Technology as a medium for elementary preteachers' problem-posing experience in mathematics.

This article attempts to extend current research and development activities related to the use of technology in problem posing, to early grades mathematics. It is motivated by the authors' work with elementary preservice teachers toward this goal, both at the graduate and undergraduate levels. New York State Learning Standards for K-4 mathematics serve as a background for technology-enabled learning. Spreadsheet-based environments designed by the authors (using Microsoft Excel 2004) are introduced from a tool kit perspective, enabling a meaningful combination of manipulative and computing activities by elementary preteachers and their students alike.

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One of the central tenets of the current reform movement in mathematics education holds that appropriate use of tools of technology is integral to the teaching and learning of mathematics at all grade levels. In the context of preparing teachers for the 21st century classrooms, the word "appropriate" may include the notion of teacher as a technologically minded curriculum developer, capable of exploring--and helping his/her students to explore--new avenues in mathematical content; in particular, being skillful in the use of technology for posing and solving problems. This puts mathematics educators involved in the preparation of teachers for elementary schools in a unique position because such technology-enabled changes in pedagogy must be feasible from the very outset in the chain of children's educational experiences.

It has been more than a decade since the National Council of Teachers of Mathematics (1991) suggested that technology has the potential "to enhance and extend mathematics learning and teaching" and that "the most promising are in the areas of problem posing and problem solving in activities that permit students to design their own explorations and create their own mathematics" (p. 134). Nonetheless, as an extensive search of the literature indicates, the few existing papers that describe the use of technology as a medium for problem posing are mostly concerned with the secondary mathematics education (Abramovich & Brouwer, 2003; Abramovich & Norton, 2006; Hoyles & Sutherland, 1986; Laborde, 1995; Noss, 1986; Yerushalmy, Chazan, & Gordon, 1993).

This article attempts to extend current research and development activities related to the use of technology in problem posing, to mathematics education in early grades. It has been motivated by the authors' work with elementary preservice teachers (referred to as teachers) toward this goal, both at the graduate and undergraduate levels, using a resource guide (New York State Education Department, 1998) that provides guidance to districts and schools in New York for structuring local curricula and instruction. This curriculum document focuses on using open-ended problems with young children, something that requires special skills by the teachers. These skills may include the ability to use computers as cognitive amplifiers in exploring the open-ended nature of appropriate mathematical situations. In an open-ended environment of a technology-enhanced classroom, one can expect young children to ask unforeseen questions about familiar concepts. This, in turn, has a potential for learning to become a reciprocal process (Confrey, 1995; Steffe, 1991). Apparently, the implementation of such a dynamic perspective on the learning of mathematics begins with the preparation of teachers.

It should be noted that to make technology integration into a quality teacher education program a success, one has to make right decisions regarding the choice of software involved. One type of software, which for more than two decades has gained widespread recognition as an exploratory tool, is a spreadsheet (Baker & Sugden, 2003). Designed originally for non-educational purposes, a spreadsheet may be conceptualized in educational terms as a combination of an electronic blackboard and electronic chalk (Power, 2000). Thus, it came as no surprise that the Principles and Standards for School Mathematics (National Council of Teachers of Mathematics, 2000) recommended that spreadsheets be used with children as early as in grades 3-5. In support of such a recommendation, several authors reported successful uses of spreadsheets with young children, as well as with their future teachers in various grade-appropriate contexts (Abramovich, 2003; Abramovich, Stanton, & Baer, 2002; Ainley, 1995; Drier, 1999, 2001). This article introduces spreadsheet-based environments to be used both with the teachers and their students in the context of situated addition and subtraction as a medium for both problem posing and problem solving. It focuses on multiple issues, both practical and theoretical, associated with the use of technology as a scaffolding device for teachers' open-ended problem posing experiences as well as young children's ability to explore that type of problem. As will be shown, problem posing, by definition, includes a problem-solving phase as an important part.

Open-Ended Mathematics Pedagogy

It has been three decades since the effectiveness of using an open-ended approach in facilitating and evaluating one's higher order thinking in mathematics was emphasized by educational researchers (Becker & Selter, 1996; Becker & Shimada, 1997; Shimada, 1977). Open-ended pedagogy does not require only one correct answer. Rather, it requires a "multiplicity of correct answers or approaches to provide experience in finding something new in the process, through combining children's own knowledge, skills and mathematical ways of thinking" (Becker & Selter, 1996, p. 526). This suggests the importance of providing teachers with experiences in developing higher order thinking skills among young children through technology-enhanced use of open-ended problems. The following is an example of a problem with a hidden open-ended structure (New York State Testing Program, 1998):

    Michael has two quarters, two nickels, and two pennies, while Tara
    has a quarter, a nickel, and two dimes. Which coins could Michael
    give Tara so that they both have the same amount of money?

To reveal an open-ended pedagogical potential of this testing problem, note that if one is allowed to alter its numerical structure as well as rules of actions involved, many interesting questions can be explored, among them:

* Does the problem have only one correct answer? Why or why not?

* Would an answer to this question be different if Michael and Tara were allowed to exchange coins?

* Having the same amount of money but in different coins, could Michael and Tara not find a solution to this problem?

* Does the answer to the previous question depend on the rules of action involved?

* If Michael and Tara each have two coins, what are the coins that allow equal sharing? Does this question have one and only one answer, both in terms of rules of action and coins involved?

* Given the amounts of money each of them have, what is the minimum number of coins that can be used to solve this money-sharing problem?

* For which combination of coins could Tara share money with Michael so that after sharing, Tara still has twice (three, four, five, etc. times) as much money as Michael?

Apparently, the variation provided in the last question is beyond mathematical abilities of fourth graders--for whom the test was written. Yet the case of Tara and Michael having, respectively, 13 cents and 5 cents can easily be modeled at an even lower grade level under a teacher's guidance by using pennies. For a teacher to be capable of providing such guidance, he or she should have experience in designing open-ended situations of this kind. To help teachers formulate such problems and, consequently, to identify coins involved as a way of creating conditions for problem solving, the authors designed a spreadsheet-based environment with multiple worksheets that, in addition, have the potential of fostering mathematical reasoning and thinking skills of young children. This experience for teachers should include the ability to solve a problem, perhaps in more than one way, by using grade-appropriate strategies and techniques. Such an approach turns the original problem into an open-ended one in which numbers involved become parameters that can be altered and tested in a problem-solving situation and then chosen to signify the completion of the problem-posing phase of this intellectual activity.

The Money-Sharing Environment

A spreadsheet-based environment designed to support problem-formulating in the context of money sharing consists of two types of worksheets--computational and manipulative. The former type includes a single worksheet designed to develop a numerical part of the problem; the latter type includes multiple worksheets. The use by teachers of a manipulative worksheet (MW) is to ensure that problem with numerical data chosen can, indeed, be solved under a given rule of actions (i.e., either without or with money exchange). The environment has the potential to be used by young children provided that teachers understand three didactical objectives that structure such a use. The first objective is to situate one's learning of addition and subtraction in context. The second objective is to provide young children with the experience of arriving at more than one correct answer in open-ended, contextually familiar problematic situations. The third objective is to enhance one's comprehension of the concept of money.

The spreadsheet pictured in Figure 1 includes three slider-controlled cells--D3, E3, and C3--the first two of which (problem-posing sliders) enable the parameterization of difficulty of a problem structure in terms of properties of numbers representing money (e.g., relationship between the last digits, mutual proximity of numbers, the size of numbers). The third (problem solving) slider allows for a computational solution through trial-and-error that the spreadsheet accepts as the correct one through the message "EQUAL!" appearing in cell F3. Simultaneously, it triggers the display of the amount of money after sharing and provides an evaluative comment "WOW!" in cells H6 and H2 respectively.

[FIGURE 1 OMITTED]

MW 1 pictured in Figure 2 is linked to the computational referent through cells B19 and H19. It includes a four-coin storage from which the coins can be retrieved through the use of the corresponding macros. In addition, the coins are arranged in decreasing order of their denominations and are put into a one-to-one correspondence with macro buttons labeled according to the coins' names. MW 2 (not pictured here) differs from MW 1 in the coins' appearance while preserving the one-to-one correspondence between a coin and its label. Finally, in MW 3 (not pictured here) the coins are turned with heads up, are not ordered by denominations, and have had the one-to-one correspondence between coins and labels removed. Besides being useful for teachers in the context of problem posing, the three manipulative worksheets have practical applications to early childhood mathematics.

[FIGURE 2 OMITTED]

Implications for Early Childhood Mathematics

The development of standards for early childhood mathematics has been a focus of research in recent years (Clements & Sarama, 2004). The appropriate use of technology brings new opportunities for teaching and learning at that level. These standards should focus on big ideas of children's mathematics defined as "mathematical, central and coherent, consistent with children's thinking and generative of future learners" (Clements, 2004, p. 13). The appropriate use of technology brings new opportunities for teaching and learning at that level. Facilitating ideas related to numbers and operations is one of the most important goals for mathematics in the early grades and, therefore, creating technological tools that support both professional development of teachers and cognitive growth of young children in this area is a useful direction in mathematics education. As mentioned by Kamii (2004), one of the pillars of young children's mathematical knowledge is the concept of the conservation of numbers that develops through activities in which the notion of correspondence plays an important role.

Recall that the concept of conservation deals with children's understanding that a set of objects in a collection remains the same regardless of whether they are changed about or altered to look different. The notion of correspondence refers to the cognitive capability or action to pair (or match) objects in one collection with objects in another collection. Piaget's (1961) studies on conservation of number and correspondence between two sets of objects show the relationship between conservation of quantities and the development of one-to-one correspondence. Corresponding or pairing objects is a self-checking way for a child to see if a change in the formation of objects alters their total number. As far as qualitative aspects of objects and their relation to correspondence are concerned, "in order that the correspondence shall be exact (i.e., each term being counted once and once only), the different terms must be ordered in a sequence in which each element is distinguishable from all the others" (Piaget, 1961, p. 96). This explains the design of MW 1 (Figure 2), in which coins (and buttons alike) are arranged in the decreasing order of their denominations.

Using the concept of conservation, the goal of activities for young children using the manipulative worksheets could be to facilitate the development of their "logico-mathematical knowledge" (Piaget, 1971) as they learn "to conserve" coins through creating conditions for problem solving. For example, the environment allows children "to conserve" coins regardless of the appearance (heads or tails) and to learn one-to-one correspondence between coins and buttons. To this end, Macros that generate coins were written in such a way that if a coin's name and its value (or image) do not coincide in one's cognitive space, this discordance would become apparent because it is the button that generates a coin. Thus by trying to click the buttons, a child sees the results of his/her actions in terms of the response of the environment. Moreover, a child's activity on MW 1 is a complex endeavor. It involves not only the need to correspond the image of a coin and its name (in other words, "to conserve" the coin), but also to create conditions for successful problem solving (creating a set of coins which total value is given). Note that such conditions might be erroneous as the example of Figure 2 indicates. By using the Coin Eraser, one can delete ill-chosen sets of coins and then try another combination.

To conclude this section, note that children can be offered activities according to their levels of reading skills and understanding the coin values and images. To this end, MW 2 in the money-sharing environment was designed to enable one's ability to match the images (heads-tails) and the names of the coins without any clues. To solve the problems, one has to know and identify the values, names, and images of individual coins. Thus, MW 2 (as well as MW 3) has the potential to contribute to one's basic understanding of the values and images of coins as well as to the development of problem solving skills.

Money-Sharing Problem as "a Text within a Text"

In the context of the theory of semiotic mediation, the word "text" refers to any meaningful verbal and nonverbal semiotic structure. In the money-sharing environment, a number, a set of coins, an evaluative comment can be viewed as text. A problem itself is a text. According to Lotman (1988), any text may simultaneously serve at least two basic functions--univocal and dialogic. The univocal function of text is to communicate constant information; its dialogic function is to generate new meaning. A text capable of producing new semantic effects is characterized by an open-ended organization that allows for multiple interpretations and thus, in educational contexts, creates conditions conducive to the development of higher order thinking skills. However, one's ability to extract multiple meanings from text, that is, to recognize and then put to work its open structure can not be taken for granted. Such ability develops through appropriate pedagogical mediation.

Consider the original money-sharing problem. In a traditional learning environment, its text has rigid boundaries that are not expected (and often not allowed) to have been crossed. The role of the problem's text in this situation is to fulfill the univocal function; that is, to request a correct answer. However, when approached from a dialogic perspective, the problem has the potential to emerge as text with flexible boundaries, the crossing of which is the rule rather than an exception. With such a rule being in place, a routine problem becomes a thinking device or a generator of new meaning that can animate problem posing followed by problem-solving activity.

It has bee argued that a worthwhile mathematical problem evolves from the awareness of mathematical significance of a situation that one is attempting to explore (Kilpatrick, 1987). The notion of a worthwhile problem includes an open-ended problem. In turn, what delineates an open-ended problem is a rich mathematical structure, which permeates the problem situation and connects it to a new conceptual domain. This connection may become apparent for a learner when suddenly, through the problem-solving phase of a problem-posing activity, a new insight into a familiar concept is generated. The extent to which one gains such insight indicates the extent to which one's problem-posing skills have been developed.

Indeed, a technology-enhanced experimentation with coins can reveal different layers of the problem structure that are disconnected when its text serves the univocal function. When exploring this structure in an open way and making connections among its seemingly disconnected layers, one searches for new meanings and thus enables text to serve its second function. Of course, not any text allows for a worthwhile dialogic interanimation. Yet, one's ability to recognize internal heterogeneity of text is akin to the act of sagacity defined by Aristotle as "a hitting by guess upon the essential connection in an inappreciable time" (cited in Polya, 1945, p. 58).

More specifically, when the rules of action allow for the exchange of coins, one can come up with the following simple questions: In how many ways can one make a certain amount of money out of pennies and nickels, or nickels and dimes, or pennies, nickels and dimes? A new twist given to the original money-sharing problem highlights its semiotically heterogeneous layers and allows for an open-ended problem to be conceptualized as what Lotman (1988) called "a text within a text" structure. Figure 3 shows an example of a problem formulated by a teacher. The teacher demonstrated how, through a problem-posing activity, one can hit upon a new conceptual domain concerned with the partition of numbers into a sum of other numbers--a branch of mathematics bordering number theory and combinatorics.

[FIGURE 3 OMITTED]

Apparently, as the number of partitions grows larger, one has to reason systematically in order to handle the multiplicity of answers. The next section will show how a spreadsheet can be used as a scaffolding device for problem posing in the context of partitions, thereby allowing for the development of system in intuitive strategies.

From Problem Posing to Systematic Reasoning

Partitioning problems permeate the K-12 mathematics curriculum of New York State (New York State Education Department, 1998) starting in early grades. Interestingly, the following problem (cf., questions in the previous section) was found in the curriculum of a small elementary school in rural upstate New York: "In how many ways can one make a quarter out of pennies, nickels, and dimes?" It appears that young children would not likely be able to find all solutions that the three coins provide. However, whereas it might not be important for the children to solve the problem completely, it would be a reasonable expectation for the teachers to do so because some children may want to know how close their efforts are to the complete solution.

With this in mind, the spreadsheet pictured in Figure 4 was designed. Its text has the potential to serve both functions. The univocal function highlights 12 ways to change a quarter into pennies, nickels, and dimes. The dialogic function reveals a system through which solutions are generated: fix the number of dimes used (this number varies from zero to two) and find all combinations of other two coins that comprise a quarter.

[FIGURE 4 OMITTED]

Learning to reason systematically is an important component of the mathematical preparation of elementary preteachers. This reasoning can be enhanced by physically creating all partitions of a quarter in the manipulative environment. The ability to reason systematically, as a result of technological amplification of mathematical thinking, indicates the emergence of residual mental power that can be used in the absence of technology (Abramovich & Norton, 2006).

A didactical power of this environment is that its slider-controlled variability of numerical data interactively generates the corresponding solution to a problem, thereby allowing one to formulate an open-ended problem with a reasonable number of correct answers. Furthermore, the environment enables an alteration of a problem-solving context while preserving its mathematical structure. Following is an example of a problem posed by an elementary preteacher using this environment (Figure 5).

    Sarah was on her way to class and decided she was thirsty. Upon
    finding a soda machine, she saw that it would cost her 80 cents to
    buy a drink. In her pocket she has nickels, dimes, and quarters. How
    many ways can she use her change to purchase her soda?

[FIGURE 5 OMITTED]

The teacher goes on to explain how one can solve this problem through a system:

    First, we should see how many times we could subtract 25 from 80.
    Secondly, we could see how many times we could subtract 10 from 80.
    Third, we could see how many times we could subtract 5 from 80. Once
    those three numbers are figured out, then we could work with them.
    The number 10 could also be two 5s. The number 25 could be five 5s,
    three 5s and a 10, as well as two 10s and a 5. From here on, it is a
    matter of grouping numbers together in order to equal 80.

In such a way, the appropriate use of a spreadsheet by teachers in the context of problem posing can serve both functions, univocal and dialogic, and thus allow for the meaning-making process to occur as a result of "an interaction between semiotically heterogeneous layers of text that are mutually untranslatable relative to one another" (Lotman, 1988, p. 43). Indeed, the text of Figure 5 has at least two seemingly disconnected layers dealing with the multiplicity and geometry of answers. The fact that the teacher was able to make a connection between the two layers is testament to the dialogicality of the text of the spreadsheet.

Concluding Remarks

One of the hidden messages of teacher education is that the way teachers learn affects the way that they will teach. This is especially true for mathematics teacher education. Many efforts of mathematics education reform are aimed at the development of new intellectual activities in support of classroom pedagogy enhanced by constructivist applications of educational technology. As this article has demonstrated, spreadsheet-based environments have the potential to be used by teachers for posing and solving grade appropriate problems. By being engaged in these activities, teachers learn to use technology for constructing worthwhile extensions of the existing curriculum.

Although the experience in technology-enabled mathematical problem posing is a relatively new pedagogical notion, being grounded in professional standards for teaching, it has the potential to enhance significantly early childhood teacher preparation course work. It provides teachers with research-like skills in the development of instructional materials for early childhood mathematics. Such skills are critical for making intelligent decisions under the demands of standards-based curricula.

Note that basic familiarity with a spreadsheet is often treated as one of the components of computer literacy. That is why the software can be construed as a new generation of educational technology, the utilization of which in the elementary classroom is not dependent on financial constraints and commercial availability. The proficiency of teachers in using a spreadsheet as a tool for conceptual development and educative growth of young children becomes an important factor in developing and implementing standards for early childhood mathematics (1). Through technology-enabled problem posing, the learning of mathematics can become a reciprocal process that advances intellectual diversity in consistently heterogeneous community of learners.

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Note

(1) In the context of problem-posing environments discussed in this article, such proficiency can be developed within a special course such as Using Spreadsheet in Teaching School Mathematics (Abramovich, 2006). At this website, one can find worksheets pictured in Figures 1 and 2 under the title "Money Sharing." Those who are interested in the project may contact the authors and discuss the ways to utilize the environments in more detail.

SERGEI ABRAMOVICH AND EUN KYEONG CHO 

State University of New York, College at Potsdam

USA

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