Kai Lin Yang

Students' performance in reasoning and proof in Taiwan and Germany: Results, paradoxes and open questions

In different international studies on mathematical achievement East Asian students outperformed the students from Western countries. A deeper analysis shows that this is not restricted to routine tasks but also affects students’ performance for complex mathematical problem solving and proof tasks. This fact seems to be surprising since the mathematics instruction in most of the East Asian countries is described as examination driven and based on memorising rules and facts. In contrast, the mathematics classroom in western countries aims at a meaningful and individualised learning. In this article we discuss this “paradox” in detail for Taiwan and Germany as two typical countries from East Asia and Western Europe.

Defining a rectangle under a social and practical setting by two seventh graders

Regarding defining as a mathematical activity bridging informal to formal proof, two seventh graders will reinvent the definition of rectangles under a social and practical setting based on their informal argumentation. Their apprehensions of figures, implicit concepts/theorems and the cognitive architecture of defining are discussed in this paper.

Designing a Competence-Based Entry Course for Prospective Secondary Mathematics Teachers

The goal of this study aims at introducing an entry course of a 3-year sequential courses module for a secondary mathematics teacher education program in Taiwan. This module is a reformed teacher education curriculum planned for Prospective Secondary Mathematics Teachers (PSMTs) to learn how to teach with the field-study approach. The field-study approach provides abundant opportunities for PSMTs to cultivate their competencies in teaching. In this chapter, we take the first year course to deliberate why the Psychology of Mathematics Learning is selected as an entry course for the teacher education program and how it works. Considering the importance to raise PSMTs’ awareness of students’ mathematical thinking and to cultivate their competencies of sensitizing students’ mathematical thinking, and ultimately to bear the competencies as the habitus in their future teaching professional, the mission of the course focuses on PSMTs’ learning of understanding students’ mathematical thinking through the process of cyclic learning. The quality of PSMTs dynamic learning in the field study can be evaluated by their study work. This chapter provides one example of PSMTs’ survey study in one complete learning cycle, and summarizes several criteria of evaluating how PSMTs conduct a study to understand students’ mathematical thinking in a holistic perspective.

The Effects of PISA in Taiwan: Contemporary Assessment Reform

Taiwan has always been one of the top ranked countries in PISA, so initially interest in PISA was mainly concerned with standards monitoring, with some analysis of how instruction could be improved. However, from 2012, PISA became a major public phenomenon as it became linked with proposed new school assessment and competitive entrance to desirable schools. Students, along with their parents and teachers, worried about the ability to solve PISA-like problems and private educational providers offered additional tutoring. This chapter reports and explains these dramatic effects. Increasingly, the PISA concept of mathematical literacy has been used, along with other frameworks, as the theoretical background for thinking about future directions for teaching and assessment in schools. This is seen as part of an endeavour to change the strong emphasis on memorisation and repetitive practice in Taiwanese schools.

Investigating mathematics teachers' thoughts of statistical inference

Research on statistical cognition and application suggests that statistical inference concepts are commonly misunderstood by students and even misinterpreted by researchers. Students’ misconceptions of confidence intervals (CIs) detected by Fidler (2006) include: (1) CIs are a range of plausible values for the sample means; (2) CIs are a range of individual scores; (3) CIs are a range of individual scores within one standard deviation; (4) the width of a CI increases as the sample size increases; (5) the width of a CI is not affected by sample size; and (6) a 90% CI is wider than a 95% CI for the same data. Although some research has been done on students’ misunderstanding or misconceptions of CIs, few studies explore either students’ or mathematics teachers’ thoughts underlying these misconceptions. In order to investigate mathematics teachers’ underlying thoughts about CIs, we developed an instrument for collecting teachers’ thoughts on the CI-related concepts of binomial and normal sampling distributions and CIs. 35 multiple-choice test items were included in this instrument. An illustrative item about binomial proportion is shown in Figure 1. 24 in-service mathematics teachers taking a graduate course in introduction to statistics participated in this study. Their average teaching experience was about six years, and 16 participants had learned the CI concept when they studied in university. A statistician taught them about CIs, illustrating how a confidence interval was calculated, and explaining the central limit theorem. Subsequently, the teachers took about 50 minutes to answer the 35 test items individually, and about 50 minutes to discuss their answers in class. We did not intervene, except to ask them to explain their thoughts more clearly. After their discussion, the statistician took about 50 minutes to give the correct answers, and asked the teachers whether they agreed with the correct answers, and why. Six teachers were selected for interview to clarify their understanding further during the discussion. The data analysis aimed to investigate the thoughts underlying these misunderstandings. Thus, we decided to explore qualitative data without a pre-determined theoretical or descriptive framework; incorrect answers and the thoughts underlying them were our main concern. We found that mathematics teachers knew about means, variances and some properties of a random variable, about what a CI for a proportion measures, and about the relationships between sample sizes, confidence levels and the width of a CI for a proportion. In addition, they were able to calculate a CI for a proportion.