations about the relationships
among the pieces. Randomly select students to
share their observations. Be sure to record the
observations for student reference. Direct students’ attention to the fractional value of a rod
relative to the different sizes of whole rods; for
example, the yellow rod is one-half the length
of the orange rod (see fig. 1). Additionally, students could make wholes by placing a train of
rods end-to-end. For example, the yellow rod
is one-third the length of a train composed of
a blue rod and a dark green rod (see fig. 2). Ask
students to justify their statements by drawing
them or by demonstrating with the rods in front
of the class. Reinforce the expectation that students should justify their reasoning by asking
such questions as, “How do you know?” or “Can
you prove that using the rods?”
Now you are ready to begin. Direct students
into groups of four with the goal of working
together to use the clues to build Jesse’s train.
Give each group member one of the initial
clues and remind everyone that all members
are responsible for making sure the group’s
solution fits each person’s clue. An individual
may read her clue to her group members, but
February 2014 • teaching children mathematics | Vol. 20, No. 6
Copyright © 2014 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
www.nctm.org
Extending student thinking
Tasks that lead to multiple correct solutions
allow students the opportunity to “communicate their thoughts effectively, justify their
arguments, and examine issues from different
perspectives” (Cohen et al. 1999, p. 83). This
task has several correct solutions based on the
interpretation that the train consists of exactly
four rods, but many more solutions can be constructed if the number of rods is greater than
four. Encouraging students to explore other
possible solutions allows them to engage in
repeated reasoning and to look for and make
use of structure (CCSSM 2010).
You could extend this task by asking students
to find the fractional part of their train that
each rod represents. Discussions could explore
www.nctm.org
Figu r e 1
Fractional values of a Cuisinaire Rod are relative to different
sizes of wholes; for example, the yellow rod is one-half the
size of the orange rod.
11
11
22
1
111
222
1
2
why one rod might have one fractional value
for one train and another value for a longer
or shorter train. Students could then create
number sentences to represent their train. For
each possible solution, the entire train would
represent the whole, and each rod would
represent a fractional part of the whole (see
fig. 3). To promote a discussion, ask, “Why does
each rod represent a different fractional value?
Will this always be true?” Students should
recognize that because each rod is a different
length, it would represent a different part
of the whole. Yet if more than one rod of the
F ig ure 2
she may not hand her clue to another member.
Circulate around the room as students work;
note which clues students find most useful.
The clue, “Jesse used four different colored
rods to build a train,” often leads to interesting discussions and debates among students.
Some students will interpret this to mean that
exactly four different, colored rods are in Jesse’s
train. Others may interpret the clue to mean
that Jesse’s train must have four colors but that
the number of rods is unlimited. This ambiguity in the clue allows students to develop viable
arguments, critique one another’s reasoning,
and decide which interpretation their group
solution will represent.
If students finish early, you could extend
the interaction by providing extra clues, which
may require the group to change their original
solution. Encourage students to record their
original solution and then work toward a solution that aligns with the new clues. Additional
clues could also be used to support groups that
are struggling to find a solution. The clue that
removes all purple rods will help students narrow their options.
When each group has a solution, ask students to walk around and view other groups’
solutions. If no group has received additional
clues, ask students if the other groups’ trains
have successfully met all the given clues. After
students have returned to their seats, they may
want to discuss the different interpretations of
the four-color clue or other discrepancies they
have noticed.
Values for a whole can be made with a train of rods placed
end-to-end. For example, the yellow rod is one-third the size
of the “train” composed of a blue rod and a dark green rod.
1
311
3
3
1
1
1
1
1
1
11
31
33
3
1
3111
33
3
11
33
Vol. 20, No. 6 | teaching children mathematics • February 2014
347
Figu r E 3
problem solvers: problem
Students must interpret each clue in the context of the
problem as well as reason and think critically about other
solutions and representations of these solutions.
1
1
1
11 6
6
66
1
1
3
1
131
3
33
1
1
11
1
9
1
3
1
11
9
99
1 1 1 7
+ + +
=1
61 311 9 1118 77
1
7
18
7
77
18
18
1 18
3
1+
+1 +
+1 +
+7 =
=11
+
+
+
6
18= 1
66 3
33 9
99 18
18
same color is represented in a group’s solution,
students should recognize that each rod of the
same color will have the same fractional value
for that train.
Share your students’ work
Try this problem in your classroom. We are
interested in how your students responded to
the problem, which problem-solving strategies
Where’s the math?
This task allows students the opportunity to explore the magnitude of
fractions compared with different-size wholes. For example, in the solution
shown in figure 3, the light green rod represents one-half the length of the
dark green rod, fulfilling the second criterion. Another representation of
one-half is also evident in that the light green and dark green rods together
represent one-half the length of the entire train, meeting the requirements
of the third criterion. You could ask students to consider the question, “How
can the light green rod represent one-half if the light green rod and the dark
green rod together also represent one-half?” This should lead to a discussion
of the need to identify the whole when referring to any fractional value.
Additionally, students must interpret each clue in the context of the
problem as well as reason and think critically about other solutions and
representations of these solutions. For example, students could lay six light
green rods end-to-end to prove that the light green rod represents one-sixth
the length of the entire train in figure 3. Alternatively, they may compare
each rod with a certain number of the white cubes to derive their fractions.
As students share solutions, encourage them to be precise by referring to the
appropriate parts and wholes. Be sure to encourage students to listen to one
another by asking that they repeat a presenter’s argument or idea.
348
February 2014 • teaching children mathematics | Vol. 20, No. 6
they used, and how they explained or justified
their reasoning. Send your thoughts and reflections—including information about how you
posed the problem, samples of students’ work,
and photographs showing your problem solvers in action—by April 15, 2014, to either Problem Solvers department editor Signe Kastberg,
Purdue University, 100 North University St.,
West Lafayette, IN 47907-2098, or Erin Moss,
Millersville University, P.O. Box 1002, Millersville, PA 17551-0302; or e-mail skastber@
purdue.edu or [email protected].
Selected submissions will be published in a
subsequent issue of TCM and acknowledged by
name, grade level, and school name unless you
indicate otherwise.
R E F E RE N C E S
Cohen, Elizabeth G., Rachel A. Lotan, Beth A.
Scarloss, and Adele R. Arellano. 1999. “Complex
Instruction: Equity in Cooperative Learning Classrooms.” Theory into Practice 38 (Spring): 80–86.
Common Core State Standards Initiative (CCSSI).
2010. Common Core State Standards for Mathematics. Washington, DC: National Governors
Association Center for Best Practices and the
Council of Chief State School Officers. http://
www.corestandards.org/assets/CCSSI_Math%20
Standards.pdf
Erickson, Tim. 1996. United We Solve: 116 Math
Problems for Groups. Oakland, CA: Eeps Media.
Horn, Ilana S. 2005. “Learning on the Job: A Situated
Account of Teacher Learning in Two High School
Mathematics Departments.” Cognition & Instruction 23 (2): 207–36.
Julie James, [email protected], is the professional
development coordinator for the Center for Mathematics and Science Education at the University of Mississippi
in Oxford. Alice Steimle, [email protected], is
Associate Director for the Center for Mathematics and
Science Education at the University of Mississippi in
Oxford. Edited by Signe E. Kastberg, a teacher of
prospective elementary teachers at Purdue University
in West Lafayette, Indiana; and Erin Moss, an assistant
professor in the math department at Millersville University of Pennsylvania. Each month, this section of the
Problem Solvers department features a new challenge
for students. Readers are encouraged to submit problems to be considered for future columns. Receipt of
problems will not be acknowledged; however, those
selected for publication will be credited to the author.
www.nctm.org
➺ problem solvers activity sheet
Jesse’s Train Clue Card Template
Jesse’s train
Jesse’s train
Use your clue and your Cuisenaire Rods
to help your group build Jesse’s train.
Use your clue and your Cuisenaire Rods
to help your group build Jesse’s train.
Jesse used four different
colored rods to build a train.
One rod in Jesse’s train represents onehalf of one other rod in the train.
Jesse’s train
Jesse’s train
Use your clue and your Cuisenaire Rods
to help your group build Jesse’s train.
Use your clue and your Cuisenaire Rods
to help your group build Jesse’s train.
Two rods in Jesse’s train
can be placed end-to-end to
represent half of the entire train.
Jesse did not use
the white rod
in his train.
Jesse’s train
Jesse’s train
Use your clue and your Cuisenaire Rods
to help your group build Jesse’s train.
Use your clue and your Cuisenaire Rods
to help your group build Jesse’s train.
***EXTRA CLUE***
One rod in Jesse’s train represents onethird of another rod in Jesse’s train.
***EXTRA CLUE***
Jesse did not use
the purple rod.
From the February 2014 issue of