Visualization and art in the mathematics classroom

In this paper we summarize our concepts and practice on computer-aided mathematical experimentation, and illustrate them byMathematica projects that we have developed for our research and the courses “Computer-aided mathematical modelling” and “Computer Algebra I–II” held for students of life sciences at University of Szeged and computational engineering at TFH Berlin, University of Applied Sciences.     

Two classroom proofs concerning composition operators

We consider composition operators on spaces of analytic functions. First we give an elementary proof for the well-known fact that these operators are bounded on the usual Hardy spaces. Then, for more general spaces, we give two results concerning the question when composition operators are injective or onto, and get a complete answer for a class of spaces, including the Hardy spaces.     

Investigations and explorations in the mathematics classroom

In Portugal, since the beginning of the 1990s, problem solving became increasingly identified with mathematical explorations and investigations. A number of research studies have been conducted, focusing on students’ learning, teachers’ classroom practices and teacher education. Currently, this line of work involves studies from primary school to university mathematics. This perspective impacted the mathematics curriculum documents that explicitly recommend teachers to propose mathematics investigations in their classrooms. On national meetings, many teachers report experiences involving students’ doing investigations and indicate to use regularly such tasks in their practice. However, this still appears to be a marginal activity in most mathematics classes, especially when there is pressure for preparation for external examinations (at grades 9 and 12). International assessments such as PISA and national assessments (at grades 4 and 6) emphasize tasks with realistic contexts. They reinforce the view that mathematics tasks must be varied beyond simple computational exercises or intricate abstract problems but they do not support the notion of extended explorations. Future developments will show what paths will emerge from these contradictions between promising research and classroom reports, curriculum orientations, professional experience, and assessment frameworks and instruments.     

Ancient accounting in the modern mathematics classroom

Florida State University owns a collection of twenty-five cuneiform tablets, acquired from Edgar J Banks in the 1920s. We describe their rediscovery, present an edition of one of them (a twenty-first century BC labour account from the Sumerian city of Umma), and discuss their potential for use in undergraduate mathematics education.1 ¹We are very grateful to Steve Garfinkle, Denise Giannino, John Larson, Lucia Patrick, Plato L Smith II, and Giesele Towels for their help in the research and writing of this article.      

Counter-examples for refinement of conjectures and proofs in primary school mathematics

The purpose of this study is to explore how primary school students reexamine their conjectures and proofs when they confront counter-examples to the conjectures they have proved. In the case study, a pair of Japanese fifth graders thought that they had proved their primitive conjecture with manipulative objects (that is, they constructed an action proof), and then the author presented a counter-example to them. Confronting the counter-example functioned as a driving force for them to refine their conjectures and proofs. They understood the reason why their conjecture was false through their analysis of its proof and therefore could modify their primitive conjecture. They also identified the part of the proof which was applicable to the counter-example. This identification and their action proof were essential for their invention of a more comprehensive conjecture.     

The mathematics of medical imaging in the classroom

The article presents an integrated exposition of aspects of secondary school mathematics and a medical science specialty together with related classroom activities. Clinical medical practice and theoretical and empirical literature in mathematics education and radiology were reviewed to develop and pilot model integrative classroom topics and activities. The techniques of computer-axial tomography (CT) and positron emission tomography (PET) are discussed, followed by a presentation of accessible mathematical applications in numeration and linear algebra for use in a high school classroom. This discussion of the mathematics of a medical speciality, and the related activities, might not only offer teachers and students specific examples of the connections between their everyday study and a professional discipline, but also might foster further investigation into the importance and relevance of mathematics in other technology-based careers.     

MobileMath: exploring mathematics outside the classroom

Computer games seem to have a potential for engaging students in meaningful learning, inside as well as outside of school. With the growing availability of mobile handheld technology (HHT), a number of location-based games for handheld mobile phones with GPS have been designed for educational use. The exploitation of this potential for engaging students into meaningful learning, however, so far remains unexplored. In an explorative design research, we investigated whether a location-based game with HHT provides opportunities for engaging in mathematical activities through the design of a geometry game called MobileMath. Its usability and opportunities for learning were tested in a pilot on three different secondary schools with 60 12–14-year-old students. Data were gathered by means of participatory observation, online storage of game data, an online survey and interviews with students and teachers. The results suggest that students were highly motivated, and enjoyed playing the game. Students indicated they learned to use the GPS, to read a map and to construct quadrilaterals. The study suggests learning opportunities that MobileMath provides and that need further investigation.     

Transformation of the mathematics classroom with the internet

ZDM – Mathematics Education - Growing use of the internet in educational contexts has been prominent in recent years. In this survey paper, we describe how the internet is transforming the...     

Cultural distance and identities-in-construction within the multicultural mathematics classroom

In this paper we present and exemplify our interpretation of some theoretical constructs that have proved useful to our understanding of the complexity of multicultural mathematics classrooms. Constructs such as culture, cultural distance, cultural conflict and identities-in-construction have oriented our study of the complexity of highly multicultural mathematics classrooms in Barcelona. The purpose of this paper is to discuss how cultural distance arising from the different meanings that students, being local or immigrant, inevitably bring to the mathematics classroom may turn into cultural conflicts when cultural interaction is not facilitated through classroom discourse. The lack of cultural interaction and communication may give rise to strong negative feedings and refusal to participate on the side of the students. Students' nonparticipation can be understood as an active response to cultural distance and negative opinions in order to safeguard the identities they (wish to) construct within a context that they perceive as hostile.     

Potential scenarios for Internet use in the mathematics classroom

Research on the influence of multiple representations in mathematics education gained new momentum when personal computers and software started to become available in the mid-1980s. It became much easier for students who were not fond of algebraic representations to work with concepts such as function using graphs or tables. Research on how students use such software showed that they shaped the tools to their own needs, resulting in an intershaping relationship in which tools shape the way students know at the same time the students shape the tools and influence the design of the next generation of tools. This kind of research led to the theoretical perspective presented in this paper: knowledge is constructed by collectives of humans-with-media. In this paper, I will discuss how media have shaped the notions of problem and knowledge, and a parallel will be developed between the way that software has brought new possibilities to mathematics education and the changes that the Internet may bring to mathematics education. This paper is, therefore, a discussion about the future of mathematics education. Potential scenarios for the future of mathematics education, if the Internet becomes accepted in the classroom, will be discussed.     

Problem solving in the mathematics classroom: the German perspective

In Germany, problem solving has important roots that date back at least to the beginning of the twentieth century. However, problem solving was not primarily an aspect of mathematics education but was particularly influenced by cognitive psychologists. Above all, the Gestalt psychology developed by researchers such as Köhler (Intelligenzprüfungen an Anthropoiden. Verlag der Königlichen Akademie des Wissens, Berlin, 1917; English translation: The mentality of apes. Harcourt, Brace, New York, 1925), Duncker (Zur Psychologie des produktiven Denkens. Springer, Berlin, 1935), Wertheimer (Productive thinking. Harper, New York, 1945), and Metzger (Schöpferische Freiheit. Waldemar Kramer, Frankfurt, 1962) made extensive use of mathematical problems in order to describe their specific problem-solving theories. However, this research had hardly any influence on mathematics education—neither as a scientific discipline nor as a foundation for mathematics instruction. In the German mathematics classroom, problem solving, which is according to Halmos (in Am Math Mon 87:519–524, 1980) the “heart of mathematics,” did not attract the interest it deserved as a genuine mathematical topic. There is some evidence that this situation may change. In the past few years, nationwide standards for school mathematics have been introduced in Germany. In these standards, problem solving is specifically addressed as a process-oriented standard that should be part of the mathematics classroom through all grades. This article provides an overview on problem solving in Germany with reference to psychology, mathematics, and mathematics education. It starts with a presentation of the historical roots but gives also insights into contemporary developments and the classroom practice.     

Gender justice,human rights and measurement in the mathematics classroom

In line with international trends, the new South African mathematics curriculum implores mathematics educators to realize a pedagogy in their classrooms that is more practical, activity-oriented, and connected to their learners' lives. Drawing on data from a larger study that explores theory–practice relations in mathematics education, this paper shows how such progressive practices, when interpreted with respect to the teaching of measurement which required learners to use different measuring instruments for measuring the school grounds in learning about length and perimeter, were found to be deeply gendered. In two different contexts of an ‘African' township school and a predominantly ‘Indian' suburban school, girls in a grade 6 mathematics classroom faced direct sexism as they struggled to take the opportunity to participate in the activity and learn how to measure – an important mathematical competence and everyday knowledge and skill. The article analyses the data with reference to the human rights imperatives of the new national curricula and approaches to addressing disadvantage and discrimination for girls in mathematics classrooms.     

Envisioning my mathematics classroom: Validating the Draw-a-Mathematics-Teacher-Test Rubric

Teachers need the opportunity to reflect, rethink, and adapt as they continually develop their image of their role in their mathematics classrooms. Thus, the purpose of this research was to examine how the Draw-a-Mathematics-Teacher-Test (DAMTT) and rubric can be used to assess preservice elementary teachers’ images of and beliefs about their future mathematics classrooms and validate the Draw-a-Math-TeacherTest-Rubric (DAMTT-R). Results suggest that the DAMTT-R is a valid measure and yields consistent results. Additionally, analysis of preservice elementary teachers’ (PETs) DAMTT revealed that only slightly more than one-third (36.9%) drew a picture and described their classroom in such a way that it reflected beliefs aligned with student-centered pedagogic practices. While mathematics educators may aim for the majority of PETs to leave their programs having developed beliefs aligned with and supportive of student-centered pedagogic practices, the results of this study revealed that 25% of PETs held beliefs that align with teacher-centered pedagogic practices. Lastly, 38.1% of the PETs reflected beliefs about their pedagogic practices, as measured by the DAMTT and the DAMTT-R, aligned with a transition between teacher-centered and student-centered.     

Undergraduate mathematics majors’ writing performance producing proofs and counterexamples about continuous functions

In advanced mathematical thinking, proving and refuting are crucial abilities to demonstrate whether and why a proposition is true or false. Learning proofs and counterexamples within the domain of continuous functions is important because students encounter continuous functions in many mathematics courses. Recently, a growing number of studies have provided evidence that students have difficulty with mathematical proofs. Few of these research studies, however, have focused on undergraduates’ abilities to produce proofs and counterexamples in the domain of continuous functions. The goal of this study is to contribute to research on student productions of proofs and counterexamples and to identify their abilities and mathematical understandings. The findings suggest more attention should be paid to teaching and learning proofs and counterexamples, as participants showed difficulty in writing these statements. More importantly, the analysis provides insight into the design of curriculum and instruction that may improve undergraduates’ learning in advanced mathematics courses.     

Validating proofs and counterexamples across content domains: Practices of importance for mathematics majors

Validating proofs and counterexamples across content domains is considered vital practices for undergraduate students to advance their mathematical reasoning and knowledge. To date, not enough is known about the ways mathematics majors determine the validity of arguments in the domains of algebra, analysis, geometry, and number theory—the domains that are central to many mathematics courses. This study reported how 16 mathematics majors, including eight specializing in secondary mathematics education, who had completed more proof-based courses than transition-to-proof classes evaluated various arguments. The results suggest that the students use one of the following strategies in proof and counterexample validation: (1) examination of the argument's structure and (2) line-by-line checking with informal deductive reasoning, example-based reasoning, experience-based reasoning, and informal deductive and example-based reasoning. Most students tended to examine all steps of the argument with informal deductive reasoning across various tasks, suggesting that this approach might be problem dependent. Even though all participating students had taken more proof-related mathematics courses, it is surprising that many of them did not recognize global-structure or line-by-line content-based flaws presented in the argument.     

Emergent pedagogies and the changing role of the teacher in the TI-Nspire Navigator-networked mathematics classroom

It is generally accepted that the introduction of networked technologies to the mathematics classroom can stimulate an irreversible change within the classroom concerning: the role of the teacher; the nature of the classroom tasks; and the way in which students engage in the process of learning mathematics. This article will use the context of a classroom-based study into teachers’ developing practices with the TI-Nspire Navigator-networked system of handhelds to explore the nature of these practices and the implications for the mathematics classroom. The emergence of a range of formative assessment practices is described and the implication of these practices on desirable learning opportunities (as described by the teachers themselves) is discussed.