What are Billy's chances? computer spreadsheet as a learning tool for younger children and their teachers alike.

This article demonstrates how multiple features of a computer spreadsheet extended motivational activities with M&Ms and enhanced mathematical thinking of younger children in the context of data analysis and probability. Technology-enabled childhood pedagogy described in this article was implemented by two elementary preteachers who, by working as apprentices to their graduate professor, developed a high appreciation of the pedagogy and acted as agents of change in a P.D.S. setting.

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The National Council for Accreditation of Teacher Education (1997) Task Force on Technology and Teacher Education assessed the situation with the integration of technology in teacher education. It was found that preservice teachers rarely have occasion to apply technology in their courses and are not engaged in role models of faculty teaching with technology. This applies to all content areas, mathematics included. To improve the situation in the context of secondary mathematics, a set of guidelines for the use of computers m the preparation of teachers and appropriate technology-enhanced activities were developed at the University of Virginia's Curry School of Education's Center for Technology and Teacher Education (Garofalo, Shockey, Harper, & Drier, 1999; Garofalo, Drier, Harper, Timmerman, & Shockey, 2000). It was suggested that teacher education programs should provide learning experiences for preteachers of mathematics in using a computer as an exploratory tool in both the theoretical and applied mili eus.

The guidelines could be extended to include the preparation of elementary teachers with the focus on "learning with technology, not about technology" (Shaw, 1997, p.7). In other words, all preteachers of mathematics should become familiar with educational uses of computers and best practices of technology-enabled pedagogy during all stages of their education including regular coursework and student teaching.

As Browning and Klespis (2000) have pointed out, in addition, preteachers should be given authentic experiences in developing technology-enabled activities for a precollege classroom. This position has been further advanced by Willis (2001) who suggested that preteachers should be given opportunities for professional growth including teaching their own technology-enhanced lessons. These tenets combined with the Technology Principle (National Council of Teachers of Mathematics, 2000) that views the elementary classroom as the appropriate place for computer-enhanced instruction, suggest that all efforts should be made to actively engage preteachers in incorporating computers into the childhood mathematics curriculum. This article addresses the recommendations and concerns by presenting the authors' work on developing spreadsheet-based environments for younger children and incorporating these environments into the 3rd-grade classroom in the framework of an elementary mathematics methods course with a student te aching component.

THREE APPROACHES TO TECHNOLOGY-ENABLED PEDAGOGY FOR PRETEACHERS

Three approaches to providing elementary preteachers with an experience in applications of technology to teaching and learning mathematics that currently have been used in the teacher education department of the State University of New York at Potsdam are outlined. The first approach is to introduce preteachers to technology-enabled mathematics pedagogy through a computer-enhanced mathematics methods course. Usually, in such a course a computer is available for demonstration purposes only. In an egalitarian classroom, however, the notion of demonstration may be extended from a teacher-centered to a student-centered environment to include the students who feel comfortable sharing emerging ideas and practicing computer skills in front of the class. For example, in a mathematics methods course such student-centered activities may include playing the game of chance with a follow-up discussion of mathematical concepts behind the game (Drier, 2001; Dugdale, 2001), using the application KidPix (Broderbund Software Inc., 1998) as a problem-solving tool (Abramovich, 2001), and employing Internet-based manipulatives (Bulaevsky, 1999) while helping elementary preteachers construct meanings for arithmetical operations, to name just a few activities.

The second approach is to offer a course that focuses on the design of technology-enabled lessons of mathematics (Drier, 2001; Harper, Schirack, Drier, & Garafalo, 2001; Abramovich, 2000). Through active participation in such a course, preteachers can acquire the ability of enhancing the teaching of mathematics topics with technology, develop skills in formulating questions to be explored on a computer, and learn to decide on directions in which original explorations can be extended. Ultimately, these skills and abilities help preservice teachers gain intellectual courage in making pedagogical and curricular decisions in a technological paradigm. A shortcoming of this approach is in the absence of an actual application of technology to the elementary classroom by preteachers.

The third approach to the introduction of technology into a mathematics teacher education program is grounded in preteachers' participation in a methods course with a student teaching component. In the first approach, such a course is computer-enhanced and preteachers have the opportunity to be introduced to the best practices of using technology in mathematics education. In addition, preteachers are encouraged to incorporate these practices into their own (student) teaching. In such a way, the third approach has a potential to bring about the apprenticeship model of learning to design technology-rich curriculum materials with a subsequent implementation of the materials into the fabric of real classroom. It is this approach that is the focus of this article.

OBSTACLES IN THE WAY OF BRINGING TECHNOLOGY TO YOUNGER CHILDREN

Many obstacles may stand in the way of implementing the third approach. One deals with a quite common situation when a preteacher's field work ends up with a traditional teaching (or its observation) with no technology integration. Data collected recently by the first author in the span of four semesters indicates that only 15% of all preteachers involved in field work have observed some use of computers in the teaching of mathematics in elementary classroom. This situation, however, is not surprising because, as research has suggested (Pieper, 1995; Hanszek-Brill, 1997; Olive, 2000), teachers' beliefs about mathematics pedagogy and the comfort level with available technology greatly affect the learning environment of their own classroom.

Often, elementary teachers hold very strong beliefs that using computers with younger children may serve the purpose of a reward for a successful performance in a traditional learning environment. Also, the reluctance of teachers to integrate computers in elementary mathematics classrooms may stem from the lack of content knowledge that, in turn, creates math anxiety (Peterson & Barnes, 1996). In addition, a common perspective on the use of computers as tools for drill and practice hinders their use in exploratory and investigative contexts that promote conceptual understanding of mathematics.

In such a way, host teachers' comfort level with technology, inadequate mathematical content knowledge, and their pedagogical beliefs about technology use may be identified as the major obstacles in integrating computers into elementary school mathematics curriculum through student teaching. Yet, one can find a way around these obstacles as the following pages will demonstrate.

THE M&M PROJECT

As it was previously mentioned, this article is about using computers with younger children (aged 8-9 years old). It tells the story of two preservice elementary teachers (hereafter referenced as teachers) who participated in an authentic preservice teaching experience that introduced spreadsheet-based mathematics lessons to the children. The teachers worked in apprenticeship with a graduate professor of mathematics education (hereafter referenced as professor). The team designed a computational environment to be used in a 3rd-grade classroom. This setting was designed in accordance with the Principles and Standards for School Mathematics (National Council of Teachers of Mathematics, 2000) that singled out a spreadsheet as an appropriate tool for doing mathematics as early as in grades 3-5. According to this document, students at this grade level are expected to acquire the ability to set up a simple exploratory spreadsheet in their mathematical pursuits and "use it to pose and solve problems, examine data, and investigate patterns" (ibid., p.207).

Prior to the project's commencement, the teachers themselves lacked confidence in their technological skills. They had limited knowledge of using a spreadsheet as a mathematical/pedagogical tool and no experience in constructing spreadsheet-based computational learning environments. However, working one-on-one with the professor throughout the duration of the semester, the teachers learned a great deal about the opportunities that exist when computers are used in a classroom. In fact, they now consider this to be one of the most valuable lessons they learned during their preservice teacher education.

The project took place in a small rural school in upstate New York. The goal of the project was to put together a set of activities that initially included hands on mathematics lessons and then progressed towards a series of lessons enhanced by a spreadsheet-based computer environment; the goal being to extend the hands on activities. The principal of the host school volunteered eight advanced third graders (hereafter referenced as students) for the project but was initially skeptical about the use of computers to teach mathematics to younger children. He felt that a computer should be supplemental rather than the main focus of a lesson.

Such an attitude was expected. In fact, the implicit intention behind this project was to change similar beliefs commonly found in the elementary education community. Nevertheless, the principal was very cooperative and putting his skepticism aside, allowed the teachers to communicate their vision of technology as a natural extension and cultural amplifier of a conventional lesson. A compromise was struck whereby spreadsheets and M&M mathematics (in other words, the use of multicolored candies both as manipulatives and incentives) were integrated. This union forged a bridge between alternative practices put forth by the National Council of Teachers of Mathematics (1989, 1991, 2000) and the already traditional hands on mathematical activities.

CLASSROOM SETTING

Once a project outline was agreed upon, the teachers and professor began working on a computer program that used spreadsheet-enabled M&M mathematics. It was decided to make this experience a memorable one for the students, therefore M&M candies were chosen as the data to be processed by spreadsheets. The end product resulted in a manipulative-computational environment using spreadsheet sliders as graphing tools. This significantly reduced the students' need for advanced computer skills. Ultimately, it was hoped that this environment would serve the purpose of providing the children with a visual manipulative that would aid comprehension in the context of data analysis and probability.

The students were working in the groups of two; each group with their own laptop computer supplied by the school--a setting similar to the one described by Ainley (1995) except that the activities were not extended to using laptops at home overnight. The students were incredibly bright and eager; yet their attention span needed to remain one of the top priorities for the teachers. This is where the use of M&M mathematics proved valuable. All four groups started at the same time and received the same amount of assistance. Because the students did not differ significantly from a mathematical perspective, the teaching team was able to work with everyone simultaneously. However, due to the differences in comfort level with computers, it eventually became necessary to address individual needs of the students at different times and stages of the project. This detracted from some of the benefits derived from the small intimate environment prevalent at the onset of the project. Even though the students were excited a bout the project some became restless while waiting for their questions to be answered.

The physical environment was quite conducive for the purposes the teachers had in mind. In the groups of two, the students sat around one large table (Figure 1) creating an ideal situation where individual assistance could be immediately offered upon request. Also, such an arrangement allowed for the sharing of students' ideas, an occurrence not always possible in larger educational environments. Overall, and compared with many elementary schools in this area and rural Ontario (the residence of one of the teachers) the school was extremely well equipped for the use of advanced technology as an aid for the enhancement of a teacher-led instruction.

FROM TEDIOUS TO INTERACTIVE GRAPHING

The first meeting with the students entailed a brief lesson on graphing concepts. This included the idea of the Cartesian plane and the types of graphs that can be constructed on this plane. The focus, however, was on the use of bar graphs. To this end, each student was given two bags of plain and peanut M&Ms respectively, worksheets with instructions and questions, and a stylized (computer-generated) grid paper to record their data in the form of bar graphs. The second meeting with the students included the discussion of individual graphs constructed at home.

Following are the types of hands on tasks that the students were given: Count the number of each color (among red, yellow, green, blue, and orange) in your bags of plain and peanut M&Ms and represent the numbers found in the form of bar graphs. Eat all M&Ms that do not belong to your data. Use the graphs to decide: (a) which color has the most and the least candies in the plain (peanut) M&Ms? and (b) what combination of two colors when added together have the most and the least candies in the plain (peanut) M&Ms? Note that these tasks were completed as a combination of class and home activities. In particular, the students were not allowed to take their candies home, and the motivational element (i.e., eating candies) was limited to class time only. An example of one student's graph is depicted in Figure 2.

After the idea of graphing as a visual representation of data collected had been internalized by the students, the introduction of the computer program was highly conducive for the students' further conceptual development. Their ability to manipulate a physical environment (i.e., drawing graphs) was significantly below their intellectual potential to understand mathematical concepts involved. In fact, the design of the computer program was aimed at liberating the students from the tedium of extensive physical manipulation enabling them to concentrate their attention on mathematical tasks and to work on increasingly difficult problems without being hindered by the lack of sophisticated fine motor skills required.

On the third day the students were very excited about using computers to plug in their numerical data and to interactively visualize the friendly production of familiar graphs in accustomed colors (Figure 3). The use of spreadsheet sliders incorporated into the computer program made it possible to keep this task at the click-and-see level of physical manipulation. The students remained eager and focused on the task of creating a computer copy of their hand made graphs because they knew they would be printing this copy on a color printer, and taking it home to share their computer skills with their family.

INTUITIVE COMPARISON OF CHANCES IN A COMPUTER ENVIRONMENT

The M&M mathematics helped the teachers to balance motivation with challenge. Principles and Standards for School Mathematics (National Council of Teachers of Mathematics, 2000) stress the importance for all students, including those in primary grades, not only to collect, organize, display, and analyze data, but to use the data to discuss the concept of probability and apply it in realistic problem solving situations. With this in mind, two spreadsheet-based assignments were given to the students at the fourth meeting.

Assignment 1. Billy wants to eat red M&Ms only. There are two bags of M&Ms available. If there are 4 red plain out of 10 total in the first bag and 10 red peanut out of 20 total in the second bag, for which bag does Billy have more chances to get a red M&M? What are these chances? Use sliders to create the bags of M&Ms. You may use the bar graphs to answer the first question.

Assignment 2. Billy ate (by using sliders) one red M&M from each bag. How many more red peanut M&Ms does Billy have to eat in order to give Mary more chances to get a red plain M&M than a red peanut M&M? Use sliders to create new bags of M&Ms. What are Mary's chances for getting a red M&M for both bags?

From the quantitative perspective, Assignment I requires the comparison of two fractions, 4/10 and 10/20. The project was conducted in the fall when the students did not have at their disposal a formal means for comparing such fractions. An intuitive grasp of fractions as quantities can be developed using their iconic representations. To this end, spreadsheet-based bar graphs can be employed for an intuitive, visual comparison of fractions. Such graphs are an analogue of those used to represent and compare whole numbers. In such a way, students can acquire an informal, intuitive understanding of quantitative relationships between fractions before they actually learn about formal ways of comparing fractions. In other words, the comparison of chances can be developed first using mathematical visualization in the iconic environment of a spreadsheet. The use of fractions as symbolic representations of the chances develops later on the foundation provided by visual skills. That is, students can build their leanin g of formal operations with fractions on the knowledge they received in a fun and informal, yet mathematically meaningful, way. Figure 4 presents one student's (correct) response to questions posed in Billy's problem (Assignment 1).

Analysis of the students' work indicated that all of them completed Assignment 1 correctly whereas, only three students successfully completed Assignment 2 (Figure 5). However, the very fact that the three students were able to deal correctly with Billy and Mary's problem (Assignment 2) is itself quite impressive. Without formal knowledge of mathematical concepts behind the assignment, these students were able to use technology as the amplifier of their mathematical thinking in a meaningful context.

Providing foundation for algebra learning

Note that Assignment 2 is, indeed, a challenge for a 3rd-grade student. It deals with new quantitative descriptions that may stem from the M&M mathematics and allows the student using a spreadsheet to intuitively explore complex ideas that are not normally accessible otherwise. On a formal, decontextualized level this assignment concerns the property of proper fractions to decrease as both its numerator and denominator decreases by the same number. In more general (algebraic) terms this property means that for any two positive numbers a and b, such that a<b, the function f(x)=(a-x)/(b-x) decreases monotonously as x grows larger and Figure 6 represents this statement graphically. When put into the context of M&M mathematics, however, this abstract statement can be described in the following friendly and less mathematically challenging form: By eating red peanut candy, Billy makes chances of getting it from the bag smaller on each turn.

In such a way, Assignment 2 provides the students with a strong foundation for learning important mathematical ideas at the upper elementary and secondary levels through intuitive experience with the ideas at a lower level. It gives the students an appropriate context to analyze change and builds up to an understanding of functions and their behaviors in high school.

Interaction with the students in the technology-rich environment provided the teachers with a strong belief in the power of a spreadsheet as a learning tool and the abilities of the students to handle the complexity of the tasks when alternative approaches to the learning of mathematics are introduced. It encouraged the teachers to advance their own knowledge about mathematics taught with technology in context. This resulted in the construction of new computational environments that can be used in exploring the complex web of ideas about fractions at an upper elementary level. An example of such an extended exploration with fractions in context is provided in the next section.

ADVANCED SCENARIOS FOR LEARNING FRACTIONS IN CONTEXT

This section contains ideas about explorations with fractions in the context of M&M mathematics that may lead to the study of algebraic inequalities using the combination of software, something that is appropriate for the advanced upper elementary, or secondary and college (teacher education) levels only. Although the main focus of this article is on using technology with younger children, there are other important issues that unfolded in the course of the project development. These include the usefulness of a context as a scaffolding device for connecting mathematical ideas at different levels of sophistication, the use of technology as an amplifier of mathematical thinking, and the importance of fostering reflective thinking in the absence of technology. With this in mind, consider the following

Assignment 3. Part 1. Billy wants to eat red M&Ms only. There are two bags of M&Ms available. There are 6 red plain out of 32 total in the first bag and 4 red peanut out of 20 total in the second bag. For which bag does Billy have more chances to get a red M&M?

Part 2. Billy ate one red M&M from each bag. For which bag does Billy have more chances to get a red M&M now?

Part 3. Using sliders create another two bags with the same property (that is, two bags with more chances for a peanut M&M on the first turn and with more chances for a plain M&M on the second turn).

The situation described in Assignment 3 is different from that of Assignment 2. Indeed, by eating just one red candy from both bags, Billy reverses chances in the favor of the first bag with plain M&Ms. Arithmetically, this means the following: whereas 6/32<4/20, the opposite inequality holds true for fractions 5/31 and 3/19, namely 5/31>3/19. Apparently, such a pair of bags (or, alternatively, fractions) is not easy to find. However, using a spreadsheet as a computational environment enables one to generate data for problems similar to the one described in Assignment 3.

Figure 7 shows the slider-controlled spreadsheet that displays three such pairs. In particular, the spreadsheet generated the fractions 5/26 and 4/20 such that 5/26<4/20, yet 4/25>3/19. In terms of M&M mathematics, this numeric relationship can be presented through the following situation:

When the first bag has 5 red plain M&Ms out of 26 total, and the second bag has 4 red peanut out of 20 total, then there are more chances to get a red plain than a red peanut M&M. However, after eating one red M&M from each bag, the chances change in the favor of red peanut M&Ms.

As mentioned elsewhere (Abramovich & Strock, 2002) an important role that the computer as an intellectual tool can play in one's cognitive development is the emergence of so-called residual mental power that can be used in the absence of the computer. To assess this mental power in the context of M&M mathematics, one may be presented with the following off-computer task: Analyze what is special about the four numbers involved in the above situation and try to generalize a pattern discovered. A rather simple analysis of the four numbers may lead into the following algebraic "generalization":

Let n and m be any whole numbers; then two opposite inequalities m/nm>n/([n.sup.2]+J) and (m-I)/(nm-1)<(n-1)/[n.sup.2] hold true.

Indeed, for any whole number n the inequality 1/n> n/([n.sup.2]+1) holds true and the inequality 1/5>5/26 serves as a numeric example. In addition, m/nm>n/([n.sup.2]+1) because 1/n and m/nm are equivalent fractions. Numerical evidence provided by the bags of M&Ms in the last situation suggests to conjecture that (m-l)/(nm-l)<(n-l)/[n.sup.2]. Is this conjecture true for all n and m? Apparently, if m>n this conjecture is not true and the pair (n,m)=(5,6) may serve as a counterexample. Would it then be true for all m [less than or equal to] n?

With the goal to explore this conjecture, a new piece of technology--a computer-based graphing calculator (Avitzur, Gooding, Robbins, Wales, Zadrozny, & Apple Computer, Inc., l998)-was introduced to the teachers. Figure 8 shows the graph of the inequality (m-l)/(nm-l)<(n-l)/[n.sup.2] in the plane (m,n) along with the graphs of the straight lines n=m (passing through the origin) and n=m+l (penetrating the shaded region). Mathematical visualization provided by the graph allowed the teachers to comprehend that no points with whole number coordinates belong to the graph below the line n=m+l. In other words, the teachers were able to recognize that the above conjecture breaks down. Computer graphics enabled for the teachers' alternative insight into the nature of fractions and it brought about a clear and simple algorithm for generating problems similar to that presented in Assignment 3.

This example shows that the use of technology in the mathematics classroom, be it the elementary, secondary, or college level, has to be a mixture of computing activities and mental tasks that are true reflections on the activities. To conclude this section, note that the environment in Figure 7 allows for another interesting investigation. Namely, given the quadruple of whole numbers (n, m, l, k) such that n<m, l<k, n/m<zk/l, find the smallest whole number k (k<a, k<c) for which (a-k)/(b-k)>(c-k)/(d-k). In other words, what is the minimum number of red M&Ms that Billy can eat from each bag so that the chances would become opposite?

For example, if there are two bags with 9 red plain and 8 red peanut out of 21 and 18 M&Ms respectively, Billy has more chances to get a red peanut M&M from the first bag. However, he can equalize chances in three steps and then make the chances in the favor of a plain M&M on the fourth step. Numerically, this means that starting with the inequality 9/21<8/18 and subsequently decreasing each numeral in the fractions by one, yields first the equality 6/18=5/15 and then inequality 5/17>4/14. The teachers wondered: Can Billy always equalize the chances in such a situation? Why or why not?

CONCLUDING REMARKS

The M&M project described in this article was successful in the sense that multiple educational objectives were achieved. These included:

* the use of technology with younger children;

* using computers in elementary teacher education;

* learning mathematics and technology in context;

* affecting elementary teachers' beliefs about technology; and

* initiating change in a professional development school setting.

The following remarks are in support of the previous statement.

Indeed, the philosophy behind the project was to demonstrate that computers do enhance any mathematical topic taught to younger children. As was mentioned above, the school principal suggested graphing as the topic and the teachers and professor then chose a spreadsheet as a (interactive) computer graphics environment. This environment was a natural extension of traditional hands-on activities. It liberated the students from the tedium of pencil-and-paper graphing and enabled them to concentrate their attention on mathematical tasks and the interpretation of graphs as chances. Spreadsheets allowed the students to surpass physical and developmental limitations associated with their age. This was accomplished by providing a mathematical environment that focused on their mental and intuitive capacities rather than on fine motor development. As a result, a high level of success in children's performance on a nontraditional assignment was observed.

In this regard, the authors agree with Ainley (1995) who conjectured that relatively low level of success among middle and high school students in interpreting graphs reported by researchers in USA (e.g., Padilla, McKenzie, & Shaw, 1986) and UK (e.g., Sharma, 1993) may be due to an inadequate pedagogy rather than cognitive barriers. In fact, this conjecture has recently been advanced to suggest that from a didactical perspective, the interpretation of graphs and not their production should be seen as a purpose of graphing activity at least at the elementary level (Ainley, Nardi, & Pratt, 2000).

From the perspective of using computers in elementary teacher education, the project demonstrated how instructor-student relationship in a mathematics method course with a student teaching component originates on the expert/novice plane and quickly moves to the master/apprentice plane. The master/apprentice metaphor is used here in the sense that the teachers were given an opportunity to learn to construct computational environments working as apprentices to the professor. The teachers were able to overcome their own limitations regarding computer skills, knowledge of mathematical content and technology-enabled pedagogy. The apprenticeship model of learning developed the teachers' confidence in using computers with younger children. Didactical situations that emerged during this work were new for the teachers and professor alike. In that setting, both parties learned about spreadsheet-enabled childhood pedagogy in the applied context.

Teaching and learning mathematics with technology in context enabled the discovery of advanced mathematical activities that furthered the teachers' knowledge about properties of fractions and brought about instructional scenarios for upper elementary levels not generally known. The teachers developed an appreciation for an appropriate context as a scaffolding device for connecting mathematical concepts at different levels of complexity. Also, they learned to value reflective mathematics pedagogy in the absence of technology.

Regarding an elementary teacher's reluctance to integrate computers m childhood education, the project led to a beneficial conclusion. Using a computer as a natural extension and cultural amplifier of a conventional lesson for younger children has a potential to forge a bridge between alternative teaching strategies and the already traditional hands on activities. It is by moving along such a bridge that attitudes change towards technology as an appropriate learning tool for children and their teachers alike.

Finally, from the professional development school perspective, the authors were extremely cautious about imposing their views on the school stakeholders, although holding a PDS title commonly implies "an institutional commitment to a particular vision of change" (Valli, Cooper, & Prankes, 1997). As the project demonstrated, open-minded perspectives of all parties involved were conducive to a mutually beneficial compromise whereby spreadsheets and M&M mathematics were integrated into the childhood curriculum. The authors believe that the project enabled the teachers to act as agents of change in the PDS environment and helped them develop a disposition towards the fieldwork classroom "as a site for inquiry" (Ebby, 1999, p. 71).

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