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Developing technology-mediated entries into hidden mathematics curriculum as a vehicle for "good learning" by elementary pre-teachers.

 This paper suggests that knowledge of the school mathematics
 curriculum, as one of the key components of teachers' education, can
 be extended to include concepts and structures that generally belong
 to hidden domains of the curriculum. Motivated by work done with
 elementary pre-service teachers in a mathematics course employing a
 hidden mathematics curriculum framework (Abramovich & Brouwer,
 2003a), this paper shows how computing technology, including The
 Geometer's Sketchpad and spreadsheet environments, facilitate the
 development of entries into advanced mathematical ideas dealing with
 the partition of unit (Egyptian) fractions. The combination of
 Freudenthal's pedagogy of learning mathematics as advancement of the
 culture of mankind and Vygotskian theory of learning in a social
 context provides theoretical underpinning for this framework.


Since the beginning of the present mathematics education reform movement in the U.S. (National Council of Teachers of Mathematics, 1980), the impact of curriculum on the preparation of teachers of mathematics has been the focus of both national and international research (Howson, Keitel, & Kilpatrick, 1981; Fennema & Franke, 1992; Clarke, Clarke, & Sullivan, 1996; Fuson & Li, 2002). The need for prospective teachers to understand the underlying conceptual structure of school mathematics activities across grades has brought about the quest for an integrated mathematics education curriculum, something that is both mathematically rich and yet connected to traditional topics that teachers have been preparing to teach. Curricular knowledge, which in particular includes experience in appropriate uses of technology, was one of the three categories proposed by Shulman (1986) for analyzing teacher's knowledge. This category can be extended to include concepts and structures generally hidden in the curriculum such as an implicit interpretation of number as a linear combination of its face and place values, an example provided by the Mathematical Education of Teachers report (Conference Board of the Mathematical Sciences, 2001) in connection with the preparation of elementary teachers (1).

This paper is an extension of the authors' earlier work that introduced the notion of hidden curriculum in mathematics teacher education (Abramovich & Brouwer, 2003a, 2003b). The general notion of hidden curriculum can be traced back more than three decades (Jackson, 1968) and has received much attention in foundational educational research (Ginsburg and Clift, 1990). While a hidden curriculum framework in a traditional sense explores tacit features that structure life in schools, hidden mathematics curriculum includes tacit concepts and structures that underlie a variety of school mathematical activities. The notion of hidden mathematics curriculum is based on the observation that many mathematical activities across the K-12 curriculum, seemingly disconnected from a naive perspective, are, in fact, permeated by a common mathematical concept or structure, traditionally hidden from learners because of its complexity. Such complexity may be either procedural or conceptual in nature. The authors' approach to investigating the idea of hidden curriculum in mathematics teacher education is to find and work with a series of problems scattered across the curriculum that from a deeper perspective may be described by a common mathematical concept. Technology has great potential to enhance this approach.

It is often observed that teachers of mathematics do not have sufficient mathematical background to see the general concepts behind particular phenomena. This lack of understanding contributes to the communicating of mathematics to learners in a disconnected fashion, so that elementary students believe that the problem solving work that they are doing is limited to their grade level. By the same token, secondary students cannot see the connection between problem solving they are currently engaged in and their earlier mathematical experience. Thus, the knowledge of hidden concepts and structures in the mathematics curriculum can be used to extend the curriculum in both directions. This suggests the importance of exploring a hidden curriculum framework across all levels of mathematics teacher education.

The afore mentioned imperative may have a profound impact on mathematics teacher education provided that prospective teachers (hereafter referred to as pre-teachers) are given the opportunity to learn advanced ideas in a social context of competent guidance enhanced by appropriate technology tools. As argued elsewhere (Abramovich & Brouwer, 2003a) the use of pedagogy informed by a technology-mediated hidden mathematics curriculum framework supports the advancement of Freudenthal's (1983) theoretical construct of the didactical phenomenology of mathematics as "a way to show the teacher the places where the learner might step into the learning process of mankind" (p. ix). In other words, technology-enabled learning in a social milieu of expert-novice relationships opens windows into the hidden meanings of, otherwise perceived as elementary, mathematical concepts.

Such a focus on expertly assisted learning of mathematics by pre-teachers brings to mind one of the basic tenets of Vygotskian pedagogy which considers social interaction as the primary educative mechanism and conceptualizes learning as a transactional process of developing informed entries into a culture with the support of more capable members, or agents of this culture (Bruner, 1985). As far as a mathematical culture is concerned, the above notion of hidden curriculum may serve as a powerful intellectual link between the two concepts of Freudenthal's didactical phenomenology of mathematics and Vygotsky's zone of proximal development (ZPD) that learning by transaction creates. The concept of ZPD emerged as an alternative to the three theoretical positions in educational psychology at the time: (i) that learning and development are independent; (ii) that learning is development; and (iii) that learning and development overlap (Vygotsky, 1978). This alternative was based on the assumption that human learning is essentially a social process in the sense that what one can do with the assistance of a more knowledgeable other fully characterizes one's cognitive development. In other words, ZPD is a dynamic characteristic of cognition that, in a problem-solving situation, measures the distance between the two levels of one's development as determined by independent and assisted performances. Vygotsky argued that learning by transaction creates the ZPD and proposed "a new formula, namely that the only 'good learning' is that which is in advance of development" (ibid, p. 89).

At this confluence of pedagogical and psychological theories, the combination of Freudenthal's pedagogy of learning mathematics as advancement of the culture of mankind and Vygotskian theory of learning in a social context provides theoretical underpinning for the didactical framework of hidden mathematics curriculum. More specifically, the pedagogy of revealing hidden curriculum messages to pre-teachers in a social milieu of computer-enabled learning creates the ZPD that, in turn, provides the basis for one's profound understanding of elementary mathematics. It is through educated assistance at the points of the zone where such assistance is needed (Tharp & Gallimore, 1988) that one develops the skills and confidence needed for its competent teaching. Therefore, it appears that a hidden curriculum framework enhanced by technology has potential to become a vehicle for what Vygotsky has called "good learning." In what follows, the authors will show how this potential can be realized.

Technology, Curriculum and Pedagogy

It should be noted that technology, as a cultural amplifier of mathematics teaching and learning or more generally, in Bruner's (1985) words, a prosthetic device needed to master the world around us, is not self-sufficient in the sense that it can not create a ZPD in the same way a teacher or peer can (unless conditions are created for its dialogic interaction with the learner). Instead, the use of technology requires from the learner certain skills, and in contemporary discourse, stressing the social aspect of learning, technology is considered "both as a tool-to-be-taught and as a tool-for-teaching" (Rivera, Galarza, Entz, & Tharp, 2002, p. 182). As an object of teaching, technology can motivate learning of the mathematical concepts that enabled its development. As a teaching tool, technology plays an important role in facilitating entries into mathematical culture by functioning as a scaffolding device in one's transition from a present level of mathematical understanding to a higher ground. In other words, through the integration of social tools such as expertly assisted learning with the tools of technology, one may reach the potential for full development of higher cognitive processes. Additionally, technology alters social interactions among learners of mathematics and has potential to affect various psychological phenomena, including math anxiety and self-confidence (Tobias, 1993). In fact, on the surface, this paper is a reflection on a mathematics education course designed for prospective elementary teachers with a goal of bringing positive change in the aforementioned phenomena. In designing this course, technology was developed to enable pre-teachers to master fundamental topics from traditional mathematics curriculum at a conceptual level. This technology required minimal skills from the pre-teachers, something that was taught as the course went along.

An early analysis of the impact of technology-related research on mathematics education curricula suggested the importance of revealing conceptual linkages among topics and ideas within the traditional curriculum through the use of technology (Kaput & Thompson, 1994). It was further argued (Balacheff & Kaput, 1996) that technology is having an ever-deepening impact on the curricular dimension of mathematics education. This suggests the importance of exploring the interplay between technology and mathematics curriculum, particularly the use of the former in revealing hidden aspects of the latter in the context of teacher education.

The Partition of Unit Fractions as Hidden Curriculum

As detailed elsewhere (Abramovich & Brouwer, 2003a), one fruitful mathematical concept that can be explored as a hidden curricular structure is the partition of integers. By using problems from standards-based curricula, it was demonstrated how this simply formulated, yet remarkably profound, concept was implicitly present at all levels of the curriculum and may be viewed as a hidden thread. To support this demonstration, technology tools were developed that made it possible to engage pre-teachers in mathematical investigations leading to their conceptual growth.

The present paper aims to extend this type of technology-enhanced work with pre-teachers within a hidden curriculum framework to a related but distinct topic--the partition of unit fractions (often referred to as Egyptian fractions [e.g., Borowski & Borwein, 1991]) into sums of like fractions. There are problems both within and outside mathematics in which the importance of such representations arises, and this paper will provide some examples of these types of problems. It should be noted that there has been a recent increase of interest in problems involving Egyptian fractions (Eggleton, 1996; Guy, 1994; Hoffman, 1998; Martin, 1999) and this helped motivate the authors to incorporate the ideas into mathematics teacher education through a hidden curriculum framework.

To the authors' knowledge, partition of unit fractions has never been highlighted as a unifying theme in the school mathematics curriculum. This may be due to the complexity of the mathematics behind the concept. By the same token, it often takes a genuine insight into a mathematical situation to discover its hidden meaning. Nevertheless, it is possible to lead pre-teachers on an informal journey into such advanced mathematics.

Indeed, this paper (as well as a part of the course associated with it) was prompted by an episode observed in a mathematics methods class when a pre-service elementary teacher, referring to her observation of an elementary classroom in rural upstate New York, shared with the class the activity of covering one-half of a circle with other fraction circles. (2) The mathematical depth of this activity was not immediately apparent to the class; yet it may be used in mathematics teacher education as a springboard to exploring advanced mathematical ideas. The following pages will illustrate how one may do this with pre-teachers as appropriate, proceeding from the following possible formal description of the above-mentioned activity.

Brain Teaser. Completely cover the fraction circle representing one-half of a circle using at most three other fraction circles, in all possible ways.

Below, this problematic situation will be split into several smaller problems including partitioning one-half into two unit fractions using a number of different models, then partitioning one-half into three unit fractions, and then extending the situation to include other unit fractions. Finally, computational tools will be developed to count the total number of representations in each of the cases explored.

Note that because of the natural limitations of both physical and electronic manipulatives, partitions with more than three fractions are not considered in this paper. Another reason for such a constraint is contextual as related to the geometric domain. As will be shown below, a partition of a unit fraction into a sum of either two or three like fractions may be given geometric interpretations in terms of plane and solid geometrical situations respectively. However, the geometric interpretation of partitioning a unit fraction into more than three like fractions requires moving beyond three dimensions.

Partitioning a Unit Fraction into Two Like Fractions

To begin, consider the following:

Problem 1. Completely cover the fraction circle representing one-half of a circle using two other fraction circles in all possible ways.

This activity can be supported either by using traditional hands-on manipulatives or with a computer environment using a dynamic geometry program. Figure 1 shows a representation of Problem 2 using custom tools created within The Geometer's Sketchpad (GSP).

When Problem 1 was given to the class of elementary pre-teachers, it appeared to them that one-half can only be covered in two ways: 1/2 = 1/3 + 1/6 and 1/2 = 1/4 + 1/4, as Figure 1 illustrates. At that point several pre-teachers inquired: "How do we know that these are all the ways?" The course instructor appreciated the inquiry for it naturally generated an interesting follow-up question that the pre-teachers were then asked to explore: How can one use mathematical reasoning to explain the correctness of this hands-on finding? Generally, in agreement with a philosophy of reformed teaching practices that encourages and promotes student's ideas and judgments (Remillard, 1999), such formal inquiry is more successful when motivated by genuine student curiosity.

Interestingly enough, such a formal justification can be provided geometrically in terms of fraction circles. Indeed, an even partition of one-half yields two one-quarter fractions. But between one-half and one-quarter there exists only one unit fraction, namely one-third, which together with one-sixth makes one-half. As one pre-teacher put it in a written form: "There are two different ways. The two different ways are 1/4+1/4 and 1/3+1/6. The reason why there are only two ways is because you can not go smaller than 1/6, because the other one would have to be bigger than 1/3. You can not have 1/5 because if you did it would require three fraction circles to equal 1/2 instead of two. And you can also not use 1/4 with anything else because 1/4 is half of 1/2." While the pre-teachers were not accustomed to writing mathematical proofs, they were surprised to see how simple formal reasoning might be.

[FIGURE 1 OMITTED]

Before fully addressing the solution to the Brain Teaser (i.e., partitioning one-half into three unit fractions), the pre-teachers were introduced to another geometric situation in which the partition of one-half into two unit fractions emerges.

Problem 2. Find all rectangles with integral sides whose areas are numerically equal to their perimeters.

Pre-teachers were asked to explore this problem experimentally by using computational features of The Geometer's Sketchpad enabling the calculation of area and perimeter of specified polygons. Figure 2 shows such a solution in which two rectangles satisfying the problem were found by a pre-teacher. Once again, the pre-teachers were unsure if the problem has been solved completely. This motivated the introduction of an algebraic approach which turned out to be equivalent to representing one-half as the sum of two unit fractions. Indeed, by setting x and y to be the integral dimensions of a rectangle, one can see that the equations xy = 2(x+y) and 1/2 = 1/x + 1/y are equivalent.

At that point another pre-service teacher asked if there was a similar geometric situation leading to the representation of a unit fraction other than one-half as a sum of two like fractions. Note that it is unlikely that such a question would have arisen had the hidden curriculum framework described above not been followed. In other words, the learners were placed in a position where they were able to see the broader mathematical picture and interchangeably use the formal and informal tools at their disposal to connect different mathematical ideas and answer their own questions. Indeed, the following problem may serve as an answer to the above question by the pre-teacher.

[FIGURE 2 OMITTED]

Problem 3. Find all rectangles whose area, numerically, is three times as much as its semi-perimeter.

By using notation developed in the solution of Problem 2, the pre-teachers found that the equations xy = 3(x + y) and 1/3 = 1/x + 1/y are equivalent which led them to the realization that the dimensions x and y of the rectangles sought are precisely the denominators in the corresponding partitioning problem. This partitioning problem was solved by the teachers using The Geometer's Sketchpad (Figure 3) and, once again, the need for formal justification--that 1/3 = 1/4 + 1/12 and 1/3 = 1/6 + 1/6--are the only possible partitions was brought to light.

At this time the pre-teachers understood a way to approach this justification (proof-oriented) task--in much the same way as a partition of one-half an even partition of one-third yields two one-sixth fractions and there are only two unit fractions that fall between these two values, namely, one-quarter and one-fifth. As another pre-teacher stated this argument: "There are only two equations to answer this question because we can not use a fraction which is smaller than 1/12 (mustard) and larger than 1/4 (purple) in order to make a 1/3 fraction circle with only two fraction circle manipulatives. There can not be any other solution because you can not use a fraction circle larger than 1/6. If we use 1/5, then we need 2/15 to make one third, which does not work." While the colors mentioned by the pre-teacher reflect the unique appearance of each fraction circle generated by the GSP environment, the fraction 2/15 results from a numeric computation derived from a geometric situation created through virtual concrete activity. In fact, by showing numerically that the difference of 1/3 and 1/5 does not yield a unit fraction, the pre-teacher grew through three modes of knowledge representation: enactive, iconic, and symbolic (Bruner, 1964).

[FIGURE 3 OMITTED]

Note that some pre-teachers wondered if there would always be just two ways of covering a fraction circle. That is, pre-teachers were attempting to make an inductive leap based on a limited number of observations. Of course, such induction is only true for a special class of fraction circles and one-fourth (that gives three partitions: 1/5 + 1/20, 1/6 + 1/12, 1/8 + 1/8) can serve as a counterexample. In general, as mentioned elsewhere (Abramovich & Nabors, 1998), (3) the partition of a unit fraction 1/n with a prime denominator n into two unit fractions results in the even partition 1/n = 1/2n + 1/2n and a second one based on the identity 1/n = 1/(n+1) + 1/n(n+1), known to Fibonacci who was interested in investigations involving Egyptian fractions (Hoffman, 1998).

A generalization of Problem 3 can lead to an interesting computer investigation, namely, finding the total number of rectangles with integral sides whose area, numerically, is n times as much as its semi-perimeter. Indeed, this number (for which the notation [R.sub.2](n) will be used below) co-incides with the number of partitions of 1/n into two unit fractions and depends on the prime factorization of n. More specifically, if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[R.sub.2](n) = [(2[k.sub.1] + 1)(2[k.sub.2] + 1) ... (2[k.sub.l] + 1) + 1]/2 (1)

Formula (1), whose proof is given in the Appendix, confirms the above-mentioned fact; that is, when n is a prime number, [R.sub.2](n) = 2. Furthermore, as the pre-teachers themselves observed working with fraction circles, the number of ways to partition 1/n into two unit fractions is not greater than n itself; in other words, [R.sub.2](n) [less than or equal to] n with equality at n = 2.

Using spreadsheet techniques described elsewhere (Abramovich, 1999; Abramovich & Brouwer, 2003a), one can construct a table representation of the function [R.sub.2](n), graph it, and by applying spreadsheet conditional formatting (Abramovich & Sugden, 2004) to this table, highlight those n for which [R.sub.2](n) = 2. This process introduces an alternative representation of prime numbers--associated with local minima of the function [R.sub.2](n)--different from the commonly known Sieve of Eratosthenes.

[FIGURE 4 OMITTED]

An interesting extension of the above generalization of Problem 3 is to find the rectangle with the least (greatest) area. Such an extension makes it possible to introduce pre-teachers to new concepts and ideas. Algebraically speaking, the problem is to find the least (greatest) value of the product xy provided that, given n, the sum of reciprocals of its factors equals to 1/n. This problem can first be solved numerically for special cases of n (e.g., n = 2, 3, 4) using the results of above explorations with fraction circles and then algebraically without recourse to calculus (see Appendix). It turns out that the least and the greatest areas are equal to 4[n.sup.2] and n(n + 1[).sup.2] respectively, a trivial conclusion in the case of n prime.

Partitioning into Three Unit Fractions

The next step in pre-teachers' activities was to explore the case when the solution to the Brain Teaser might involve three different fraction circles. That is, using the numerical results of Problems 1 and 3, the pre-teachers were asked to find as many representations as possible of one-half as the sum of three different fractions and interpret these representations through fraction circles using The Geometer's Sketchpad. With the guidance of the instructor and using a spreadsheet-based partitioner of a unit fraction into a sum of two like fractions (Figure 5), six such representations were found in both symbolic and iconic forms (Figure 6). Note that using such a tool, the programming of which is based on formula (A3) given in the Appendix, was necessary for pre-teachers to avoid having to do extensive paper-and-pencil computations in the case of the partitioning of 1/6. Once again, a question related to the contextual roots of this mathematical investigation was raised and the following problem was presented to illustrate how the arithmetic of fractions and topics in high school geometry can be connected.

[FIGURE 5 OMITTED]

Problem 4. How many right rectangular prisms are there with different integral sides and volume numerically equal to surface area?

[FIGURE 6 OMITTED]

The equivalence of this problem to that of partitioning one-half into a sum of three unit fractions becomes clear if one sets x, y, and z to be the integral dimensions of a right rectangular prism, and constructs the equation xyz = 2xy + 2(x + y)z that describes a prism with the property sought. Dividing both sides of the last equation by 2xyz yields 1/2 = 1/x + 1/y + 1/z. Thus, the reciprocals of any three different Egyptian fractions that add up to one-half are the dimensions of such a prism.

To conclude this section note that for the complete solution of the Brain Teaser one has to find a few more cases involving repetition of fraction circles. A pure computational approach to partitioning one-half (or any unit fraction) into a sum of three unit fractions will be discussed in the next section. This approach will be based on the use of a spreadsheet as a three-dimensional modeling tool.

Using a Spreadsheet as a Three-Dimensional Partitioner

As the above generalization of Problem 3 led to computer exploration of the function [R.sub.2](n), the following generalization of Problem 4, namely finding the total number of right rectangular prisms with integral sides whose volume, numerically, is n times as much as the half of its surface area leads to a new computationally challenging problem of finding such a number (for which the notation [R.sub.3](n) will be used below). This time, however, the problem is equivalent to counting all partitions of 1/n into a sum of three unit fractions. Indeed, by setting x, y, and z to be the integral dimensions of a right rectangular prism, one can see that by dividing both sides of the equality xyz =n(xy + yz + xz) by nxyz yields 1/n = 1/x + 1/y + 1/z.

However, unlike the case of [R.sub.2](n), this paper provides a spreadsheet-based tabulation of the function [R.sub.3](n) only. It appears that no closed formula for [R.sub.3](n) similar to formula (2) is known. As n grows larger the number of computations required to tabulate [R.sub.3](n) grows as [n.sup.3]. Thus the authors provide results for n [less than or equal to] 15 only. Figure 7 shows a spreadsheet that tabulated values of [R.sub.3](n) in the range B3:B16. For the case n = 2 (cell A1), cells C1, B2, and C2 display, respectively, the largest values of x, y, and z satisfying the inequalities z [greater than or equal to] y[greater than or equal to] x. Data on the right part of the spreadsheet shows all 10 representations of one-half as a sum of three fractions. For example, the triple (8, 3, 24) displayed in cells D10, G2, and G10, respectively, corresponds to one of the representations pictured in Figure 6. One may wonder: Which representation of one-half as a sum of two unit fractions (Figure 3) is responsible for the triple (5, 5, 10)?

Connecting Mathematics Across the Curriculum

Partitioning of one-half into a sum of three unit fractions has yet another connection to geometry and it deals with the problem of tiling (or tessellating) the plane by regular polygons, something that itself can be viewed as a partition of the plane into regions (Grunbaum & Shepard, 1987). The edge-to-edge tiling may involve fitting together several kinds of regular polygons in a cyclic order, vertex-to-vertex (or edge-to-edge), in such a way that the same polygons surround each vertex. For example, a ternary tessellation in which three polygons share the same point as vertex requires that the sum of the three interior angles must be 360[degrees]. In the case of the edge-to-edge tiling by three regular polygons of sides x, y, and z respectively, this yields a three-variable equation 180[degrees](x - 2)/x + 180[degrees](y - 2)/y + 180[degrees](z - 2)/z = 360[degrees] which turns out to be equivalent to the equation 1/x + 1/y + 1/z = 1/2 studied in the previous section. Thus, each of the partitions of one-half into a sum of three unit fractions can be interpreted as the edge-to-edge tiling by three corresponding regular polygons. Figure 8 shows one such tessellation in which a square, regular hexagon, and regular 12-gon are involved.

[FIGURE 7 OMITTED]

A follow-up activity on hands-on tessellation with regular polygons that pre-teachers were engaged in was the use of The Geometer's Sketchpad in creating geometric constructions like the one pictured in Figure 8. The goal was to use the software as a scaffolding tool to help pre-teachers to achieve conscious control over the conceptual system of rotation in the context of geometry of regular polygons. This, in particular, required grasping the meaning of the above three-variable equation both conceptually and procedurally. Comparing off-and-on computer activities related to this topic resulted in the following observation: when physically manipulating polygons, one consciously uses rotation skills; yet one's attention is not directed towards possessing formal knowledge of geometry. However, by using dynamic geometry software one develops conscious control over the conceptual system of geometry of regular polygons and can then use it as a tool. In other words, in a technology-mediated intellectual milieu, achieving control over a concept occurs in a socially created ZPD where intuitive understanding of the concept meets the logic and formalism needed for its representation through a computational medium. The product of this human-computer interaction aided by competent tutelage is a solution (in this case, a correct geometric construction), which, once internalized, becomes a part of one's consciousness.

[FIGURE 8 OMITTED]

A combination of informal and formal mathematical activities in the ZPD furthered a dialogue between pre-teachers and the instructor through which several interesting questions were raised:

* Is there any connection between a triple of polygons and that of fraction circles?

* Can the edge-to-edge tiling be extended to cover the whole plane like in the case of triangles or quadrilaterals?

* Does the number of edges of the polygons involved affect such an extension?

Whereas answers to these questions require formal mathematics that may be beyond the grasp of elementary pre-teachers (and thus attempts to provide answers may stretch individual ZPDs too far), the very fact of the emergence of sophisticated inquiry in the class of pre-service elementary teachers indicates that a computer-mediated framework of hidden mathematics curriculum is conducive to keeping the teachers within their ZPDs. In full agreement with Vygotsky's (1986) theory of development of scientific concepts, this episode demonstrates that turning points at which one develops mathematical insight cannot be set in advance by the curriculum for they are located in its hidden areas. Expertly aided and technology-enhanced forays into those areas encourage making connections among mathematical ideas--an important condition of seeing their beauty (Cuoco, Goldenberg, & Mark, 1995)--and play a major role in one's mathematical development. However, such forays into hidden areas of mathematics curriculum should be flexible so that their depth depends on learners' interest and intellect. To conclude this section, note that Ward (2003) reported success in conducting similar activities with K-6 pre-teachers through a Vygotskian approach using the combination of hands-on and computer software (TesselMania!) explorations.

DISCUSSION

It has been demonstrated repeatedly by many authors how the appropriate use of computers in teacher education supports meaningful inquiry into mathematical structures, provides a venue for visualization, allows for the observation of patterns, stimulates making and testing conjectures, and helps discover commonalities among seemingly disconnected concepts (e.g., Hatfield, 1982; Hoyles, 1992; Hitt, 1994; Abramovich, 2000; Dugdale, 2001; Harper, Schirack, Stohl, & Garafalo, 2001; Lingefjard, 2002). Computers appear to be particularly valuable when more than one software tool can be brought to bear on a particular problem in order to deepen understanding and promote connection building through multiple representations (Dugdale, 1999). Instructional computing offers empirical confirmation of analytical results and provides conceptual foundation for further inquiries into the problem at hand (Relf & Almeida, 1999).

In making a choice for software to be used in the preparation of elementary teachers, one helpful consideration is that general software tools have great potential to support grade appropriate and powerful inquiry into the subject. Indeed, as this paper has demonstrated, using partitions of unit fractions as a context, a multiple-application environment, including a spreadsheet and dynamic geometry software can be employed effectively in a mathematics education course for elementary pre-teachers within a hidden curriculum framework. For example, the two-dimensional partitioner, by hiding some of the procedural complexity and formal structure of the arithmetic of fractions involved, allowed the pre-teachers to engage new content without being intimidated by its, perhaps pre-conceived, complexity. In such a learning environment, the pre-teachers were able to focus on computer-generated arithmetic and not be distracted by computational details that they find difficult.

In order to enhance the pre-teachers' skills and confidence of mathematics required for successful teaching at the elementary level, one can foster their ability to discover and internalize hidden mathematical meanings (or, alternatively, 'messages' [Ginsburg & Clift, 1990]) in the elementary curriculum with easy-to-understand informal representations of formal mathematics. The use of technology makes it possible not only to make these curricular messages more explicit, but better still, to provide both informal and formal inquiry into their underlying mathematical structure. In doing so, one can extract new concepts from the conceptually rich information of the messages, and further explore their educational potential. This creates an appreciation of mathematics and its different pedagogies, including those technology-enabled and play-oriented, as an integrated whole by the pre-teachers.

It is important for the teachers to experience learning episodes where inquiry into the problem drives the need for more formal mathematical reasoning. As was described early in this paper, pre-teachers were able to partition fraction circles into other fraction circles, largely through trial and error. However, they were genuinely curious about whether all such partitions were found. This curiosity naturally led to an opportunity to engage these elementary pre-teachers in formal mathematical reasoning without them expressing the typical reluctance associated with this kind of mathematical activity. They were then, with some prompting, able to actually prove in a systematic way that the solutions found represented the complete set. In doing so, perhaps to their own surprise, the pre-teachers were employing an analytical/transformational proof scheme, one in which the argument is general (that is, applicable beyond the particular case) and reasoning is formal (Sowder and Harel, 1998).

The sense that mathematics proof is concerned with the "public acceptability of the knowledge being discovered" (Bell, 1979, p. 368) is consistent with the Vygotskian notion of learning as a social activity. Asking pre-teachers to communicate their proof schemata through written speech (as part of portfolio assessment in the case of this course) is a form of creating ZPD. Writing proofs may be seen as an elevation to higher ground what Vygotsky (1978) called "second-order symbolism, which involves the creation of written signs for the spoken symbols of words" (p. 115). Vygotsky argued that such symbolism can be developed through a meaningful play which in the case of pre-teachers involved the use of multicolored electronic fraction circles, a kind of toys for adult learners.

To conclude the discussion, it is fruitful to summarize the impact of this new notion of hidden curriculum within the realm of technology-enhanced mathematics teacher education. First, the notion of hidden curriculum enables pre-teachers' learning of advanced mathematical concepts within the context of a mathematics education course. Pre-teachers develop an understanding of how these advanced concepts provide common structure for the explicit curriculum and connect its different ideas and representations. Additionally, the notion of hidden curriculum provides a structure through which one can systematically address and meaningfully incorporate the NCTM process standard "Connections" (National Council of Teachers of Mathematics, 2000).

Second, the notion of hidden curriculum enables one to introduce technology into mathematics teacher education programs in more than one way. One option is to introduce various tools of technology into a program to support student mathematical investigation and connection building without teaching special courses on technology. Another option is to develop a follow-up course on technology grounded in its educational application in which the pre-teachers concurrently learn technology and mathematics in the context of creating computational environments already familiar to them. As one pre-teacher remarked in a class assessment: "I would like to learn how the tools we used were created." Such a remark indicates that the above idea has merit.

Finally, the notion of a hidden curriculum approach that identifies deep concepts and structures of mathematics makes it possible to elevate elementary pre-teachers' learning of mathematics to "higher ground" (Bruner, 1985, p. 23), so that "the new higher concepts in turn transform the meaning of the lower" (Vygotsky, 1986, p. 202). Climbing to this new height creates in pre-teachers greater self-confidence in their abilities to teach mathematics, a content area they typically perceive to be difficult due to previous learning experiences. In this situation, traditionally poorly understood topics, like formal arithmetical operations with fractions, when highlighted from a different, sometimes advanced, perspective in which the pre-teachers experience success, leads to a greater understanding of and confidence in those topics.

Indeed, classroom observation of pre-teachers working through a series of carefully graduated combination of hands-on and technology-enhanced investigations indicated that their confidence steadily grew as the activities went along. Using technology, the pre-teachers were able to make significant progress in connecting their informal explorations with formal symbolic mathematics. In this setting, they were undertaking genuine mathematical investigations and developing true insights, perhaps for the first time in their lives with these particular topics. To put it differently, experiencing success at a high conceptual level expanded their ZPDs, thus empowering them to tackle procedural details with confidence--a skill expected from an elementary teacher. It is through the expansion of this zone that the notion of a hidden mathematics curriculum has the potential to significantly broaden pre-teachers' content knowledge and bring positive change in various teaching-related psychological phenomena, including self-confidence and mathematical anxiety. This, in turn, can affect the way that mathematics is taught in the schools. As the Glenn commission states, "High-quality teaching requires that teachers have a deep knowledge of subject matter. For this there in no substitute." (United States Department of Education, 2000, p. 22).

APPENDIX

Solution to extension of Problem 3.

As it was shown above (Problem 2), if x and y are integral sides of a rectangle whose area, numerically, is n times as much as its semi-perimeter, then 1/x + 1/y = 1/n (A1)

From this equality it follows that y = nx(x - n), thus the area of the rectangle is

xy = n[x.sup.2]/(x - n) (A2)

Simple algebraic transformations and the arithmetic-mean-geometric-mean inequality applied to the right-hand side of relation (A2) yields

n[x.sup.2] / (x - n) = n(x - n + n[).sup.2]/(x - n) = n(x - n + 2n + [n.sup.2]/(x - n)) [greater than or equal to] n(2n + 2((x - n)[n.sup.2]/(x - n)[).sup.0.5) = 4[n.sup.2].

Hence xy [greater than or equal to] 4[n.sup.2] with equality when x = y = 2n. In other words, a rectangle with the above-mentioned property and the least area is the square of side 2n.

As far as the greatest area is concerned, one can graph the function f(x) = n[x.sup.2]/(x - n) for different values of n using an appropriate graphing software (e.g., The Graphing Calculator - 3.2 [Avitzur, Gooding, Herrmann, Piovanelli, Robbins, Wales, & Zadrozny, 2002]) The graph shown in Figure 9 enables one to see that the function f(x) assumes maximum value at each of the end points of the segment [n + 1, n(n + 1)] (cf. the above-mentioned identity known to Fibonacci). Thus the largest value of area of a rectangle whose area, numerically, is n times as much as its semi-perimeter, equals to n(n + 1[).sup.2]).

Proof of formula (1)

In order to prove formula (1), note that equality (A1) can be transformed to

y = n + [n.sup.2]/(x - n) (A3)

One can see that the number of integral pairs (x,y) satisfying equation (A1) depends on how many time x - n divides [n.sup.2]. Because n + 1 [less than or equal to] x [less than or equal to]2n, the denominator in (A3) varies 1 [less than or equal to] x - n [less than or equal to]n and thus the problem of finding [R.sub.2](n) is equivalent to finding how many divisors among 1 through n does [n.sup.2] have. One can also see that each such divisor k has a counterpart [n.sup.2]/k and in order to find [R.sub.2](n) one can find the number of all divisors of [n.sup.2], increase this number by one and divide the sum by two. Assuming that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], one can note that each factor [p.sub.i] can be chosen in 2[k.sub.i] + 1 ways, thus formula (1).

[FIGURE 9 OMITTED]

Acknowledgement

The authors wish to thank John Olive for useful comments regarding using a spreadsheet as a three-dimensional partitioner.

Notes

(1.) It is worth noting that this hidden nature of a number system was recognized more than 70 years ago by Vygotsky (1987): "In school, the child does not learn the decimal system as such. He learns to write numbers, add, multiply, and solve problems. Nonetheless, some general concept of the decimal system does develop" (p. 207). Clearly, the development of this general concept occurs through instruction making explicit the hidden curricular message.

(2.) A fraction circle, commonly used in elementary classrooms to illustrate the idea of fraction, is a manipulative representing a unit fraction and shaped as a sector of a whole circle.

(3.) This paper also notes another contextual interpretation of the number of partitions of a unit fraction into two other like fractions formulated in terms of work problems.

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SERGEI ABRAMOVICH

State University of New York College at Potsdam

USA

abramovs@potsdam.edu

PETER BROUWER

State University of New York College at Potsdam

USA

brouweps@potsdam.edu
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