|
Article Excerpt
To integrate technology into mathematics teaching and learning
effectively, teachers could create a technology-based learning environment that provides students with opportunities to experience the process of mathematical investigation. These opportunities range from exploring using mathematical ideas to making and testing conjectures, as well as extending their conjectures to a general form, if possible. Additionally, the learning environment should support students in ways that encourage them to articulate not only what they know about the mathematical ideas in their exploration, but also how they arrive at their conjectures and how they generalise the ideas. This article offers a framework that encompasses the processes of exploring, conjecturing, verifying, and generalising to help mathematics teachers plan and design effective technology-based lessons to create an environment that engages students in meaningful learning in the mathematics classroom. An interactive spreadsheet template, based on a popular mathematics problem commonly found under the topic of calculus and involving finding the maximum area of a rectangular enclosure given a fixed perimeter, was designed to illustrate the framework.
INTRODUCTION
The Ministry of Education in Singapore has introduced two Master-plans for Information Technology (IT) in Education since 1997. The intent was to encourage teachers to harness the numerous benefits of IT in teaching and learning. However, anecdotal evidence gathered through interactions with local practising mathematics teachers suggests that the potential of IT seems to remain unrealised in the classroom. So, why are local teachers still not tapping into the potential of IT despite the fact that they have been provided with training to equip them with the basic proficiency to integrate IT into the curriculum? Reasons they typically cite, which are consistent with those reported in the research literature (Heid, 1997; Oppenheimer, 1997; Kaur & Yap, 1998; Manoucherhri, 1999), include a lack of curriculum time to allow pupils to learn through exploration and investigation, a lack of outside curriculum time for teachers to plan and design appropriate technology-based lessons, and the inadequacy of teacher training. Although the teachers' reasons may appear to be rational justifications, the main reason seems to indicate a lack of pedagogical knowledge and confidence in implementing technology-based lessons. Teachers often feel reluctant or uncomfortable because their pedagogical knowledge perhaps does not include a framework for conducting technology-based activities in their lessons.
Drawing on this problem faced by teachers, this paper aims to offer a framework to help mathematics teachers plan and design effective technology-based lessons. The paper begins with a discussion of the framework and follows with a description of the mathematics problem selected to illustrate it as well as a typical approach for solving it. A pre-designed Excel template is developed to guide pupils through this problem. A description of the tasks, which pupils are expected to perform in this activity by using this template, and an elaboration of how the framework underpins the teaching and learning of these tasks, are provided in the subsequent sections.
THE FRAMEWORK
Characterised as a psychological perspective on knowledge and learning (Jaworski, 1994), constructivism refers to the idea that learners generate meaning as they make sense of their world. Although opinions are diverse within the philosophy of constructivism, all seem to share one common view that knowledge is not passively attained by learners from their teachers, but rather, it is constructed by learners through their own experiences (Mestre, 1989; Olivier, 1989; Davis, Maher, & Noddings, 1990; Cobb, 1994). Therefore, it is logical to state that learners need to explore and test ideas through relevant activities in order to construct new ideas. This belief underpins the essence of the framework in this paper. The framework, developed from a constructivist point of view, aims to create a technology-based learning environment where students can construct their mathematical knowledge through interactive activities with computers. Such a learning environment can also provide students with opportunities for social interaction where they share and discuss ideas with their peers as well as their teachers. The social context constructed in the course of their interaction helps to enhance the students' thinking and learning in the classroom (Vygotsky, 1978; Cobb, Wood, & Yackel, 1990).
The framework involves four key components: exploring, conjecturing, verifying, and generalising. Of the four components, the last three are fundamental processes of mathematical thinking. Given that the current literature considerably advocates the exploring and conjecturing processes (Schoaff, 1993; Olive, 1998; Leong & Lim-Teo, 2002), these processes are perceived to be important for developing technology-based mathematics lessons. The exploring process can promote pupils' inquiry and investigation of the task and the conjecturing process provides a means for pupils to construct their own mathematical knowledge. The verifying component is deemed essential because pupils should be encouraged to cultivate a good habit of testing and checking the appropriateness and reasonableness of their...
|
