EXPLORING QUADRATIC-LIKE SEQUENCES WITH SPREADSHEETS
THROUGH A TOOL KIT APPROACH
Sergei Abramovich
State University of New York at Potsdam, NY 13676-2294
abramovs@potsdam.edu
Andrew Brantlinger
Department of Mathematics, Statistics, and Computer Science
University of Illinois at Chicago, IL 60607-7045
abrant1@uic.edu
Anderson Norton
Department of Mathematics Education
University of Georgia at Athens, GA 30602
anorton@coe.uga.edu
Introduction
In the context of technology-mediated mathematics pedagogy, the metaphor of a tool kit
means an array of representational formats which mediate students’ mathematical
thinking and conceptual development.
The variety of qualitatively different
representational formats (sign systems) generated by this environment affects students’
acquisition of mathematical concepts in different ways. This is a major claim of the tool
kit approach to teaching and learning mathematics. A rich computational environment is
typically comprised of different computer applications. These applications provide
iconic, numeric, symbolic, graphic and geometric notations for the development of
mathematical concepts. Such a technology-rich environment makes it possible to
integrate the above notations in the teaching of mathematics. This is in full agreement
with current trends in collegiate mathematics education.
For a collegiate mathematics classroom of preservice and inservice teachers, the use of
the tool kit has a twofold objective: (i) to enhance learning through introducing
mathematical concepts in different representational formats and (ii) to provide training in
creative uses of these tools. Whereas the former objective is not concerned with the
choice of software, the latter objective makes this choice a crucial factor in developing
teacher education programs. The best way to structure technology-oriented teacher
preparation is to use those technology tools that are available to teachers for their own
teaching. A way to boost the second objective of the technological component of
mathematics teacher education courses is to shift the emphasis from specific computer
programs to a broader and more sophisticated use of general purpose software. In
particular, spreadsheets have become increasingly available in all educational settings.
Such a shift in emphasis makes it necessary to re-think the notion of a technology-based
tool kit from multiple computer applications to the spreadsheet program alone. Due to its
complex and heterogeneous semiotic structure, the spreadsheet can be singled out as a
tool kit, in itself. This paper is a continuation of [1] which, in turn, stems from ideas
discussed in [2]. While the latter suggests using a multiple-application medium for the
study of linear iteration sequences, the former introduces a spreadsheet-based tool kit as
an effective environment for the exploring a similar topic in secondary teacher education
classroom. As this paper will demonstrate, the same tool kit can be utilized to explore
quadratic-like iteration sequences — sometimes referred to as quadratic maps. These
sequences arise commonly as mathematical models of biological processes. When treated
as dynamic systems, they can exhibit an extremely complicated and intriguing behavior
called chaos. The present paper reflects on activities associated with a computer course
for secondary mathematics education majors with an emphasis on using a spreadsheet as
a tool kit. In particular, a spreadsheet’s computational capacity made it possible to bring
secondary teachers to the forefront of knowledge about quadratic maps. Students were
able to explore such remarkable phenomenon as the period-doubling route to chaos
through a tool kit approach. For an alternative information on the use of spreadsheets in
this context see [3-5].
Introduction to spreadsheet constructions and use
The particular quadratic-like sequence that we will demonstrate in this environment are
given by the recursion xn+1 = xn(1-xn), for  values between 0 and 4. (The same
computational environment can be used to explore the behavior of another remarkable
sequence xn+1=sin(πxn)). For a fixed parameter  a spreadsheet-based tool kit allows
us to represent the behavior of quadratic-like sequences in the following ways: (i) a
cobweb diagram; (ii) analytic representation; (iii) numeric representation of iterations;
(iv) graphic representation of the iterations; (v) and finally to construct a bifurcation
diagram as the parameter  varies over any range that one wishes to explore. When  is
restricted to the interval [0,4], this diagram maps the interval [0,1] to itself, best
illustrating the period doubling which gives way to chaos. While the proof of period
doubling bifurcation phenomena is very complicated, the use of computers in exploring
complex behavior of quadratic-like sequences, is well described in [6]. Here, we restrict
our focus to specific uses of the following three constructions: cobweb, convergence, and
bifurcation diagrams. Also, in the descriptions of these constructions, we use terms
associated with Microsoft Excel, though the procedures generalize for other spreadsheet
programs.
Cobweb diagram construction
The cobweb representation of the quadratic-like sequence xn+1=xn(1-xn) for a given 
can be constructed as follows. Take any starting point and mark it x1 on the x-axis. Draw
a vertical line segment from (x1,0) to the point (x1,x1(1-x1))=(x1,x2) which belongs to
the parabola y=x(1-x). Draw a horizontal segment from the point (x1,x2) to the point
(x2,x2), which is on the bisector y=x. Continue by drawing a vertical segment from (x2,x2)
to (x2,x2(1-x2))=(x2,x3), which is again on the curve y=x(1-x). Draw a segment from
(x2,x3) to (x3,x3), and so on. The resulting geometric construction has the appearance of a
cobweb which wraps around the bisector y=x and the parabola y=x(1-x).
It is not immediately apparent how a spreadsheet can be used to graph the process
described above. A simple spreadsheet program described below can be used to this end.
First, we use cells A1 and B1 to identify the  and x0 values of choice. Let by
entering that value in A1 and enter .5 for x0 in cell B1. In column A, from cell A3 to cell
A23, we enter numbers 0, 0.05, .10, ... , 1. This data is used to construct the line y=x.
Cells C3 and D3 are entered, respectively, with the formulas =A1 and =$B$1*A3(1-A3)
which are then filled down to row 23. This data is used to construct the parabola
described by y=x(1-x). Note that $B$1, in the formula, instructs our spreadsheet to
always refer to B1 for the value of . To construct data for the cobweb itself, the range
A25:B99 is used. Cell A25 is entered with the starting point =B1 (x0), cell B25 is entered
with 0 (as the cobweb stems from the x-axis). Cells A26, B26, A27, and B27 are
entered, respectively, with the formulas =A25, =1.5*A26*(1-A26), =B26, and =A27.
The entire quadruple A26:B27 is then replicated down to row 99. To construct the graph,
one should highlight the range $A$3:$D$99, and choose the XY-Scatter Format #2 in the
ChartWizard menu.
Convergence diagram construction
With the cobweb construction described above, we are able to geometrically view the
nature of convergence for a given quadratic-like  sequence. On a new worksheet in the
same workbook, we can construct a tool for exploring this convergence further. The idea
here is to consider the values of a given sequence after each iteration. We can graph these
values against their corresponding iteration numbers to determine whether a particular
sequence converges or whether it enters a cycle. We can further test the nature of any
cycle analytically.
To begin, we again reserve cells A1 and B1 for the specific values of  and x0,
respectively. We include a cell, C1, for a third constant , which will serve as a tolerance
in testing cycles. We begin the iterative sequence in cell A3 with the value in B1, by
entering =B1. Now enter =A$1*A3*(1-A3) in cell A4 and copy this cell down to A250
(‘A$1’ instructs the program to always use the first value in the A column). The
appropriate chart is created by selecting A3:A250 and using ChartWizard’s line graph #2.
A second chart, using only the last 50 values might prove useful as well. These charts
will allow us to identify cycles graphically, but a numeric tool will be more precise in
testing for cycles.
We will use the B, C, D, E, F, G and H columns to test for 2, 3, 4, 5, 6, 8 and 16-cycles,
respectively. In the case of 2-cycles, we enter IF(ABS($A3-$A5)<$C$!, “*”, “”) in cell
B3. This code instructs the program to place a * in a cell if the difference of the
corresponding A cell and its second successor differ by a value less than . This code can
be copied down to B250, and similar codes can be used in the other cycle columns.
Last 50 Iterations
1
0.5
Xn value s
0
0
10
20
30
40
50
Figure 1.
Bifurcation diagram construction
A third worksheet may be used to view convergences or cycles for an interval of -values
(sections of the period-doubling bifurcation diagram). We reserve A1, B1 and C1 for the
3 constants , 0 and x0, respectively. The values for the A column are constructed
much as they were in the last worksheet. Use =B1 for A3 and =A3*$C1*(1-$C1) for A4.
The first entry simply places 0 in A3 and the second entry finds the first iteration of the
resulting sequence, using C1 for the x0 value. The next cell, A5, will contain
=A$3*A4*(1-A4) (keeping  constant) and may be copied down to A250. The iterations
for other -values can be found by entering =A3+$A1 in cell B3 and copying it over to Z.
Now A4 and A5 can be copied over to Z and the whole block can be copied down to 250
to give an array of values. These values will be used to construct a portion of the
bifurcation diagram as follows. Using the chart wizard, select the first row and the last
fifty rows ($A$1:$Z$1, $A$201:$Z$250) and select line graph #2. Make sure to set the
options to “data in rows” and “use first row for x-axis” (it will also help to leave out the
legend). The resulting graph displays the last 50 iterations for each -value (as in Figure
2).
Figure 2.
Suggestions for exploration
Students should begin with a review of recursive formulas and convergence using the
web construction, where the nature of an input-rule-output-feedback process is depicted
geometrically. Convergence can be further explored using the line graphs on the second
sheet. The graph of the last fifty iterations might also make clear the concept of cycles
(Figure 1 appears to illustrate a 4-cycle). In order to verify the existence of such cycles,
the analytical approach, using a given tolerance , may be employed on this same
spreadsheet. Incrementing the  values in this tool can also help students identify the
lowest value for which a given cycle occurs. This kind of exploration leads to the
discovery of the Feigenbaum number (4.669...), which is the ratio between consecutive
differences of consecutive period doublings. Students can use this number to predict
when sequential doublings occur, and then use the cobweb and convergence constructions
to support their conjectures. In addition, students might be asked to predict the sizes of
cycles for larger lambda values. They can test these conjectures with the bifurcation
construction (Figure 2), which plots the last 50 iterations for each lambda value in the 0,
 defined interval. If these iterations are close in value, the 50 points appear as one and
the corresponding sequence is assumed to converge. Likewise, n-cycles are displayed as
n separate points. If the graph displays many such points, we might assume the existence
of chaos. In exploring the bifurcation diagram, students may notice “windows” within
the chaos where a lambda value appears to produce a finite cycle again; indeed, Figure 2
illustrates at least one such example at =3.83, which yields a 3-cycle!
Conclusion
The proposed toolkit provides a rich environment for studying quadratic-like sequences.
Through such explorations, students can acquire mature concepts of such phenomena as
convergence, divergence, n-cycles, period doubling and chaos. The concepts are
developed through student interaction with the various representations within the toolkit.
The cobweb construction provides a geometric representation of convergence and
divergence (in which n-cycles and chaos arise). The convergence diagram offers a
graphic representation of each iteration while, on the same sheet, students can test for ncycles analytically. Finally, the bifurcation diagram represents, graphically, the state of
convergence or divergence for a range of  values. As suggested above, explorations can
lead to discussions and discoveries of the Feigenbaum number and “windows” of stability
within chaos. But, the true power of this toolkit lies in its accessibility. Being that
spreadsheets are widely available, this dynamic environment for exploring quadratic-like
sequences is both versatile and feasible.
References
1. Abramovich, S., & Brantlinger, A. (1998). Tool kit approach to using spreadsheets in
secondary mathematics teacher education. In S. McNeil, J.D. Price, S. Boger-Mehall, B.
Robin, J. Willis (Eds.) Technology and Teacher Education Annual, 1998 (pp. 573-77)
Charlottesville, VA: AACE.
2. Abramovich, S., & Norton, A. (1998). Exploring infinite sequences in a multipleapplication environment. Manuscript submitted for publication.
3. Abramovich, S., & Levin, I. (1993). Some New Aspects of Using Computer
Technology in the Simulation of Biological Processes. Paper presented at the
International Conference Science Education in Developing Countries. Jerusalem.
4. Durkin, M.B., & Nevils, B.C. (1994). Using Spreadsheets to See Chaos. Journal of
Computers in Mathematics and Science Teaching, 13(3), 321-338.
5. Iseke-Barnes, J.M. (1997). Enacting a Chaos Theory Curriculum through Computer
Interactions. Journal of Computers in Mathematics and Science Teaching, 16(1), 61-89.
6. Bruce, J.W., Giblin, P.J., and Rippon, P.J. (1990). Microcomputers and Mathematics.
Cambridge: Cambridge University Press.