Yeo, Joseph B. W.

The effect of computer-based learning using LiveMath, an interactive computer algebra system, on Singapore Secondary Four students' conceptual and procedural knowledge of exponential and logarithmic curves, was investigated in this study. Sixty-five students from two middle-ability Express classes in an independent school were taught using a guided discovery approach to explore the characteristics of the graphs of exponential and logarithmic hnctions. The experimental class used LiveMath for their exploration whereas the control class used worksheets that contained pre-printed graphs. The findings show that the students in the experimental group performed significantly better than the students in the control group in both the conceptual and the procedural knowledge tests.

The study investigated the effects of using an interactive computer algebra system called LiveMath on the cognitive development and attitudes of First Year Junior College students. The teacher used the software to engage the students in the experimental class by guiding them to explore mathematical concepts involving topics such as reciprocal curves, Maclaurin’s expansion and applications of integrations. There were two control classes. The first control class used the same guided discovery approach as the experimental class but without the help of technology. The second control class underwent traditional teacher-directed teaching. This paper will discuss the pros and cons of using LiveMath to engage the students and it will present the findings of the study and their implications for teaching and learning.

This article presents an exploratory study to find out whether high-ability secondary school students in Singapore were able to deal with open mathematical investigative tasks. A class of Secondary One (or Grade 7) students, who had no prior experience with this kind of investigation, were given a paper-and-pencil test consisting of four open tasks. The results show that these students did not even know how to begin, despite sample questions being given in the first two tasks to guide and help them pose their own problems. The main difficulty was the inability to understand the task requirement: what does it mean to investigate? Another issue was the difference between searching for any patterns without a specific problem to solve, and searching for patterns to solve a given problem. The implications of these findings on teaching and on research methodologies that rely on paper-and-pencil test instruments will also be discussed.

In this paper, the differences between mathematical tasks such as problem-solving tasks, investigative tasks, guided-discovery tasks, project work, real-life tasks, problem-posing tasks, open tasks and ill structured tasks will be contrasted. Such clarification is important because it can affect how and what teachers teach since the diverse types of tasks have different pedagogical uses, and it can also help researchers to define more clearly the tasks that they are investigating on. A framework to characterise the openness of mathematical tasks based on task variables such as the goal, the method, the answer, scaffolding and extension will be described. The tasks are then classified according to their teaching purpose: mathematically-rich tasks, such as analytical tasks and synthesis tasks, can provide students with opportunities to learn new mathematics and to develop mathematical processes such as problem solving strategies, analytical thinking, metacognition and creativity; and non-mathematically-rich tasks, such as procedural tasks, can only provide students with practice of procedures. Rich assessment tasks that teachers can use to assess students’ conceptual understanding, mathematical communication and thinking processes will also be discussed. The clarification of terminologies and the classification of mathematical tasks will help teachers to understand more about the purpose and characteristics of the diverse types of tasks so that they can choose appropriate tasks to develop the different facets of their students’ mental structures and to assess the various aspects of their learning.

This paper describes a research study on how and what secondary school students investigate when faced with an open investigative task involving an interesting game that combines magic square and tic-tac-toe. It will examine the strategies that the students use and the mathematical thinking processes that they engage in when doing their investigation. The findings will be used to inform a theoretical model that we have devised to study the cognitive processes of open mathematical investigation, which include understanding the task, posing problems to investigate, specialising, formulating and testing conjectures, generalising, looking back and extending the task.

The study investigated the effect of exploratory computer-based instruction on pupils’ conceptual and procedural knowledge of graphs. Many previous studies compared the effect of computer-assisted instruction with traditional teacher-directed teaching and any difference in performance might be due to a different pedagogical approach instead of the use of information technology (IT). In this study, both the experimental and control classes were taught using a guided discovery method to explore the characteristics of the exponential and logarithmic curves. One class used an interactive computer algebra system called LiveMath, while the other did not have access to IT. The findings indicated a significant difference in pupils’ conceptual and procedural knowledge. This seemed to suggest that there was an inherent advantage of using IT to explore mathematical concepts.

Many educators believe that mathematical investigation is open and it involves both problem posing and problem solving, but some teachers have taught their students to investigate during problem solving. The confusion about the relationship between investigation and problem solving may affect how teachers teach their students and how researchers conduct their research. Moreover, there is a research gap in studying the thinking processes in mathematical investigation, partly because it is not easy to define these processes. Therefore, this article seeks to address these issues by first distinguishing between investigation as a task, a process and an activity; and then providing an alternative characterisation of the process of investigation in terms of its core cognitive processes: specialising, conjecturing, justifying and generalising. These will help to clarify the relationship between investigation and problem solving: an open investigative activity involves both problem posing and problem solving; but the problem-solving process entails solving by the process of investigation and/or by using "other heuristics". In other words, mathematical investigation does not have to be open. The article concludes with some implications of this alternative view of mathematical investigation on teaching and research.

Mathematics education research has focused on developing teachers’ knowledge or other visible aspects of the teaching practice. This paper contributes to conversations around making a teacher’s thinking visible and enhancing a teacher’s pedagogical reasoning by exploring the use of pedagogical documentation. In this paper, we describe how a teacher’s pedagogical reasoning was made visible and highlight aspects of his thinking in relation to his instructional decisions during a series of lessons on division. Implications for professional learning are discussed.

Equivalence is one of the eight clusters of big ideas proposed for the 2020 secondary school mathematics syllabus in Singapore. In this chapter, I will unpack the meaning of equivalence: the idea of an equivalence relation, the differences between equivalence and equality, and the usefulness of equivalent equations and equivalent statements in solving mathematical problems. In particular, I will examine the solutions of some types of equations which do not seem to produce equivalent equations in subsequent steps, e.g. the introduction of an extraneous solution when solving some equations involving surds or logarithms, and the elimination of a variable when solving a pair of simultaneous equations in two variables, and explain how the solutions of these equations can still produce equivalent equations in subsequent steps. In other words, the transformation or conversion from one equation to another equivalent equation is still the basis of the method for solving any kinds of equations. In addition, this chapter will discuss how to teach secondary school students the big idea of equivalence without teaching the abstract idea of an equivalence relation.