Making sense of mathematics : supportive and problematic conceptions with special reference to trigonometry

This thesis is concerned with how a group of student teachers make sense of trigonometry. There are three main ideas in this study. This first idea is about the theoretical framework which focusses on the growth of mathematical thinking based on human perception, operation and reason. This framework evolves from the work of Piaget, Bruner, Skemp, Dienes, Van Hiele and others. Although the study focusses on trigonometry, the theory constructed is applicable to a wide range of mathematics topics. The second idea is about three distinct contexts of trigonometry namely triangle trigonometry, circle trigonometry and analytic trigonometry. Triangle trigonometry is based on right angled triangles with positive sides and angles bigger than 0 [degrees] and less than 90 [degrees]. Circle trigonometry involves dynamic angles of any size and sign with trigonometric ratios involving signed numbers and the properties of trigonometric functions represented as graphs. Analytic trigonometry involves trigonometric functions expressed as power series and the use of complex numbers to relate exponential and trigonometric functions. The third idea is about supportive and problematic conceptions in making sense of mathematics. This idea evolves from the idea of met‐before as proposed in Tall (2004). In this case, the concept of ‘met‐before’ is given a working definition as ‘a trace that it leaves in the mind that affects our current thinking’. Supportive conception supports generalization in a new contexts whereas problematic conception impedes generalization. Furthermore, a supportive conception might contain problematic aspects in it and a problematic conception might contain supportive aspects in it. In general, supportive conceptions will give the learner a sense of confidence whereas problematic conceptions will give the learner of sense of anxiety. Supportive conceptions may occur in different ways. Some learners might know how to perform an algorithm without a grasp of how it can be related to different mathematical concepts and the underlying reasons for using such an algorithm

EXPLORING THE IMPLEMENTATION OF AN INTERVENTION FOR A PUPIL WITH MATHEMATICAL LEARNING DIFFICULTIES: A CASE STUDY

This study presents a single case study of how a remedial mathematics teacher incorporated an instructional intervention into her teaching practices in order to teach counting to a pupil with mathematical learning difficulties. This new theory-driven intervention was developed by the authors of this study. Dyscalculia is a term which refers to a wide range of mathematical learning difficulties or disabilities. Dyscalculic pupils have a specific mathematics learning disorder with a core deficit in representing and processing of numerosity. They might not be able to recognise numerical quantities, performing counting and so on. Early supports such as interventions have a great potential in helping dyscalculic pupils to improve mathematical skills. However, there remains a lack of appropriate instructional scaffolds to help dyscalculic pupils to organise their learning structures by addressing both cognitive deficits and mathematical skills. The present study involves a primary school remedial teacher, Daisy, and an at-risk dyscalculic pupil, David, both pseudonyms. Data were collected through interviews, lesson observations, and reflective journals. The findings revealed that the proposed intervention improved the counting ability of the pupil

Knowing and Grasping of Two University Students: The Case of Complex Numbers

This paper aims to introduce the notions of knowing and grasping of a mathematics concept. We choose the concept of complex numbers to illustrate how these notions can be used to describe the understanding of students for this particular topic. As students develop their knowledge, supportive conceptions and problematic conceptions may occur in different ways. A student may know how to perform an algorithm or how to use a particular concept, without ‘grasping’ the meaning of the idea in a manner which enables him to comprehend more sophisticated ideas in an extended context. Grasping a concept means the ability to see the different aspects of an underlying concept, manipulate it and use it in different ways for different purposes. On top of that, one should be able to speak of it as a meaningful entity in its own right. In this paper, we will report the data which were collected through questionnaires and follow-up interviews of two third year undergraduate mathematics students to illustrate the subtle distinctions between “knowing” and “grasping”. The results reveal that both participants, M1 and M2, couldn’t grasp the concept of complex numbers as a coherent whole due to the problematic conceptions which arose from real numbers. There might be other factors which contributed to this phenomenon such as the theoretically abstract nature of complex numbers and surrounding factors that affect learning. This study leads us to realise the importance of creating the necessary experience for learners to make sense of complex numbers so that learners can build from their existing knowledge in real numbers which may conflict with complex numbers. Too often, we have focused on a particular context, teaching the content that learners should know without helping them to grasp the essential ideas for further leaning

An investigation of students algebraic proficiency from a structure sense perspective

Structure sense can be interpreted as an intuitive ability towards symbolic expressions, including skills to perceive, to interpret, and to manipulate symbols in different roles. This ability shows student algebraic proficiency in dealing with various symbolic expressions and is considered important to be mastered by secondary school students for advanced study or professional work. This study, therefore, aims to investigate students algebraic proficiency in terms of structure sense. To reach this aim, we set up a qualitative case study with the following three steps. First, after conducting a literature study, we designed structure sense tasks according to structure sense characteristics for the topic of equations. Second, we administered an individual written test involving 28 grade XI students (16-17 year-old). Third, we analyzed students written work using a structure sense perspective. The results showed that about two-Thirds of the participated students lack of structure sense in which they tend to use more procedural strategies than structure sense strategies in solving equations. We conclude that the perspective of structure sense provides a fruitful lens for assessing students algebraic proficiency

AN INVESTIGATION OF STUDENTS’ ALGEBRAIC PROFICIENCY FROM A STRUCTURE SENSE PERSPECTIVE

Structure sense can be interpreted as an intuitive ability towards symbolic expressions, including skills to perceive, to interpret, and to manipulate symbols in different roles. This ability shows student algebraic proficiency in dealing with various symbolic expressions and is considered important to be mastered by secondary school students for advanced study or professional work. This study, therefore, aims to investigate students’ algebraic proficiency in terms of structure sense. To reach this aim, we set up a qualitative case study with the following three steps. First, after conducting a literature study, we designed structure sense tasks according to structure sense characteristics for the topic of equations. Second, we administered an individual written test involving 28 grade XI students (16-17 year-old). Third, we analyzed students’ written work using a structure sense perspective. The results showed that about two-thirds of the participated students lack of structure sense in which they tend to use more procedural strategies than structure sense strategies in solving equations. We conclude that the perspective of structure sense provides a fruitful lens for assessing students’ algebraic proficiency

Making sense of mathematics through perception, operation & reason: The case of divisibility of a segment

The conception of infinity as a process (potential infinity) or as an object (actual infinity) is important for students to acquire understanding in many other related areas in mathematics. This study attempts to describe the infinite divisibility thinking of mathematics student teachers in an Institute of Teacher Education in Malaysia by making sense of mathematics through perception, operation and reason. Data were collected through a self-reporting questionnaire that was administered to 238 elementary school pre-service teachers from selected Teacher Education Institutes in Malaysia. Researchers categorised qualitatively different types of thinking and reported them by using descriptive statistics. The result revealed that the percentage of respondents who conceived infinity as an object was just slightly lower as compare to the percentage of respondents who conceived infinity as a process. Additionally, this study found that there were respondents with problematic conceptions as shown by their inconsistent answers. The open-ended explanations given by all the respondents revealed that most of the pre-service teachers used perception to make meaning on finite and infinite divisibility

Students' attitudes to learning mathematics with technology at rural schools in Sabah, Malaysia

The purpose of this study was to investigate students’ attitudes to learning Mathematics with Technology at rural Secondary Schools in Sabah, Malaysia. This study involved 17 Secondary rural and non-rural Secondary Schools in Sabah. A total of 613 Form 4, Form 2, and Form 1 students were randomly chosen as respondents. Descriptive and inferential statistics were used to analyze the collected data. The reliability of the instrument was analyzed by using the Statistical Packages for Social Sciences (SPSS) version 13.0 for Windows. Descriptive statistical analysis showed that only 13.0% of rural Secondary School students possessed positive attitude to learning Mathematics with Technology as compared to 21.5% of students from non-rural Secondary Schools. Results of independent sample t-test has indicated that there was a significant difference (t = -2.424, df = 543, p < 0.05) in attitudes to learning Mathematics with Technology between rural and non-rural school students. Students from non-rural Secondary Schools possessed higher Confidence with Technology compared to students from rural schools. Inferentional statistical analysis also showed that there was no significant difference in students’ attitudes to learning Mathematics with Technology based on gender, streaming, and level of schooling. Therefore, Mathematics teacher is the main factors in how technology is used in classroom. Finally, school administrators should encourage Mathematics teacher to use ICT (Information and Communication Technology) widely to enhance their teaching