Yeo, Joseph B. W.

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.

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 writers believe that mathematical investigation is open and it involves both problem posing and problem solving. However, some teachers feel that there is a sense of doing some sort of investigation when solving problems with a closed goal and answer but they are unable to identify the characteristics of this type of investigation. Such confusion will affect how teachers teach their students and how researchers conduct their research on investigation. Therefore, this article seeks to clarify the relationship between investigation and problem solving by providing an alternative characterisation of mathematical investigation as a process involving specialisation, conjecturing, justification and generalisation. It also distinguishes between mathematical investigation as a process and as an activity: investigation, as a process, can occur when solving problems with a closed goal and answer, while investigation, as an activity involving open investigative tasks, includes both problem posing and problem solving. Implicit support for this alternative characterisation of mathematical investigation is gathered from some existing literature as these writers did not state this perspective explicitly. The article concludes with some implications of this alternative view on teaching and research.

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.

This paper reports how 30 experienced and competent Singapore secondary mathematics teachers attempted to imbue desired attitudes in their students and some possible factors that might have influenced the teachers’ choice of instructional approaches. It was found from the analysis of lesson observations of these teachers that most of those teaching lower-ability students tended to build their students’ confidence and perseverance, while those teaching higher-ability students were more inclined to help their students appreciate the relevance of mathematics. Only a minority of the teachers tried to make lessons fun by using mathematics related resources or telling non-mathematics-related jokes. It was also discovered from the teacher interviews that two factors appeared to influence the teachers’ choice of the types of positive attitudes to develop in their students: the abilities of their students and the beliefs of the teachers on what mathematics is.

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 illstructured 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 problemsolving 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 reports on the types of problems that high-achieving students posed when given investigative tasks that were constructed by opening up some mathematical problems. A teaching experiment was conducted to develop the students’ thinking processes during mathematical investigation, and each student was videotaped thinking aloud during a pretest and a posttest. The findings show that some students were unable to pose the original intended problems and what Krutetskii (1976) called problems that ‘naturally follow’ from the task, including extending the task to generalise. The implications of the difficulty encountered by these students for teaching and research 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.

The underlying basis of the self-determination theory (SDT) is that people are inherently motivated to learn if their basic psychological needs of autonomy, competence and relatedness are met. The theory also provides a comprehensive taxonomy on the different types of extrinsic and intrinsic motivations. For example, identified and integrated extrinsic motivations are based on a sense of value while intrinsic motivation is based on interest. In this article, I will put the theory into practice, suggesting in more concrete terms how teachers could motivate their students to learn mathematics. First, I will describe some applications within mathematics and in the real world which could be used to motivate students extrinsically by helping them see the value of what they are studying. Moreover, real life examples might also help students relate mathematics to their own experiences. Then I will provide some examples of catchy mathematics songs and amusing videos which could be used to motivate students intrinsically by arousing their interest. I will also discuss how to build up students’ competence in mathematics by developing concepts using examples and not definitions, and by using guided discovery learning and guided proofs, which could also provide autonomy support for the students. I will examine how the practice of procedural skills could be structured more effectively and how mathematics puzzles and gamification could make such practice more enjoyable. Lastly, I will draw on a research study to inform what Singapore teachers are doing to motivate their students.

This paper describes a research study to find out the ability of Singapore secondary school students in attempting open investigative tasks. The results show that most high-ability students had no experience in open mathematical investigation and they did not even know how to start. Providing sample problems in the tasks for students to investigate did not seem to help them understand the requirements of the tasks. The implication of these findings on research methodology using paper-and-pencil tests will be discussed.