Promoting
Algebraic Reasoning
Using Students’
Thinking
JOYCE BISHOP, [email protected], teaches at Eastern Illinois
J O Y C E W. B I S H O P, A L B E R T D. O T T O,
A N D C H E R Y L A. L U B I N S K I
508
University, Charleston, IL 61920. She is interested in
teacher preparation and the development of algebraic reasoning in grades K–8. ALBERT OTTO, [email protected], and
CHERYL LUBINSKI, [email protected], are colleagues at
Illinois State University, Normal, IL 61790-4520. Otto is
interested in the development of algebraic reasoning in students; Lubinski’s research focuses on instructional decisionmaking in mathematics.
MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
Copyright © 2001 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.
One of the authors taught the class while the
other two observed and occasionally questioned
students or made suggestions about the direction
of the lesson. The lesson was videotaped to provide
an accurate account of what occurred.
The Lesson
WE BEGAN THE LESSON WITH A REVIEW OF
perimeter. Students were told that each side of a
pattern block has a length of 1 unit. Students
viewed the pattern formed by the blue rhombus
pieces and readily volunteered appropriate answers
for the perimeters of the first four shapes. They
were then asked to explain how they determined
their answers, as the following dialogue illustrates:
Teacher. OK, how would you count the second
one? [See fig. 1.] I want somebody to come up and
show me how you would do that.
Jason. You just count these two sides. Two
[pointing to the top two sides of the rhombuses];
and then these two, which would be four [pointing
to the bottom two sides of the rhombuses]; and
then these two sides would be six [pointing individually to the two ends of the set of two rhombuses]
[see fig. 2].
Fig. 1 Sequence of geometric shapes formed using rhombus
regions
PHOTOGRAPH BY BRIAN BRAYE, ILLINOIS STATE UNIVERSITY FTSS; ALL RIGHTS RESERVED
PHOTOGRAPH BY BRIAN BRAYE, ILLINOIS STATE UNIVERSITY FTSS; ALL RIGHTS RESERVED
T
HE CHANGING APPLICATIONS OF
mathematics have contributed to a shift
from the perception that mathematics is a
fixed body of arbitrary rules to the realization that the discipline is “a vigorous active science
of patterns” (National Research Council 1989, p. 13).
NCTM’s Curriculum and Evaluation Standards for
School Mathematics (1989) recommends using patterns to promote mathematical understanding and,
in particular, algebraic reasoning. A number of
other mathematics education reform documents
make similar recommendations (e.g., AAAS [1989];
National Research Council [1990]; Steen [1990];
NCTM [2000]). Researchers have begun to identify
different approaches that students use to reason
about patterns (Bishop 1997; MacGregor and
Stacey 1993; Orton and Orton 1996; Stacey 1989).
Research also shows that using students’ thinking
about patterns can help them develop a better understanding of mathematical concepts and the representations that reflect those concepts (Carey
1992; Fennema, Carpenter, and Peterson 1989).
This article illustrates how students’ thinking about
geometric patterns can be used to help them develop algebraic reasoning and to make sense of
mathematical notation and symbols.
To better understand the process, we conducted
an investigation with a seventh-grade-mathematics
class to find the perimeters of sequential geometric
shapes formed with pattern blocks. The sequence
was constructed by adding one blue rhombus region at a time to form a zigzag shape (see fig. 1).
We used translucent pattern blocks with an overhead projector to allow the entire class to view the
sequence. This format also allowed students to use
the overhead projector to share their thinking with
others in the class.
Posing the initial question to the students was an
essential component of the investigation. We first
asked students to tell us the perimeter of each of
the first four shapes in the sequence shown on the
overhead display and to explain their strategies for
finding the perimeters. The students were then
asked to form an arithmetic expression that reflected their reasoning. We then extended the activity to shapes that were not displayed, including a
situation in which the number of shapes was not
specified and, therefore, required using a variable.
Finally, we presented the number of units in a
perimeter and asked the students to find the number of blue rhombus regions that would have this
perimeter. One major goal of the investigation was
to encourage students to base their reasoning on
the physical features of the pattern rather than on a
series of numbers, demonstrating that algebraic expressions could represent a physical situation.
Fig. 2 Jason explains his reasoning.
VOL. 6, NO. 9 . MAY 2001
509
At this point, we used symbolic representation to
show the mathematics in the counting process and
to lay the foundation for using variables to represent
the more general situation. The lesson continued:
Teacher. Did you see how he counted? What he
did first, second, and third? How would I write how
he counted? Say it one more time. I want to represent what he said with symbols.
Jason. You count the two.
Teacher. Two [writes the number 2 on the chalkboard]. OK.
Jason. Two tops and then the two bottoms
[teacher writes a second 2 on the chalkboard] . . .
then the two sides.
Teacher. [Writes two 1’s on the chalkboard.]
What would I put up there to show what he did to
get his answer?
Ryan. Maybe 2 × 2 + 1 or + 2.
Teacher. Do you think that what he said was represented by 2 × 2? Did he say two 2s?
Ryan. Yeah.
Teacher. Did you say two 2s [pointing to Jason]?
Jason. I said count the two sides.
Teacher. Two, two, one, one. How would I represent that symbolically?
Stephanie. Maybe by putting a plus sign in between both 2s, and then a plus sign in between
those numbers.
Teacher. [Writes 2 + 2 + 1 + 1.] OK. Do you see
the difference? The difference is that he didn’t say
two 2s. He said, “Two plus two more.” He didn’t say
“plus,” but “and two more and one more and one
more.” And “plus” represents what he said. It’s different than saying two 2s [pointing to the two sides
of the shape on the overhead], which would be 2 × 2
and 1 and 1 [2 × 2 + 1 + 1].
Clarifying this point was important to show the
students that although the numerical expressions
have the same value, the representation that used
multiplication did not connect with the reasoning that
Jason expressed. In other words, both the teacher
and the others in the class needed to listen carefully
to what the student said and represent precisely what
he explained. As teachers, we sometimes make the
mistake of simplifying or reading our own thinking
into students’ explanations and, thus, lose the reasoning that the students are expressing. An alternative
510
MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
approach could have been to have the student write
on the chalkboard his own arithmetic expression for
what he explained. This approach encourages a
strong connection between students’ reasoning and
their own representations of their reasoning.
The teacher continued by asking for other ways
to count the perimeter of the second shape. David
counted the two top sides, the right end, the two
bottom sides, and the left end. When asked what
numbers would be used to show his counting, he
responded as follows:
David. Two, one. . . .
Teacher. [Writes 2 and hesitates.] Plus 1?
David. Yeah . . . [teacher writes + 1] . . . plus one
only because it’s the very last one you’d put there.
Teacher. And now what else would you do?
David. You’d count your other two [pointing to
the bottom sides of the rhombuses] and then + 1.
Teacher. OK, so for the two [and] one, I do 2 + 1,
and then how do I get the rest of the perimeter?
David. 2 + 1 and then 2 + 1.
Another student, Ryan, explained, “Since there’s
two on the top and two on the bottom, you take 2 × 2,
and then you add 2 for each of the sides.” Barbara
asked, “Can you count it as 3s? Then she showed her
method by counting the number of sides contributed
by each rhombus shape to the perimeter. Note that
each counting explanation was followed by an expression to represent the strategy used by the student, and the teacher was again careful to write the
arithmetic expression that represented the student’s
reasoning. For Barbara, the teacher wrote 3 + 3 on
the chalkboard; for David, she wrote 2 + 1 + 2 + 1.
For the third and fourth rhombus shapes (see
fig. 1), some students used strategies that extended previous counting strategies, for example:
Tara. You can take these three [pointing to the
sides of the rhombus on the right] right here and
these two [pointing to the top and bottom of the
middle rhombus], and then these three right here
[pointing to the sides of the rhombus on the left].
Teacher. [Writes 3 + 2 + 3.] Does that represent
what you did? [Student nods.]
Tara’s approach extended the strategy used by Barbara of “using 3s.” Greg’s approach extended the
strategy used by Jason, as follows:
Greg. You could take, since there’s three pieces,
3 × 2 + 2, for the ends.
Teacher. Come and show me what your 3 describes.
Greg. Three pieces [as he separates the three
rhombuses], so 3 × 2 because here’s two sides [referring to the top and bottom of each rhombus],
and then + 2 for the two ends.
Using someone else’s strategy to develop your own
approach to another problem is an important component of both problem solving and developing generalizations, which enhances algebraic reasoning.
Denise tried a different approach: “I start[ed] on
the left [pointing to the top of the left-most rhombus]
and [counted] one, two, three, four, five, six, seven,
eight [pointing to each side in a clockwise direction
around the set of three rhombuses].” This directmodeling approach was represented arithmetically
by the expression 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1. Explaining the approach was important to encourage students to become involved and to be flexible in their
reasoning strategies when counting the number of
sides in the perimeter. The teacher made a conscious instructional decision not to pursue further
the direct-modeling strategy, however, because it
does not lend itself to generalization.
For the fourth rhombus shape, students again offered explanations for their counting strategies. For
example, Jeff said, “There’s three there [pointing to
the sides of the left-most rhombus] and three there
[pointing to the right-most rhombus], so it would
be 3 × 2, and then, + 2 × 2 [pointing to the sides of
the top and bottom of the two middle rhombuses in
the set of four].”
After this explanation, the teacher began to have
the students find perimeters of shapes that were
not modeled on the overhead projector. Many students simply extended the strategies that they had
used for the first four shapes in the sequence. Careful probing by the teacher was required to ensure
that their strategies connected with the shapes
rather than with their descriptions of the numerical
patterns. For example, when the teacher asked
what the perimeter of the eleventh shape would be,
Brenden reasoned by using the perimeter of the
fourth shape, which is 10, and added 2 units for
each of the seven additional rhombuses:
Brenden. You could take 7 × 2, since you’re always adding 2 for each one you put on, and that
would be 14.
Teacher. Why did you multiply 7 × 2?
Brenden. Because you’re adding seven blocks,
and each block adds 2 to your total.
Brenden identified a recursive relationship between
successive perimeters, namely, that each perimeter is
2 greater than the previous one. The student is looking
at the relationships among the numbers describing
the perimeter rather than describing how the perimeter can be determined from the figure itself. Although
this recursive reasoning is correct, it does not easily
lead to a general symbolic expression for the perimeter in terms of the number of rhombuses. Nevertheless, this recursive approach is important and could
easily be explored further with calculators or spreadsheets. The teacher made an instructional decision not
to pursue this line of reasoning, though, in the hope of
eliciting more generalized reasoning. She continued
by asking, “What do the rest of you think about that?”
Tom. I think there’s a pattern where you times
however many there are by 2 and you add 2 to get
the perimeter.
VOL. 6, NO. 9 . MAY 2001
511
Teacher. Why do you think that works?
Tom. Because if you’re looking at the 4, you
times . . . if the perimeter is 10, so if you times 4 by
2, then add 2, it’d be 10, and if it’s 5, it’s 12. So if you
times 5 by 2, it would be 10 and plus 2 would be 12.
Teacher. What are you looking at when you think
of that, multiplying by 2 and adding 2?
Tom. I’m looking at the numbers [referring to
the number of rhombuses in a shape] . . . how many
there are and, like, how many sides there are after
so many numbers.
To further probe their reasoning, the teacher asked
the students what the perimeter would be if the
shape had one hundred rhombus regions. Students
suggested that 2 × 100 + 2 would be the answer and,
with additional probing, generated an explanation
that connected the expression with the shape. An
example follows:
Amanda. If there’s one hundred strings . . . I’m
looking at the bottom one right there . . . there’s
two . . . think of it as two lines going, and you could
only show two in there, so each diamond is two
sides and that’s where you get . . . the × 2. If there’s
one hundred of them, you times it by 2 because 100
is one diamond, but then the two sides, and then
you add 2 for the ends.
Perimeter: 3n + 2
Perimeter: 2n + 3
Perimeter: 4n
Fig. 3 Shapes for investigating perimeters
512
MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
Teacher. Ah, now I see what you are saying. You
multiply by 2 because each of these shows two
pieces, and then you add 2; would you say that again?
Amanda. For the sides.
Teacher. For which sides?
Amanda. The two ends. . . .
Note that this discussion lays a foundation for
many students to develop strategies for other problems. For example, the teacher next asked, “How
would you find the perimeter of a shape no matter
what number of rhombus regions you have, where
n is the number of rhombus regions?” Many students used the previously developed strategies to
build an algebraic expression for the perimeter of
the nth shape, for example:
Brian. Take the number of shapes, like, say, you
have fifty or something like that, you do that times 2.
Teacher. [Writes 2n on the chalkboard.] That
will give you what? What did you count?
Brian. That will give you the top and bottom.
Teacher. The top and the bottom.
Brian. And then you do plus 2, which will be like
the two sides.
Teacher. . . . plus 2, which would be like the two
sides [writes 2n + 2 on the chalkboard]. There’s one
way to count it. How would you count it using 3s?
Cheryl. You could do 3 × 2 for each of the ends.
Teacher. OK, 3 × 2 [writes 3 × 2 on the chalkboard] and, now, I’m going to go 2 × 3 because it’s
two 3s [erases 3 × 2 and writes 2 × 3]. And that
would get you your end pieces. [Note that the
teacher caught herself in changing what she wrote
because her original expression did not represent
the thinking of the student.]
Cheryl. And then you could just count by 2 for
the middle pieces.
Teacher. How many middle pieces will there be
in relation to n? If n is the number of pieces I put
out there, how many middle pieces do you have?
It’s not n anymore.
Jen. I think n – 2.
Teacher. Why?
Cheryl. Because you have already used the two ends.
Teacher. You have already used the two ends. Do
you see that? So your number, the number of
pieces, is going to be 2 less than all of them [writing
n – 2 on the chalkboard and pointing to the expression]. What does this count?
Joyce. The middle pieces.
To give students another opportunity to think
about the relationships in the figures instead of the
calculations, the teacher showed the class the expression 2(n + 1). Students were asked whether
they could imagine how the student who wrote the
expression was thinking about the perimeter of the
shapes. Because this expression did not represent
any of the previous strategies, it became an assessment tool. The students were being challenged to
make sense of a new algebraic expression in relation to the geometric shapes.
In response to the teacher’s question, Jennifer
initially said, “The 2 is next to the parentheses, so
do the addition first and then multiply. It tells us
what order.” The discussion focused on interpreting the calculations indicated by the expression.
Only when the teacher rephrased the question to
“What are we counting?” did students begin to look
at the parts of the shape.
Tom. Because if you subtract first, you take off
the end pieces, so all you have to do is take that and
divide by half.
Teacher. OK, so the subtracting 2 is taking off
the two end pieces, and why, then, do you divide by
2 after that?
Tom. Because each one of them is worth 2, so in
order to find each one, you divide by 2.
Nick. First you get the top and one side.
Teacher. Then why do we multiply by 2?
Rachel. To get the bottom and the other side.
THIS CLASS SESSION SUGGESTS THAT AN INITIAL
Rachel recognized that n + 1 can represent the
top edges of all the rhombus regions and one end;
multiplying by 2 accounts for the bottom edges of
the shape and the other end.
Next, we reversed the question, asking students
how many rhombuses a shape would have if the
perimeter equaled 62. The students in this seventhgrade classroom had not studied formal algebra
and so had to use strategies other than solving linear equations.
Teacher. Suppose the perimeter is 62. How many
rhombi would we have? [After some initial discussion, a student volunteers.]
Tom. I think that, um, it’s 30 because of you take
62, subtract 2, and then divide by 2, because you’d
be doing the opposite of what you did to get it,
you’d get 30.
Teacher. OK, you say 30, because if you subtract
2 and divide by 2, you’d be doing the opposite of
what you get. Why does it work to do that?
Tom. ’Cause if you . . . if you times something
and you want to check it, you want to divide the answer by whatever number. . . like, do the reverse,
you do the opposite of what you did to get it.
Here we see an example of a student’s solving a
linear equation, 2n + 2 = 62, informally by recognizing the order in which the operations were performed. Next, the students needed to connect this
procedure with the shape having a perimeter of 62.
The teacher continued with the discussion:
Teacher. If we’re looking at this picture, can you
say you want to subtract first? Why would you want
to subtract first?
This discussion helped connect the numerical explanation with the physical setting so that the numerical calculations took on meaning from the original problem.
Discussion
emphasis on methods of counting, followed by the
recording of an accurate representation of each
counting method, helps students connect symbolic
representation with their counting actions. When
students had the opportunity to demonstrate their
counting methods, many more remained engaged
in the discussion.
The teacher can model the method for recording
the representations first, then have students write
their own expressions. At every stage, the teacher
should emphasize the need to have the expression exactly fit the counting strategy and challenge students
when discrepancies arise between reasoning and representations. Students should be encouraged to write
as many representations as they can. They should
explain how their expressions connect with precise
counting strategies when the resulting expressions
are shared and discussed. Using an algebraic expression for a perimeter that has not been previously
brought up by the students enables the teacher to assess the development of their algebraic reasoning.
The use of pattern-block shapes puts powerful
mathematical concepts in reach of students with a
wide range of ability levels. The classroom activities
described in this article provide a context for helping
students construct meaning for mathematical expressions by relating them to the physical action of
counting perimeter. These activities also offer a format for implementing the process standards (NCTM
2000), because students develop problem-solving
strategies for finding perimeter, communicate their
strategies in both words and symbols, reason about
the relationships between counting strategies and
symbolic representations, and connect the counting
methods of arithmetic with the generalization of algebra. Using students’ thinking to investigate the
perimeters of pattern-block shapes presented a
solid opportunity to cultivate algebraic reasoning.
(See fig. 3 for additional shapes.)
VOL. 6, NO. 9 . MAY 2001
513
References
American Association for the Advancement of Science
(AAAS). Science for All Americans: A Project 2061
Report on the Literacy Goals in Science, Mathematics,
and Technology. Washington, D.C.: AAAS, 1989.
Bishop, Joyce W. “Middle School Students’
Understanding of Mathematical Patterns and Their
Symbolic Representations.” Ph.D. diss., Illinois State
University, 1997.
Carey, Deborah A. “Research into Practice: Students’
Use of Symbols.” Arithmetic Teacher 40 (November
1992): 184–86.
Fennema, Elizabeth, Thomas P. Carpenter, and Penny
Peterson. “Teachers’ Decision Making and
Cognitively Guided Instruction: A New Paradigm for
Curriculum Development.” In School Mathematics:
The Challenge to Change, edited by N. F. Ellerton and
M. A. (Ken) Clements, pp. 174–87. Geelong, Victoria,
Australia: Australian University Press, 1989.
MacGregor, Mollie, and Kaye Stacey. “Seeing a Pattern
and Writing a Rule.” In Proceedings of the Seventeenth
International Conference for the Psychology of
Mathematics Education, vol. 1, edited by I.
Hirayabashi, N. Nohda, K. Shigematsu, and F.
–I Lin, pp. 181–88. Tsukuba, Japan:
International Group for the Psychology of
Mathematics Education, 1993.
National Council of Teachers of Mathematics
(NCTM). Curriculum and Evaluation
Standards for School Mathematics. Reston,
Va.: NCTM, 1989.
———. Principles and Standards for School
Mathematics. Reston, Va.: NCTM, 2000.
National Research Council. Reshaping School
Mathematics: A Philosophy and Framework for
Curriculum. Washington, D.C.: National
Academy Press, 1990.
Orton, Anthony, and Jean Orton. “Students’
You
Perception and Use of Pattern and
have probably
Generalization.” In Proceedings of the
Eighteenth Conference of the International
spent much time and efGroup for the Psychology of Mathematics
fort encouraging your stuEducation, vol. 3, edited by J. P. da Ponte and
dent teachers to write about
J. F. Matos, pp. 404–14. Lisbon: University of
their thinking. The Editorial Panel
Lisbon, 1994.
of Mathematics Teaching in the MidStacey, Kaye. “Finding and Using Patterns in
dle School invites you, a teacherLinear Generalizing Problems.” Educational
educator who specializes in middleStudies in Mathematics 20 (1989) 147–64.
grades mathematics, to do the same and
Steen, Lynn A. “Pattern.” In On the Shoulders of
share your ideas with your colleagues
Giants: New Approaches to Numeracy, edited
by writing for the journal. Teacherby Lynn A. Steen. Washington, D.C.: National
educators have a special role to play in
Academy Press, 1990.
Calling All
TeacherEducators
To find out more
about writing for
the journal, contact
Kathleen Lay at
[email protected] and
ask for the “MTMS
Writer’s Packet.” If
helping to translate the theory of good
you have a manuscript ready to go,
practice into pedagogical ideas that
send it directly to
teachers can employ in their classMathematics Teaching
rooms. Fur thermore, many
in the Middle School,
teacher-educators read the
NCTM, 1906 Association
journal to get new ideas
Drive, Reston, VA
for their teaching.
20191-9988. All submissions
must include five doublespaced copies of the manuscript.
514
MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
The preparation of this article was supported in
part by a grant from the National Science
Foundation (grant no. DUE-9250044) on the
“Influences on Preservice Teachers’ Instructional
Decision Making.” Any opinions expressed herein
are those of the authors and do not necessarily
reflect the views of the National Science
Foundation. We acknowledge the assistance of
Barb Metroff in the preparation of this article. C