Paul Drijvers

The Co-Emergence of Machine Techniques, Paper-and-Pencil Techniques, and Theoretical Reflection: A Study of Cas use in Secondary School Algebra

This paper addresses the dialectical relation between theoretical thinking and technique, as they co-emerge in a combined computer algebra (CAS) and paper-and-pencil environment. The theoretical framework in this ongoing study consists of the instrumental approach to tool use and an adaptation of Chevallard’s anthropological theory. The main aim is to unravel the subtle intertwining of students’ theoretical thinking and the techniques they use in both media, within the process of instrumental genesis. Two grade 10 teaching experiments are described, the first one on equivalence, equality and equation, and the second one on generalizing and proving within factoring. Even though the two topics are quite different, findings indicate the importance of the co-emergence of theory and technique in both cases. Some further extensions of the theoretical framework are suggested, focusing on the relation between paper-and-pencil techniques and computer algebra techniques, and on the issue of language and discourse in the learning process.

Integrating Technology into Mathematics Education: Theoretical Perspectives

The central question at stake in this chapter is: What theoretical frames are used in technology-related research in the domain of mathematics education and what do these theoretical perspectives offer? An historical overview of the development of theoretical frameworks that are considered to be relevant to the issue of integrating technological tools into mathematics education is provided. Instrumental approaches and the notion of semiotic mediation are discussed in more detail. A plea is made for the development of integrative theoretical frameworks that allow for the articulation of different theoretical perspectives.

Students Encountering Obstacles Using a CAS.

The paper describes a pilot study on the use of computer algebra at upper secondary level. A symbolic calculator was introduced in a pre-examination class studying for advanced pre-university mathematics. With the theoretical framework of Realistic Mathematics Education and Developmental Research as a background, the study focused on the identification of obstacles that students encountered while using computer algebra. Five obstacles were identified that have both a technical and a mathematical character. It is the authors belief that taking these barriers seriously is important in developing useful pedagogical strategies.

Effects of attitudes and behaviours on learning mathematics with computer tools

This mixed-methods study investigates the effects of student attitudes and behaviours on the outcomes of learning mathematics with computer tools. A computer tool was used to help students develop the mathematical concept of function. In the whole sample (N=521), student attitudes could account for a 3.4 point difference in test scores between individuals on a 10-point scale. General attitude towards mathematics positively predicted test scores. However, more able students who were well-disposed towards mathematical computer tools achieved lower scores. Self-reported behaviours were unrelated to test scores. Detailed observation of a small number of students (N=8) revealed that positive attitudes towards mathematics and mathematical computer tools augmented exhibited learning behaviours, and that both a positive attitude to mathematical computer tools and exhibited learning behaviours benefited tool mastery. Although tool mastery and test scores are intimately related, reflective processes appear to mediate this relationship. Promoting learning with mathematical computer tools needs to take several factors into account, including improving student attitudes, raising levels of learning behaviours, and giving sufficient opportunity for constructing new mathematical knowledge within meaningful mathematical discourse.

Computer Algebra as an Instrument: Examples of Algebraic Schemes

In this chapter, we investigate the relationship between computer algebra use and algebraic thinking from the perspective of the instrumental approach to learning mathematics in a technological environment, which was addressed in the previous chapter.

Digital Tools for Algebra Education: Criteria and Evaluation

In the Netherlands, as in many other countries, the algebraic expertise of students graduating from secondary education is an issue. The use of digital tools for algebra education is expected to change epistemologies, activity structures and student achievement. Therefore, a study was set up to investigate in what way the use of ICT in upper secondary education might enhance the algebraic expertise of students. One of the first decisions to be made concerned the choice of appropriate digital tools. This paper describes the process of designing and using an instrument for evaluating digital tools. The conceptual framework guiding this process includes notions on symbol sense, instrumental genesis and formative assessment. The evaluation instrument is designed through a Delphi method and provides a blueprint of tool features that are relevant for the purpose of this study. The results show that such an evaluation instrument is valuable both for choosing appropriate digital tools and for making concrete the aims and expectations that researchers have on the issue of integrating technology in algebra education. The final instrument is presented and illustrated through examples implemented in different digital algebra tools.

Learning mathematics in a computer algebra environment: obstacles are opportunities

Using computer algebra is not as easy it may seem. Students often encounter obstacles while working in a computer algebra environment. In this paper, global and local obstacles are distinguished, and obstacles from both categories are identified. The theory of instrumentation provides a framework for interpreting an obstacle as an unbalance of the conceptual and technical aspects of an instrumentation scheme. It is argued that making the obstacles explicit and trying to overcome them leads to conceptual development. Therefore, obstacles are opportunities for learning.

Digital Technology in Mathematics Education: Why It Works (Or Doesn't).

The integration of digital technology confronts teachers, educators and researchers with many questions. What is the potential of ICT for learning and teaching, and which factors are decisive in making it work in the mathematics classroom? To investigate these questions, six cases from leading studies in the field are described, and decisive success factors are identified. This leads to the conclusion that crucial factors for the success of digital technology in mathematics education include the design of the digital tool and corresponding tasks exploiting the tool’s pedagogical potential, the role of the teacher and the educational context.

Technology-Driven Developments and Policy Implications for Mathematics Education

The advent of technology has done more than merely increase the range of resources available for mathematics teaching and learning: it represents the emergence of a new culture--a virtual culture with new paradigms--which differs crucially from preceding cultural forms. In this chapter, the implications of this paradigm shift for policies concerning learning, curriculum design, and teacher education will be discussed. Also, the ubiquitous possibility of emergence of ever-new forms of technology brings about both new opportunities for learning and collaborative work (involving students and teachers), as well as potential dangers. Policy measures may give priority to technological access and developments, over the intellectual growth of learners and the professional development of teachers--which should be more demanding goals of mathematics education. Such policy issues will be discussed.

Teachers Transforming Resources into Orchestrations

Nowadays, mathematics teachers are confronted with a myriad of resources that are available through technological means and through the internet in particular. However, teachers may perceive difficulties in orchestrating education which makes use of technological tools and resources. With instrumental orchestration as a theoretical lens, the main question addressed in this chapter is into which types of orchestrations teachers transform the technological resources. In the setting of a pilot teaching sequence in grade 12 on using applets for practicing algebraic skills, this question is investigated, through a case study of one participating mathematics teacher, through questionnaires among the 69 participating teachers, and through interviews with six of them. The results show that teachers privilege orchestrations in which students work individually or in pairs, at the cost of whole-class orchestration types. Within the performance of these student-centered orchestrations, we recognize elements already known from previously identified orchestrations. Compared to their regular teaching practices and their expectations before the pilot, the involvement in the pilot causes teachers to adapt their orchestrations during the pilot.