Irrational Numbers: The Gap between Formal and Intuitive Knowledge

This report focuses on prospective secondary mathematics teachers’ understanding of irrational numbers. Various dimensions of participants’ knowledge regarding the relation between the two sets, rational and irrational, are examined. Three issues are addressed: richness and density of numbers, the fitting of rational and irrational numbers on the real number line, and operations amongst the elements of the two sets. The results indicate that there are inconsistencies between participants’ intuitions and their formal and algorithmic knowledge. Explanations used by the vast majority of participants relied primarily on considering the infinite non-repeating decimal representations of irrationals, which provided a limited access to issues mentioned above.     

Teacher education through the history of mathematics

In this paper I consider the problem of designing strategies for teacher education programs that may promote an aware style of teaching. Among the various elements to be considered I focus on the need to address prospective teachers’ belief that they must reproduce the style of mathematics teaching seen in their school days. Towards this aim, I argue that the prospective teachers need a context allowing them to look at the topics they will teach in a different manner. This context may be provided by the history of mathematics. In this paper I shall discuss how history affected the construction of teaching sequences on algebra during the activities of the ‘laboratory of mathematics education’ carried out in a 2 year education program for prospective teachers. The conditions of the experiment, notably the fact that our prospective teachers had not had specific preparation in the history of mathematics, allow us to outline opportunities and caveats of the use of history in teacher education.     

Yup’ik Cosmology to School Mathematics: The Power of Symmetry and Proportional Measuring

This article shows how Yup’ik cosmology, epistemology, and everyday practice have implications for the teaching of school mathematics. Math in a Cultural Context (MCC) has a long–term collaborative relationship with Yup’ik elders and experienced Yup’ik teachers. Because of this long–term ethnographically–oriented relationship, the authors – both insiders and an outsider – have been able to understand the mathematical implications of everyday Yup’ik practice. As the article demonstrates, body proportional measuring and symmetry/splitting are two generative solution strategies used by Yup’ik elders in solving everyday problems. We argue that proportional measuring coupled with symmetry/splitting can provide school mathematics with an alternative pathway to the teaching of some aspects of geometry and rational number reasoning.     

New Avenues for History in Mathematics Education: Mathematical Competencies and Anchoring

The paper addresses the apparent lack of impact of ‘history in mathematics education’ in mathematics education research in general, and proposes new avenues for research. We identify two general scenarios of integrating history in mathematics education that each gives rise to different problems. The first scenario occurs when history is used as a ‘tool’ for the learning and teaching of mathematics, the second when history of mathematics as a ‘goal’ is pursued as an integral part of mathematics education. We introduce a multiple-perspective approach to history, and suggest that research on history in mathematics education follows one of two different avenues in dealing with these scenarios. The first is to focus on students’ development of mathematical competencies when history is used a tool for the learning of curriculum-dictated mathematical in-issues. A framework for this is described. Secondly, when using history as a goal it is argued that an anchoring of the meta-issues in the related in-issues is essential, and a framework for this is given. Both frameworks are illustrated through empirical examples.     

Beyond motivation: history as a method for learning meta-discursive rules in mathematics

In this paper, we argue that history might have a profound role to play for learning mathematics by providing a self-evident (if not indispensable) strategy for revealing meta-discursive rules in mathematics and turning them into explicit objects of reflection for students. Our argument is based on Sfard’s theory of Thinking as Communicating, combined with ideas from historiography of mathematics regarding a multiple perspective approach to the history of practices of mathematics. We analyse two project reports from a cohort of history of mathematics projects performed by students at Roskilde University. These project reports constitute the experiential and empirical basis for our claims. The project reports are analysed with respect to students’ reflections about meta-discursive rules to illustrate how and in what sense history can be used in mathematics education to facilitate the development of students’ meta-discursive rules of mathematical discourse.     

EXAMINING EIGHTH GRADE KUWAITI STUDENTS’ RECOGNITION AND INTERPRETATION OF REASONABLE ANSWERS

This research documents Kuwaiti eighth grade students’ performance in recognizing reasonable answers and the strategies they used to determine reasonableness. The results from over 200 eighth grade students show they were generally unable to recognize reasonable answers. Students’ performance was consistently low across all three number domains (whole numbers, fractions, and decimals). There was no significant difference in students’ performance on items that focused on the practicality of the answers or on items that focused on the relationships of numbers and the effect of operations, or on both. Interview data revealed that 35% of the students’ strategies were derived from two criteria for judging answers for reasonableness: the relationships of numbers and the effect of operations, and the practicality of the answers. They used strategies such as estimation, numerical benchmarks, real-world benchmarks, and applied their understanding of the meaning of operations. However, over 60% of the students’ strategies were procedurally driven. That is, they relied on algorithmic techniques such as carrying out paper-and-pencil procedures. Additionally, some of the students’ strategies reflected misunderstandings of how and when to apply certain procedures. Given these findings, mathematics education in Kuwait should shift the emphasis from paper-and-pencil procedures and provide systematic attention to the development of number sense and computational estimation so Kuwaiti students will be more adept at recognizing reasonable answers.     

Employing genetic ‘moments’ in the history of mathematics in classroom activities

The integration of history into educational practice can lead to the development of activities through the use of genetic ‘moments’ in the history of mathematics. In the present paper, we utilize Oresme’s genetic ideas – developed during the fourteenth century, including ideas on the velocity–time graphical representation as well as geometric transformations and reconfigurations – to develop mathematical models that can be employed for the solution of problems relating to linear motion. The representation of distance covered as the area of the figure between the graph of velocity and the time axis employed in these activities, leads on naturally to the study of problems on motion by means of functions, as well as allowing for the use of tools (concepts and propositions) from Euclidean geometry of relevance to such problems. By employing simple geometric transformations, equivalent real life problems are obtained which lead, in turn, to a simple classification of all linear motion-related problems. When applied to a wider range of motion problems, this approach prepares the way for the introduction of basic Calculus concepts (such as integral, derivative and their interrelation); in fact, we would argue that it could be beneficial to teach the basic concepts and results of Calculus from an early grade by employing natural extensions of the teaching methods considered in this paper.     

Feeling number: grounding number sense in a sense of quantity

Drawing on results from psychology and from cultural and linguistic studies, we argue for an increased focus on developing quantity sense in school mathematics. We explore the notion of “feeling number”, a phrase that we offer in a twofold sense—resisting tendencies to feel numb-er (more numb) by developing a feeling for numbers and the quantities they represent. First, we distinguish between quantity sense and the relatively vague notion of number sense. Second, we consider the human capacity for quantity sense and place that in the context of related cultural issues, including verbal and symbolic representations of number. Third and more pragmatically, we offer teaching strategies that seem helpful in the development of quantity sense coupled with number sense. Finally, we argue that there is a moral imperative to connect number sense with such a quantity sense that allows students to feel the weight of numbers. It is important that learners develop a feeling for number, which includes a sense of what numbers are and what they can do.     

Aspects of negative numbers in the early 17th century

This paper argues that the questions, posed by researchers in the field of didactics of mathematics, require new historical research which mainly concerns the problems related to the emergence and evolution of concepts. Motivated by recent historico-didactical studies on negative numbers, the author explores two different types of problems through which these numbers started being used systematically in mathematics. The first problem deals with the correspondence between the terms of an arithmetical and a geometrical progression, which constitutes the theoretical basis of logarithms; the second deals with the application of algebraic syntactical ruies in the theory of equations. In the specific context of these problems, concepts, such as negative logarithm or negative root, were established in the early 17th century, long before the appearance of a general concept of negative quantity in mathematical textbooks. The analysis of these problems reveals the conventional character of negative numbers and poses certain questions about the meaning of the various concrete models, traditionally employed in their teaching (via temperature, debits and credits, etc.). Recent, large-scale empirical research has shown a major percentage of failure in understanding negative numbers and their operations; this fact is related to the meanings attributed to negative numbers during their introduction at school. The matter of revising traditional teaching models is considered in connection with a constructive learning hypothesis; there is a need for new problem-situations, which entirely justify the meaning of the concept that must be used and constructed by the pupil and allow a fruitful interaction with it. The case of negative numbers provides an illuminating example of the role historical problems can play in the creation of situations like these.     

The gap between expectations and reality: integrating computers into mathematics classrooms

As a result of dramatic changes in mathematics education around the world, in Turkey both elementary and secondary school mathematics curriculums have changed in the light of new demands since 2005. In order to perform the expected change in newly developed curriculum, computer should be integrated into learning and teaching process. Teachers’ beliefs play a key role in this integration process. Negative beliefs against using computer in mathematics teaching may lead to failure of this process. With the help of this study, it is aimed to detect mathematics teachers’ beliefs concerning Computer Assisted Mathematics Instruction (CAMI). Within the scope of this aim, the conducted questionnaire (The opinions of teachers about using computer in Mathematics Instruction) has been carried out on 91 mathematics teachers in the city of Trabzon. The acquired results have shown that mathematics teachers have developed negative opinions against CAMI. This state has revealed that there is a huge inconsistency between curriculum’s positive expectations arising from computer usage and teachers’ convictions.     

THE ROLE OF TEACHER CHARACTERISTICS AND PRACTICES ON UPPER SECONDARY SCHOOL STUDENTS�� MATHEMATICS SELF-EFFICACY IN NYANZA PROVINCE OF KENYA: A MULTILEVEL ANALYSIS

The study identified two dimensions of teacher self-efficacy and practices and five dimensions of students’ mathematics self-efficacy and sought to determine the extent to which teacher characteristics and practices can enhance secondary school students’ self-efficacy. Data were collected from 13,173 students in 193 teachers’ classrooms from 141 schools in the 10 districts of Lake Victoria Region of Kenya. Two-level hierarchical linear model revealed that teachers’ frequent use of mathematics homework, their level of interest and enjoyment of mathematics, as well as their ability and competence in teaching mathematics were found to play a key role in promoting students’ mathematics self-efficacy. Teachers’ ability and competence in teaching were also found to be effective in narrowing the gender gap in students’ self-confidence and competence in mathematics. The study recommends that teacher training colleges emphasize such teacher practices and values in order to enhance students’ mathematics self-efficacy, reduce their level of anxiety and fear of mathematics, and consequently, enhance their achievement in mathematics. Professional development opportunities should also be made available to in-service teachers to continually update their knowledge and skills and develop new strategies for teacher effectiveness.     

Teachers’ innovative change within countrywide reform: a case study in Rwanda

This article presents practical perspectives on mathematics teacher change through results of collaborative research with two mathematics secondary school teachers in order to improve the teaching and learning of mathematics in Rwanda. The 2006 national mathematics curriculum reform stresses pedagogies that enhance problem-solving, critical thinking and argumentation. Teachers need to use new teaching strategies. This article is a case study looking at issues around developing teachers’ use of interactions in mathematics classrooms independently of the national programme. Outputs of the study include teachers’ awareness of the need for change and their increased flexibility to accept learners’ autonomy in shifting from teacher-centred to learner-centred pedagogy. Geometer’s Sketchpad challenged teachers’ practice and then provoked reflection to improve student learning.     

Exploring mathematics teacher knowledge to explain the gap between theory-based recommendations and school practice in the use of connecting tasks

This study is grounded in the theoretical position that solving problems in different ways creates mathematical connections when learning and teaching mathematics. It acknowledges the central role teachers play in providing students with learning opportunities, and it is based on the empirical finding that mathematics teachers are reluctant to solve problems in different ways in the classroom. In this paper we address the contradiction between theory-based recommendations and school mathematics practice. Based on analysis of individual interviews and two group meetings with 12 Israeli secondary school mathematics teachers, we demonstrate that in the context of multiple-solution connecting tasks this discrepancy is caused by the situated nature of the teachers’ knowledge. We also reveal the complex relationship between different types of teacher knowledge and argue the significance of developing a common language between members of the mathematics education community, including teacher educators and researchers. The names of the teachers have been changed to protect their privacy.     

Mathematical thinking of kindergarten boys and girls: similar achievement, different contributing processes

The objective of this study was to examine gender differences in the relations between verbal, spatial, mathematics, and teacher–child mathematics interaction variables. Kindergarten children (N = 80) were videotaped playing games that require mathematical reasoning in the presence of their teachers. The children’s mathematics, spatial, and verbal skills and the teachers’ mathematical communication were assessed. No gender differences were found between the mathematical achievements of the boys and girls, or between their verbal and spatial skills. However, mathematics performance was related to boys’ spatial reasoning and to girls’ verbal skills, suggesting that they use different processes for solving mathematical problems. Furthermore, the boys’ levels of spatial and verbal skills were not found to be related, whereas they were significantly related for girls. The mathematical communication level provided in teacher–child interactions was found to be related to girls’ but not to boys’ mathematics performance, suggesting that boys may need other forms of mathematics communication and teaching.     

A Framework for Analysis of Teachers’ Geometric Content Knowledge and Geometric Knowledge for Teaching

Current reform-driven mathematics documents stress the need for teachers to provide learning environments in which students will be challenged to engage with mathematics concepts and extend their understandings in meaningful ways (e.g., National Council of Teachers of Mathematics, 2000, Curriculum and evaluation standards for school mathematics. Reston, VA: The Council). The type of rich learning contexts that are envisaged by such reforms are predicated on a number of factors, not the least of which is the quality of teachers’ experience and knowledge in the domain of mathematics. Although the study of teacher knowledge has received considerable attention, there is less information about the teachers’ content knowledge that impacts on classroom practice. Ball (2000, Journal of Teacher Education, 51(3), 241–247) suggested that teachers’ need to ‘deconstruct’ their content knowledge into more visible forms that would help children make connections with their previous understandings and experiences. The documenting of teachers’ content knowledge for teaching has received little attention in debates about teacher knowledge. In particular, there is limited information about how we might go about systematically characterising the key dimensions of quality of teachers’ mathematics knowledge for teaching and connections among these dimensions. In this paper we describe a framework for describing and analysing the quality of teachers’ content knowledge for teaching in one area within the domain of geometry. An example of use of this framework is then developed for the case of two teachers’ knowledge of the concept ‘square’.     

中小学"数学情境与提出问题"教学的实验研究

This research tends to make the experimental study on the mathematics teaching model of “situated creation and problem-based instruction” (SCPBI), namely, the teaching process of “creating situations—posing problems—solving problems—applying mathematics”. It is aimed at changing the situation where students generally lack problem-based learning experience and problem awareness. Result shows that this teaching model plays a vital role in arousing students’ interest in mathematics, improving their ability to pose problems and upgrading their mathematics learning ability as well.      

The meaning of mathematics instruction in multilingual classrooms: analyzing the importance of responsibility for learning

In the multilingual mathematics classroom, the assignment for teachers to scaffold students by means of instruction and guidance in order to facilitate language progress and learning for all is often emphasized. In Sweden, where mathematics education is characterized by a low level of teacher responsibility for students’ performance, this responsibility is in part passed on to students. However, research investigating the complexity of relations between mathematics teaching and learning in multilingual classrooms, as well as effect studies of mathematics teaching, often take the existence of teachers’ responsibility for offering specific content activities for granted. This study investigates the relations between different aspects of responsibility in mathematics teaching and students’ performance in the multilingual mathematics classroom. The relationship between different group compositions and how the responsibility is expressed is also investigated. Multilevel structural equation models using TIMSS 2003 data identified a substantial positive influence on mathematics achievement of teachers taking responsibility for students’ learning processes by organizing and offering a learning environment where the teacher actively and openly supports the students in their mathematics learning, and where the students also are active and learn mathematics themselves. A correlation was also revealed between group composition, in terms of students’ social and linguistic background, and how mathematics teaching was performed. This relationship indicates pedagogical segregation in Swedish mathematics education by teachers taking less responsibility for students’ learning processes in classes with a high proportion of students born abroad or a high proportion of students with low socio-economic status.     

Audrey's acquisition of fractions: A case study into the learning of formal mathematics

National standards for teaching mathematics in primary schools in the Netherlands leave little room for formal fractions. However,a newly developed programme in fractions aims at learning formal fractions. The starting point in the development of this curriculum is the students’ acquisition of `numeracy infractions’. In this case study we describe the growth in reasoning ability with fractions of one student in this newly developed programme of 30 lessons during one whole school year. In the study we found indications that the programme and its teaching stimulated the progress of an average performer in mathematics. Moreover we found arguments as to what extent formal operations with fractions suits as an educational goal. This revised version was published online in July 2006 with corrections to the Cover Date.     

Inversion in mathematical thinking and learning

Inversion is a fundamental relational building block both within mathematics as the study of structures and within people’s physical and social experience, linked to many other key elements such as equilibrium, invariance, reversal, compensation, symmetry, and balance. Within purely formal arithmetic, the inverse relationships between addition and subtraction, and multiplication and division, have important implications in relation to flexible and efficient computation, and for the assessment of students’ conceptual understanding. It is suggested that the extensive research on arithmetic should be extended to take account of numerical domains beyond the natural numbers and of the difficulties students have in extending the meanings of operations to those of more general domains. When the range of situations modelled by the arithmetical operations is considered, the complexity of inverse relationships between operations, and the variability in the forms that these relationships take, become much greater. Finally, some comments are offered on the divergent goals and preoccupations of cognitive psychologists and mathematics educators as illuminated by research in this area.