Yeo, Joseph B. W.

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.

The study investigated the effect of exploratory computer-based instruction on pupils' conceptual and procedural knowledge of graphs, and the affective issues towards the use of computers in mathematics. 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 although there was no significant difference in their affect towards mathematics in general and towards the topic in particular. The pupils in the experimental class also showed a moderately positive affect towards the use of IT. This seemed to suggest that there was an inherent advantage of using IT to explore mathematical concepts.

The purpose of this research study is to examine the nature and development of cognitive and metacognitive processes that students use when attempting open investigative tasks. Mathematical investigation is important in many school curricula because many educators think that school students should do some real mathematics, the mathematics which academic mathematicians do in their daily and working lives, investigating and solving problems to discover new mathematics. They believe in the benefits of the processes that these mathematicians engage in, e.g. problem posing, specialising, conjecturing, justifying and generalising. Thus it is vital to understand the nature of these processes (i.e. the types of investigation processes and how they interact with one another), and how they can be developed, so that the teachers are better informed to cultivate these processes in their students. Currently, there is a research gap in this field, as there are few empirical studies on processes in mathematical investigation. Therefore, this research study could add value to the advancement of mathematics education in this area.

The sample for the main study consisted of 10 Secondary Two (equivalent to Grade 8) students from a high- performing Singapore school. They went through a teaching experiment consisting of a familiarisation lesson and five developing lessons. The duration of each lesson was two hours. They sat for a pretest at the end of the familiarisation lesson, and a posttest at the end of the last developing lesson. Each student was separately videotaped thinking aloud while working on two open investigative tasks (one from Type A and the other one from Type B) in each test. The verbal protocols were transcribed and coded using a coding scheme, which had passed an inter-coder reliability test. The coded transcripts were then analysed qualitatively to validate and refine the two theoretical investigation models for cognitive and metacognitive processes formulated for this research, to study the effect of these processes on the investigation outcomes, and to examine the development of these processes. A scoring rubric was also devised to score the pretest and the posttest in order to study the effect of the teaching experiment on the development of the investigation processes quantitatively using descriptive statistics.

The findings indicated that the two types of investigative tasks tend to elicit different types of investigation processes and investigation pathways: for Type A, students set out to search for any pattern by specialising, conjecturing, justifying and generalising; for Type B, students posed specific problems to solve by using other heuristics, such as reasoning, and then they extended the task by changing the given in order to generalise. Some new cognitive and metacognitive processes and outcomes were also found, which resulted in the refinement of the two theoretical investigation models. Data analysis showed that there was no direct relationship between the completion of an investigation pathway and the types of investigation outcomes produced. The study also identified the processes that had helped the students to produce significant or non-trivial outcomes in their investigation, the processes that were developed more fully in the students during the teaching experiment, and the processes that were still lacking in the students. The implication was that it is possible to develop investigation processes by teaching the students these processes and providing them the opportunity to develop these processes when they attempt suitable investigative tasks. The research also revealed which processes took a longer time to develop, so more attention should be paid to cultivate these processes during teaching.

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.

This paper introduces a new framework to model the interactions of the processes of specialising and conjecturing when students engage in mathematical investigation. The framework posits that there is usually a cyclic pathway alternating between examining specific examples (specialising) and searching for pattern (conjecturing), instead of a linear pathway as in many other theoretical models. The framework also distinguishes between observing a pattern and formulating it as a conjecture, unlike most models that treat an observed pattern as a conjecture to be proven or refuted. I will then use the framework to analyse and explicate a secondary school student's specialising and conjecturing processes while he attempted an open investigative task.

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.

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.

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.

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.