Young Children Investigating Advanced Mathematical Concepts with Haptic Technologies: Future Design Perspectives

1. IntroductionIn this chapter, we focus on how new technologies can be used with young children to investigate mathematical ideas and concepts that would normally be introduced at a later age. In particular, we focus on haptic technologies that allow learners to touch and feel objects through force feedback in addition to visual images on a screen. The main purpose of this paper is to describe how these technologies can be used to enable young learners to construct meaning about geometric shapes and surfaces as well as attributes of particular mathematical constructions in multiple dimensions (particularly 2D and 3D for purposes of this chapter). Such learning environments enable various forms of mediation both through the devices and software used as well as socially, as students work together to develop meaning and create models of complex ideas.We begin by describing how and why young learners in particular should be working in such learning environments in order to provide a rationale for our work. In Section 2, we provide some background on how these technologies have evolved and their use in other disciplines and how we have built on prior research in the use of dynamic geometry in mathematics education. Section 3 presents how relevant these new learning environments can be with some specific examples from preliminary work at the Kaput Center. The section also contains some theoretical reflections on how we can begin to analyze and understand how students work and construct meaning in such environments. Section 4 then concludes by offering some design principles for future research and development.How Should Young Children be Doing Mathematical Problem-Solving in the Future?We believe that the answer to this question lies in three areas that focus on the early introduction of mathematical ideas, the use of technology, and engagement.Early introduction. Several researchers have promoted the idea of introducing mathematical ideas earlier in the curriculum and even introducing the foundation of advanced mathematical thinking in the early grades (Kaput, Carraher & Blanton, 2008; Kaput, 1994). If not then, many children will never be exposed to important mathematics and engage in fruitful and relevant investigations. This can have detrimental effects throughout a child's educational career, reducing their desire to want to learn mathematics because of its lack of relevance or inaccessible representations.Technology use. Technology is often not a major part of elementary school classroom teaching due to a lack of resources and perception of its role and use. The predominant form of technology use in most elementary school classrooms in the U.S. is PowerPoint presentations. Some researchers (Carraher & Schliemann, 2000) believe that the introduction of technology is not enough:It is important to provide a social analysis in consonance with a cognitive one. Because technology does not act directly on learners, but only exerts an influence on the social activities and contexts in which it is employed, introducing technology into the mathematics classroom ultimately entails questions such as the following: What is the teacher's role; what are the students trying to achieve in the tasks ... . (p. 174)While we agree that these questions are important, new technologies can have a more participatory and collaborative role rather than be a prosthetic device to prop up existing pedagogical practices. New technologies can actually re-structure interaction in the classroom and allow the introduction of advanced mathematical ideas through radically new mathematical representation systems. The interactions of teachers, students and technologies within a learning environment can modify and transform activity structures (Jonassen, 2000).Technological affordances can also be mathematical affordances providing a symbiotic link between how mathematical activity can occur. Mathematizing technological affordances is an important step and one we discuss in detail later. …