Designing Effective Software

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Designing Effective Software

Douglas H. Clements The State University of New York-Buffalo

Michael T. Battista Kent State University

There is little doubt that technology will have a major impact on the teaching and learning of mathematics and science. However, we contend that results from the enterprise of designing effective software have fallen short of their potential. We take "effective" to mean simultaneously pedagogically efficacious and fecund in the development of theoretical ani empirical research. In this chapter, we describe a model for integrated research and software ani curricula development that we believe will help to fulfill the potential of software development and correlated research. What of present practice limits software from fulfilling its promise? In the majority of cases, testing the software with target users is rare (which may account for the generally low quality of the software). Often, there is only minimal formative research, such as a polling of easily accessible peers, rather than any systematic testing with an appropriate target audience. "Beta" testing is done sometimes, but late enough in the process that changes are minimal,

given the time and resources dedicated to the project already and the limited budget ani pressing deadlines that remain (Char, 1989). Such testing is more summative than formative (Schauble, 1990). Even when conducted, most summative evaluations are limited in scope. The majority of studies have used traditional quantitative designs in which the "computer" was the "treatment." The general conclusion drawn was that such treatments lead to moderate but statistically 761

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significant learning gains, especially in mathematics (H. J. Becker, 1992; D. H. Clements & Nastasi, 1992; Kelman, 1990; Roblyer, Castine, & King, 1988). However, this conclusion must be tempered with the realization that most of the software used presented drill ani practice exercises. Therefore, the potential of software based on different approaches to learning (as well as the appropriateness of the methodology for evaluating such different types of software) has not been addressed adequately. What of software based on other approaches to, philosophies of, and goals for learning? Many pieces of software are created from a vision that computers can provide experiences far beyond such drill and practice work. This recalls Duckworth's analysis of how Piagetian thought circa 1970 did and did not inspire good curricula. Simply put, details of children's learning and development obtained from Piagetian research failed to guide curricula development in direct useful ways (Duckworth, 1979). However, Piaget's constructivist vision did suggest, in general, the validity of encouraging children to "have wonderful ideas" (Duckworth, 1973). Similarly, some software is developed based on a vision of encouraging children to have and explore wonderful ideas with technology. Many developers proceed from intuition and unbridled creativity; often, their products are fascinating and may be worthwhile educationally, but typically they are unconnected to theory and research. Other developers are familiar with mathematics education research and theory and have extensive experience doing informal research with students and computers. Their products, then, are connected to theory and research, albeit indirectly, and can be of high quality (W. D. Crown, personal communication, December 4, 1995). In contrast to either of these approaches, we argue that the time is ripe for research ani software design to be more intimately connected, mutually supporting processes. Compared to Duckworth's era, the state of the art is such that we have models with sufficient explanatory power to permit design to grow concurrently with the refinement of these models (Biddlecomb, 1994; D. H. Clements & Sarama, 1995; Hennessy, 1995). Thus, software design can and should have an explicit theoretical and empirical foundation, beyond its genesis in someone's intuitive grasp of children's learning, and it should interact with the ongoing development of theory and research-reaching toward the ideal of testing a theory by testing the software. To maximize its contributions to both mathematics and science curricula and to theory and research, software development must take a large number of issues into consideration and proceed through a variety of phases.

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A MODEL FOR INTEGRATED RESEARCH AND CURRICULA AND SOFTWARE DEVELOPMENT Capitalizing fully on both research and curriculum development opportunities requires the maintenance of explicit connections between these two domains and formative research with users throughout the development process (cf. Laurillard & 1. Taylor, 1994). The complexity of computer-enhanced teaching and learning, including the need to examine both the processes and the products of learning and the variety of relevant social transactions, necessitates the use of multiple methodologies (see, e.g., D. H. Clements & Sarama, 1995; P. W. Thompson, 1992). The research design model described herein moves through phases in a sequence that is as much recursive as linear. The phases begin with a draft of initial values and educational goals, proceed through a series of investigations with small numbers of children (e.g., clinical interviews and teaching experiments), and end (for the time being) in more summative, classroom implementations. Each phase is described and illustrated with examples from projects that used the model or aspects of it.

Phase 1: Draft the Initial Goals Mathematical Learning Goals. The first phase begins with the identification of a problematic domain of mathematics. This domain should be significant in two ways: The learning of the domain would make a substantive contribution to students' mathematical development, and learning about students' mathematical activity in the domain would make a substantive contribution to research and theory. For example, one team designed rational numbers microworlds (RNMs) (Biddlecomb, 1994; Olive, 1996; Steffe & Wiegel, 1994). We designed Logo environments for the learning of elementary geometry and the use of geometric models in other areas of mathematics (Battista & D. H. Clements, 1991; D. H. Clements & Battista, 1991). In establishing mathematical learning goals, intense study of reform recommendations (National Council of Teachers of Mathematics, 1989, 1991b; National Research Council, 1989), the history of the curriculum domain (see Dennis, chap. 28, this volume), and equity issues is recommended. Equity issues (see Confrey, chap. 5, this volume) imply that considerable thought be given to the students who are envisioned as users and who participate in field tests; a convenience sample may be inappropriate. Systemic sociocultural issues, and, finally, technical issues should be considered as well.

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CLEMENTS AND BArnSTA The Advantages and Disadvantages of Technocentrism. Technocentrism refers to the

tendency to place a technical object, such as a computer, in a central position (Papert, 1987). This leads to questions such as, "What is the effect of the computer on mathematics development?" There are palpable disadvantages to this perspective when considered in such a simplistic manner. It belies the complexity of the educational situation, omits considerations of types of software and types of mathematics (Skemp, 1976), and ignores the classroom culture. It also implies that the computer is an entity separate from other curricular considerations. In contrast, most designers of educational software who follow procedures consistent with the model described herein consider themselves creators of activities or curricula, not merely of software (Char, 1990; D. H. Clements & Sarama, 1995). They view software as one, albeit critical, piece of a situation involving planning to use the computer, using the computer, making sense of their experiences, and reasoning about them both at ani away from the computer (P. W. Thompson, personal communication, November 29. 1995). Their goal is that students learn concepts and processes that are worthwhile with and without their experiences with the computer. They do not invent a tool first without asking, early in the process, for what it will be used (R. Lehrer, personal communication, November 30, 1995). Can a focus on technology be advantageous? Reflecting on the actions and activities that are enabled by a new technology can catalyze a reconceptualization of the nature and the content of the mathematics that could and should be learned. Also, the developer can focus designs by thinking about how software might provide tools (that enhance students' actions and imagination or that suggest an encapsulation of a process) or constraints (obstacles they meet by design that force them to grapple with an important idea or issue); (Kaput, 1994). Finally, the flexibility of computer technologies allows the creation of a vision less hampered by the limitations of traditional materials and pedagogical approaches (cf. Confrey, in press). For example, computer-based communication can extend the model for science ani mathematical learning beyond the classroom, and computers can allow representations ani actions not possible with other media. Such thinking anticipates Phase 3 and illustrates the nonlinear nature of the design model. The product of this first phase is a description of a problematic aspect of mathematics. This description should be quite detailed, especially if the purpose is developing innovative software and if the mathematics differs from that in common use.

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Phase 2: Build an Explicit Model of Students' Knowledge and Learning in the Goal Domain The next step is to build a model of students' learning that is sufficiently explicit to describe the processes involved in the construction of the goal mathematics. Frequently, extant models are available, although they vary in degree of specificity; some highly specific models can be used directly (e.g., Steffe & Cobb, 1988), whereas others are general and, therefore, mostly suggestive (e.g., D. H. Clements & Battista, 1991; van Hiele, 1986). Especially in the latter case, the developer uses clinical interviews and observations to examine students' knowledge of the content domain, including conceptions, strategies, and intuitive ideas used to solve problems. Students may use paper and physical materials (although these can be configured to anticipate the software experiences that will be designed). In these experiments, the teacher tries to set up a situation or task that will elicit pertinent concepts and processes. The interviewer asks follow-up questions to ascertain in as much detail as possible what and how the students are thinking about the situation. Pairs of children may be observed working on problems, because they can be more relaxed and expressive about their thinking and problemsolving activity with a peer. Once a (static) model has been partially developed, it is tested ani extended with exploratory teaching (see Steffe & Thompson, chap. II, this volume). This approach contrasts with that of the "wonderful ideas" perspective, with its initial focus on what the computer can do and succeeding attempts to use these technological capabilities to teach mathematics. If specific activities are planned as a component of the software, similar techniques are

used to assess the relative appropriateness of proposed problems. That is, there must be teaching that is exploratory and intuitive, followed by analyses. The researchers' mental models of students' thinking in the domain guide their design of the activities. As is true at each phase of development, the researchers' further experiences with new students provide feedback that leads to modification of that mental model. Other approaches are possible. For example, Char (1989) analyzed major textbook activities, examining such dimensions as curriculum goals, types of problem levels, nature of presentation, and nature of user response. The end result of this phase is an explicit model of students' learning of mathematics in the target domain. Ideally, such models specify knowledge structures, the development of these structures, and mechanisms or processes of development. To repeat a caveat: Activities ani results of this phase are connected, even overlapping, with those in the following phases, with the model of students' learning developing constantly.

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Phase 3: Create an Initial Design for Software and Activities Plan the Design. Our descriptions and illustrations of this phase become more specific. We continue our focus on the design of theoretically based and research-based microworlds. However, there are many other possibilities, such as producing: • A cognitive tool (specifically, a computer enactment of lower level operations. obviating the need for novices to achieve automatization of these skills prior to carrying out higher order thinking; Lajoie. 1990). • A simulation (Hennessy, 1995). • A programming language (e.g., Logo). • Specialized programming language (e.g., StarLogo). • Scriptable, modular components that may represent any of these types. The software might be embedded in a student-centered design. Though details of the design model would need to be modified for such diverse types of software, most of the components of our model apply. Based on the model of students' learning generated in Phase 2, developers create a basic design to describe the objects that will constitute the software environment and the actions that may be performed on these objects. These actions-on-objects should mirror the hypothesized mathematical activity of students as closely as possible. The design also describes how the actions are to be instantiated (including the questions of what choices ani options are available to the learners, and what should be decided by the developers). The rationale for these and all of the other design decisions is critical to the development ani research process (Kaput, 1994). For example, the RNM team (Biddlecomb, 1994; Steffe & Wiegel, 1994) included actions for creating objects ("Toys" in one program, line segments or "Sticks" in another), copying objects, uniting and disembedding objects. and hiding objects. In the Sticks microworld. for instance, students can segment sticks into parts or combine them into higher order units. Methods of production also illustrate these developers' theory-design connections. Objects can be produced by clicking (producing single objects) or (in Sticks) by dragging (producing an entire cluster that is constituted as a unit). "Covers" that hide some objects were included to encourage students' internalization of perceptual counting schemes. Finally. the authors wanted students to create and reason with composite units, so, they incorporated such objects as "strings," "chains," and "stacks" of toys.

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Such actions on objects-in this case, creating, copying, uniting, disembedding, ani hiding both individual units and composite units-are in line with the mental actions that the developers wanted students to construct (D. H. Clements & McMillen, 1996). Further, the provision of objects and actions on these objects is consistent with the Vygotskian theory that mediation by tools and signs is critical in the development of human cognition (Steffe & Tzur, 1994). The "Turtle Math" (D. H. Clements & Meredith, 1994) software was designed to include the traditional "turtle" object and actions on that object (e.g., entering commands on the computer such as "fd 30" or "rt 120") because children's initial representations of space are based on action, rather than on the passive "copying" of sensory data (Piaget & Inhelder, 1967). That is, we hypothesized that the use of Logo commands that correspond to the motions necessary for constructing geometric figures (e.g., moving forward and turning) increases the salience of the critical components of the figure. We also went beyond this by including objects that resulted from that activity (e.g., line segments or polygons) and actions on those objects (e.g., motions, scaling, and other transformations). Our early research indicated that such transformations are consistent with elementary students' mental motions (e.g., in "proving" congruency) and can extend these motions to more general and abstract levels (Johnson-Gentile, D. H. Clements, & Battista, 1994). Finally, we designed dynamic and consistent relationships between different representations of the same object (D. H. Clements & Battista, 1992). In other words, there is a dynamic link between the commands in the command center and the geometry of the drawn figure. Any change in the commands in "Turtle Math" leads to a corresponding change in the figure, so that the commands in the command center reflect the geometry in the figure precisely. In this way, command center code constitutes what we call a proleptic procedure. Thus, the Logo code in the command center stands halfway between traditional, immediate-mode records and procedures created in an editor, helping to link the symbolic and visual representations. The second basic issue is the directionality of the visual-symbol connection. One of Logo's main strengths has been its support of linkages between visual and symbolic representations. One of its limitations has been the lack of bidirectionality between these modes (Noss & Hoyles, 1992). That is, one creates or modifies symbolic code to produce visual effects, but not the reverse. "Turtle Math" provides a "draw commands" tool that allows the student to use the mouse to tum and move the turtle, with corresponding Logo commands created automatically. The student creates continuously first a tum ("rt" or "It") command, then a movement ("fd" or "bk") command

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until he or she clicks outside the window. "Turtle Math" also provides a "change shape" tool that allows the student to use the mouse to alter the geometric figure directly, with corresponding changes made automatically to the Logo commands. Thus, Logo students benefit from the direct manipulation of geometric figures (cf. de Villiers, 1995). Such direct manipulation is consistent with students' mental operations (see, e.g., D. H. Clements & Battista, 1991, 1992; Johnson-Gentile et aI., 1994) and, through dynamic connections, aids children in the abstraction and generalization of these operations. Designs based on objects and actions force the developer to focus on explicit actions or processes and what they will mean to the students. This characteristic mirrors the benefit attributed to cognitive science models of human thinking; they did not allow "black boxes" to hide weaknesses in the theory. It must be emphasized that designs are not detennined fully by this line of reasoning: Intuition and the art of teaching play critical roles in the design of software (cf. Confrey, in press). That is, this phase often involves the developers, as individuals and groups, decentering and imaging the projected activities and interactions.

Consider Unique Characteristics of Computers. Expanding on a theme introduced previously, developers should consider the potential unique contributions of the computer in planning objects and actions. For example, the dynamic aspect of Toys and Sticks (Biddlecomb, 1994; Steffe & Wiegel, 1994) captured students' interest, according to the developers. Students used their schemes in ways not possible when working in a mathematics textbook or with structured learning material such as Cuisenaire rods. From the students' perspective, acting in the microworlds created experiential situations that metamorphosed with every accommodation. That is, as the children UIed structure to the situations through their actions, the situations changed correspondingly and new possibilities for action emerged. 1he microworlds opened pathways for mathematical activity (Steffe & Wiegel, 1994). Additional conceivable characteristics of computer manipulatives include (D. H. Clements & McMillen, 1996, includes a discussion): •

Offering flexibility.

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Changing arrangement or representation.

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Storing and later retrieving configurations.

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Recording and replaying students' actions.

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Linking the concrete and the symbolic with feedback.

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Dynamically linking multiple representations.

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Focusing attention and increasing motivation.

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Changing the nature of the manipulative.

In addition to the aforementioned characteristics, the following have been identified as unique contributions of Logo and "Turtle Math" in particular (i.e .. more so than previous versions of Logo): • Promoting the connection of formal representations with dynamic visual representations, supporting the construction of mathematical strategies and ideas out of initial intuitions and visual approaches. • Encouraging the manipulation of specific shapes in ways that help students to view them as mathematical representatives of a class of shapes. • Encouraging wondering about and posing problems by providing an environment in which to test ideas and receive feedback about them • Facilitating students' reflection on and modification of their Logo C