Promoting
Problem-Posing
Explorations
y fourth-grade class had just completed an
exploration of pentominoes (polygonal
shapes with an area of five square units).
Finding all twelve shapes gives children valuable
geometric problem-solving practice by highlighting transformations (flips, slides, and turns) and
congruence (shapes can be differently oriented, yet
congruent). Before moving on to another lesson, I
realized that the students might use the same
twelve shapes to examine perimeter and area.
Eleven of the shapes have a perimeter of twelve
units. Only one shape yields a different perimeter,
ten units (see fig. 1). The children had limited
experience with perimeter and area; I doubted that
they understood that shapes with a fixed area could
have perimeters of different lengths. Because they
were so familiar with the pentominoes, I felt that
this material would give them a good opportunity
to address these concepts in more detail. Although
I did expect them to calculate the perimeters and
areas of the twelve shapes, I did not foresee that the
children’s follow-up discussion would open an
opportunity for problem-posing explorations. This
article describes my evolving curricular decision
making, the children’s investigations, and what I
learned from this unanticipated experience.
As part of my originally planned “last day” of
working with pentominoes, I reviewed the definitions of perimeter and area and asked the children
to describe their calculations and observations in
their journals. As the students worked, many
expressed surprise and curiosity as they compared
the twelve perimeters. At the close of the period, I
gathered the children together and asked, “Why
M
By Phyllis Whitin
Phyllis Whitin, [email protected], is an associate professor of elementary education
at Wayne State University. She is particularly interested in the role of communication in developing mathematical understanding.
180
does only one shape have a perimeter of ten units,
and eleven shapes have a perimeter of twelve
units?” What follows is a portion of their discussion of the shape with the unique perimeter:
Jonathan. Four [square units] are stuck together.
Mark. You’ve moved two from the bottom and
put them on the side (reducing outside edges).
Jessie. With the 10 [perimeter] piece, there are
two squares covered up. It would probably be different if we had six squares.
Lisa. There’s a line in the middle; you can’t
count it.
Kara. You don’t count the inner bar lines.
Tricia. Why is the perimeter twelve units long
for all of the other shapes? I expected that the
straight line [1 × 5 rectangle] would have the most.
I admitted to the children that I was surprised
and intrigued by their many insights and questions,
and that we might find ways to pursue their ideas.
The Tensions of Curricular
Decision Making
Reflecting on this conversation, I recalled a statement from Principles and Standards for School
Mathematics: “Teaching mathematics well is a complex endeavor, and there are no easy recipes”
(NCTM 2000, p. 17). I faced several of the perennial
challenges of teaching. First was the issue of time:
Do I summarize the conversation and move on to
another mathematical topic, or do I provide time for
further investigation? Judging from the tone in the
children’s voices and the interested look on their
faces, I sensed that the initial problem had served to
“pique students’ curiosity and draw them into mathematics” (NCTM 2000, p. 18). Jessie’s comment—
“It would be different if we used six squares”—
directly suggested next steps. Furthermore, the
Teaching Children Mathematics / November 2004
Copyright © 2004 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.
Photograph by David J. Whitin; all rights reserved
children’s explanations of “bunching up” and what
one can “count” in calculating perimeter suggested
that many were developing an understanding of the
meaning of perimeter and area. Tricia, however,
courageously admitted that her results did not match
her predictions, and now she wanted to know why.
Spending more time exploring perimeter and area
might deepen students’ understanding of these
often-confused mathematical ideas.
Next, I wondered what kind of follow-up lesson
design would encourage the children to “think, quesTeaching Children Mathematics / November 2004
tion, solve problems, and discuss their ideas, strategies, and solutions” (NCTM 2000, p. 18). I wanted
the children to take responsibility for planning and
carrying out personally meaningful and challenging
investigations. I knew, however, that too much choice
could be overwhelming for the children and difficult
for me to manage. After discussing my choices with
a colleague, I decided on the following steps:
• Engage the children in examining another
example closely (first day).
181
• Take notes as children talk about, write about,
and draw their findings.
• Create problem-posing extensions from their
observations.
• Ask children to choose one question and predict
the results (a second day).
• Provide time for children to compare and discuss their findings (a portion of third day).
The Foundation: A
Common Experience
Many of the children’s initial comments focused
on the idea that the perimeter of “bunched up”
shapes were less than the perimeter of more linear
shapes with the same area. To highlight this idea, I
decided to focus only on rectangles with fixed
areas, instead of including other types of polygonal
shapes. (I realized that this decision would eliminate some potentially rich investigations.) I there-
fore gave the children centimeter grid paper and
the following directions:
1. Make rectangles that have an area of 12 square
centimeters.
2. Calculate the perimeter of each rectangle.
3. Analyze and explain your findings.
4. State a theory, hypothesis, or “wonder” (“I wonder if . . . ”).
The children found three possible rectangles:
1 × 12 (p = 26), 2 × 6 (p = 16), and 3 × 4 (p = 14)
(see fig. 2). Even though the children all calculated
the same results, their written explanations provided a variety of connections and raised new
considerations:
Lisa. You keep taking half away [half of the
square units] and putting them up against each
other. My theory is that when you have more than
Figure 1
The perimeters and area of the 12 pentominoes
Pentominoes with an area of five square units and a perimeter of twelve units:
One pentomino with an area of five square units and a perimeter of ten units:
182
Teaching Children Mathematics / November 2004
one line in the middle, you have less perimeter,
but when you have less lines, you have more
perimeter.
Troy. Area is how much you can hold in it.
Perimeter is the width all the way around it. The
perimeter got shorter because we bunched the
squares together.
Jeffrey. When there were more groups, the
perimeter got smaller.
Susannah. It reminds me of a person running
around a field and it’s showing you each and every
one of the steps you take. I predicted that the third
rectangle would be less than 16 because the rectangle is almost like a square.
Libby. They’re all even numbers [perimeters].
The children were developing a deeper understanding of perimeter and area by investigating
relationships involving the two for rectangles with
a fixed area. In past experiences with children, I
have often found that they confuse the terms
perimeter and area. Susannah’s definition of “running around a field and showing each and every
one of the steps” and Troy’s definition, “Area is
how much you can hold in it,” incorporated
“everyday, familiar language” (NCTM 2000, p.
63) that their classmates could relate to as well.
I devoted a portion of the next day to an oral
sharing session. I was surprised by the interest
that many children expressed in Libby’s idea. I
had thought it was obvious that all perimeters of
rectangles are even, but raising this topic for
exploration proved to be one of the most fruitful
activities of the next part of the children’s followup investigations. The children were now ready to
pursue problem-posing extensions.
Figure 2
Perimeters of rectangles with 12 square
units
p = 26
p = 16
p = 14
tions would make the problem-posing process
more explicit to them (Vygotsky 1978). I therefore
listed three observations on chart paper, leaving
room below each. As the lesson began, I showed
the list to the children and asked, “What questions
could we ask about each statement?” When completed, our chart looked like this:
1. The perimeters are all even numbers (26 – 16
– 14 units).
• What about other numbers [of square units]?
• Are perimeters always even?
2. The perimeter changes quickly, then slowly:
-10
Problem-Posing Extensions
In reviewing some of the students’ comments that
evening, I developed a short representative list of
observations. I wanted the list to encompass all the
children’s ideas in order to preserve their sense of
ownership, yet be short enough to be manageable
for both the children and me. I carefully retained
the children’s own descriptive language as I wrote
the list. Although the children’s observations were
statements, they suggested problem-posing extensions (Whitin 2004). Their statements identified
important attributes of the problem (Brown and
Walter 1990). The next step would be to pose questions for further investigation. I realized that
involving the children in framing extending quesTeaching Children Mathematics / November 2004
-2
26
16
14.
• What about other numbers [of square units]?
• Is this always true?
3. The more the rectangles get bunched up, the
less the perimeter is.
• What about other numbers [of square units]?
• Is this always true?
Next, I gave the children the following directions, which I had written on chart paper:
1. Choose a question for your investigation.
2. Record it in your journal.
3. Make a prediction and give a reason for this
prediction.
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Figure 3
The pattern that Jenna discovered
Array
1 × 12
2×6
3×4
Perimeter
26
> 10
16
>2
14
4. Investigate your problem.
A sampling of the children’s plans included the
following:
Tricia. I’m going to investigate if all perimeters
are even. I predict that they are all even because we
have tried it and even if the area is odd (5) you will
come out even.
Troy. My plan is to see how much the perimeter
drops when the square centimeters get bunched up.
I predict that the perimeter will be around 30 on a
1 × 18.
Lisa. I predict that 18 will drop just like 12 and
the last two numbers will have just a few numbers
in between them [number 2 above, using a fixed
area of 18].
Examining Odd and
Even Numbers
After making their plans, the children worked for
the remainder of the class period, and they discussed their findings on the following day. Several
children investigated whether perimeters of rectangles are always even. Will and Susannah used
examples for their “proof.” Will tried areas of 18
and 24. By studying the rectangles, he concluded
that odd numbers added to odd numbers equaled
even numbers. He explained:
The perimeter is always even because when you
add two odd numbers [for example, 1 + 1] you
get an even number, and when you add two even
numbers [24 + 24], you get an even number.
1 × 24, p = 50
2 × 12, p = 28
3 × 8, p = 22
4 × 6, p = 20
Yes, the perimeter is even! My prediction was
correct!!
184
Array
1 × 18
2×9
3×6
Perimeter
38
> 16
22
>4
18
Susannah investigated rectangles with an area
of 100 (for example, 50 × 2, 25 × 4) and found that
their perimeters were even. She wrote, “I found
that the 100s are even too [the perimeters]. I have a
theory that out of all the numbers in the world,
there are not many ones that you can make an odd
perimeter.” She claimed that to prove her conjecture she needed to test “all the numbers in the
world.” Until she did so, she was still doubtful. Her
doubt hinted at the need for a proof of the generalization.
Tricia’s argument better approached the generalization for adding odd numbers. She wrote, “If
you add even and odd two times, you might get an
even—like 6 × 3, you add 6 two times and 3 two
times, you get 18, and 18 is even. Like 6 + 3 = 9,
and then add 6 + 3 again and you get 9, and so 9 +
9 is 18.” Tricia was using the commutative and
associative laws to show how the arrangement of
odd and even numbers would still yield an even
sum: 6 + 6 + 3 + 3 = (6 + 3) + (6 + 3).
Jeffrey had another question about adding odd
and even numbers. He noticed that because we
were working with rectangles, odd numbers were
always added in pairs (3 + 3). He then mused,
“What if you add two odd numbers that are not the
same? Three plus five is even.” Jeffrey’s comment
helped the class consider that in all cases, odd plus
odd equals even. By bringing all these children’s
ideas to the class sharing session, the students were
better able to address the generalizations for
adding odd and even numbers.
Exploring Other Questions
Other children pursued different questions on the
problem-posing list. Jenna and Lisa were interested in looking at the numerical differences
among perimeters (“Is it always true that perimeters change quickly, then slowly?”). Jenna noticed
that the rectangles for an area of 12 and an area of
18 had “the same shape, but the 18s are bigger”
(they have more area). In both instances, the
Teaching Children Mathematics / November 2004
biggest drop in perimeter occurred when the long
array was divided in half. The third rectangle in
both examples showed a less dramatic change.
Even though she did not reach a mathematical generalization, she discovered a pattern that occurred
across several rectangles. Jenna helped show that
after this initial halving, the perimeter did not
change as dramatically (see fig. 3).
Lisa, on the other hand, was most intrigued with
the 6 × 3 array. She noticed that the number of linear units (p = 18) was the same as the number of
square units (a = 18). Her writing opened a new
question to pursue: “I wonder if any other number
[rectangle] has the same number for perimeter and
area.” (Of course, it is important to clarify that the
same “number” applies to two different units of
measure.) Lisa’s speculation demonstrated once
again that raising new questions is part of the problem-posing cycle.
Troy and Jessie were interested in the relationship between the shape of a rectangle and its perimeter (“Is it always true that the more rectangles get
bunched up, the less the perimeter is?”). They each
found that an area of 18 yielded three different rectangles with decreasing perimeters. Troy explained
his discovery by writing, “I noticed that the more
bunched up the square cm get, the littler the perimeter gets because all the square cm are inside and
you aren’t able to count part of their perimeter” (see
fig. 4). Jessie used the square as a reference point for
her description: “The closer a shape is to a square,
the smaller the perimeter.” These insights are important because they build a foundation for discovering
that a square is the rectangle that yields the least
perimeter for any given area.
Kara also chose the “bunched up” question, but
her observations addressed an issue that I had taken
for granted. She made one 1 × 18 rectangle, and
one 3 × 6, but decided to show the 2 × 9 twice—
one oriented vertically on the page, and the other
horizontally. She carefully marked each segment of
the 2 × 9 arrays in order to calculate the perimeter
of 22, and she wrote consecutive numbers in the
squares to show the area of 18. Then she wrote, “I
noticed that when I put 2 × 9 two times but in different directions, they might have the same perimeter. So I counted it and they were the same!” Kara’s
writing gave me a window into her thinking. I had
assumed that all the children understood that orientation does not affect size and shape, yet apparently Kara had not been so sure. I realized that as
teachers, we cannot assume what is clear to children. Problem posing allows children to test out
Teaching Children Mathematics / November 2004
Figure 4
Troy’s work
what is intriguing to them. In this context, Kara had
the freedom to make sense of this fundamentally
important idea for herself.
In reflecting on the children’s work and the discussions of odd and even numbers, I considered the
benefits of this kind of problem-posing investigation. This work involving perimeter and area incorporated several mathematical ideas: factors and
products, odd and even numbers, mathematical patterns, properties of rectangles, spatial orientation,
and, of course, the skill of calculating perimeter and
area. These choices gave students of various abilities the opportunity to challenge themselves by pursuing what was most intriguing to them.
Reflecting on the Experience
This extended experience with perimeter and area
helped me clarify several aspects of my role as a
teacher. First, that the children wrote about their
observations was important. Providing some openended prompts such as “What do you notice?” and
“What do you predict?” encouraged these fourth
graders to identify a range of attributes that
described relationships involving perimeter and
area for different types of rectangles. Their oral and
written statements gave me an opportunity to
assess their understandings and doubts, as well as
their interests. Their observations, such as
“Perimeters are even,” raised issues that I had not
envisioned and guided my curricular decisions.
Second, this experience highlighted the role of
185
teachers in planning, organizing, and developing
tasks with their students. It is teachers who must
decide “what aspects of a task to highlight, how to
organize and orchestrate the work of the students,
what questions to ask to challenge those with varied levels of expertise, and how to support students
without taking over the process of thinking for
them and thus eliminating the challenge” (NCTM
2000, p. 19). The children’s initial conversation
indicated that continued work with perimeter and
area would be worthwhile. They showed interest in
the topic, and their observations highlighted several interconnected mathematical ideas. Engaging
them in the systematic process of turning their
observations into problems could strengthen the
disposition to “analyze situations carefully in
mathematical terms and to pose problems based on
situations they see” (NCTM 2000, p. 53).
This experience also illustrated that by highlighting only certain attributes of the problem, I eliminated other potentially rich investigations. In the
opening conversation, Tricia expressed a keen interest in the fact that most of the different shapes (with
an area of five square units) had the same perimeter.
186
Jessie’s observation that six square units would
“probably be different” also suggested work with
hexominoes or even other types of polygonal
shapes. I was aware that by having the children work
only with rectangles, I would sacrifice these excellent ideas, but I felt unable to manage too many
choices. The explorations that the class did pursue
within these parameters encouraged me to look for
future opportunities in which I would be able to support an even broader range of student investigations.
References
Brown, Stephen, and Marion Walter. The Art of Problem
Posing. Hillsdale, N.J.: Lawrence Erlbaum, 1990.
National Council of Teachers of Mathematics (NCTM).
Principles and Standards for School Mathematics.
Reston, Va.: NCTM, 2000.
Vygotsky, Lev. Mind in Society. Cambridge, Mass.: Harvard University Press, 1978.
Whitin, David J. “Building a Mathematical Community
through Problem Posing.” In Perspectives on the
Teaching of Mathematics, 66th Yearbook of the
National Council of Teachers of Mathematics
(NCTM), edited by Rheta N. Rubenstein, pp.
129–40. Reston, Va.: NCTM, 2004. ▲
Teaching Children Mathematics / November 2004