Yeo, Joseph B. W.

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.

The study investigated the effects of using an interactive computer algebra system called LiveMath on the cognitive development and attitudes of First Year Junior College students. The teacher used the software to engage the students in the experimental class by guiding them to explore mathematical concepts involving topics such as reciprocal curves, Maclaurin’s expansion and applications of integrations. There were two control classes. The first control class used the same guided discovery approach as the experimental class but without the help of technology. The second control class underwent traditional teacher-directed teaching. This paper will discuss the pros and cons of using LiveMath to engage the students and it will present the findings of the study and their implications for teaching and learning.

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.

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.

Mathematics education research has focused on developing teachers’ knowledge or other visible aspects of the teaching practice. This paper contributes to conversations around making a teacher’s thinking visible and enhancing a teacher’s pedagogical reasoning by exploring the use of pedagogical documentation. In this paper, we describe how a teacher’s pedagogical reasoning was made visible and highlight aspects of his thinking in relation to his instructional decisions during a series of lessons on division. Implications for professional learning are discussed.

Joseph B. W. Yeo describes an interactive algebra computer system to help students explore algebra and calculus.

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.

Many teachers have experienced at one time or another the frustrations of trying to impart their knowledge to their students but the latter somehow seem not to grasp the full meaning of the content taught. This may be due to the constructivists’ belief that knowledge cannot be transmitted from teachers to learners but is actively constructed by the learners themselves as they attempt to make sense of their experiences. So this paper attempts to look at how mathematics teachers can redesign their pedagogy by taking into account new teaching methods that are made possible by technology. The paper will also give a few examples of how to use various mathematical software to guide pupils to explore mathematical concepts so that they can construct their own knowledge.

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.