take time for action
Shelby P. Morge and Eleanor L. Pusey
Step Up to Nonlinear Functions
d
Edited by Janet Hart Frost, frost@wsu
.edu, of Washington State University, Spokane, Washington, and Lynda R. Wiest,
wiest@unr.edu, a mathematics education
professor in the Department of Educational Specialties, University of Nevada,
Reno. “Take Time for Action” encourages
active involvement in research by teachers
as part of their classroom practice.
310
Do your students think that all functions are linear? Do they have experience working with nonlinear functions? If so, how do they think about
those functions?
Middle school students are often
asked to generate a story to explain a
graph of a piecewise function. Unfortunately, they often have had little
experience translating among various
representations of nonlinear functions.
Van de Walle, Karp, and Bay-Williams
(forthcoming) emphasize the importance of developing fluency moving
among the various representations of
functional relationships.
Step functions are a completely
different kind of pattern that may
be unfamiliar to most middle-grades
students. For this reason, these functions can be problematic when set
in a context. However, Billings and
McClure (2005) state that context is
important for middle school students
to develop a deep understanding of
step-wise functional relationships.
Working with these types of functions
in middle school will help students
develop a stronger understanding of
functional relationships and their various representations.
We invite you to work through the
following task on your own before
trying it with your students.
THE TASK: SNEAKING
UP THE LINE
To learn more about how students
think about nonlinear functions
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Vol. 15, No. 6, February 2010
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This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
and their representations, we asked
students to solve the Sneaking Up
the Line task (EDC 2005, p. 348). It
requires students to explore a pattern about how many jumps a sneaky
sheep must make to get to the front
of the line, depending on how many
sheep start out ahead (fig. 1). The
students completing the task included
twelve middle school students and five
university students.
We presented the situation in
figure 1. To help students clarify the
pattern, we asked these questions:
• How often does Eric move ahead?
• What makes the number of sheep
in front of Eric change?
We then asked students to predict the
number of sheep that will be shorn
before Eric and describe the strategies they used to find the number.
Before completing it on their own, we
worked through simpler versions of
the task with the whole class by asking
students if they could find how many
sheep would be shorn before Eric if
there were 6 or 11 sheep in front. All
students were encouraged to use twocolor counters or color tiles to model
these situations. Some students volunteered to act out the situation in front
of the class. We recommend using the
information from these simpler tasks
to solve the original problem.
Students were then placed in pairs
to work. We monitored their progress
and asked these questions:
1. Do you see any pattern in your
answers to the simpler tasks?
2. How are you thinking about solving the problem?
3. What is happening in your picture,
expression, or table?
4. How did you decide on your answer?
Students were asked to describe a
rule or expression that they could use
to find the number of sheep shorn
before Eric for any number of sheep
in front. The university students did
not immediately generate tables and
graphs depicting this relationship; we
inferred that making the translation
into a rule or expression was challenging for them. As a result, a table and a
blank graph were added to the handout to provide additional scaffolding
for the middle school students. After
the task was completed, the class
discussed solution strategies. Students
were asked how the numbers in this
pattern compared with other patterns
they had studied.
STUDENT PERFORMANCE
Both the middle school and university
students used different strategies to
predict the number of sheep shorn before Eric. Although the middle school
students used an assortment of strategies, the most common was to divide
by 2 (because Eric skipped 2 sheep
each time). All university students
took the number 3 into account for
their predictions because Eric skipped
2 sheep and 1 was removed from the
line. Their most common solution
strategy was to divide 49 by 3.
When working out the actual solution to the task, the middle school students used three common strategies:
• Modeling with 49 counters (a
continuation of physically acting
out the task)
• Drawing a diagram and marking
off the shorn sheep as Eric moved
up the line
Fig. 1 This sheep-jumping task asks students to consider a nonlinear function.
PART 1: MAKING PREDICTIONS
Eric the sheep is at the end of a line of sheep waiting to be shorn. But being
an impatient sort of a sheep, every time the shearer takes a sheep from the
front to be shorn, Eric sneaks up the line two places.
Think about how long it will take Eric to reach the head of the line. Before
you begin to work, make a prediction. If there are 49 sheep ahead of him,
how many of the sheep will be shorn before Eric?
Prediction: __________________
Describe the strategies you used to predict the answer. How could you
predict the answer for any number of sheep in the line? Is your method for
predicting “algebraic”? Why, or why not? Please explain.
“Sneaking Up the Line,” from MathScape: Seeing and Thinking Mathematically, Course 1,
© 2005 McGraw Hill Glencoe. Reprinted with permission.
• Using a table to extend the pattern
to 49 sheep
Most middle school students who
used one or more of these strategies
found the correct answer, which was
that 17 sheep were shorn ahead of
Eric. The next most popular answer
was 16, as students struggled to keep
an accurate count once Eric was
within 3 sheep of being at the front
of the line. In other words, when
Eric became third in line, sheep 1
was shorn, but only 1 sheep remained
to jump over.
Similar confusion surfaced when
these students acted out the problem in class. A remainder became
an unanticipated source of difficulty.
How does one extend a pattern when
it cannot be fully continued, and what
does a remainder mean? It represented the number of sheep in front
of Eric, but students interpreted it to
mean a fraction of a sheep. In contrast, the university students did not
have the same difficulty in deciding
how to handle the remainders, as
evidenced in the rules they recorded
both symbolically (as shown in fig. 2)
and verbally:
Vol. 15, No. 6, February 2010
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The rule I used was the number of
sheep divided by 3. If there was a
remainder of 1 or 2, I rounded up.
Most students discussed dividing the
number of sheep ahead of Eric by 3
and rounding up when the number
was not evenly divisible by 3. However, two university students reasoned
incorrectly about why they needed to
round to “account for partial sheep”:
Mathematics Teaching in the Middle School
311
Fig. 2 A university student gave this correct symbolic rule for any number of sheep.
Solve the problem about Eric the Sheep using whatever methods make sense
to you. Describe a rule or expression you could use to find the number shorn
before Eric for any number of sheep in front of him.
Fig. 3 In (a), a student described the pattern with subtraction and drew a picture three
times; in (b), a student used counters to find a solution, but the diagram was nonsensical.
Solve the problem about Eric the Sheep using whatever methods make sense
to you. Explain how you reached your solution.
My prediction was correct, but the expression is more specific (i.e., rounding
up to account for partial sheep).
In addition, most university students
seemed to be overconfident in their
ability to obtain a correct solution,
so they jumped from their physical
models to symbolic and verbal rules
without generating tables and graphs
of the pattern.
None of the middle school students described the pattern in terms
of division, but rather in terms of
subtracting 1 and then subtracting 2
(fig. 3a). We believe that this is one
reason the younger students had
trouble finding a rule for any number of sheep. Simply recording their
strategy with a picture was difficult for
some (fig. 3b). In one case, a student
made three drawings that resulted in
three different answers (fig. 3a). We
conjecture that this occurred because
it was challenging to represent a
dynamic pattern on paper with a static
diagram. Those students who successfully used a table strategy continued to
have difficulty when it came to writing a rule. For example, one student
described his process this way:
We use a chart to get 17 and I took
the chart and went from 9−49 and it
gave me 17.
This student was not able to “write
down” a rule, starting with “y =.” We
attributed this result to middle school
students’ inexperience with the patterns of a step function and how to
represent it. The context of shearing
sheep was accessible for them, but
the numerical pattern was unfamiliar
(e.g., 3, 3, 3, 4, 4, 4, 5, . . .).
(a)
IMPLICATIONS FOR TEACHING
We believe the choice to expose both
middle-grades students and university
students to a more difficult pattern
than they were familiar with was both
(b)
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Vol. 15, No. 6, February 2010
insightful and interesting, in part, for
these reasons:
1. Both groups of students were able
to experience some success because
the task was something they found
appealing and engaging.
2. Being able to act out the task and
model with counters made it accessible for the students.
3. Those who chose to represent the
problem in multiple ways tended
to have the most success, most
likely because they were able to
“see” the pattern within the various
representations and cross-check
their answers.
In the future, we need to be better
prepared to help students wrestle with
the issue of the remainder and methods for communicating their results.
We feel that it would be appropriate
to discuss how to handle the remainder after students have wrestled with a
few simplified problems on their own.
This would give them some prior
knowledge in thinking about what to
do with the remainder before applying
it to a more difficult problem.
We also recognize the importance
of structuring the middle school
students’ task to include tables and
graphs, which were not evident in the
university students’ solutions. When
those representations were not included, the university students found
it difficult to generate a rule. This led
us to include both a table and a blank
graph in the revised handout for the
middle-grades students.
If the pattern in this task had been
continuous, it is likely that students
would not have had as much trouble
identifying or extending the pattern.
They have seen linear and quadratic
patterns before and these patterns are
more intuitive to extend. However,
the skipping pattern generated by a
step function is unfamiliar. A regular
45-minute class period was not enough
time. We recommend allowing ample
time for students to work through this
task because it may take them longer to
process a pattern of this nature.
REFERENCES
Billings, Esther M., and Melanie S.
McClure. “Mailing a Publication:
Exploring Linear and Step Functions
in a Real-World Context.” Mathematics Teaching in the Middle School 10
(March 2005): 349−55.
Education Development Center (EDC).
MathScape: Seeing and Thinking Mathematically, Course 1. Columbus, OH:
Glencoe, 2005.
Van de Walle, John A., Karen S. Karp, and
Jennifer M. Bay-Williams. Elementary
and Middle School Mathematics: Teaching Developmentally. Boston: Allyn &
Bacon, forthcoming.
Shelby P. Morge, morges@uncw.edu,
is an assistant professor in the Department of Elementary, Middle Level, and
Literacy Education at the University of
North Carolina Wilmington. She is
interested in gender differences in
mathematics, mathematics-related
beliefs, using instructional technology
in a problem-based learning environment, assessment, and teacher professional development. Eleanor L. Pusey,
epusey@earthlink.net, is pursuing a
doctorate in mathematics education
through the ACCLAIM (Appalachian
Collaborative Center for Learning,
Assessment, and Instruction in Mathematics) program. She also works with
middle school and high school teachers
as a mathematics coach for the Columbus County Schools in southeastern
North Carolina.
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