Yeo, Joseph B. W.

Engaged learning is an integral part of the ‘Teach Less, Learn More’ initiative. This paper will describe various strategies to engage secondary schools students in their learning of mathematics. It will provide concrete examples of how to use worksheets to guide students to discover certain mathematical concepts, and how to use real-life investigative tasks to engage the minds of the students and to develop in them an inquiry attitude towards mathematics. All these strategies were taught in an inservice course for secondary school teachers and the paper will present their views and attitudes towards engaged learning for their students.

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.

For decades, students have been experiencing problems in understanding mathematical concepts. The use of technology has helped to overcome some of these problems. This paper attempts to share some of the technologies used by secondary school teachers in Singapore to facilitate students’ learning, e.g. the use of dynamic geometry software such as the Geometer’s Sketchpad (GSP) to help students visualize, experiment and construct new knowledge about geometric properties, and the use of Excel to reduce the tediousness of constructing graphs and enhance students’ understanding of functions and graphs.

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.

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, I will discuss the use of mathematically-rich games to develop in students certain skills and processes that are important in their daily and future workplace life. For example, students will learn through these games how to pose relevant and important questions when faced with a problem, how to formulate conjectures to solve the problem, what strategies or heuristics to use, and how to monitor their progress and their own thinking. The context is very real for these students because the outcome, whether they win or lose, matters to them.

Many mathematics educators in Singapore secondary schools are aware that The Geometer’s Sketchpad, a dynamic geometry software, can be used to explore geometry. But most of them do not know of any computer algebra system (CAS) that can be used to explore algebra and calculus. Traditionally, most mathematicians, scientists and engineers have always used a CAS, such as Maple, to perform symbolic manipulations in order to solve algebraic and calculus problems. However most educators do not see any purpose in their pupils learning a CAS to perform symbolic manipulations, such as factorisation, differentiation and integration, when formal assessments still require them to perform such skills by hand. But with the advance of LiveMath (previously known as Theorist and MathView), an intriguing CAS that provides “a unique user interface that allows one to perform ‘natural’ algebraic maneuvers even more ‘naturally’ than one can achieve them on paper” (Kaput, 1992), there is now another way of using a CAS in the teaching and learning of mathematics, i.e., to explore algebraic and calculus concepts. Moreover the capability of LiveMath templates to be interactive even on Web pages opens up an exciting chapter in online mathematics learning. This paper looks at some examples of how educators can use LiveMath as an interactive tool for their pupils to explore algebra and calculus. It also provides some research evidence to suggest that the use of LiveMath for exploring mathematics may enhance pupil learning.

Despite recent calls to adopt practice-embedded approaches to teacher professional learning, how teachers learn from their practice is not clear. What really matters is not the type of professional learning activities, but how teachers engage with them. In this paper, we position learning from teaching as a dialogic process involving teachers’ pedagogical reasoning and actions. In particular, we present a case of an experienced teacher, Mr. Robert, who was part of a primary school’s mathematics professional learning team (PLT) to describe how he learned to teach differently, and how he taught differently to learn for a series of lessons on division. The findings reiterate the complexity of teacher learning and suggest possible implications for mathematics teacher professional development.

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.