The appearance of computers as alternative instructional tools has created serious problems in undergraduate pedagogy. This is particularly the case for mathematics teacher education, because technology as a mathematical/pedagogical tool is not always used appropriately in schools. Indeed, some teachers who attempt to incorporate technology into the curriculum, limit its use to routine computations only due to a lack of experience with this technology. The mathematics education community views this problem as a great challenge to educational reform.

The recently published Discussion Document for ICMI Study on the Teaching and Learning of Mathematics at University Level (International Commission on Mathematics Instruction, 1998) recognizes the increasing use of computing technology in mathematics instruction. Yet, one of the important questions raised in the Document is: In what ways can computing technology be used to enhance understanding and enable learning beyond what can be done without it? Special attention is given to the preparation of pre-service teachers. In order to promote the use of computers in undergraduate mathematics instruction beyond computations, the Document points to the importance of collecting examples in which technology enhances studentsí experience of mathematics and leads to better understanding and learning. Such examples can be found in computer-based instruction which incorporates mathematical modeling.

One of the major advantages of modeling mathematics with computers in the undergraduate curriculum is that it enables students to explore mathematical topics which are beyond their grasp without the use of such tools. In particular, mathematical modeling in the preparation of teachers is aimed at the exploration of mathematical concepts and structures through activities mediated by the tools of technology, creation of a variety of persuasive representations of mathematical ideas and connections among them in alternative environments, establishing of a cognitive heterogeneity in mathematics classroom via computer investigations.

Modeling with computers in the context of mathematics education can broadly be conceived in two ways: (i) modeling by constructing an abstract mathematical model through the decontextualization of computer-generated manipulatives which portray mathematical concepts on a concrete, intuitive, action-oriented level, and (ii) modeling by constructing a set of computer-generated representations of an abstract mathematical structure or concept which enables one to understand underlying relationships among the elements of this structure or concept. This is not to say that the two approaches to modeling are mutually exclusive. On the contrary, they can be used as two phases in the larger process of presenting and developing significant mathematical ideas through bouncing back and forth between abstract and concrete representations. This paper shows how this kind of mathematical modeling can be implemented through the use of spreadsheets in the teaching of number theory.

More specifically, in the context of this paper modeling number theory with spreadsheets means the representation of an abstract mathematical structure through the elements of a spreadsheet-based tool kit. As discussed elsewhere (Abramovich & Brantlinger, 1998), the metaphor of a tool kit means an array of semiotically heterogeneous representational formats which mediates studentsí mathematical thinking in a technology-rich environment. As far as a spreadsheet-based learning environment is concerned, such an array includes iconic, numeric, geometric and graphic notations. By using these notations both elementary and advanced ideas of number theory can be presented in a meaningful way.

To illustrate this approach, consider one of the fundamental concepts of mathematics ? the Pythagorean equation. By using a spreadsheet, its solution (a Pythagorean triple) can come into play in five different ways:

(i) by constructing a square which can be split into two smaller squares ? an enactive solution in a manipulative environment;

(ii) by locating a perfect square among the sums of two squares ? a numeric solution in a computational environment;

(iii) by constructing a Pythagorean triangle on a numeric template with vertices at the elements of a corresponding Pythagorean triple ? an enactive geometric solution in a computational environment;

(iv) by using the general solution to the Pythagorean equation? a numeric representation in a computer environment; and

(v) by generating the so called "brideís chair" (Boyer, 1968, p. 119) on a spreadsheet chart ? a graphic solution in a computational environment.

By using the triple (3, 4, 5) as the simplest solution to the Pythagorean equation all semiotically different representations of the solution ? iconic, numeric, geometric and graphic can be introduced to students through the tool kit approach. The paper describes these approaches to modeling Pythagorean triples and discusses some extensions of the topic to more advanced ideas of number theory made possible by the use of technology.

2 Spreadsheet as a manipulative

Action with manipulatives is considered as an important mediator of childrenís mathematical thinking and contemporary method courses for teachers promote the use of counting cubes, cuisenaire rods, pattern blocks, colored chips, and other mediational means for modeling mathematical concepts and structures. Familiarity with manipualtive approaches to the teaching of mathematics helps the teachers feel conformable in using these tools in their own teaching. The advent of electronic information technology can advance enactive strategies in the teaching of mathematics. The point is that physical manipulatives can be simulated on a computer. In particular, a spreadsheet has a potential to be used as a manipulative environment with interactive linkability to numeric notation. That is, any action in such an environment can be reflected interactively in a symbolic domain, an idea behind many content specific mathematics education software. What is special about a spreadsheet-based simulation of manipulative learning environments is that one of the most commonly available general purpose software can be used for enactive modeling of mathematical concepts. Mathematics teachers can use manipulatives with children not only for the discussion of concepts in arithmetic but also of those which are mathematically more advanced. The Pythagorean equation is an example of an advanced concept despite of its rather simple structure.

Figure 1 shows an enactive solution to the Pythagorean equation which is linked to its numeric form ? the Pythagorean triple (3, 4, 5). How can the Pythagorean triple (3,4,5) be constructed through action? Geometrically the problem here is to find a square which can be split into two squares of a smaller size. The environment shown in Figure 1 incorporates a Macro which links icons to numbers so that a spreadsheet can recognize colored icons as numeric entities. Two buttons shown on the top part of Figure 1 serve for the following purpose: by applying the "smile face" button to a cell one can incorporate a hidden zero with color into this cell; by applying the "sad face" button one can discard the result of applying the smile face button. The spreadsheet logical function

=IF(INT(SQRT(COUNT(A5:G12)))=SQRT(COUNT(A5:G12)),SQRT(COUNT(A5:G12))," ") defined in cell B2 (Figure 1) displays the size of the constructed square and leaves cell blank if a shaded area does not represent a square. Similar formulas are defined in cells I2 and O2. When the contents of the three cells give a Pythagorean triple, the nested logical function =IF(OR(B2=" ",I2=" ",O2=" ")," ",IF(B2^2=I2^2+O2^2,"TRIPLE!"," ")) generates the message "TRIPLE!" in row 11.

Unfortunately, the ease of enactive solution has another side, its limitation. Indeed, attempts to find more solutions to the Pythagorean equation within a manipulative environment would most likely fail; the environment becomes inefficient as squares grow large. This shows the need for a technique that goes beyond the use of the enactive solution. However, an intuitive idea of finding a square that can be represented as the sum of two squares may be productive when a numeric approach comes into play. Such approach requires the use of a two-dimensional thinking as one has to coordinate two parameters ? the legs of a right triangle in order to find a corresponding hypotenuse. This brings about the idea to use a spreadsheet as a two-dimensional computational environment. As the authors shall demonstrate below, in this environment the Pythagorean equation can be explored effectively from numeric perspective enabling conjecturing of many powerful mathematical ideas dealing with additive representations of whole numbers.

3 Spreadsheet as two-dimensional modeling tool

The multiplication table is an example of a two-dimensional representation of the products of two positive integers, an arithmetic structure which is well known from the school mathematics. In the interior of this table each cell is an output of an arithmetic procedure applied to two numbers (inputs) on its border. In much the same way one can construct the table with the same inputs but apply another arithmetic operation to them, namely, the summing of the squares of the inputs. In order not to duplicate the sums which differ in the order of summands the table can be generated in the a triangular form. Once again, this implies the use of a logical function IF. The resulting numeric template is shown in Figure 2(a) in which cell B2 is entered with the spreadsheet formula =IF(B$1>=$A2,B$1^2+$A2^2," "). It is interesting to note that the interior of the table still contains equal numbers: for example, one can see that 50 and 65 appear twice here. However, this is not a trivial duplication of the sums ? the appearance of the same numbers in the table indicates the possibility of multiple representation of integers as the sums of two squares. The extension of the discussion to this property of integers will be provided below.

The next step in generating Pythagorean triples is to look for perfect squares in this table, generate square roots of the perfect squares, and leave cells blank otherwise. The spreadsheet function

=IF(AND(B$1-$A2>=0,INT(SQRT($A2^2+B$1^2))=

SQRT ($A2^2+B$1^2),SQRT ($A2^2+B$1^2)," ")

is defined in cell B2 and is replicated across all columns and down all rows. This logical function with the function AND embedded into it enables a computer to do two tests at once ? the first test that prevents trivial duplications of the sums of two squares, and the second test that provides the search for perfect squares. The results of numerical modeling of Pythagorean triples is shown in Figure 2 (b).

Figure 2. (a) the sums of two squares; (b) Pythagorean triples

As far as the conjecturing of a general solution to the Pythagorean equation is concerned, the advantage of using a spreadsheet as a two-dimensional modeling tool is in the possibility to visualize two arrays of numbers on a computer display ? the sums of two squares and Pythagorean triples. This visualization enables one to recognize a profound characteristic feature of the largest element of any primitive Pythagorean triple. Such recognition, however, was beyond the grasp of students in a traditional, non-computerized collegiate mathematics classroom (Schoenfeld, 1992). Yet, the experience of one of the authors with pre-service and in-service teachers (Abramovich, 1995) indicates that when asked to find commonalties between numbers on the two templates, the teachers are able to make the following observation: For any primitive Pythagorean triple displayed on the template of Figure 2(b) its largest element is always an element of the array of Figure 2(a). What does this mean? The answer is in the nature of numbers of Figure 2(a). The numbers here are the sums of two squares. Consider, for example, 13 ? an element of the triple (5, 12, 13). This number can be found on both templates which yields the representation 13=32+22. That is, the number 13 can be generated in terms of 3 and 2 as the sum of their squares. Can the same generators be used to represent other two elements of the triple, that is, 12 and 5? The reasonable guess: 5=32-22, and 12=2·3·2 turns out to be correct and exploring other cases justifies this result through numerical evidence. In other words, the statement about a general form of Pythagorean triples can be formulated as follows:

4 Pythagorean triples and the Euclidean algorithm

The above general formulas for Pythagorean triples can be utilized in a computational environment for generating non-trivial solutions to the Pythagorean equation. This new environment should be capable of coordinating rather delicate properties of the generators m and n, namely their relative primality and opposite parity. This coordination requires the use of another famous number theoretical structure known as the Euclidean algorithm. This ancient algorithm is a recursive procedure of successive divisions based on the relation of fundamental importance in number theory ? the remainder equals the dividend minus the divisor times the quotient. Setting two whole numbers as a dividend and divisor the last non-zero remainder so obtained is the greatest common divisor between the two numbers. A spreadsheet is particularly amenable for such recurrent counting.

Figure 4 shows such an environment with the Euclidean algorithm hidden between columns F and BF enabling the algorithm to be performed on all 2-digit numbers of opposite parity. The environment has a single input ? an entry in cell A1, which is assigned for the generator m. Upon entering any whole number in cell A1, the spreadsheet generates in columns B and C all corresponding relatively prime generators m and n of opposite parity and corresponding triples a, b, c in columns D, E, and F respectively. Each positive integer in cell A1 yields its own screen with a unique set of generators and triples.

An educative value of this computational environment is not only in the systematic presentation of all primitive Pythagorean triples related to a given generator m, but also in the fact that its structure invites new observations and stimulates new profound number theoretical inquiries. For example, for a given m, how many n exist? Figure 4 shows that when m =15 (cell A1), there are 4 values for n (cell A2); by changing m to 16 the spreadsheet displays the number 8 in cell A2. Through such computational experiments one can observe that when the value of m in cell A1 is even, the content of cell A2 is the number of relatively prime numbers less than m , when m is odd the content of cell A2 is half that number. If students are familiar with the Euler phi-function j(m) they may come up with the following conjecture in terms of this function: For a given generator m, there are j(m) generators n, when m is even, and j(m)/2 generators n, when m is odd. Familiarity with advanced concepts, however, is not an imperative. Rather, through modeling Pythagorean triples the Euler phi-function can be introduced in context. By the same token, familiar concepts and structures become connected to each other from a larger perspective when utilized meaningfully in mathematical modeling. Due to lack of space the authors have chosen not to include all the formulas that produced the environment of Figure 4. More information, however, is available upon request.

Finally, in columns BF and BG (Figure 4) the numbers 0.5(c-a) and 0.5(c+a) are computed for different primitive Pythagorean triples. Here new questions can arise from spreadsheet modeling. What do these numbers have in common? Do they have factors in common? How do these numbers depend on generators of corresponding triples? Note that these questions are aimed at developing a formal demonstration of the general solution to the Pythagorean equation. The following is a sketch of a proof driven by numerical evidence.

Modeling shows that and are relatively prime perfect squares. Denoting =n2 and =m2 infers ()2 = m2n2. Addition and subtraction result in formulas a=m2-n2, b=2mn, c=m2+n2 which have been already discovered through a data-driven induction.

5 Generating Brideís chair

Spreadsheet graphics can be used for constructing yet another representation of Pythagorean triples which goes back to the ancient tradition of the geometrization of mathematical structures. The well-known diagram, associated with the Pythagorean Theorem and shown in Figure 5, is often referred to as a Brideís Chair. The construction of the Brideís Chair brings students full circle to the enactive approach to the Pythagorean equation, of breaking a square into two smaller squares. One can see the resemblance between Figures 1 and 5, both generated by a spreadsheet. However, whereas the iconic representation (Figure 1) of the triple (3, 4, 5) came as a result of physical manipulation on a computer screen, its computer graphics representation (Figure 5), enabled by theoretical knowledge, is not constrained by action and therefore is not limited to one triple. In this new environment, students are no longer concerned with finding solutions, rather they can examine how the Pythagorean equation relates to right triangles and areas of corresponding squares.

In order to generate the Brideís Chair related to the triple (a, b, c) we enter values for a and b in cells A2 and B2 respectively. These values are then used to create a sequence of 13 ordered pairs in columns B and C which when plotted on the XY-Scatter selection from the Chartwizard menu forms the edges of the three squares. Those points are; (0,a), (-a,a), (-a, 0), (0, 0), (b, 0), (b, -b), (0,-b), (0,0), (0, a), (a, b+a), (b+a,b), (b,0), (0,a). Figure 5 shows the particular case when a=5 and b=12. Unfortunately, these points alone are not enough to create a diagram where the squares indeed appear as squares and not rectangles. In order to get around this technical obstacle we add 4 more points, which are hidden from view, but force the diagram to be square.

6 Representation of integers as the sum of perfect powers

There are problems in the elementary theory of numbers which deal with a representation of an integer as a finite sum of positive integers from a given set. Because sums are involved such problems constitute the so-called additive number theory. Many additive properties of integers were first discovered inductively; that is, through reasoning based on observations. One relevant problem deals with the representation of integers as the sum of two squares. The template of Figure 2(a) shows many examples in which positive integers are the sums of two squares. Yet, not all integers have such representation whereas some integers, like 50 or 65, have more than one representation as the sum of two squares. Once this fact is recognized through numerical evidence, the following inquiries come out naturally:

ï What is special about a number which can be represented as the sum of two squares in more than one way ?

ï Given a number, how does one know this number can be represented as the sum of two squares ?

ï How does one know that a number can be represented as the sum of two squares in more than one way?

With respect to the first question consider the number 50. In particular, 50 is the sum of two equal squares. Is it true that every number which is the sum of two equal squares can be represented as the sum of two squares in more than one way? One may note that these numbers are located along the main diagonal on the template of Figure 2(a). However, the number 32 is not the sum of two different squares. Clearly, the number 50 is the smallest such number. What is special about the number 50? What is the difference between 32 and 50 in this sense? What is the next number on the main diagonal which can be represented by the sum of two different squares? What is special about numbers that have a unique representation as the sum of two squares? These explorations can help students grasp the meaning of the following famous theorem discovered by Fermat and proved a hundred years later by Euler: Every prime number of the form 4n+1 can be represented as the sum of two squares in one and only one way.

One can recognize that the numbers 3, 7, 11, 15 can not be represented as the sum of two squares. These are numbers of the form 4n+3. How many squares are required to represent these numbers as the sum of squares? For example, 3=12+12+12; 7=22+12+12+12; 11=32+12+12; 15=32+ 22+12+12. It appears that numbers of the form 4n+3 are expressible either as the sum of three squares, or as the sum of four squares. This is a remarkable property (known as Lagrangeís Theorem): Every integer is the sum of at most four squares. In order to see properties of whole numbers from this advanced perspective one should be introduced to a computational environment, like spreadsheet, which is conducive to such thinking. The acquisition of an advanced perspective generates a new inquiry: Can integers be represented through the sum of positive cubes and if so, what is the most number of cubes can be involved in such a representation? The inquiry can be extended to the sum of fourth powers. These questions touch upon a famous Waringís problem according to which every integer is the sum of not more that 9 positive cubes and 19 fourth powers.

7 Concluding remarks

Because of the increasing availability of spreadsheets in a variety of educational settings the authors believe that mathematics teacher education courses may consider re-thinking the concept of a tool kit from multiple programs to this single type of program. This paper demonstrated how the complex semiotic structure of the spreadsheet enabled a single computer program to be used as a tool kit. Such use of the software creates an environment that allows students to see number theoretical relationships that are otherwise not so simple to encounter. It stimulates natural curiosity, investigation of number patterns, and conjecturing of non-trivial results. Furthermore, the environment is conducive to the continuos extension of an original situation through subsequent inquiry. In such a way in-service and pre-service mathematics teachers can by empowered by profound ideas from number theory which stimulated the development of mathematics from the antiquity to present time through the appropriate use of technology.

References

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Abramovich, S. (1995). Pythagorean Triples and Technology. Paper presented at the
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