Douglas H. Clements
State University of New York at Buffalo
Technology can have a major impact on the teaching and learning of mathematics. Educational software will not fully realize that potential, however, as it is presently designed. Software development is not currently progressive -- we are not learning from each other. We need new models for integrated research and software development.
Most of present software development is limited in at least five ways.
1. Software ideas are not based on accumulated knowledge. Most software is generated based on narrow personal opinions, intuitions, review of other products, and brainstorming based on these factors (Battista & Clements, in press; Clements & Battista, in press). None of these bases is inherently unwise, but alone they are inadequate. They replace other considerations, the most important of which we turn to next.
2. Software is not based on accumulated knowledge of children's thinking. Decades ago, theories such as those of Piaget constituted major contributions, but they were too general to be used directly to built good curricula (Duckworth, 1979). Now we have models with sufficient explanatory power to guide design of software and to permit that design to grow concurrently with the refinement of these models (Biddlecomb, 1994; Clements & Sarama, 1995; Hennessy, 1995). These theory- and research-based models specify how students thing about specific curriculum content and what learning trajectories we can expect students to follow. They specify the mental concepts children construct and the processes they use in acting on those concepts as they move through these trajectories. But the vast majority developers do not use these models, contradicting the prevalent agreement that instructional decisions should be based on how students learn particular content (Baroody & Ginsburg, 1990; Battista & Clements, in press; Clements & Battista, in press; Cobb et al., 1991; Hiebert & Wearne, 1992).
3. Testing of software with target users is rare and limited in breadth. Often, there is only minimal formative research, such as a polling of easily accessible peers and children, 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, the limited budget and pressing deadlines that remain (Char, 1989). Most problematic, most of this testing is limited to surface interface and usability issues. It does not carefully track whether the software actually helps students move through the learning trajectories in the domain.
4. Summative evaluations are rare and limited in scope. Few well-controlled studies are conducted of most software. Most existing studies used traditional quantitative designs in which the computer was the treatment. The general conclusion drawn was that such treatments lead to moderate but statistically significant learning gains, especially in mathematics (Becker, 1992; Clements & Nastasi, 1992; Kelman, 1990; Roblyer, Castine, & King, 1988). However, most of the software used presented drill and practice exercises. Therefore, the potential of software based on different approaches to learning has not been addressed adequately.
5. The development process is not reviewable. There is little or no documentation of the software development process. Thus, there is little reflection, inspection, and critique of the process. It is, then, no wonder that there are no established standards for the development of effective educational software. The time is ripe for research and software design to be more intimately connected, mutually supporting processes. Software design can and should have an explicit theoretical and empirical foundation, beyond its genesis in someone's intuitive grasp of children's learning. Design should be intimately connected with both development of theory and research, so that the ideal of testing a theory by empirically testing the software (Clements & Battista, in press).
This is the basis for our project, "Building Blocks—Foundation for Mathematical Thinking, Pre-Kindergarten to Grade 2: Research-based materials Development."1 Our approach is finding the mathematics in, and developing mathematics from, young children's activity. In this approach, children extend and mathematize their everyday activities, from art to songs to puzzles to building blocks. Mathematization emphasizes representing and elaborating mathematically -- creating models of everyday activity with mathematical objects, such as numbers and shapes; mathematical actions, such as counting or transforming shapes; and their structural relationships. Our software embodies these actions-on-objects in a way that mirrors what research has identified as critical mental actions -- children's cognitive building clocks. These cognitive building blocks include specific ways of creating, copying, and combining objects such as shapes or numbers. Activities are sequenced to guide students through research-based learning trajectories.
We are developing this software based on a model for integrated research and software development (Clements & Battista, in press). This model was generated by abstracting previous work by ourselves and others (Battista & Clements, 1991; Biddlecomb, 1994; Clements & Battista, 1991; Olive, 1996; Stefe & Wiegel, 1994). Capitalizing fully on both research and curriculum development opportunities requires maintenance of explicit connections between these two domains and formative research with users throughout the development process. Thus, it is essential that the entire process be carefully documented.
Our design model moves through phases in a sequence that is as much recursive as linear. The methodologies are complex and interwoven. Phases of the 9-step design process model include: draft curriculum goals; build an explicit model of children's thinking and learning in the goal domain including cognitive objects and actions; create an initial design including on-screen objects and actions that mirror their cognitive counterparts; investigate these components; assess prototypes and curriculum to ensure that children are developing the intended mathematical concepts and processes; conduct pilot tests; conduct field tests in multiple settings; recourse; and publish and disseminate. These phases include a close interaction between materials development and a variety of research methodologies, from clinical interviews, to teaching experiments, to ethnographic participant observation (for details, see http://www.gse.buffalo.edu/org/buildingblocks and Clements & Battista, in press). We are currently investigating components of our first of our first two software environments; we invite readers who are interested in helping assess the forthcoming prototype to contact us through the Building Blocks web site.
Connecting research and software design does not mean we find the "best or "one right" way or even that various parties agree. Many decisions involve values and are a matter of social policy (Hiebert, 1999). There will continue to be disagreements and different points of view, as there is in all branches of science. Connecting research and design does mean that these disagreements will be discussed, and sometimes resolved, based on coherent arguments supported by empirical evidence.
In summary, technology and research have the potential to significantly improve mathematics education. They have not done so to date. I believe that they will not until they are synthesized. Software development is presently a haphazard enterprise in which the few successes are not built upon. Using a research-based approach to software development will allow the scientific accumulation of knowledge necessary to realize the potential of computers in mathematics education. Conversely, the research process benefits from the tools and activities technology affords. We need integrated research and software development to effectively reform mathematics education.
1 "Building Blocks—Foundation for Mathematical Thinking, Pre-Kindergarten to Grade 2: Research-based materials Development." is being developed at the State University of New York at Buffalo and Wayne State University, where Julie Sarama is the co-director. See http://www.gse.buffalo.edu/org/buildingblocks.
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Battista, M. T. & Clements, D. H. (1991). Logo geometry. Morristown, NJ: Silver Burdett & Ginn.
Battista, M. T. & Clements D. H. (in press). Mathematics curriculum development as a scientific endeavor. In E. Kelly & R. Flesh (Ed.), Handbook of innovative research design in mathematics and science education. Englewood Cliffs, NJ: Lawrence Erlbaum Associates.
Becker, H. J. (1992). Computer-based integrated learning systems in the elementary and middle grades: A critical review and synthesis of evaluation reports. Journal of Educational Computing Research, 8, 1-41.
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Char, C. A. (1989, March). Formative research in the development of mathematics software for young children. Paper presented at the meeting of the American Educational Research Association, San Francisco, CA.
Clements, D. H. & Battista, M. T. (in press). Developing effective software. In E. Kelly & R. Lesh (Ed.), Handbook of innovative research design in mathematics and education. Englewood Cliffs, NJ: Lawrence Erlbaum Association.
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Clements, D. H. & Sarama J. (1995). Design a Logo environment for elementary geometry. Journal of Mathematical Behavior, 14, 381-398.
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Duckworth, E. (1979). Either we're too early and they can't learn it or we're too late and they know it already: The dilemma of "applying Piage". Harvard Educational Review, 49, 297-312.
Hennessy, S. (1995). Design of a computer-augmented curriculum for mechanics. International Journal of Science Education, 17(1), 75-92.
Hiebert, J. & Wearne, D. (1992). Links between teaching and learning place value with understanding in first grade. Journal for Research in Mathematics Education, 23, 98-122.
Hiebert, J. C. (1999). Relationships between research and the NCTM Standards. Journal for Research in Mathematics Education, 30, 3-19.
Kelman, P. (1990, June). Alternatives to integrated instructional systems. Paper presented at the meeting of the National Educational Computing Conference, Nashville, TN.
Olive, J. (1996, July 14-21). Constructing Multiplicative operations with fractions using tools for interactive mathematical activity (TIMA) microworlds. Paper presented at the meeting of the Eighth International Congress on Mathematical Education (ICME-8), Topic Group 19, Seville, Spain.
Boblyer, M. D., Castine, W. H. & King, F. J. (1988). Assessing the impact of computer-based instruction: A review of recent research. New York: Haworth Press.
Steffe, L. P. & Wiegel, H. G. (1994). Cognitive play and mathematical learning in computer microworlds. Journal of Research in Childhood Education, 8 (2), 117-131.
Time to prepare this material was partially provided by National Science Foundation Research Grant ESI-9730804, "Building Blocks—Foundation for Mathematical Thinking, Pre-Kindergarten to Grade 2: Research-based Materials Development." Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the author and do not necessarily reflect the views of the National Science Foundation.
Douglas Clements teaches early childhood, elementary mathematics, and computer education courses at the State University of New York at Buffalo, Buffalo, NY 14260.
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