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). 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