Yeo, Joseph Kai Kow

This study sets out to investigate the effects of two modes of instruction on the learning of quadratic curves among the secondary two students. The two modes of instruction are Computer-Assisted Instruction (CAI) and Traditional Approach (TA). These two approaches were examined in terms of their effects on achievement and attitudes, both toward the topic and the subject. A total of 192 students from the Express stream of an independent school in Singapore participated in the study. Six intact classes were chosen. Two high, two medium and two low ability classes were selected. One of each of the ability group was randomly assigned as experimental group (CAI) and the other as the control group (TA). All the six groups were taught by the same teacher and used the same amount of time to learn the same concepts, with similar assignments set as homework. The experiment was conducted over a period of two weeks with a total of 10 periods each for the six groups. The duration of each period was 35 minutes.

In the CAI group, a specially designed program on Quadratic Curves was written to meet the specific instructional objectives of the topics. Worksheets were constructed to supplement the software.

Before the main study was conducted, a Quadratic Curves Achievement Test (QC1) was administrated to the six groups of students at the end of April 1993. This was to determine if the students in each ability group differed in terms of their achievement in Mathematics. Similarly, the Attitudes toward Mathematics Questionnaire (MA1) was also given to the students to respond. This was to find out if the six groups differed in their attitudes toward Mathematics prior to the treatments.

At the end of the experiment, a 40-item objective type Quadratic Curves Achievement Test (QC) was given to six groups to confirm the effects of both modes of instruction. The Attitudes toward Mathematics Questionnaire (MA2) was also administered as a posttest to determine if there were any changes in attitudes toward Mathematics for the six groups. In addition, a specially designed Attitudes toward Quadratic Curves Questionnaire was also given to the six groups to determine if the different groups differed in their attitudes toward the content topic covered.

The independent t-test was used to compare the means of pretest, QC scores, QCA scores, mean gain scores of QC1 and QC2 for the different ability group and mean gain scores of MA1 and MA2. An one way analysis of variance was also used to compare the means of gain scores of QC1 and QC2 and gain scores of MA1 and MA2 for the three different groups.

The main findings of the study were:

The analysis of Quadratic Curves Achievement pretest indicated that there were no significant differences in mathematical background before treatment between students in the CAI and TA groups. Analysis of posttest data on achievement (QC) test showed that medium ability CAI group seemed to perform significantly better than the medium ability TA group. However, The high and low ability CAI group did not differed significantly from the high and low ability TA group. When the results were further examined between the three CAI groups, the mean gain scores indicated that the three CAI groups differed in their quadratic curves achievement after the treatment. Similarly, the three TA groups mean gain scores also showed that the three TA groups differed in their quadratic curves achievement.

In terms of attitudes toward Mathematics as measured by the MA, the analysis of the pretest indicated that there were no significant difference in their attitudes toward Mathematics before treatment between students in the CAI and TA groups. On the other hand, analysis of mean gain scores showed that the high, medium and low ability students in the CAI group had more positive attitudes than the high, medium and low ability TA groups after the treatment.

In the attitudes toward the particular topic, the three CAI groups had significantly more positive attitudes than the three TA groups.

Assessment is an essential part of the teaching and learning process. Since assessment plays such a significant part in the educational process, it is imperative that we examine closely the individual questions that make up the assessment paper. This paper is based on a preliminary investigation into about 30 sets of secondary one semestral examination papers. In this paper only a sample of problematic short-ended and structured / long-answer questions have been highlighted and discussed. From this sample, it could be seen that short-ended and structured / long-answer questions are challenging and demanding to design. The questions were scrutinized based on the following criteria involved in the designing of test/examination questions: (1) Questions with incorrect Mathematical Concept, (2) Questions which test trivial facts, (3) Questions with ambiguous verbal communication, (4) Questions with inconsistent symbols and notations, (5) Questions with misleading diagrams, (5) Questions with impracticality of scenarios, and (6) Questions with imbalanced learning objectives.

Researchers have highlighted the significance of affective factors and their influences on problem-solving performance. Moreover, it was been reported that research on the cognitive processes in mathematical problem solving has been over-represented resulting in a neglect of non-cognitive factors such as emotions and anxiety. Therefore, the purpose of this study was to identify interrelationships among mathematics anxiety, test anxiety and problem-solving performance. A total of 621 Secondary 2 students from ten secondary schools in Singapore participated in the study. The data showed a positive relationship between mathematics anxiety and test anxiety. In addition, mathematics anxiety and performance on a nonroutine mathematical problem solving test showed a marginal linear relationship whereas test anxiety had almost no relationship with the performance on the nonroutine mathematical problem solving test.

Mathematics education in Singapore schools in the twenty-first century is still going through the period of change and innovation that began in the early 1990s: changes in emphasis from rote memorisation to meaningful understanding of concepts and problem-solving; from a dependence on paper and pencil and manipulative calculations and skills to mental computations and thinking strategies; and from teaching by telling to activity-based learning, group work, and communication in mathematics. This chapter makes an attempt to describe how the mathematics curriculum in Singapore has responded to such changes and innovations. At the same time, the chapter has chosen three out of many major innovations in Singapore mathematics education and discusses them in relation to school mathematics: problems in real-world contexts (PRWC), Learning Support for Mathematics (LSM), and Improving Confidence and Achievement in Numeracy (ICAN). These innovations are selected because the ways Singapore has approached them might be of theoretical and practical interest to international readers.

The use of scientific calculators will be first allowed in Singapore Primary School Leaving Examination (PSLE) for all primary level mathematics subjects from the year 2009 onwards. All Primary 5 and 6 mathematics teachers will be expected to explore the use of calculators into their mathematics lessons from 2008 onwards. To meet the new assessment requirements, primary school mathematics teachers are required to be proficient in using the calculator and adept at facilitating pupils’ usage of the calculator. Evidence from literature review and research has showed that calculator is an effective tool for enhancement of mathematical concepts, development of mental arithmetic skills, pattern recognition, mathematical investigation, solving real-life problems and improving problem-solving ability. The purpose of this paper is to review what research says about outcome of calculator use in the learning of primary mathematics. This paper also describes six appropriate calculator activities that can be incorporated in the teaching and learning of mathematics at the primary level.

In this study, addition and subtraction word problems based on the semantic structures are analysed in the sole Primary Two mathematics textbooks and accompanying workbooks used in Singapore. Based on a conceptual coding framework, the word problems were coded accordingly. The results revealed a significant representation of combine and compare structures across all contents in both the textbooks and accompanying workbooks. In particular, the lack of word problems involving change structure suggests an unequal distribution of the semantic structures. Based on the findings, it is recommended that educators and textbook authors to be aware in providing students the opportunity to be equally exposed to the various semantic structures in the teaching and learning of both the addition and subtraction word problems.

This exploratory study attempted to identify interrelationships between and among mathematics anxiety, test anxiety and problem-solving performance of Secondary 2 students in Singapore, categorise the mathematics-anxiety levels of these students into five levels, and explore their mathematical problem-solving performance in each level. The research also studied the heuristics and mathematical problem-solving framework used by the students in each of the mathematics-anxiety levels to solve problems. It delved further into characteristics of high mathematics-anxiety students and explored their reasons and feelings with regards to choices of problems as well as the difficulties they faced when solving problems.

A total of 621 Secondary 2 students from Singapore schools participated in Phase I of the study. Of these 621 students, 112 high mathematics-anxiety students were selected to participate in Phase II of the study. The sample was representative of the general student population in Singapore schools.

The design involved the development and use of paper and pencil instruments to collect data from the 621 Secondary 2 students during Phase I and from a sub-sample of 112 students during Phase II. During Phase I, the individual reflections of the 621 students on their problem-solving processes were recorded. Interviews were carried out with 56 students during Phase II of the study.

The results of the study showed that there was a positive correlation between test- and mathematics-anxiety scales while test anxiety did not associate with non-routine mathematical problem-solving test. The mathematics anxiety and Problems Test scores showed a marginal linear relationship. The varied performance of the Secondary 2 students on the five problems items also suggests that different mathematics-anxiety levels students may perform differently on different problems. It appears that the students at the low mathematics-anxiety level performed better on a non-routine mathematical problem-solving test than the high mathematics-anxiety students. Particular mathematical problem-solving heuristics were found to be used by students from mathematics-anxiety levels 1 to 5 to solve non-routine mathematical tasks. Although they were found to differ in the repertoire of heuristics, the difference was only marginal. The Secondary 2 students were found to rely on individual problem-solving frameworks to guide them when solving problems. The framework of the different mathematics-anxiety levels students was found to be similar, brief, and specific in nature.

The reasons given by high mathematics-anxiety students when choosing a problem to solve first were: "easiest problem", "familiar problem", "minimum working required", and "understand the problem". The main reasons for choosing to solve a particular problem last were: "difficulty of the problem", "multiple steps required", "more time required", and "lack of understanding". The feelings manifested by high mathematics-anxiety students when choosing a problem to solve first were more positive. However, with a problem that they chose to solve last, they felt anxious, stressed, tensed, irritated, frustrated, angry, fearful, and bewildered. It was found that the difficulties experienced by high mathematics-anxiety students when obtaining a solution were : (a) lack of comprehension of the problem posed, (b) lack of strategy knowledge, (c) inability to translate the problem into mathematical form, and (d) inability to use the correct mathematics.

Probability is often an immensely challenging topic for students at the secondary level. The main difficulty for these students often lies in their inability to relate the theories learnt in classroom to their everyday lives. It is for this reason that lessons on the topic of Probability, especially those at the introductory level, rely heavily on hands-on activities. As empirical probabilities will only converge to the results predicted by theory, implementation of hands-on activities is, inherently, ridden with problems and difficulties. The problem is so acute that ‘mistake-prone’, ‘repetitive’ and ‘time consuming’ are but a few phrases which students have associated with traditional hands-on activities. This apparent lack of meaningful hands-on activities has motivated the authors to develop an IT-based discovery activity for the topic of Probability at the secondary level. Microsoft’s Excel has been embraced by the authors in automating the counting and calculations involved in obtaining experimental probabilities which would otherwise be cumbersome to obtain. To enhance the appeal of the discovery, the authors have used the game of Monopoly as the theme of analysis. This paper provides pertinent details of the discovery activity. It also describes the benefits and possible barriers to successful implementation of the activity.

Teachers tend to design worksheets to assist their students to develop their spatial skills. Many secondary schools students may not have sufficient concrete experiences to tackle abstract reasoning in geometrical concepts competently. In addition, students who are nonvisual and spatial learners may have difficulties in learning geometrical topics. The aim of this study was to use the software, ProDesktop, to facilitate the teaching and learning of Plan and Elevation Geometry in a computer laboratory. A total of forty students from the Express stream of a neighborhood school in Singapore was selected to participate in the study. The results suggested that the use of ProDesktop had enhanced the learning of Plan and Elevation geometry both in terms of interest and proficiency of the topic.