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Article Excerpt
Teaching mathematics through multiple representation frameworks
presents an open-ended but complex learning structure. This article elaborates on teaching mathematics for conceptual understanding in the Information and Communication Technologies (ICT)-based environment, by exploring conceptual understanding through multiple representations and addressing some effects of the cognitive tools to teaching and learning mathematics in such environments.
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Jl. of Computers in Mathematics and Science Teaching (2003) 22(4), 381-399
... the defining characteristic of knowledge workers is that they are themselves changed by the information they process. (Kidd, 1994, p. 186)
Mathematics curriculum development in ICT-based environment is a complex task requiring ongoing feedback and refinement. (Dreyfus, 2002). For many teachers understandings of key mathematical concepts and phenomena are grounded in the ways they have learned them before emerging technologies have been accessible to them. And even if they had some exposure to ICT integration, they are now teaching new generations, born and educated surrounded by emerging technologies. Bridging this digital divide requires revisiting roles that representations, and translations among representations, play in mathematical learning and problem solving (Alagic, & Langrall, 2001).
This article brings together teaching mathematics for conceptual understanding in the technology-based environment, by considering mathematics teaching and ICT in general terms, exploring conceptual understanding through multiple representations and addressing some effects of the cognitive tools to teaching and learning mathematics. It is grounded in existing research and reflects the author's experiences from ongoing collaborations with teachers learning how and integrating ICT in their teaching and learning of mathematics (Alagic, 2002a, 2002b, 2002c; Alagic & Langrall, 2002).
ICT INTEGRATION: MATHEMATICS
... what theoretical reflection we need if we want to really help teachers to adequately use technological tools ... (LaGrange, 2002, p. 15)
The National Council of Teachers of Mathematics (NCTM, 2000) identified the Technology Principle as one of six principles of high quality mathematics education. It stated: "Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students' learning" (p. 24).
Successful technology integration in mathematics curriculum/classroom depends on many factors such as mathematics teachers, mathematical connections, and appropriate integration approach.
Mathematics Teachers
The research on use of technology reveals the complexities of the interplay of technology and teaching in the learning of mathematics. One thing remains constant. It is ultimately the mathematics teachers, not the technological tools that continue to be the key to the success of the mathematical learning environment. Their own perspective on the nature of mathematics, on the potential of the technology, and the training that they receive determine their effectiveness in integration of the technology in mathematics teaching and learning. (Garofalo, Drier, Harper, Timmerman, & Shockey, 2000; Kaput, 1992; NCTM, 2000). The external world is interpreted according to one's own experiences, beliefs, and knowledge and therefore, each person visualizes the external world at least slightly differently. As learners, teachers are able to comprehend a variety of interpretations and use them. But they cannot map their own interpretations of the world directly onto their students, because they do not share a set of common experiences and understandings. Some students have an exceptional fluency with technology tools, and some are very far from it, challenging their teachers even further. Yet another essential piece of teacher knowledge for building a technology-based learning environment is how to teach for transfer. Teaching practices congruent with a metacognitive approach to learning include those that focus on sense making, self-assessment, and reflection on what worked and what needs improving. These practices have been shown to increase the degree to which students transfer their learning to new settings and events (Schoenfeld, 1992).
Mathematical Connections
The use of ICT in mathematics teaching should support and facilitate conceptual development, exploration, reasoning, and problem-solving, as described by the NCTM (2000). Technology enables users to explore topics in more depth and in more interactive ways. By removing computational constraints it makes accessible the study of mathematics topics that were previously impractical, such as recursion and regression. Technology-augmented activities should take advantage of these capabilities of technology, and hence should extend beyond or significantly enhance what could be done without technology. These activities can facilitate mathematical connections in a variety of ways as: (a) linking multiple representations, (b) interconnecting and integrating mathematical topics and ideas, and (c) connecting mathematics to real-world phenomena. Therefore a collection of representations for a certain concept is richer, providing more access points for diverse learners at different levels of their understandings.
Studies in complex domains such as solving science problems (Heller & Reif, 1984) have suggested that conceptual understanding is associated with connections--connections between science concepts and everyday life and connections among the different concepts in a discipline. Someone who is good at solving transfer problems does not randomly connect concepts (which might occur when using memorized algorithms to solve problems) but rather integrates the concepts into a well-structured knowledge base in the context.
Integration Guidelines ICT
What further factors determine the success of the use of ICT in learning mathematics? According to Flick and Bell (2000), the following interconnected guidelines provide the essential ideas for strengthening mathematics instruction while integrating technology. ICT should (a) be introduced in the context of mathematics content, (b) address worthwhile mathematics with appropriate pedagogy, and (c) make scientific views more accessible. Furthermore, ICT instruction in mathematics should take advantage of the unique features of technology and develop students' understanding of the relationship between technology and mathematics. Preparing teachers to integrate technology appropriately, with above guidelines, requires professional development that focuses on both conceptual and pedagogical issues as well as ongoing support.
TEACHING MATHEMATICS: CONCEPTUAL ORIENTATION
Students do not necessarily interpret results in the manner that is obvious (to the mathematics teacher). (Dreyfus, 2002, p. 26)
When students attain understanding, what have they achieved? What students do, in response to the questions that put understanding...
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