Order out of chaos: A spreadsheet excursion into a mathematical frontier.

Technology offers new pathways of exploration and ready access to new domains of mathematical inquiry. When students' exploration of a familiar model expanded into apparently chaotic behavior, the spreadsheet environment provided both the entrance to this mathematical frontier and the tools to pursue the new problem to conclusion. The students' investigation combined tabular and graphical spreadsheet representations, and the relationships among several of these representations were essential to the line of reasoning that shed light on the function's behavior and eventually established order in the chaos.

During the past decade computer spreadsheets have gained widespread attention as tools to support and enhance mathematics teaching and learning. Spreadsheets have proven particularly effective for exploring mathematical patterns and building conceptual understanding of variables and functional relationships. Examples are plentiful across a wide range of grade levels and mathematical topics. Sutherland and Rojano (1993) discussed the role of a spreadsheet environment in improving 10-11-year olds' understanding of functional relationship. Feicht (1999) advocated use of a spreadsheet to make three-dimensional graphing concepts accessible to middle school students. Hoeffner, Kendall, Stellenwerf Thames, and Williams (1990) described the use of spreadsheet activities to help middle school children develop conceptual understanding of relationships between variables. Masalski (1990), Russell (1992), and Whitmer (1992) offered collections of spreadsheet applications appropriate throughout the middle school and second ary curriculum. Abramovich and Nabors (1998) suggested that the iconic and computational features of a spreadsheet could enhance the transition from natural language sentences to pictorial and numerical representations of algebraic word problems. Garofolo, Shockey, Harper, and Drier (1999) presented spreadsheet models of applied mathematics concepts in a multiple representation environment. Lesser (1999) used a spreadsheet to explore probability in an introductory statistics course. Cornell and Siegfried (1991) and Maxim and Verhey (1991), pointed out the advantages of using spreadsheets to make recursively defined functions accessible to students from beginning algebra through high school. Pagon (1998) recommended spreadsheet modeling of matrix operations. Abramovich (1995) and Neuwirth (1995) applied the spreadsheet's facility for recursive definition of functions to explore combinatorics. Durkin and Nevils (1994) used spreadsheets to introduce dynamical systems and chaos. PinterLucke (1992) and Glidden (19 93) advocated spreadsheet techniques for root finding in precalculus.

Each of the reports cited suggested compelling applications for spreadsheets in mathematics classrooms, and together they exemplify the large and growing body of research and practice that supports spreadsheets as valuable and versatile tools for investigation and problem-solving from the middle school level through undergraduate mathematics. A spreadsheet environment allows functions to be defined recursively and investigated dynamically. Once a model is set up, the initial values can be altered to investigate any number cases.

One notable benefit of technology in mathematics education is its ability to progress beyond facilitating a traditional curriculum. This article discusses students' solution paths when a spreadsheet exploration evolved from a familiar topic into a new mathematical domain. Beginning with an iterated function for approximating square roots, students used a spreadsheet environment for dynamic exploration and visualization of mathematical relationships. When the familiar topic of square roots expanded into an apparently chaotic situation, students pursued a variety of paths in search of order within the chaos.

A SPREADSHEET MODEL

One of the early collections of spreadsheet ideas for mathematics teaching (Masalski, 1990) included a familiar divide-and-average square root algorithm. Widely recognized as Newton's Method for approximating square root, this algorithm can be stated formally as the iterative process of approximately solving the equation [x.sup.2] = A by repeatedly computing 

[x.sub.n] + - A/

[x.sub.n] + 1 = [x.sub.n]/2

given an initial estimate, [x.sub.0], of the square root of A. There is evidence of this method having been used also in ancient Babylonian texts (Bunt, Jones, & Bedient, 1976).

Figure 1 shows a spreadsheet implementation of this divide-and-average algorithm. The spreadsheet accepts a number in Cell A3 and an initial estimate of the square root of that number in Cell B3. The spreadsheet then calculates the square root by successive approximation and graphs the sequence of approximations from Column B. From either the graph or the successive values in Column B, it is apparent that the sequence of approximations converges rapidly. Formulas for the display in Figure 1 are shown in Figure 2. The spreadsheet displays in this paper were created using Excel (Microsoft, 1996), though any spreadsheet with good graphing capability would be adequate.

The two steps, divide and average, are computed separately in Columns C and D. Although the algorithm is as easily done in one column, separating the division and the averaging into two columns can make the procedure more accessible for some students, and it facilitates building the spreadsheet with a group of students one step at a time, first dividing and then averaging. Further, the students' investigation reported in this article actually used the relationship between Columns B and C. This particular progression of reasoning would not have occurred if the algorithm had been condensed to one column.

Suggested activities for exploring this spreadsheet model for square root at the middle school and high school levels appear in Masalski (1990) and Dugdale (1994, 1998). The present article focuses on an extension of the model into an apparently chaotic case and the quest for order there. This extension is best approached after thorough exploration of the basic model, for example, entering different values, noting patterns, and discussing the algorithm, to establish a basis of understanding about how the model works.

ENCOUNTERING A FRONTIER

By entering a negative number into Cell A3 of the spreadsheet in Figure 1, students can find themselves confronted with seemingly chaotic behavior as the spreadsheet's successive calculations no longer converge to a square root (Figure 3). After substantial exploration of the model for calculating square roots of positive numbers, the author usually introduces this special case by having students first predict what will happen if a negative number is entered in Cell A3. Based on students' prior experience with trying to take the square root of a negative number, predictions typically include calculator-type error responses, and sometimes students suggest the possibility of the spreadsheet responding with a complex number. The apparently chaotic behavior evident in the graph for a negative input can be startling, and students usually want to experiment with various combinations of negative numbers and initial square root estimates in search of possible patterns.

Explorations have taken a variety of directions, with individuals, pairs and small groups of students pursuing different paths and sometimes coming back together to exchange ideas. The following discussion represents one productive path of exploration that developed in class sessions with preservice teachers. The discussion captures one train of thought, and in the interest of brevity, omits several side trips and proceeds in a more linear fashion than what actually transpired. Although the participants' mathematical backgrounds varied, the parts of their investigation included here required only high school algebra and a healthy curiosity.

ANALYZING THE FRONTIER AND DEVELOPING CONNECTIONS

Students' initial efforts to understand the seemingly chaotic behavior focused on entering a variety of values in Cells A3 and B3 in hopes of identifying some consistent behavior. Many students also extended the spreadsheet to study longer sequences of values, but still found no convincing evidence of regularity.

While searching for patterns in the table and graph in displays similar to Figure 3, some students tried graphing the sequences of values in Columns B and C together, in case this might reveal a useful pattern in Column C or in the relationship between the two sequences. However, as shown in Figure 4, graphing the sequences in Columns B and C on the same axes revealed little beyond what is apparent from the tabular representation: for any given iteration of the function, the values in Columns B and C have opposite signs and plot on opposite sides of the horizontal axis. The reason for this is apparent from the formula used in Column C: for each row, n, of the spreadsheet, Cn = A3/Bn. Hence, when A3 is negative, Cn and Bn must have opposite signs.

A Scatter Plot Perspective

Although graphing Columns B and C together (Figure 4) was not particularly enlightening in itself, a productive result of this line of reasoning was the recognition that Bn x Cn = A3, for each row of the spreadsheet. Hence, each pair of entries (Bn, Cn) defines a point on a hyperbola of the form xy = k. Following up on his observation, students graphed the pairs of values in Columns B and C as a scatter plot and were rewarded by the familiar shape shown in Figure 5.

After having grappled at some length with the persistent disorder illustrated in Figure 3, students were encouraged by the sight of a familiar and predictable result! They noted that the relationship observed between Columns B and C holds for either a positive entry or a negative entry in Cell A3. That is, regardless of the sign of the entry in Cell A3, the number pairs (Bn, Cn) lie on a hyperbola of the form xy = k. Further, in either case, the graph of successive "estimates" in Column B represents the sequence of x-values of this hyperbola plotted in order of occurrence. Given this relationship between the line graph in Figure 3 and the scatter plot in Figure 5, students addressed the question of whether the scatter plot of (Bn, Cn) pairs could offer any insight into why the sequence in Column B appears so orderly when A3 is positive, as in Figure 1, and so disorderly when A3 is negative, as in Figure 3.

Through experimentation and examination of the spreadsheet formulas, students identified a fundamental difference between the two cases. When the value in Cell A3 is positive, a positive initial estimate leads to all positive values in Columns B and C (as in Figure 1), and a negative initial estimate leads to all negative values in Columns B and C. Hence, for a positive initial estimate each number pair (Bn, Cn) corresponds to a point in the first quadrant, and for a negative initial estimate each number pair corresponds to a point in the third quadrant. In short, when A3 is positive, the (Bn, Cn) scatter plot stays neatly in one quadrant. In contrast, when the value in Cell A3 is negative, as shown in Figure 3, Columns B and C each include both positive and negative entries, and the number pairs correspond to points in both the second and fourth quadrants, as in Figure 5. In fact, it is apparent from the table of values in Columns B and C of Figure 3 that graphing the (Bn, Cn) pairs sequentially will result in a graph that flips frequently between quadrants H and IV.

Stringing It All Together

Students hypothesized that the scatter plot's frequent flipping between quadrants could be related to the disorder observed in Figure 3, as compared to the orderly sequence observed in Figure 1. To pursue this possibility, they displayed their scatter plot with points connected in sequential order. The result was a pattern suggestive of "string art," where strings connect nails on a board to create artistic designs. Students tried an assortment of number entries in Cells A3 and B3 and noted a characteristic regularity to the string design. When a long sequence was graphed, a pattern similar to Figure 6 appeared consistently for negative entries in Cell A3.

Strings that crossed between quadrants were of particular interest, and curiosity about the places where the graph changed quadrants led to plotting selected shorter sequences of the (Bn, Cn) pairs to examine more closely the behavior of the graph before and after it changed quadrants. Figure 7 shows a typical sequence for a section of the graph with three points in Quadrant IV, followed by three points in Quadrant II.

Another effort that proved informative involved distinguishing "incoming" and "outgoing" strings in each quadrant. The pattern in Figure 6 includes strings going both directions: from quadrant II into quadrant IV, as well as from quadrant IV into quadrant II. Looking at only those strings originating in one quadrant simplified the picture and made it easier to detect an explainable pattern in the strings. Figures 8 and 9 show the strings originating in quadrants II and IV, respectively (1). Note that the graph in Figure 6 includes a composite of the strings in Figures 8 and 9.

Some Patterns

From the displays in Figures 8 and 9, it appears that the quadrant-crossing strings all originate from points either to the right of the vertex of the second quadrant branch (Figure 8) or to the left of the vertex of the fourth quadrant branch (Figure 9). Further, the slope of the strings increases as the originating point approaches the vertex.

These observations prompted a close look at the spreadsheet algorithm for points (Bn, Cn) on each side of the vertex in each quadrant. For example, focusing on Figure 8, all points in Quadrant II have Bn < 0 and Cn > 0. If(Bn, Cn) is to the left of the vertex, then \Bn\>\Cn\, and Bn+Cn/2<0, so the next point is also in the second quadrant. However, if (Bn, Cn) is to the right of the vertex, then, \Bn\<\Cn\ and Bn + Cn /2 > o, so the next point is in the fourth quadrant. Further, when Bn and Cn are very close in absolute value (i.e., (Bn, Cn) is close to the vertex), their average is very close to zero, and the next point plots close to the vertical axis far from the origin. Hence the cluster of strings from the neighborhood of each vertex fanning out to points off the screen in the second and fourth quadrants.

Figure 9 shows a similar pattern for strings originating in Quadrant IV. Given these observations and the graph in Figure 7, the group concluded that successive points (Bn, Cn) in Quadrant IV follow the hyperbola from right to left until one point plots to the left of the vertex, then the next point jumps to Quadrant II, and the process begins again: successive points in Quadrant II follow the hyperbola from left to right until one point plots to the right of the vertex, then the next point jumps to Quadrant IV, and so on.

Reviewing this conclusion, someone questioned whether we knew that the sequence always proceeds in orderly fashion toward the vertical axis until it changes quadrants. The group resolved this question by noting that Bn and Cn have opposite signs, so if \B\ > \Cn\, then (Bn+Cn)/2 is between Bn and zero, and the sequence of points proceeds along the hyperbola toward the vertical axis until \Bn\[less than or equal to] \Cn\. Then, if \Bn\ <\Cn\, the next point will be in the opposite quadrant, or if \Bn\ = \Cn\, a division by zero error will result in the next iteration.

Relating the Line Graph and Scatter Plot

Having identified patterns in the scatter plot and some regularity in the sequence of points in that plot, students returned to the line graph in Figure 3 to see if the patterns deduced from Figures 7, 8, and 9 could shed light on the disorderly graph in Figure 3. It was a simple matter to relate the quadrant flips in Figures 8 and 9 to the axis crossings in Figure 3. Each time the sequence of values in the spreadsheet's Column B changes sign, the scatter plot in Figures 8 and 9 flips between quadrants, and the line graph in Figure 3 crosses the horizontal axis.

Relationships between other aspects of the two graphs quickly followed. As the points (Bn, Cn) in the scatter plot proceed along the hyperbola until one point plots past the vertex, the line graph of the Bn values in Figure 3 approaches the horizontal axis until one point plots within the range of [+ or -] [square root of (\Bn\ Cn\)], then the next point plots on the other side of the axis. (See Figure 10 for a copy of the Figure 3 graph with this range shaded.)

Points near a vertex of the hyperbola in the scatter plot correspond to points in the line graph near [+ or -] [sqare root of (\BnCn\) , where Bn and Cn are close in absolute value. When Bn and Cn are close in absolute value, the next point in the scatter plot is close to the vertical axis and far from the origin, that is, small \Bn\ and large \Cn\; in the line graph a small \Bn\ corresponds to a point near the horizontal axis. The closer this last point is to the axis, the farther from the axis the next point will be as it starts the next sequence approaching the axis.

From this perspective it became possible to see the graph in Figure 3 as a chain of successive segments, each of which exhibits orderly behavior similar to that shown in Figure 1. Note that each time the function value in Figure 3 is far from the horizontal axis (i.e., large in absolute value), the next several values become successively closer to the axis in a pattern that looks much like the first several points of the graph in Figure 1.

In Figure 1 the graph converges to the square root, where Bn = Cn. Similarly, the graph in Figure 3 tends toward the axis. However, with Bn and On opposite in sign, the sequence does not converge, and as soon as the sequence passes the point where \Bn\ = \Cn\ , the next Bn value changes sign, and the graph flips across the axis.

CONCLUSION

Technology offers new pathways of exploration and ready access to new domains of mathematical inquiry. Students working with spreadsheet-based mathematics can be just a keypress away from mathematical frontiers, and the spreadsheet environment can be instrumental not only in having unexpected results arise in the course of a familiar topic, but also in providing the tools necessary to examine new problems and pursue them to conclusion. This article has presented a case where exploration of a familiar model expanded naturally into a mathematical frontier. When students working with the familiar divide-and-average square root algorithm found themselves confronted with surprising results, they proceeded in a variety of directions. Some of their explorations merged to shed light on the function's behavior and suggest order in the chaos.

Students' explorations combined five representations of the problem: (a)the spreadsheet formulas, (b) the spreadsheet table, (c) a line graph connecting successive entries in Column B, (d) a scatter plot of points (Bn, Cn), and (e) a connected scatter plot. Frequent examination of the relationships among these representations was essential to the line of reasoning that eventually established an understanding of the functional behavior. Of course, the investigation could have proceeded along other paths, and it often has. See, for example, Dugdale (1993) for quite different (although compatible) results that drew upon a spreadsheet's three-dimensional graphing capability.

To some, the line of reasoning pursued by the students in this article may seem a rather roundabout way to get to what, in retrospect, might be explained using the table and graph (or even just the table) in Figure 3. However, experience suggests that the table and graph in Figure 3 simply do not lead directly to a convincing recognition of any order in the sequence of values. The alternate representation of the scatter plot provided a more recognizable regularity and an implicit invitation to focus on those places where that regularity was interrupted. With the connected scatter plot, the picture became one of orderliness alternating from one quadrant to the other, and given this, students were able to identify a corresponding orderliness in the line graph in Figure 3. The pursuit of alternate representations and the examination of relationships among those representations were key steps in the students' analysis of the situation. These steps made important contributions to the richness of the investigation and to the understandings gleaned from the investigation.

Publisher's Note: re-published from JCMST 20(3) due to an error in printing the equations. Please refer to this corrected document when citing material. We regret any inconvenience this may have caused.

[Figure 4 omitted]

[Figure 5 omitted]

[Figure 8 omitted]

[Figure 7 omitted]

[Figure 8 omitted]

[Figure 9 omitted]

[Figure 10 omitted]

Figure 1.

A spreadsheet model for calculating square root

Square Root Calculator

        Square Root   Quotient     Average of
Number   Estimate    (Num./Est.)  Est. & Quot.

 125        30        4.1666667    17.0833333
        17.0833333    7.3170732    12.2002033
        12.2002033   10.2457309    11.2229671
        11.2229671   11.1378746    11.1804208
        11.1804208   11.1802589    11.1803399
        11.1803399   11.1803399    11.1803399
        11.1803399   11.1803399    11.1803399
        11.1803399   11.1803399    11.1803399
        11.1803399   11.1803399    11.1803399
        11.1803399   11.1803399    11.1803399
        11.1803399   11.1803399    11.1803399
        11.1803399   11.1803399    11.1803399

Note: Table made from line graph
Figure 2.

Formulas for the spreadsheet model in Figure 1

Square Root Calculator

        Square Root     Quotient     Average of
Number    Estimate    (Num./Est.)   Est. & Quo.

 125         30         =$A$3/B3     =(B3+C3)2
            =D3        Fill Down     Fill Down
         Fill Down    [down arrow]  [down arrow]
        [down arrow]
Figure 3.

Entering a negative number in Cell A3 produces unexpected results

Square Root Calculator

         Square Root   Quotient      Average of
Number    Estimate    (Num./Est.)    Est. & Quot.

-125      30          - 4.1666667      12.9166667
          12.9166667  - 9.6774194      1.6196237
          1.6196237   - 77.1784232   - 37.7793998
        - 37.7793998    3.3086815    - 17.2353592
        - 17.2353592    7.2525324    - 4.9914134
        - 4.9914134     25.0430068     10.0257967
          10.0257967  - 12.4678371   - 1.2210202
        - 1.2210202     102.3734111    50.5761955
          50.5761955  - 2.4715184      24.0523385
          24.0523385  - 5.1969999      9.4276693
          9.4276693   - 13.2588443   - 1.9155875
        - 1.9155875     65.2541321     31.6692723

Note: Table made from line graph

Note

(1.) Creating Figures 8 and 9 involved editing the spreadsheet table to remove all strings originating in one quadrant in order to view all strings originating in the other quadrant. For example, in Excel (Microsoft, 1996), one way to remove all strings originating in the second quadrant is to (a) replace the formulas in Columns B and C with values (select Columns B and C, select "copy" then "paste special" from the edit menu, then select "values"), (b) delete from the table any second quadrant (Bn, Cn) pairs that are not immediately preceded by a fourth quadrant pair (leave the cells blank so that the graph does not connect across the deleted pairs), then (c) insert a blank row after any remaining second quadrant pairs.

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