© Carolyn A.
Maher & Regina D. Kiczek
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We have engaged in a research program of teaching
experiments designed to create classroom environments within
which sense-making is a cultural norm for over a decade. In
the course of these experiments, a particularly striking
outcome of the culture has been the emergence of
argumentation, justification, and proof-making in students'
discourse. Such processes have developed in the context of
coherent strands of mathematics -- combinatorics,
probability, and mathematical modeling. We have been
particularly interested in observing the development of
representations and explanations provided by the students,
as well as the reasoning offered to support their ideas. The
arguments, justifications, and proof making contributions of
a focus group of students who have been participants in a
study, now in its twelfth year, will be traced in this
paper. This paper describes the building of mathematical
ideas over a seven-year period, fifth through eleventh
grade. Four episodes that come from videotapes of
problem-solving sessions in grades 5, 10, 11, and 12 are
reported. The "Pizza Problem" , a task originally explored
by the children in the fifth grade (10-11 year olds), became
an important metaphor that guided students' later
justification of the addition rule of Pascal's triangle and
the generalization of that rule.
Episode 1: Grade 5,
4/2/93. Romina, Jeff, Brian and Ankur worked
as a group on the Pizza Problem as fifth graders and used
a variety of strategies and representations to produce
the sixteen combinations. These included a partial tree
diagram, lists, and an organization that systematically
controlled for variables. Michael worked in parallel with
the others and represented the various pizzas by drawing
circles and labeled each "pizza" with its toppings. All
of the students created codes using letters or
abbreviations to represent the four toppings (for
example, pe = pepperoni; m = mushrooms) and all decided
to code for a pizza with no toppings (pl = plain or c =
cheese). The pizzas were categorized as "whole" (plain
and one-topping pizzas) and "mixed" (two or more
toppings). The students found all sixteen pizzas,
justifying their solution by the way they organized their
results.
Episode 2: Grade 10,
12/12/97. Four years later, the same five
students were again given the "Pizza Problem." They
talked aloud about combinations of toppings and of the
patterns they were observing, using a code of letters to
represent the toppings. Some of the students shared their
lists of combinations; they switched their notation,
using the numerals one through five, and concluded that
if five toppings were available, thirty different pizzas
could be made with at least one topping, plus one plain
cheese pizza, for a total of thirty-one. Meanwhile
Michael worked alone for at least fifteen minutes quietly
developing his own solution. He found there were
thirty-two pizzas when choosing from five toppings and
challenged the solution of the other students. He used a
representation based on a binary-coding scheme to account
for all possibilities, and explained what the zeros and
ones meant in his representation and how they were used
to write base ten numbers in base two notation. He then
related his coding scheme to the pizza problem,
summarizing his conclusion as follows:
Okay, here's what I think. You know like a
binary system we learned a while ago? Like with the
ones and zeros - binary, right? The ones would mean a
topping; zero means no topping. So if you had a
four-topping pizza, you have four different places -
in the binary system, you'd have - the first one would
be just one. The second one would be that [He
wrote 10]; that's the second number up. You
remember what that was? This was like two, and this
was three [He wrote 11]
.Well, you get, I
think&emdash;I have a thing in my head. It works out
in my head
You've got four toppings. This is
like four places of the binary system. It all equals
up to fifteen. That's the answer for four
toppings
So you go from this number [He
pointed to 0001], which is in the binary system,
it's one, to this number [He pointed to 1111],
which is fifteen, and that's how many toppings you
have. There's fifteen different combinations of ones
and zeros if you have four different places.
I
don't know how to explain it, but it works out. That's
in my head&emdash;these weird things going on in my
head. And if you have an extra topping, you just add
an extra place and that would be sixteen, that would
be thirty-one.
Michael's representation using binary numbers did not
include the representation for a plain cheese pizza [0 0
0 0]. He later corrected for this by adding one to the
fifteen combinations (for making pizzas when selecting from
four toppings) and to the thirty-one combinations (for
making pizzas with five toppings available), thus accounting
for all possibilities. Jeff interjected: "And then you add
the cheese?", to which Michael acknowledged: "Plus the
cheese would be thirty-two." With his classmates, Michael
presented his binary coding scheme to the
teacher/researcher, saying, "This is the way I interpret it
into the pizza problem." When the teacher/researcher probed
further about Michael's solution, the other students joined
Michael in his justification, pointing out, for example that
the difference between one, zero, zero, zero [1 0 0
0] and zero, one, zero, zero [0 1 0 0] would
represent "the difference between an onion pizza and a
pepperoni pizza". Then, the five students considered the
case where ten toppings were available and recognized that a
string containing all zeros represented a plain cheese
pizza. When asked to generalize to the case of n toppings,
after a few minutes, determined that there would be 2n
different pizzas when there are n topping choices.
Episode 3:
Grade 11, 12/14/98.
One year later, in a task-based interview, Michael was
invited to explain his binary representation with respect
to pizzas with three toppings. In this interview, Michael
referred to the list of binary numbers he had written,
Michael explained that for pizzas, 000 was the code for
no toppings, while 111 was the code for a pizza with
"everything on it." When asked if he could describe the
distribution of kinds of toppings, Michael referred to
the rows of Pascal's triangle. He demonstrated the
addition rule by using the pizza problem as a metaphor.
Michael was then invited to present his ideas in writing.
In a subsequent e-mail message, Michael provided a
written justification for why the numbers in Pascal's
Triangle mapped to the variety of choices for making
pizzas, and illustrated this by moving from pizzas with
four toppings to pizzas with five.
Episode 4: Grade 11,
5/12/99. In a subsequent after-school session,
when invited to explain his ideas to the other four
students in the group, Michael gave an argument for
moving from pizzas with n toppings to pizzas with n+1
toppings. When presented his ideas both informally and
formally, Michael treated the numbers which appeared both
in Pascal's Triangle and in the pizza problem as the
result of underlying structures which he demonstrated, in
the general case, were isomorphic.
Michael, Jeff, Romina, and Ankur created and verified the
meaning of the expression, nCr, proposed by another student,
Robert. They explained their ideas using two examples:
arranging people in a row and building towers when selecting
from plastic cubes available in two colors (Maher &
Martino, 1996). The students referred to their notation as
"choose." They were then asked to represent the entries in
the triangle using the "choose" notation. Michael went to
the chalkboard and described how this would be done.
Referring to the fourth row of the triangle, Michael used
the pizza example once again to indicate that all the one
topping pizzas that have a new topping added plus all the
two topping pizzas that don't get a new topping added equal
all the two topping pizzas when one more topping is
available. Pointing to 4C1 and 4C2 [written in vertical
notation format], he indicated that the entries above
combine to give 5C2, explaining as follows:
Wherever this guy [4C1] goes, he's gonna
get another topping 'cause he's moving this way
[indicates to the right], so this bottom one's
gonna change to 2. This guy's [4C2] not going
anywhere &endash; this bottom number stays the same. It's
gonna be 5 - the next one's gonna be 5 and it has to be a
2 [5C2]. You understand why you add?
Jeff joined Michael in the conversation and reiterated
that they were explaining the addition rule for Pascal's
Triangle, "using chooses to fill out the triangle." On the
chalkboard, Jeff sketched the nth row [using vertical
format for nC0 nC1 ... nCx ... nCn] and explained that
the row above this would be (n-1)C0 (n-1)C1 ... (n-1)Cx ...
(n-1)C(n-1). He wrote:
He then wrote the equation:
n
+ n =
n+1
x x+1
x+1
Jeff explained that the first expression was changing but
the second was not and Michael joined him to emphasize: "The
top number changes 'cause you have one more thing ... one
more topping." Jeff continued the explanation and
indicated:
That's [n+1] the increase in n and then
the x+1 - if you added another topping onto your whole -
Say we're doing pizzas, you add another topping onto it.
... [He & Michael start speaking at the same
time.] ... When you add another topping onto it ...
if it gets a topping, it goes up to the x+1. If it
doesn't get anything, it would stay the same.
Jeff commented to another student: "So that would be
their general addition rule, in this case."
The students responded by writing an equation for the
addition rule using factorial notation on the chalkboard. As
Michael wrote, Jeff commented: "You know how intimidating
this equation must be? You just pick up a book and look at
that!"
Conclusions
The pizza task became a powerful metaphor for explaining
the addition rule of Pascal's triangle in general terms.
Michael's coding scheme enabled him to list all
possibilities. On the basis of a systematic organization,
Michael built arguments, first informally for special cases
and later in greater generality. Michael worked alone before
he shared his coding scheme with others. When he did share
it, despite apologies ("I don't know how to explain it, but
it works."), Michael showed quite clearly how the code was
used to solve the given problem. While the other students
acknowledged its usefulness in later problem situations, the
code continued to be designated Michael's "binary thing."
Over time, Michael's application of the code was more
general. For example, when Michael first used his code for
pizzas, he made a chart with topping choices as the column
headings. In later sessions, when listing possibilities,
Michael omitted column labels for a given, particular
situation. Instead, he determined how many places needed to
be filled and then listed the corresponding binary numbers
sequentially, beginning with zero. This explicit list of
numbers, without column labels, enabled him to track all
possibilities easily.
These students, both as fourth and tenth
graders, provided a justification for their solution that
took the form of a proof by cases (Muter and Maher, 1998;
Kiczek and Maher, 1998; Muter, 1999). However, their
representations and notations became increasingly more
abstract and general over time. The properties of
combinations, for these students, grew from very concrete
images, such as towers (Maher & Speiser, 1997; Maher
& Martino, 1996a & b) and pizzas (Maher, 1993). Once
these properties emerged, as they did here in Pascal's
Triangle, they linked the prior images, towers and pizzas,
into a larger framework that connected quite readily to
other ideas in algebra and combinatorics. Michael's
representation, triggered by the need to find and justify a
particular solution, served as a tool, for him and the
others, to connect mathematical situations which they
explored for a number of years. The students proposed
thoughtful and strong arguments at a young age and built
upon those ideas in later years. Their representations were
modified over time, indicating the use of symbols to
represent the objects. While the structure of the arguments
were durable over time, the particular representational
systems became increasingly more elegant and powerful.
References
Kiczek R. D., Maher C. A. (1998) Tracing
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Berenson et al. (Eds.) Proceedings of the twentieth
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International Group for the Psychology of Mathematics
Education, Raleigh: North Carolina, 1, 377-382. ERIC
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Maher C. A. (1993) "Children's Construction of
Mathematical Ideas." Invited plenary for the Sixteenth
Annual Conference of the Mathematics Education Research
Group of Australasia (MERGA), Brisbane, Australasia,
July, 1993.
Maher C.A., Speiser M. (1997) How far can you
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Maher C. A., Martino A. M. (1996a) The
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Maher C. A., Martino A. (1996b) Young children
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Muter E. M., Maher C. A. (1998) Recognizing
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