Computational Experiment Approach to Advanced Secondary Mathematics Curriculum

This chapter provides theoretical underpinning of computational experiment approach to pre-college mathematics curriculum. It reviews mathematics education research publications and (available in English) educational reform documents from Australia, Canada, England, Japan, Singapore and the United States related to the use of computers as tools for experimenting with mathematical ideas. The chapter links pioneering ideas by Euler about experimentation with mathematical ideas to the use of the word experiment in the modern context of pre-college mathematics curricula. It emphasizes the role of mathematics education reform in bringing computers first to the undergraduate level and gradually extending their use to include experimentation at the primary level. Several theoretical frameworks including signature pedagogy, Type I/Type II technology applications, parallel structures of teaching and learning, agent-consumer-amplifier framework, and collateral learning in the digital era are highlighted leading to the development of the notion of technology-enabled mathematics pedagogy (TEMP). One of the major characteristics of TEMP is its focus on the idea with ancient roots—the unity of computational experiment and formal mathematical demonstration. The relationship between technology-enabled experiment and solution-enabled experiment is introduced as a structure that makes computational experiment a meaning making process. It will be demonstrated how visual imagery can support deductive reasoning leading to an error-free computational experiment.

This chapter concerns Type II applications of computing technology to one-variable equations and inequalities. It shows how in the context of TEMP there is a need not only to justify the results of such applications by using theory but also to use theory in order to fill in missing parts left from experimentation. Consequently, the chapter argues for the importance of mathematical knowledge the teachers of mathematics need in order to provide students with Style II assistance in the context of TEMP. The chapter demonstrates how TEMP may include a transition from problem solving to problem posing so that the ACA framework can support Type II technology applications to mathematics teaching. Finally, the chapter shows how the use of technology in the context of experimentation with mathematical concepts can develop entries into the history of mathematics through which connections between the classic concepts and their representations through the modern tools can be established.

This chapter further emphasizes the importance of mathematical knowledge by teachers of secondary mathematics in the context of TEMP. Using quadratic functions as background, the chapter introduces the notion of necessary and sufficient conditions, the use of the concept of locus of an algebraic equation with a single parameter, the use of phase diagrams in the case of equations and functions depending on two parameters, the use of qualitative methods in exploring the behavior of solutions to quadratic equations with parameters. It further emphasizes the importance of seeing problem solving as a springboard into the domain of problem posing, demonstrates combinatorial connections, and investigates the behavior of solutions of two quadratic equations depending on the same as well as different parameters.

In this chapter, different types of equations with parameters will be considered. In Chap. 3, we have investigated the influence of parameters/coefficients of quadratic functions on their properties. Some problems of that kind demonstrated the need to investigate a two-variable equation of the form f(x, a) = 0 representing a family of equations in regard to the variable x whose every element corresponds to a unique value of variable a, which, in this case, is called the parameter of the equation. In this chapter, the computational experiment approach is applied to three basic types of equations with parameters: equations containing absolute values of variables and parameters, equations containing variables and parameters under radicals, and equations containing transcendental functions of variables and parameters. In addition, the approach is used for exploring simultaneous equations in two variables with parameters.

In this chapter, the equation (6.1) where a, b and c are real parameters, a ² + b ² ≠ 0, will be considered. A problem-solving technique based on reducing trigonometric Eq. (6.1) to the system of simultaneous algebraic equations, (6.2–6.3) will be explored. In the digital era, trigonometry curriculum can be made enjoyable provided that computer applications are used to illuminate mathematical ideas. Towards this end, the chapter will emphasize the importance of technology for the teachers’ understanding of connections that exist between trigonometry and geometry.

In the preceding chapters, the computational experiment approach was used to explore different models studied at the secondary level under the assumption that these models depend on one or more parameters. In the case of algebraic/trigonometric equations it was shown that depending on the value of parameter (or parameters), different types of solutions may realize. As an algebraic/trigonometric equation with a parameter may be considered as a mathematical model the behavior of which depends on the parameter, one can assume that not only the parameter varies but, in addition, its variation may depend on random factors. Therefore, one can talk about the likelihood of the event that a solution of a certain type realizes. A geometrization of problem solving in the context of equations with parameters made possible by graphing software tools leads to the construction of regions in the space of parameters the dimension of which depends on the number of parameters involved. When parameters are chosen at random, one can interpret the corresponding likelihood by measuring regions from where parameters responsible for a certain type of solution are selected.

Chapter 8 considers topics in the elementary theory of numbers that span from the time of Pythagoras to the time of Euler and connects ancient ideas about the properties of numbers that concerned Pythagoras and Euclid with more sophisticated interpretation of those ideas by Fermat and Euler. Whereas mathematical explorations that can be included in a computational experiment can be quite significant, software tools allow for hiding some of the complexity and formal structure of mathematics involved. This feature of computing is especially important for mathematics teacher education programs for it enables teacher candidates true engagement in dealing with rather advanced content without the need to have full understanding of the content and command of rigorous argument expected from future professional mathematicians.