Making sense of students’ errors in solving problems related to measures of dispersion

The importance of statistical learning has been widely discussed. However, little effort has been made to understand the difficulties of students in learning measure of dispersion, which is the key component to statistical learning. The present study sought to identify the reasons behind the errors committed in solving problems related to measures of dispersion, by examining students’ errors in the diagnostics tests, followed by in depth interviews to elicit their thinking and understanding. There were 85 grade-11 high school students involved in the first phase of the quantitative research and 10 students with weak performance were subsequently selected for the second phase qualitative research. The interviews were conducted using the contingent teaching model. The findings indicated that students’ committing errors in solving problems related to measure of dispersion due to lacking statistical vocabulary knowledge, weak symbol sense, rote learning, low statistical reasoning and statistical thinking ability. The results of the study and proper remedials are discussed

Pre-University Students\u27 Perceptual Flexibility with Mathematical Elements

Do students see things the same as what mathematics teachers have anticipated? In a light-hearted and inquisitive mode, two fundamental mathematical tasks were administered to 147 pre-university students who newly joined a private college for enrolment into various programs. One task simply required the participants to state the variables in a linear equation and the other attempted to elicit what the participants would naturally perceive from a diagram featuring a straight line crossing the coordinate axes. This study aimed to examine the participants’ perceptual flexibility with mathematical elements without explicit hints. The emergent textual responses were qualitatively classified and the frequencies of the various types of response evaluated. While the two tasks appear to be drearily ordinary to arouse excitement, the data startlingly revealed a wide range of valid and invalid responses from the participants, and some mathematical elements appeared to be more visually implicit or explicit than others to the participants. Specifically, few participants perceived composite functions embedded in the linear equation as variables and even fewer participants noticed a right-angled triangle emerging from the straight line crossing the coordinate axes. The results are discussed from the perceptual perspective of Gestalt psychology, which suggests the participants’ lack of flexibility in mentally reconfiguring perceptual elements. The potential implications for mathematics learning associated with perceptual flexibility are discussed and the instructional efforts that may enhance students’ perceptual flexibility recommended. In particular, we argue that a problem-solving process may require flexible perception of mathematical elements in order to gain access to learned concepts and hence particular solution choices

Assessing Pre-Service Secondary Mathematics Teachers' Attitude Towards Geometer's Sketchpad

The purpose of this study was to assess pre-service secondary mathematics teachers' attitude towards Geometer's Sketchpad (GSP) following an introductory GSP workshop. GSP is a dynamic geometry software program used for constructing and investigating mathematical objects. It is a dynamic tool for construction, demonstration and exploration of mathematical objects, adding a powerful dimension to the study of geometry and many other areas of mathematics. A total of 107 pre-service secondary mathematics teachers who attended a mathematics teaching methods course in a local public university participated in the GSP workshop. None of the participants had any prior experience using GSP. Analysis of their responses to the "Geometer's Sketchpad Attitude Scales" indicated that the pre-service secondary mathematics teachers showed a positive attitude towards GSP is more appropriate because the ''Geometer's Sketchpad Attitude Scales'' was used to access the participants' attitude towards GSP. The results also indicated that there was no significant difference between male and female preservice secondary mathematics teachers in their overall attitude towards GSP is more appropriate because the ''Geometer's Sketchpad Attitude Scales'' was used to access the participants' attitude towards GSP. Implications of the study for conducting GSP workshops in mathematics teaching methods courses are discussed

Enhancing Students' Geometric Thinking Through Phase-Based Instruction Using Geometer’s Sketchpad: A Case Study

I investigated Form One students’ learning of solid geometry in a phase-based instructional environment using Geometer’s Sketchpad (GSP) based on the van Hiele theory. Specifically, I examined the students’ initial van Hiele levels of geometric thinking about cubes and cuboids, and how their van Hiele levels changed after phasebased instruction with GSP. I used a case study research design and purposeful sampling to select six case study participants from a class of mixed-ability Form One students. Findings reveal that the participants’ initial van Hiele levels ranged from Level 0 to Level 2. After phase-based instruction with GSP, their van Hiele levels either increased or remained the same

Problem Solving Strategies among Primary School Teachers

The purpose of this article was to examine problem solving strategies among primary school teachers. The researchers employed survey research design to examine their problem solving strategies. The participants of this study consisted of 120 primary school teachers from a public university in Peninsula Malaysia who enrolled in a 4-year Graduating Teachers Program (Program Pensiswazahan Guru) majored in mathematics. Purposive sampling technique was used to select these participants. This article presents the analysis of the responses of the participants related to a particular problem, namely fencing problem. Result of the study suggests that 79.2% of the participants have successfully solved the fencing problem. They employed various problem solving strategies: (i) trial-and-error (also known as guess-and-check), (ii) using algebra, (iii) making tables, charts or systematic list, (iv) drawing diagrams, (v) identifying pattern, and (vi) logical reasoning. Result of the study also suggests that 85% of the participants used same strategy to check their solutions for the fencing problem without being probed. The implications of the results were also discussed. Keywords: problem solving strategies, primary school teachers, survey research design

The effects of concrete-pictorial-abstract (CPA) teaching method on the Malaysian community college students’ attitude towards Geometry

Community college is an institution that provides training and skills for post-secondary education leavers to become skilled workforce with minimum enrolment requirements. However, the admission of students who possessed low basic knowledge of Mathematics has somehow affected the Geometry teaching and learning process. These students show less interest and attitude in learning Mathematics. This quasi-experimental study aimed to investigate the effect of the CPA teaching method on Malaysian Community College students’ attitudes towards Geometry. A total of 60 students who registered their first semester Mathematics module from Pulau Pinang and Kedah Community Colleges was selected as samples. The control group used the conventional method (non-CPA), while the experimental group used concrete-pictorial-abstract (CPA) method. Questionnaires were used as the instrument. Data were analyzed using one-way ANCOVA. It was found that there are significant differences in attitude towards Geometry between the control and experimental group. This result provides positive effects on the implementation of CPA approach in the teaching and learning of Geometry. This study hence suggests that educators in Malaysia to integrate CPA teaching methods in their teaching and learning process of Geometry, which will have a better implication on the attitude of students studying Geometry

Enhancing Primary Pupils' Geometric Thinking Through Phase-Based Instruction Using The Geometer's Sketchpad

The purpose of this study was to enhance primary pupils' geometric thinking through phase-based instruction using The Geometer's Sketchpad (GSP) based on the van Hiele theory of geometric thinking. Specifically, it sought to examine Year Four pupils' van Hiele levels of geometric thinking about equilateral triangle, square, regular pentagon and regular hexagon before and after phase-based instruction using GSP, and whether there was any significant difference in the pupils' van Hiele levels of geometric thinking about the regular polygons after the intervention. The researchers employed an exploratory case study research design and purposeful sampling to select a class of 26 mixed-ability Year Four pupils from a primary school in Selangor. A van Hiele level test based on Mayberry's (1981) test and scoring criteria was devised and administered to the pupils before and after the intervention to assess their van Hiele levels of geometric thinking about the regular polygons. The results of the pre-test showed that the pupils' initial van Hiele levels were predominantly at Level 0 (Pre-recognition) for regular pentagon and regular hexagon but at Level 1 (Recognition) for equilateral triangle and square. However, the results of the post-test revealed that the pupils' van Hiele levels after the intervention were predominantly at Level 2 (Analysis) for all the regular polygons. In addition, the results of the Wilcoxon test showed that there was a significant difference in the pupils' van Hiele levels of geometric thinking for all the regular polygons after phasebased instruction using GSP. The median van Hiele level in the post-test was higher than the median van Hiele level in the pre-test for all the regular polygons, indicating that the intervention had significantly enhanced the pupils' geometric thinking about the regular polygons

Effects of discovery learning on year two pupils’ achievement in learning fractions

Understanding fractions is one of the important skills that need to be developed in Mathematics curriculum as it is essential for understanding algebra, geometry as well as other aspects of Mathematics. However, pupils around the world face difficulties in learning fractions. This quasi-experimental study examined the effects of Discovery Learning on pupils’ achievement in fractions. 66 Year Two primary school pupils in Kedah were assigned into experimental (Discovery learning) and control (conventional instruction) groups. Data was collected using pre-test and post-test. Independent samples t-test and paired-samples t-test were used to analyze the data. Results indicated that: 1) there is no significant difference in pupils’ achievement in fractions (pre-test) between the groups; 2) there is a significant difference in pupils’ achievement in fractions (post-test) between the groups; 3) there is a significant difference in pupils’ achievement in fractions between the pre-test and post-test of the experimental group; and 4) there is a significant difference in Year Two pupils’ achievement in fractions between the pre-test and post-test of the control group. This concludes that Discovery Learning was effective in improving Year Two pupils’ achievement in learning fractions. Therefore, it can play a role as a starting milestone for educators in Malaysia to implement Discovery Learning in daily teaching life especially in fractions

Primary School Mathematics And Science Teachers' Stages Of Concern About The Implementation Of Lesson Study

In this study, the Japanese model of Lesson Study was introduced as a teacher professional development programme to nine low-performing primary schools in Malaysia. The objectives of this study were to examine to what extent Lesson Study can improve low-performing primary mathematics and science teachers' stages of concern about the implementation of Lesson Study in their schools and their teaching quality as well as student learning performance. The sample consisted of 97 primary mathematics and science teachers from three types of primary schools: the National School (SK); the National Type Chinese School (SJKC) and the National Type Tamil School (SJKT). Before the implementation of Lesson Study, a workshop was conducted in every school to introduce participating teachers to the concepts of Lesson Study and the research procedure. After the workshop, the participating teachers were asked to complete the Stages of Concern Questionnaire (SoCQ) in order to identify their initial stages of concern about the implementation of Lesson Study in their schools. The SoCQ was developed based on the Concerns-Based Adoption Model (CBAM). It consists of 35 items, categorising teachers' concerns into seven stages: Stage 0 (Awareness); Stage 1 (Informational); Stage 2 (Personal); Stage 3 (Management); Stage 4 (Consequence); Stage 5 (Collaboration) and Stage 6 (Refocusing) (George, Hall & Stiegelbauer, Measuring implementation in schools: The stages of concern questionnaire (2006)). The same SoCQ was also given to the participating teachers after every Lesson Study cycle to determine whether their stages of concern changed before and after the implementation of Lesson Study cycles. In this paper, only the data collected from the first SoCQ given were analysed. The results showed that the profiles of the SK, SJKC and SJKT teachers' initial stages of concern about the implementation of Lesson Study in their schools were quite similar. The SK, SJKC and SJKT teachers' concerns were the highest in Stage 0 and the lowest in Stage 4, indicating that these teachers had a high level of concern about a number of other initiatives, tasks, and activities besides Lesson Study and they had quite a low level of concern about the consequences of implementing Lesson Study for their students, respectivel