Introducing Dynamic Number is a research and development project undertaken by KCP Technologies to extend and operationalize research in number, operation, and early algebra. The project builds on the paradigm of Dynamic Geometry (the interactive and continuous manipulation of geometric shapes and constructions) with a new technological paradigm, Dynamic Number, centered on the direct manipulation of number representations and numerical constructions. Using The Geometer's Sketchpad as a starting point, KCP Technologies is developing new software tools to deepen students' conceptions of integers, fractions, decimals, real numbers, and early algebra in grades 2-8 mathematics. These tools are accompanied by 40 supporting classroom activities, as well as a six-week online professional development course for teachers.
Both the software and learning materials are being designed from the perspective of a "bottom up/top down" analysis of the content domains. The University of Georgia, working under a subcontract, serves as the project's evaluator. Using a combination of videotaped teaching experiments and classroom observations from eight field-test schools, they are examining how Dynamic Number technology effects students' conceptions of discrete numerical quantities, continuous numerical quantities, and early algebraic principles, such as the concept of variable. They are also examining whether the software allows students to design and construct dynamic mathematical models of their own invention.
The Dynamic Number project at KCP Technologies extended the paradigm of Dynamic Geometry (the interactive and continuous manipulation of geometric shapes and constructions) with a new technological paradigm, Dynamic Number, centered on the direct manipulation of number representations and numerical constructions. Using The Geometer's Sketchpad software as the basis of our work, we focused on the following goals: Develop, field test, revise, and disseminate activities for The Geometer's Sketchpad focusing on number, operation, early algebra, and algebra. Extend Sketchpad's reach to platforms beyond its longstanding desktop home for the purpose of simplifying its interface for young learners and their teachers. In particular, focus on the development of Sketchpad for the iPad and Sketchpad for the web. Curriculum Materials In terms of our first goal—developing classroom activities for Sketchpad—we explored the following questions as we prototyped, field tested, and revised our curriculum materials: What does it mean to build and interact with discrete constructs like the integers? What do these same actions mean in the context of continuous quantities like real numbers and their decimal representations? How can Dynamic Number, which, at its core, focuses on quantitative variation, be a conceptual link to the notion of variable in early algebra? As an outcome of our curriculum efforts, we developed 70 activities for The Geometer's Sketchpad, all of which include extensive support for teachers. Most notably, every activity includes a YouTube-embedded video, providing an introduction to the mathematics and pedagogy of the lesson, in addition to an overview of the accompanying Sketchpad model. Extensive teacher notes and worksheets, downloadable as PDFs, round out the offerings, as well as connections to the Mathematical Practices and Content Standards of the Common Core. The entire set of 70 activities is available for free at the Dynamic Number website http://dynamicnumber.org. The activity topics range from addition and subtraction to multiplication, multiples and factors, fractions, place value, early algebra, algebra, and a topic we have named "Geometric Functions." Geometric Functions provide a compelling entrée to function concepts by developing geometric transformations as functions, thus emphasizing sensory-motor experiences of functions while strengthening understanding of geometric transformations. Students create points as their independent and dependent variables and label these variables using function notation. They then drag the independent variable to observe the variables’ relative rates of change and restrict the independent variable to a path to investigate domain and range. Students also compose functions by turning the output of one function into the input of another and use composition to create and understand inverse functions. Sketchpad Explorer for the iPad In August 20111, KCP Technologies released the free Sketchpad Explorer for the iPad with funding support from the Dynamic Number project and from the NSF-funded Dynamic Haptic Geometry project. Sketchpad Explorer is a free multi-touch implementation of The Geometer’s Sketchpad that can be used to view and manipulate any sketch created with the full desktop version. Sketchpad Explorer allows users to open any sketch created with Sketchpad version 4 or 5 and interact with it on the iPad. From a Dynamic Number perspective, Sketchpad Explorer meets our goal to simplify Sketchpad for young learners in grades 2–8, the target audience of our project. Specifically, Sketchpad Explorer eliminates menus entirely, requires only minimal reading ability, and allows for direct, touch-based manipulation of objects without a mouse or trackpad as an intermediary device. Web Sketchpad Web Sketchpad, developed as part of the Dynamic Number project, brings Dynamic Geometry and Dynamic Number visualization and mathematical interactivity to HTML5 web applications. It allows the creation of Sketchpad-based interactive figures and diagrams (‘web sketches’) that can be embedded easily in any HTML5 context and enhanced, optionally, by arbitrary JavaScript. Web Sketchpad users can see and interact with dynamic mathematical diagrams either in standalone web pages or as components of larger web sites, electronic texts, digital curricula, and so on. The only requirement for users is an environment that supports HTML5 execution, such as a relatively modern web browser or e-book reader. As with our iPad product, Web Sketchpad is strictly a tool for interacting with sketches, without the menus and construction tools of its desktop counterpart. This simplified interface makes Web Sketchpad accessible to young learners. And because web sketches can be embedded into websites and into digital curricula, the interactive models become a seamless part of curricula rather than an add-on component that teachers might view as optional.
