On blocks stairs and beyond

On Blocks,
Stairs, and
Beyond
Learning about
the signiicance of
representations
Laurie h. rubel and Betina a. Zolkower

T

he National Council of Teachers of
Mathematics (2000) recommends that
students at all grade levels be provided
with instructional programs that enable
them to “create and use representations
to organize, record, and communicate mathematical
ideas; select, apply, and translate among mathematical representations to solve problems; and use representations to model and interpret physical, social,
and mathematical phenomena” (p. 67). This article
describes a particular classroom activity used to highlight the significance of mathematical representations.
As mathematics teacher educators, we pose the
question: How can we support teachers in thinking

about the significance of representations in mathematics? In our teacher education courses, we do
so through cyclic processes of solving, studying, discussing, and reflecting on nonroutine mathematics
problems, with a focus on different kinds of strategies
and representations. This article shares two of these
nonroutine mathematical problems, describes four
solutions proposed by a class of beginning teachers,
and concludes with a discussion that focuses on classroom implications.

THE MATHEMATICAL TASKS
The September 2006 issue of Mathematics Teacher
contains a letter to the editor by Mark Engerman
340 MatheMatics teacher | Vol. 101, No. 5 • December 2007/January 2008
Copyright © 2007 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.

(2006) that describes a stair climbing problem and
its connection to the Fibonacci sequence as well
as to Pascal’s triangle. We present a version of this
same stair climbing problem along with an associated blocks problem (Leikin and Winicki-Landman
2001), with the goal of describing how a comparative study of these two problems can motivate

learning about the significance of representations.
We suggest that readers try to solve each of the following problems before reading further.
Staircase Problem: Suppose that a staircase comprises ten steps and that you can climb the stairs
one or two steps at a time. In how many different
ways can you climb these ten steps?

2 units
10 units
Ten of these 1 × 2 rectangles need to be
arranged in the 10 × 2 frame above.

Fig. 1 the Blocks problem

Blocks Problem: You have a 2-by-10 rectangular frame
[as shown in figure 1] as well as ten rectangular
blocks, each having the dimensions 2 by 1. Your
task is to fill the frame with the ten blocks so that no
blocks overlap and the frame is entirely filled. In how
many different ways can you arrange the ten blocks?
Following is an account of an experience from

Rubel’s teacher education course for middle and high
school mathematics teachers. These two problems
were posted on the chalkboard as participants arrived
at the session. Students were asked to form groups,
and each group had to choose one of the problems
to solve. Many participants immediately expressed
a strong preference for one problem or the other.
(Readers might want to notice whether they experienced this as well.) Each group then spent about
forty minutes working on their selected problem and
preparing a poster that summarized their work. The
instructor circulated around the room, paying careful
attention to each group’s emergent strategies. The
groups then reconvened as a whole class for the presentation and discussion of solutions.
The first group of participants selected to present their work on the Staircase problem and begin
the whole-class conversation was chosen because
they had adopted an inductive strategy. They found
the number of ways to climb a staircase comprised
of one stair, two stairs, and three stairs and then
extended these findings to find the number of
ways to climb a staircase comprising ten stairs.

They made a clear drawing that depicted how they
arrived at their combinatorial expressions for the
latter case (see fig. 2).
This group presented its specific findings for
climbing ten stairs as an organized list:
1. One could climb the stairs by taking a total of
ten steps, all single steps.
2. One could climb the stairs by taking a total of
nine steps (a combination of eight single steps

Fig. 2 the irst group’s work on the staircase problem

3.

4.

5.

6.

and one double step). This can be done in 9
ways (9C8).
One could climb the stars by taking a total of eight
steps (a combination of six single steps and two
double steps). This can be done in 28 ways (8C6).
One could climb the stairs by taking a total of
seven steps (a combination of four single steps
and three double steps). This can be done in 35
ways (7C4).
One could climb the stairs by taking a total of six
steps (a combination of two single steps and four
double steps). This can be done in 15 ways (6C2).
One could climb the stairs taking a total of five
steps (all double steps). In sum, there are 89
ways to climb the staircase.
Vol. 101, No. 5 • December 2007/January 2008 | MatheMatics teacher 341

Fig. 3 a vertical list display of the solution to the Blocks
problem

Fig. 4 student work on the Blocks problem relecting an
inductive strategy

When the first group stated this answer, a ripple
of curiosity and intrigue spread across the classroom. Although other groups that tried the Staircase problem might have been relieved to have a
matching answer, those that worked on the Blocks
problem seemed quite surprised to find that this
answer matched their answer as well.
The next group of participants selected to present their solution had worked on the Blocks problem. They too displayed their results as a vertical
list (see fig. 3):

class then was invited to consider why the two lists
corresponded. After a few moments, a participant
explained that taking a single step in the Staircase
problem corresponds to placing a vertical block in
the Blocks problem. Similarly, a double step in the
Staircase problem corresponds to a pair of horizontal
blocks in the Blocks problem. At this point, many
participants seemed surprised by the isomorphism
between the two problems, especially given that most

of them had gravitated strongly toward one problem
or the other. But before opening a discussion about
the implications of their preferences, the instructor
directed everyone’s attention to the other groups’
solutions. Although the first two solutions provided
correct answers to the two problems and aptly demonstrate the isomorphism between them, they do not
address the significance of the solution 89.
A third group, which had worked on the Blocks
problem, used an approach that differed from the
previous group’s use of the combinations model
to count all the possible arrangements. This group
of participants used an inductive strategy as well.
They experimented with a 1-by-2 frame, a 2-by-2
frame, a 3-by-2 fame, a 4-by-2 frame, and a 5-by2 frame. They noted that counting the ways to
arrange the blocks in each case resulted in a recursive rule resembling the Fibonacci sequence. They
then extended the sequence of these numbers to
arrive at 89 ways for the 10 × 2 frame (see fig. 4).
At this point in the sequencing of presentations,
the participants showed high interest in the two problems’ parallelism and their connection to the Fibonacci

1. There is 1 way to arrange all the blocks vertically.
2. We could use eight vertical blocks and one pair of
horizontal blocks. This can be done in 9 ways (9C8).
3. We could use six vertical blocks and two pairs
of horizontal blocks. This can be done in 28
ways (8C6).
4. We could use four vertical blocks and three pairs
of horizontal blocks. This can be done in 35
ways (7C4).
5. We could also arrange the blocks by taking a
combination of two vertical blocks and four
pairs of horizontal blocks. This can be done in
15 ways (6C2).
6. Finally, there is 1 way to arrange all the blocks
horizontally.
As before, the total is 89 possible arrangements.
At this point, looking at the first two groups’ solutions side by side, the class could see that the two
lists of combinations corresponded directly. The
342 MatheMatics teacher | Vol. 101, No. 5 • December 2007/January 2008

Fig. 5 a tree diagram solution to the staircase problem

sequence. The fourth group selected to present its solution strategy began by considering the options at each
stage of climbing the stairs. Participants in this group
represented these options in a tree diagram, which
they called “A Stairway to Heaven” (see fig. 5). Beginning at the first stair, one can step to either the second
or the third stair, as shown in the first level of the diagram. However, the participants had difficulty using
the tree diagram in this way to continue counting the
number of possible paths up to the tenth stair. So they
opted to consider the staircase in reverse, a process
they represented in a diagram titled “From Heaven
Back” (see fig. 6).
In this reverse representation, the group began
at the tenth stair and noted that there was only
one way to arrive at it directly from the ninth stair.
Going down one level, one could reach the eighth
stair in two ways, either by making a double step
or by passing through the ninth stair. Going back
another level, one could arrive from the seventh
stair by going through the eighth stair and using

those two previously counted ways or by going
through the ninth stair and using that one way.
In other words, to find the number of ways to get
to the top of the staircase from the seventh stair,
one can add the number of ways from the eighth
stair to the number of ways from the ninth stair.
Just to make the point clear, one can continue in
this fashion and go back an additional level to the
sixth stair. From here, one can either go to the seventh step and use one of those three ways or go to
the eighth step and use one of the two paths from
there. The fourth group continued this diagram
back to the bottom of the staircase, showing that
the 89 ways to climb the staircase are derived from

Fig. 6 student work reversing the representation begun in
igure 5

the 55 ways one can proceed from the first stair
added to the 34 ways one can proceed from the second stair. This last representation has a double significance: It not only serves as a tool for solving the
problem at hand but also reveals, quite suggestively,

the reason why the answer lies in a recursive rule
resembling the Fibonacci sequence.

IMPLICATIONS FOR TEACHING
Representation entails more than a direct or literal
translation of a problematic situation into a mathematical model such as a formula or a diagram. When
engaging in representing, problem solvers imagine
a visual story—one that is not always or necessarily
implied in the problem formulation. They impose
that story on the problem, and, acting on this representation, they derive from it the sought solution
(Arcavi 2003). In the lesson described here, the four
solution methods resulted from the unique way in
which each participant situated himself or herself
within the mathematical tasks at hand and then
schematized or abbreviated the counting processes
needed for reaching a solution.
At the end of the lesson, the groups were asked
to consider the similarities as well as the differences between the two tasks. The Staircase and
Blocks problems share three important similarities.
First, both involve an iterative process—that is, one
arranges one or two blocks at a time or climbs one or
two steps at a time. Second, both tasks also suggest
an inductive solution method, in which one proceeds
by increasing the size of the outer rectangular frame
Vol. 101, No. 5 • December 2007/January 2008 | MatheMatics teacher 343

or the total number of steps, always making use of
the previous result. Third, both tasks call for the use
of a combinations model. One could imagine posing
either task to students as a way to introduce or practice combinatorial thinking or as a means of teaching
specific problem-solving strategies, such as focusing
on smaller cases, making organized lists, searching
for patterns, or working backward.
Yet there are also important differences between
the two tasks. The Blocks problem calls for arranging objects within the constraints of a fixed space
and thus entails a static situation. The Staircase
problem, on the other hand, calls for a climbing
process that unfolds over
time and thus involves a
dynamic situation. Even
though the two problems are mathematically
isomorphic, attending
to these differences is
crucial because these
features of the tasks
make each problem more
imaginable, accessible,
or challenging for different students. Solving
and reflecting about
two mathematically isomorphic problems that
suggest a variety of representations strengthen
teachers’ ability to guide
students toward developing representations as
well as toward a disposition to build on and use
such representations while solving or formulating problems. The juxtaposition of these two tasks
allowed for an engaging and productive problemsolving experience focusing on the relationship
between the choices of representations and the
characteristics of the tasks. In addition to learning more about the significance of representations,
the participants, through the activity of comparing
and contrasting those representations, came to a
deeper understanding of the mathematics embedded therein.
After sharing and discussing alternative solution strategies and representations, participants
considered potential classroom implications of this
problem-solving experience. One immediate observation was the significance of the sequencing of
the presentations of solutions. Participants noted
the strategic manner in which the instructor made
on-the-spot decisions regarding the sequencing of
group presentations. This observation prompted
discussion about the importance of monitoring
groups as they work on a task, specifically with

attending to these
differences is
crucial because
these features make
each problem more
imaginable,
accessible, or
challenging for
different students

344 MatheMatics teacher | Vol. 101, No. 5 • December 2007/January 2008

respect to their problem-solving strategies and representations. Participants also noted that the collaborative experience of solving these two problems
in a variety of ways was a model for using these or
other nonroutine problems in a school classroom.
Although this particular lesson was conducted
as professional development for beginning teachers, the two tasks are appropriate for a wide range
of classroom implementation. Some readers might
find it productive to try the lesson as described here.
Others might want to adapt it by posing the tasks
in separate, consecutive lessons or spread over time
during a school year. And others might now be
interested in investigating other sets of isomorphic
problems, with a specific focus on identifying how
the specific features of each problem may motivate
different kinds of mathematical representations.

REFERENCES
Arcavi, Avraham. “The Role of Visual Presentations
in the Learning of Mathematics.” Educational Studies in Mathematics 52 (2003): 215–41.
Engerman, Mark. “Climbing Stairs with Fibonacci and
Pascal.” Mathematics Teacher 100, no. 2 (September 2006): 89–90.
Leikin, R., and G. Winicki-Landman. “Defining as a
Vehicle for Professional Development of Secondary
School Mathematics Teachers.” Mathematics Education Research Journal 3 (2001): 62–73.
National Council of Teachers of Mathematics (NCTM).
Principles and Standards for School Mathematics. Reston, VA: NCTM, 2000. ∞

Laurie rubeL, Lrubel@brooklyn.
cuny.edu, is an assistant professor at
brooklyn College of the City university of New York, brooklyn, NY 11210.
She enjoys working with and learning
from New York City teachers. betiNa
ZoLkower, betinaZ@brooklyn.cuny.
edu, also is an assistant professor at
brooklyn College of the City university of New
York. She is interested in how mathematics teachers working in urban heterogeneous classrooms
orchestrate whole-classroom interaction within
the context of nonroutine problem solving.