Mathematics teachers’ conceptions and constraints for changing teaching practices in Brazilian higher education: an analysis through activity theory

In recent years, changes in the Brazilian economic and social scenario have generated a growing demand for higher education in the country. In response to this new context, the federal government launched in 2007 a programme aiming at expansion of the enrolment to public higher education. In such environment of changes, several proposals have emerged to adapt the Brazilian federal universities to a new reality. Taking this context into account, the focus of our study is on proposals from the Mathematics Department of the Federal University of Minas Gerais. Their aim is to create a new model of teaching practices for the freshman lessons of the Exact Sciences area, which at first were being experimented in special classrooms of students attending their first course on differential and integral calculus. The data were collected from interviews with students and professors from the mathematics department. They were analyzed and systematized using an activity theory approach. We became instigated by a model developed by Engeström in his study on the changes in the Finnish public health system, considering it as a test (testbench) for activity theory in its application to a particular case. Following Engeström's footsteps when developing his research, we arrived at our own model showing the internal tensions in the activity of reformulation of the courses offered by the Department of Mathematics and in the conceptions of teachers that promote – and constraint – the proposals for change.     

Researcher–teacher relationships and models for teaching development in mathematics education

This article offers theoretical and analytical approaches to investigating how researchers and teachers can work together to create knowledge in mathematics education. It argues that researchers and teachers are members of separate, but related, communities of practice, which create and value different types of knowledge. However, connections between communities can be established through discrete boundary encounters, longer term boundary practices, or peripheral participation by members of one community in the practices of another community. A framework for analyzing researcher–teacher relationships is presented and then used to compare ways in which I, as a university-based researcher, worked with teachers in three different types of research projects. The analysis indicates that successful research collaborations are characterized by mutuality of researcher and teacher motivations, roles, and purposes, and complementarity of their expertise and knowledge. Such collaborations build two-way connections between communities through practices that support mutual engagement across the boundaries that define them.     

Some Uses of the Farris Transform in Mathematics and Phylogenetics—A Review

In 1970, Farris introduced a procedure that can be used to transform a tree metric into an ultra metric. Since its discovery, Farris’ procedure has been used extensively within phylogenetics where it has become commonly known as the Farris transform. Remarkably, the Farris transform has not only been rediscovered several times within phylogenetics, but also in other fields. In this paper, we will review some of its various properties and uses. The paper is divided into four parts and, altogether, 12 sections. In the first part, we introduce a standardized scheme for classifying those dissimilarity mappings to which the Farris transform can be applied—a scheme that has evolved over the years, but has apparently not been spelled out before in sufficient detail. In the second part, we will discuss how a straightforward generalization of the Farris transform naturally arises in T-Theory. The third part describes how this generalized Farris transform can be used to approximate dissimilarities by tree metrics. And in the final part, we describe some further, “non-standard” applications of the Farris transform. Received November 24, 2004     

Mathematics education graduate students’ understanding of trigonometric ratios

This study describes mathematics education graduate students’ understanding of relationships between sine and cosine of two base angles in a right triangle. To explore students’ understanding of these relationships, an elaboration of Skemp's views of instrumental and relational understanding using Tall and Vinner's concept image and concept definition was developed. Nine students volunteered to complete three paper and pencil tasks designed to elicit evidence of understanding and three students among these nine students volunteered for semi-structured interviews. As a result of fine-grained analysis of the students’ responses to the tasks, the evidence of concept image and concept definition as well as instrumental and relational understanding of trigonometric ratios was found. The unit circle and a right triangle were identified as students’ concept images, and the mnemonic was determined as their concept definition for trigonometry, specifically for trigonometric ratios. It is also suggested that students had instrumental understanding of trigonometric ratios while they were less flexible to act on trigonometric ratio tasks and had limited relational understanding. Additionally, the results indicate that graduate students’ understanding of the concept of angle mediated their understanding of trigonometry, specifically trigonometric ratios.     

On the exchange of intersection and supremum of σ-fields in filtering theory

We construct a stationary Markov process with trivial tail σ-field and a nondegenerate observation process such that the corresponding nonlinear filtering process is not uniquely ergodic. This settles in the negative a conjecture of the author in the ergodic theory of nonlinear filters arising from an erroneous proof in the classic paper of H. Kunita (1971), wherein an exchange of intersection and supremum of σ-fields is taken for granted.     

The intersection of the subgroups of finite index in Baumslag—Solitar groups

For any one-relator group in the family of Baumslag—Solitar groups, a system of its elements is indicated whose normal closure in the group coincides with the intersection of all normal finite-index subgroups. The well-known criterion for the residual finiteness of Baumslag—Solitar groups is an immediate consequence of this result. It is also shown that, if the intersection of all finite-index normal subgroups in a Baumslag—Solitar group differs from the identity subgroup (i.e., if the group is not residually finite), then this intersection cannot be the normal closure of any finite set of elements.     

Operational research and critical systems thinking—an integrated perspective

On the basis of a review of the role of systems thinking in the history of OR, Part 1 of this essay proposed a systematic understanding of OR as applied systems thinking. Further, it identified the contribution of ‘critical’ systems thinking (CST) in a combined ability of its two strands, critical systems heuristics (CSH) and total systems intervention (TSI), to enhance the conceptual sophistication of OR. Part 2 aims to translate this understanding into a framework for good professional practice. How exactly can CST strengthen the competence profile of OR professionals? Drawing on three experience-based archetypes of professional service and some basic argumentation-theoretical considerations, a new understanding of OR and applied systems thinking as argumentative practice emerges. In this new understanding CST finds a systematic place and some exemplary uses of CSH and TSI can be located—an integrated perspective.     

Micros in mathematics education—Uses of spreadsheets in CAL

In this paper it is shown that teachers and students can study the mathematical problem through commercially available powerful spreadsheet software programs without being bogged down by the programming problem. It is illustrated by two examples. One is taken from a differentiable function the derivative of which is discontinuous at the origin. The other example uses Simpson's rule to find the area under the standard normal distribution.     

Atomic intersection of σ-fields and some of its consequences

Let ${(\Omega, \mathcal{F}, P)}$ be a probability space. For each ${\mathcal{G}\subset\mathcal{F}}$ , define ${\overline{\mathcal{G}}}$ as the σ-field generated by ${\mathcal{G}}$ and those sets ${F\in \mathcal{F}}$ satisfying ${P(F)\in\{0,1\}}$ . Conditions for P to be atomic on ${\cap_{i=1}^k\overline{\mathcal{A}_i}}$ , with ${\mathcal{A }_1,\ldots,\mathcal{A}_k\subset\mathcal{F}}$ sub-σ-fields, are given. Conditions for P to be 0-1-valued on ${\cap_{i=1}^k \overline{\mathcal{A}_i}}$ are given as well. These conditions are useful in various fields, including Gibbs sampling, iterated conditional expectations and the intersection property.     

Students’ Critical Awareness of Voice and Agency in Mathematics Classroom Discourse

This account of my extended conversation with a high school mathematics class focuses on voice and agency. As an investigation of possibilities opened up by introducing mathematics students to what Fairclough (1992) Fairclough, N. 1992. Critical language awareness, London: Longman.  [Google Scholar] called “critical language awareness” (p. 2), I prompted the students daily to become ever more aware of their language practices in class. The tensions in this conversation proved parallel to the tension in mathematics between individual initiative and convention, a tension that Pickering (1995) Pickering, A. 1995. The mangle of practice: Time, agency, and science, Chicago: University of Chicago Press. [Crossref] , [Google Scholar] called the “dance of agency” (p. 21). Participant students in this classroom-based research resisted the idea of linguistic reference to human agency, although their actual language practice revealed some recognition of human agency.     

The State of Proof in Finnish and Swedish Mathematics Textbooks—Capturing Differences in Approaches to Upper-Secondary Integral Calculus

Students’ difficulties with proof, scholars’ calls for proof to be a consistent part of K-12 mathematics, and the extensive use of textbooks in mathematics classrooms motivate investigations on how proof-related items are addressed in mathematics textbooks. We contribute to textbook research by focusing on opportunities to learn proof-related reasoning in integral calculus, a key subject in transitioning from secondary to tertiary education. We analyze expository sections and nearly 2000 students’ exercises in the four most frequently used Finnish and Swedish textbook series. Results indicate that Finnish textbooks offer more opportunities for learning proof than do Swedish textbooks. Proofs are also more visible in Finnish textbooks than in Swedish materials, but the tasks in the latter reflect a higher variation in nature of proof-related reasoning. Our results are compared with methodologically similar U.S. studies. Consequences for learning and transition to university mathematics, as well as directions for future research, are discussed.     

Numerical Mathematics——A Journal of Chinese Universities

Aims and Scope： Numerical Mathematics： A Journal of Chinese Universities （English Series） published quarterly. It is an international journal covering numerical methods, analysis and applications. The journal publishes original research papers in all branches of modern computational mathematics such as numerical linear algebra, numerical optimization, numerical differential equations and computational statistics. Papers containing new ideas, creative approaches and/or innovative applications as well as invited reviews are expected to appear regularly in the journal.     

Technological Aids in the Teaching of Mathematics∗

In learning mathematics a relationship needs to be known as a detail and this also needs to be understood relative to the over‐all pattern and structure of mathematics. For some users of mathematics, techniques can be developed for presenting formulae sequentially so that they are available for term‐by‐term substitution, but the needs of creative mathematical thinkers are not met in this way. Present human resources are one mathematics graduate per 500 secondary school pupils and one mathematics graduate per 1300 students in further education. There is a pressing need to supplement these teaching resources by aids made available by educational technology, and also for research into suitable student terminals. Sequentially presented material is already available but there is a great need for visual material which can be presented synoptically. It is suggested that a steering team could mobilize many units where there are mathematicians and audio‐visual facilities to provide a large library of linking sequences to be available at computer‐controlled student terminals, relating each formula or relationship to other aspects of mathematics so that its place in the whole structure of mathematics is presented. Examples are given. There is also a need to devote some resources to the study of the effectiveness of particular diagrams and the order of presentation of visual materials since some current researches indicate these may be critical factors.