Research Reports

Attitudes have been conceptualized as "learned predisposition to respond positively or negatively to objects, situations, institution, concepts or persons (Aiken, 19XXs 2)." An attitude is said to have an element of approval-disapproval; is generalized and emotive, may not be in the full awareness of the person who possesses it, and affects the selection and organization of pertinent facts (Aiken, 19XX). Research interest in the measurement of attitudes implies a relationship between attitudes and behaviours. It is quite common that behaviours are attributed to the existence or the lack of relevant attitudes, expost facto. It is equally common, therefore, to expect attitudes to predict behaviours. In spite of the rather discouraging results of attempts at correlating attitudes and behaviours over the past half century, new theorization and methodologies have imbued new vatality into the search of more effective techniques of attitudinal measurement. This is true of research in sociology and psychology as well as in education, as evidenced by the number of attitude scales collected, for example, in Shaw and Wright (1967). Against this background, it is natural to consider the feasibility of using attitude scales for the identification and selection of potential effective teachers. Teacher education institutes have normally admitted students on the strength of their academic achievement usually supplemented by interviews, and the predictive validity of these remain largely to be proved empirically (Eng, 198..}). Two possible explanations can be advanced for this. First, students admitted to teacher education programme for highly selected groups as a result of the selection process. This yield a truncated distribution in the range of academic achievement which could have attenuated the correlation between achievement and teacher effectiveness however measured. Second, it takes much more than academic prowess to make effective teachers as teaching is as much an interaction between the teacher and pupils in the social context of the classroom and the school as it is a process of cultural transmission. It may then be safely said that academic excellence is a necessary by not sufficient condition of teacher effectiveness. It would appear then that measures of student attitudes toward education and teaching may contribute to the prediction of teacher effectivenes and supplement information gathered under the tradition of using academic achievement and interview. The pilot study to use the Minnesota Teacher Attitude Inventory (Cook et al., 1951) for predicting teacher effectiveness in the Singapore context (Eng et al., 1983) is a case in point. The research team is also looking into the predictive validity of the Sixteen Personality Factors Questionnaire (Cattell and Eber, 1968). Yet another attempt of the team is to create an indigenous scale for measuring attitudes towards education and teaching amongst students. As a matter of fact, the data analyzed in this study are those collected by the research team* and were made available to the present writer for a secondary analysis+ to complement their effort.

This paper examines some of the complexities involved in the actual work of classroom instruction by examining interactions among the goals of teaching. The research is part of a case study of teaching a Year 7 Singapore class comprising students of average mathematical ability. Among the complexities of teaching analysed here are the problems associated with trying to fulfil the multiple goals of teaching and the conflict experienced by the teacher as he attempts to carry out these goals. This provides insight into how a teacher performs the act of balancing different goals while carrying out instruction in class. The implications of these insights into teaching practice for the wider education community are also 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 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.

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.

A topological space is sober if every nonempty irreducible closed set is the closure of a unique singleton set. Sobriety is precisely the topological property that allows one to recover completely a topological space from its frame of opens. Because every Hausdor space is sober, sobriety is an overt, and hence unnamed, notion. Even in non-Hausdor settings, sober spaces abound. A well-known instance of a sober space appears in domain theory: the Scott topology of a continuous dcpo is sober. The converse is false as witnessed by two counterexamples constructed in the early 1980's: the first by P.T. Johnstone and the second (a complete lattice) by J. Isbell. Since then, there has been limited progress in the quest for an order-theoretic characterization of those dcpo's for which their Scott topology is sober. This paper provides one answer to this open problem.