Comparing German and Taiwanese secondary school students’ knowledge in solving mathematical modelling tasks requiring their assumptions

Mathematization is critical in providing students with challenges for solving modelling tasks. Inadequate assumptions in a modelling task lead to an inadequate situational model, and to an inadequate mathematical model for the problem situation. However, the role of assumptions in solving modelling problems has been investigated only rarely. In this study, we intentionally designed two types of assumptions in two modelling tasks, namely, one task that requires non-numerical assumptions only and another that requires both non-numerical and numerical assumptions. Moreover, conceptual knowledge and procedural knowledge are also two factors influencing students’ modelling performance. However, current studies comparing modelling performance between Western and non-Western students do not consider the differences in students’ knowledge. This gap in research intrigued us and prompted us to investigate whether Taiwanese students can still perform better than German students if students’ mathematical knowledge in solving modelling tasks is differentiated. The results of our study showed that the Taiwanese students had significantly higher mathematical knowledge than did the German students with regard to either conceptual knowledge or procedural knowledge. However, if students of both countries were on the same level of mathematical knowledge, the German students were found to have higher modelling performance compared to the Taiwanese students in solving the same modelling tasks, whether such tasks required non-numerical assumptions only, or both non-numerical and numerical assumptions. This study provides evidence that making assumptions is a strength of German students compared to Taiwanese students. Our findings imply that Western mathematics education may be more effective in improving students’ ability to solve holistic modelling problems.     

The changes of assessment at middle school level in Korea

Mathematics education in Korea is currently in a period of transition. The new national curriculum standard was released in 2009 and will be implemented by middle schools in 2013. Korean education has evolved from merely assessing students’ knowledge to figuring out their mathematical process using various assessment methods. The purpose of the national level assessment has been expanded to measure individual achievement. The assessment framework also has changed not only to focus on the content domain but to include the behavior domain as well. However, there were some difficulties in assessing the students’ solving process, including trying to implement various assessment methods in the classrooms. This paper aims to examine the development of assessment and to provide direction for the future application of assessment to the new Korean curriculum by analyzing middle school students’ assessment result at the national level and the class level in the changing era.     

Personal Excursions: Investigating the Dynamics of Student Engagement

We investigate the dynamics of student engagement as it is manifest in self-directed, self-motivated, relatively long-term, computer-based scientific image processing activities. The raw data for the study are video records of 19 students, grades 7 to 11, who participated in intensive 6-week, extension summer courses. From this raw data we select episodes in which students appear to be highly engaged with the subject matter. We then attend to the fine-grained texture of students’ actions, identifying a core set of phenomena that cut across engagement episodes. Analyzed as a whole, these phenomena suggest that when working in self-directed, self-motivated mode, students pursue proposed activities but sporadically and spontaneously venture into self-initiated activities. Students’ recurring self-initiated activities – which we call personal excursions – are detours from proposed activities, but which align to a greater or lesser extent with the goals of such activities. Because of the deeply personal nature of excursions, they often result in students collecting resources that feed back into both subsequent excursions and framed activities. Having developed an understanding of students’ patterns of self-directed, self-motivated engagement, we then identify four factors that seem to bear most strongly on such patterns: (1) students’ competence (broadly construed); (2) features of the software-based activities, and how such features allowed students to express their competence; (3) the time allotted for students to pursue proposed activities, as well as self-initiated ones; and (4) the flexibility of the computational environment within which the activities were implemented.     

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.     

An ‘Emergent Model’ for Rate of Change

Does speed provide a ‘model for’ rate of change in other contexts? Does JavaMathWorlds (JMW), animated simulation software, assist in the development of the ‘model for’ rate of change? This project investigates the transference of understandings of rate gained in a motion context to a non-motion context. Students were 27 14–15 year old students at an Australian secondary school. The instructional sequence, utilising JMW, provided rich learning experiences of rate of change in the context of a moving elevator. This context connects to students’ prior knowledge. The data taken from pre- and post-tests and student interviews revealed a wide variation in students’ understanding of rate of change. The variation was mapped on a hypothetical learning trajectory and interpreted in the terms of the ‘emergent models’ theory (Gravemeijer, Math Think Learn 1(2):155–177, 1999) and illustrated by specific examples from the data. The results demonstrate that most students were able to use the ‘model of’ rate of change developed in a vertical motion context as a ‘model for’ rate of change in a horizontal motion context. A smaller majority of students were able to use their, often incomplete, ‘model of’ rate of change as a ‘model for’ reasoning about rate of change in a non-motion context.     

Comparing eye movements during mathematical word problem solving in Chinese and German

Language plays an important role in word problem solving. Accordingly, the language in which a word problem is presented could affect its solution process. In particular, East-Asian, non-alphabetic languages are assumed to provide specific benefits for mathematics compared to Indo-European, alphabetic languages. By analyzing students’ eye movements in a cross-linguistic comparative study, we analyzed word problem solving processes in Chinese and German. 72 German and 67 Taiwanese undergraduate students solved PISA word problems in their own language. Results showed differences in eye movements of students, between the two languages. Moreover, independent cluster analyses revealed three clusters of reading patterns based on eye movements in both languages. Corresponding reading patterns emerged in both languages that were similarly and significantly associated with performance and motivational-affective variables. They explained more variance among students in these variables than between the languages alone. Our analyses show that eye movements of students during reading differ between the two languages, but very similar reading patterns exist in both languages. This result supports the assumption that the language alone is not a sufficient explanation for differences in students’ mathematical achievement, but that reading patterns are more strongly related to performance.     

A changing trajectory of proof learning in the geometry inquiry classroom

The purpose of this study is to explore how students changes through learning to construct mathematical proofs in an inquiry-based middle school geometry class in Korea. Although proof has long been considered as one of the most important aspects of mathematics education, it is well-known that it is one of the most difficult areas of school mathematics for students. The geometry inquiry classroom (GIC) is an experimental class designed to teach geometry, especially focusing on teaching proof, based on students’ own inquiry. Based on a 2-year participant observation in the GIC, this study was conducted to address the following research question: how has students’ practice of mathematical proof been changed through their participation in the GIC? The in-depth analysis of the classroom discourse identified three stages through which the students’ practice of mathematical proof was transformed in the GIC: ‘emergent understanding of proof’, ‘proof learning as a goal-oriented activity’, ‘experiencing proof as the practice of mathematics’. The study found that as learning evolved through these stages, so the mathematics teacher’s role shifted from being an instructor to a mediator of communication. Most importantly, this research showed that the GIC has created a learning environment where students develop their competence in constructing meaningful mathematical proof and grow to be ‘a human who proves’, ultimately ‘a person who playfully engages with mathematics’.     

How does imposing a step-by-step solution method impact students’ approach to mathematical word problem solving?

This paper presents the second phase of a larger research program with the purpose of exploring the possible consequences of a gap between what is done in the classroom regarding mathematical word problem solving and what research shows to be effective in this particular field of study. Data from the first phase of our study on teachers’ self-proclaimed practices showed that one-third of elementary teachers from the region of Quebec require their students to follow a specific sequential problem-solving method, known as the ‘what I know, what I look for’ method. These results led us to hypothesize that the observed gap may have an impact on students’ comprehension of mathematical word problems. The use of this particular method was the foundation for us to study, in the second phase, the effect of the imposition of this sequential method on students’ literal and inferential understanding of word problems. A total of 278 fourth graders (9–10 years old) solved mathematical word problems followed by a test to assess their understanding of the word problems they had just solved. The results suggest that the use of this problem solving method does not seem to improve or impair students’ understanding. From a more fundamental point of view, our study led us to the conclusion that the way word problem solving is addressed in the mathematics classroom, through sequential and inflexible methods, does not help students develop their word problem solving competence.     

Constructing Knowledge Via a Peer Interaction in a CAS Environment with Tasks Designed from a Task–Technique–Theory Perspective

Our research project aimed at understanding the complexity of the construction of knowledge in a CAS environment. Basing our work on the French instrumental approach, in particular the Task–Technique–Theory (T–T–T) theoretical frame as adapted from Chevallard’s Anthropological Theory of Didactics, we were mindful that a careful task design process was needed in order to promote in students rich and meaningful learning. In this paper, we explore further Lagrange’s (2000) conjecture that the learning of techniques can foster conceptual understanding by investigating at close range the task-based activity of a pair of 10th grade students—activity that illustrates the ways in which the use of symbolic calculators along with appropriate tasks can stimulate the emergence of epistemic actions within technique-oriented algebraic activity.     

New challenges in the 2011 revised middle school curriculum of South Korea: mathematical process and mathematical attitude

The ‘future-oriented middle school mathematics curriculum focused on creativity and personality’ was revised in August of 2011 with the aim of nurturing students’ mathematical creativity and sound personalities. The curriculum emphasizes: contextual learning from which students can grasp mathematical concepts and make connections with their everyday lives; manipulation activities through which students may attain an intuitive idea of what they are learning and enhance their creativity; and reasoning to justify mathematical results based on their knowledge and experience. Since students will not be able to engage in the intended mathematical process with the study-load imposed by the current curriculum, the newly revised curriculum modifies or deletes some parts of the contents that have been traditionally taught mechanically. This paper provides a detailed overview of the main points of the revised curriculum.     

“If You Can Turn a Rectangle into a Square, You Can Turn a Square into a Rectangle ...” Young Students Experience the Dragging Tool

This paper describes a study of the cognitive complexity of young students, in the pre-formal stage, experiencing the dragging tool. Our goal was to study how various conditions of geometric knowledge and various mental models of dragging interact and influence the learning of central concepts of quadrilaterals. We present three situations that reflect this interaction. Each situation is characterized by a specific interaction between the students’ knowledge of quadrilaterals and their understanding of the dragging tool. The analyses of these cases offer a prism for viewing the challenge involved in changing concept images of quadrilaterals while lacking understanding of the geometrical logic that underlies dragging. Understanding dragging as a manipulation that preserves the critical attributes of the shape is necessary for constructing the concept images of the shapes.     

A comparative investigation of the presence of psychological conditions in high achieving eighth graders from TIMSS 2007 Mathematics

This study is a second analysis of Trends in International Mathematics and Science Study (TIMSS) 2007 background questionnaires to investigate high achieving eighth-grade students’ possession of three elements of Krutetskii’s (The Psychology of Mathematical Abilities in Schoolchildren, The University of Chicago, 1976) psychological conditions. Analyses were made between high achieving eighth graders and their lower achieving counterparts, and between Korean high achievers and their high achieving peers across ten high performing TIMSS participating countries. After reviewing 44,643 students across selected countries, we conclude that a larger percentage of students with mathematical talent demonstrate positive attitudes toward mathematics, value mathematics, and have self-confidence in their ability to learn mathematics than their peers without high achievement in mathematics. However, a larger portion of high achieving Korean students displayed low self-confidence and valued mathematics less than their peers from other high performing countries. Findings from this study will provide insight into some educational issues in science, technology, engineering and mathematics education.     

Students’ reflections on their learning experiences: lessons from a longitudinal study on the development of mathematical ideas and reasoning

This paper characterizes the views on mathematical learning of five high school students based on the students’ reflections on their mathematical experiences in a longitudinal study that focused on the development of mathematical ideas and reasoning in particular research conditions. The students’ views are presented according to five themes about learning which describe the students’ views on the nature of knowledge and what it means to know, source of knowledge, motivation to engage in learning, certainty in knowing, and how the students’ views vary with particular areas of mathematical activity. The study addresses the need for more research on epistemological beliefs of students below college age. In particular, the results provide evidence that challenge the existing assumption that, prior to college, students exhibit naïve epistemological beliefs.     

Computational Construction as a Means to Coordinate Representations of Infinity

In this paper, we describe a design experiment aimed at helping students to explore and develop concepts of infinite processes and objects. Our approach is based on the design and development of a computational microworld, which afforded students the means to construct a range of representational models (symbolic, visual and numeric) of infinity-related objects (infinite sequences, in particular). We present episodes based on four students’ activities, seeking to illustrate how the available tools mediated students’ understandings of the infinite in rich ways, allowing them to discriminate subtle process-oriented features of infinite processes. We claim that the microworld supported students in the coordination of hitherto unconnected or conflicting intuitions concerning infinity, based on a constructive articulation of different representational forms we name as ‘representational moderation’.

Exploring college students’ mental representations of inferential statistics

We report a case study that explored how three college students mentally represented the knowledge they held of inferential statistics, how this knowledge was connected, and how it was applied in two problem solving situations. A concept map task and two problem categorization tasks were used along with interviews to gather the data. We found that the students’ representations were based on incomplete statistical understanding. Although they grasped various concepts and inferential tests, the students rarely linked key concepts together or to tests nor did they accurately apply that knowledge to categorize word problems. We suggest that one reason the students had difficulty applying their knowledge is that it was not sufficiently integrated. In addition, we found that varying the instruction for the categorization task elicited different mental representations. One instruction was particularly effective in revealing students’ partial understandings. This finding suggests that modifying the task format as we have done could be a useful diagnostic tool.     

Increasing engineering students' awareness to environment through innovative teaching of mathematical modelling

This article presents the results of two studies on using aninnovative pedagogical strategy in teaching mathematical modellingand applications to engineering students. Both studies are dealingwith introducing non-traditional contexts for engineering studentsin teaching/learning of mathematical modelling and applications:environment and ecology. The aims of using these contexts were:to introduce students to some of the techniques, methodologiesand principles of mathematical modelling for ecological andenvironmental systems; to involve the students in solving real-lifeproblems adjusted to their region emphasizing the aspects ofboth survival (short term) and sustainability (long term); toencourage students to pay attention to environmental issues.On one hand, the contexts are not directly related to engineering.On the other hand, the chances are that many graduates of engineeringwill deal with mathematical modelling of environmental systemsin one way or another in their future work because nearly everyengineering activity has an impact on the environment. The firststudy is a parallel study conducted in New Zealand and Germanysimultaneously with first-year students studying engineeringmathematics. The second study is a case study of the experimentalcourse Mathematical Modelling of Survival and Sustainabilitytaught to a mixture of year 2–5 engineering students inGermany by a visiting lecturer from New Zealand. The modelsused with the students from both studies had several specialfeatures. Analysis of students’ responses to questionnaires,their comments and attitudes towards the innovative approachin teaching are presented in the article.     

Complex variables in junior high school: the role and potential impact of an outreach mathematician

Outreach mathematicians are college faculty who are trainedin mathematics but who undertake an active role in improvingprimary and secondary education. This role is examined througha study where an outreach mathematician introduced the conceptof complex variables to junior high school students in the UnitedStates with the goal of stimulating their interest in mathematicsand improving their algebra skills. Comparison of pre- and post-testresults showed that ninth-grade students displayed a significantchange in algebraic skills while the eighth-grade students madelittle progress. The outreach mathematician lacked some awarenessof the eighth-grade students’ foundational backgroundand motivation. This illustrates the importance of working moreclosely with the participating teacher, who understands betterthe curriculum and the students’ background knowledge,levels of maturity and levels of motivation.     

A cross-national comparison of reform curricula in Korea and the US in terms of cognitive complexity: the case of fraction addition and subtraction

The overall level of conceptual understanding and mathematical proficiency of students has been a matter of increasing national interest in South Korea. Recently, a new edition of mathematics textbooks aligned with the amendment of the 7th national mathematics curriculum has become available for all elementary grade levels. To characterize the current reform efforts in South Korea, this study examined the quality of the mathematical problems in the current version of the Korean reform textbooks (KM 2) compared with the previous version (KM 1) and one representative US reform curriculum text (EM). Webb’s (Research monograph No. 18: Alignment of science and mathematics standards and assessments in four states. National Institute for Science Education, Madison, 1999) depth of knowledge framework and Son and Senk’s (Educ Stud Math 74(2):117–142, 2010) cognitive expectation feature were employed to examine the kind and level of students’ opportunities to learn along with the type of word problems presented in the three sets of materials. Analysis revealed that the KM 2 provided better opportunities for students to learn fraction addition and subtraction than the KM 1 in terms of the depth and breadth of cognitive complexity. However, there was little difference in addressing and developing the meaning of fraction addition and subtraction through word problems. Moreover, compared with the US reform curriculum materials, the KM 2 provided more problems requiring lower depth of knowledge levels than the US counterpart. Implications of these findings for curriculum developers, textbook and learning materials developers, teachers and future researchers are discussed.     

The teaching of proof at the lower secondary level—a video study

Teaching mathematical proof is one of the most challenging topics for teachers. Several empirical studies revealed repeatedly different kinds of students’ problems in this area. The results give support that students’ abilities in proving are significantly influenced by their specific mathematics classrooms. In this paper we will present a method for evaluating proof instruction and some results of a video study that describe proving processes in mathematics classrooms at the lower secondary level from a mathematical perspective.     

Assessment of teacher knowledge across countries: a review of the state of research

This review presents an overview of research on the assessment of mathematics teachers’ knowledge as one of the most important parameters of the quality of mathematics teaching in school. Its focus is on comparative and international studies that allow for analyzing the cultural dimensions of teacher knowledge. First, important conceptual frameworks underlying comparative studies of mathematics teachers’ knowledge are summarized. Then, key instruments designed to assess the content knowledge and pedagogical content knowledge of future and practicing mathematics teachers in different countries are described. Core results from comparative and international studies are documented, including what we know about factors influencing the development of teacher knowledge and how the knowledge is related to teacher performance and student achievement. Finally, we discuss the challenges connected to cross-country assessments of teacher knowledge and we point to future research prospects.