Toh, Tin Lam

This paper presents a re-design of an undergraduate mathematics content course on Introductory Differential Equations for pre-service secondary school mathematics teachers. Based on the science practical paradigm, mathematics practical lessons emphasizing problem-solving processes via the undergraduate content knowledge were embedded within the curriculum delivered through the traditional lecture-tutorial system. The pre-service teachers' performance in six mathematics practical lessons and the mathematics practical test was examined. They were able to respond to the requirements of the mathematics practical to go through the entire process of problem solving and to carry out "Look Back" at their solution: checking the correctness of their solution, offering alternative solutions, and expanding on the given problem. The use of Mathematics Practical has altered the pre-service teachers’ approach in tackling mathematics problems in a positive direction.

In this paper, we discuss the diffusion (of an innovation) and relate it to our attempt to spread our initial design of a mathematics practical paradigm in the teaching of problem solving.

This project involves the development and implementation of a problem solving package (M-ProSE) in the secondary school mathematics curriculum. It aims to induct secondary school mathematics students into the discipline of mathematics via a programme that turns well established theories of mathematical problem solving into praxis. In contrast with conventional training for mathematics competitions which tend to be restricted to a small number, M-ProSE is designed for all mathematics students Development of the project: In a pilot study conducted over two years in an Integrated Programme of a junior college, the research team observed that students were generally resistant to following the stages of Polya's model. In an attempt to 'make' the students follow the Polya model, especially when they were clearly struggling with the problem, we decided to construct a worksheet like that used in science practical lessons and told the students to treat the problem solving class as a mathematics 'practical' lesson. In this way, we hoped to achieve a paradigm shift in the way students looked at these 'difficult, unrelated' problems which had to be done in this 'special' class. Practical work to achieve the learning of the scientific processes has a long history of at least a hundred years. It is certainly conceivable that similar specialised lessons and materials for mathematics may be necessary to teach the mathematical processes, including and via problem solving. Implementation of the project: M-ProSE is an attempt to teach problem solving in 'practical' setup. Students will be taught Polya's model and problem solving in general in two or three dedicated lectures. The main mode of learning is then through a series of 'mathematics practical' lessons. Students work on usually one or at most two problems which have to be worked out on a special worksheet which requires the student to systematically and metacognitively go through the Polya model. M-ProSe is to be implemented as part of the mathematics curriculum and will be assessed. In order to implement M-ProSE, we need to build the teachers' capacity first to solve non-routine mathematics problems and thereafter to teach problem solving to their students. This involves the researchers conducting a series of workshops for the school teachers to widen their repertoire of problem solving resources. Next, we will develop with the teachers the instructional strategies to teach problem solving to their students, by means of a lesson study approach. Some of the researchers will initially teach some student classes as a model for the teachers before they take over entirely. To contribute to the understanding of teaching mathematical problem solving in general, the researchers will collect data over some cohorts which will enable them to further improve the package and make the package useful to other schools. The evidence collected will provide the basis for pedagogical practices in the mathematics classrooms.

In this paper we elaborate on the ways for assessing problem solving that goes beyond the usual focus on the products of the problem solving process. We designed a ‘practical’ worksheet to guide the students through the problem solving process. The worksheet focuses the solver’s attention on the key stages in problem solving. To assess the students’ problem solving throughout the process, we developed a scoring rubric based on Polya’s model (1954) and Schoenfeld’s framework (1985). Student response to the practical worksheet is discussed.

This paper examines the calculus curriculum in the current Singapore secondary and pre-university levels. Two concepts, (1) increasing and decreasing functions and their derivatives, and (2) the second derivative test for the nature of stationary points, are elaborated. An example of an incorrect calculus item in a national examination is brought up in relation to conditional reasoning involving calculus concepts. We reckon that the current emphasis on procedural knowledge in calculus is useful. However, we argue that formal conditional reasoning should not be introduced prematurely for school students.

Mathematical problem solving has been at the core of the Singapore mathematics curriculum framework since the 1990s. We report here the features of the Mathematical Problem Solving for Everyone (M-ProSE) project which was carried out in a Singapore school to realise the learning of mathematical problem solving and as described by Pólya and Schoenfeld. A mathematics problem solving package comprising “mathematics practical” lessons and assessment rubric was trialled in the school for Grade 8 in 2009. Responses from three students show mixed perceptions to the module, but an end-of-module assessment shows that the students were able to present their solutions along Pólya’s four stages. We also describe teacher preparation for teaching the module. After the trial period, the school adopted the module as part of the curriculum and it is now a compulsory course for all Grade 8 students in that school.