Mathematical Situations Research Articles

The importance of mathematical conceptualisation

Mathematicians typically invoke a wide range of reasons as to why their research is valuable. These reveal considerable differences between their personal images of mathematics. One of the most interesting of these concerns the relative importance accorded to conceptual reformulation and development compared with that accorded to the achievement of concrete results. Here I explore the conceptualists' claim that the scales are tilted too much in favour of the latter. I do so by taking as a case study the debate surrounding the question as to whether groupoids are significantly more powerful than groups at capturing the symmetry of a mathematical situation. The introduction of groupoids provides a suitable case as they score highly according to criteria relating to theory-building rather than problem-solving. Several of the arguments for the adoption of the groupoid concept are outlined, including claims as to its capacity for reformulating existing theory, its ability to measure symmetry more systematically, and its ‘naturalness’. This last notion is given an extensive treatment.

Activities for Students: The Archaeological Dig Site: Using Geometry to Reconstruct the Past

In secondary mathematics, students often see little connection between geometry and the real-life mathematical situations around them. When asked to describe geometric figures, their descriptions are sometimes no more than an identification of sides and angles. They have not had experience in using more than one property in a mathematical situation or in describing how two geometric properties are related. The van Hiele model of how students learn geometry proposes that students' understandings of geometry move from recognition to description to analysis (Fuys, Geddes, and Tischler 1988). For students to make this transition to analytic thinking, teachers need to create problem situations that enhance development of students' intuitive understandings. These investigations allow students to explore relationships among geometric shapes and to make conjectures about properties. The conjectures can then be stated formally as theorems.

Anatomy of an Assessment

The Assessment Standards for School Mathematics (NCTM 1995) discusses a shift in mathematical content from just mastering isolated concepts and procedures to exploring a rich variety of mathematical topics and problem situations that teach and use concepts and procedures to solve problems. The Standards document also calls for student assessment that is aligned with, and integral to, instruction. Constructing valid assessments has always been difficult, and the new demands make it even more of a challenge. What does such assessment look like, and how does it work?

The influence of overcoming fixation in mathematics towards divergent thinking in open-ended mathematics problems on Japanese junior high school students

The purpose of this study is to find the influence of overcoming fixation in mathematical problem-solving towards divergent thinking in open-ended mathematics problems. This paper presents findings of investigation of Japanese junior high school students. Students who can overcome fixation in mathematics can contribute varied and original ideas in open-ended situations in mathematics.

`Creative mathematics' – real or rhetoric?

The notion of creativity has its natural home in the fine arts, where the artist literally creates something that can be perceived by the senses. The products of mathematical activity are clearly not of this kind, yet some distinguished mathematicians have claimed that mathematics offers considerable scope for creativity. The title of the book under review, and some claims to be found in it, suggest that creativity can indeed be associated with mathematics, and that young children may experience it in the classroom. We suggest that the word ‘creative’ is being used in rather different senses in these different contexts, yet the meanings associated with the arts, say, are in danger of being applied to mathematical situations for rhetorical purposes.

The effects of an overt constructivist approach to learning mathematics and its subsequent effects on classroom teaching

A Master's unit in mathematics education was given to three practicing primary teachers. Constructivism as a theory on which to develop a flexible pedagogy was a central theme and the style of presentation of workshops was guided by constructivist principles. Contact time was 20 hours spread over 10 weeks. The course contained a number of different aspects relating to mathematics teaching and learning some of which provided research data: (1) discussion about the nature of constructivism in mathematics learning, and its implications for teaching; this centred around seminar papers on aspects of constructivism; (2) a workshop each week that focused on a rich mathematical topic designed to provoke mathematical activity and interaction and to encourage reflection on that activity; (3) an extended investigation of a mathematical situation; and (4) classroom implementation of some aspect of (3) together with a report of its implementation and an evaluation of children's understanding. A combined individual and social constructivist approach to learning mathematics by primary teachers is described. some evidence is presented of changes in belief held by teachers about the nature of mathematics and the subsequent changes in classroom practice and expectations of children's learning.

How to formalize it?: Formalization principles for information system development methods

Although the need for formalization of modelling techniques is generally recognized, not much literature is devoted to the actual process involved. This is comparable to the situation in mathematics where focus is on proofs but not on the process of proving. This paper tries to accommodate for this lacuna and provides essential principles for the process of formalization in the context of modelling techniques as well as a number of small but realistic formalization case studies.

Twenty-Five Years with Nicolas Bourbaki, 1949-1973

T he choice of dates is dictated by personal circumstances: they roughly bound the period in which I had inside knowledge of the work of Bourbaki, first through informal contacts with several members, then as a member for twenty years, until the mandatory retirement at fifty. Being based largely on personal recollections, my account is frankly subjective. Of course, I checked my memories against the available documentation, but the latter is limited in some ways: not much of the discussions about orientation and general goals has been recorded.1 Another member might present a different picture. To set the stage, I shall briefly touch upon the first fifteen years of Bourbaki. They are fairly well documented2, and I can be brief. In the early thirties the situation of mathematics in France at the university and research levels, the only ones of concern here, was highly unsatisfactory. World War I had essentially wiped out one generation. The upcoming young mathematicians had to rely for guidance on the previous one, including the main and illustrious protagonists of the so-called 1900 school, with strong emphasis on analysis. Little information was available about current developments abroad, in particular about the flourishing German school (Gottingen, Hamburg, Berlin), as some young French mathematicians (J. Herbrand, C. Chevalley, A. Weil, J. Leray) were discovering during visits to those centers.3 In 1934 A. Weil and H. Cartan were Maitres de Conferences (the equivalent of assistant professors) at the University of Strasbourg. One main duty was, of course, the teaching of differential and integral calculus. The standard text was the Traite d’Analyse of E. Goursat, which they found wanting in many ways. Cartan was frequently bugging Weil with questions on how to present this material, so that at some point, to get it over with once and for all, Weil suggested they write themselves Armand Borel is professor emeritus at the School of Mathematics, Institute for Advanced Study, Princeton, NJ.

Aligning Assessment with Classroom Practices: A Promising Testing Format

The 1990s have emerged as a decade in which the mathematics community is redefining what we teach, how we teach, and, consequently, what and how we assess. What we are teaching has a stronger conceptual orientation than in the past. How we are teaching involves ouT students more actively than ever before. And what and how we are assessing … has caused us some problems. All too often, after creating an environment wherein students have, for example, used calculators and group work to investigate challenging and meaningful mathematical situations, we assess their learning through a standard in-class test. Individuals work independently on questions that, given the short time allowed to complete them, are by necessity less demanding than the previous class work. The assessment is not consistent with the curricular or pedagogical reforms.

Explorations in Geometry: The “Art” of Mathematics

We often pose a mathematical situation, concept, or theorem and do not proceed to explore it fully. We do not experience the thrill of chasing our intuitions, the excitement of meeting the unexpected, the uplift of clarifying ideas, the feeling of enlightenment and pride upon discovering something new to us, and the rush of succinctly capturing the essence of complexity. In short, we miss the artistic experience in one of our great arts—mathematics.

THE WORLD ACCORDING TO JOHNNY: A Coping Perspective in Mathematics Education

Students in mathematical problem-solving situations often experience confusion and loss of meaning. In these situations, affective and social factors are as much part of the student's thinking and behavior as are cognitive factors. These additional factors might include, for example, the need to make sense and the need to meet expectations of the authority figure involved (e.g., teacher or researcher). In this paper we attempt to analyze students' productions, taking into account such additional affective and social factors. To this end, we have tried to reproduce the student's voice in what we call virtual monologue. It consists of a monologue in the student's voice given in first person language, in which we try to describe as vividly and as faithfully as we can our picture of what might be going on in the student's mind during such situations.

“Math and the human body”: Sharing the experiences of an activity-based learning situation

This article describes a project-based situation in mathematics, with 12- and 13-year-old students in two schools in Edinburgh, Scotland. The main theme of the project, its results, and some afterthoughts are presented, in an attempt to illustrate differences in the behavior and attitudes between the 12-year-olds (last year of primary school) and the 13-year-olds (first year of secondary school).

Children's development in solving a certain class of additive problems in mathematics: A didactic intervention based on action

Several studies have brought to light the difficulties which children experience in the representation and solution of change problems in mathematics. Analysis of these difficulties has uncovered certain implicit models used by children. These models actually interfere with the very process of number construction. A teaching experiment involving 30 third grade students (8- to 9-yr-olds) was developed on the basis of this analysis. According to research carried out in the Piagetian tradition, our didactic choice, within the form of a group sociomotor game, places the action of the subjects at the heart of the transformation of their concept of number. Observation of the notations and problem-solving procedures used throughout the experiment by the children in mathematical situations derived from a game enabled us to account for the children's development, which went from a focus on states to a type of reasoning in which changes become quantifiable “objects of awareness”.

Investigations: Vehicles for Learning and Doing Mathematics

Professional organizations of mathematics educators and mathematicians are calling for major reforms in the teaching and learning of mathematics. Among those reforms are a shift in emphasis in curriculum from mastery of lists of unrelated mathematical concepts and skills to exploration of rich mathematical topics and problem situations, and a shift in learning from memorizing and replicating algorithmic procedures to investigating and solving complex problems. To help students achieve proficiency in solving problems, the curriculum must focus on development of the major concepts of mathematics, the enhancement and enlargement of students' repertoires of problem-solving strategies and reasoning methods, and the refinement of communication and collaboration skills. Because they present intriguing problems whose solutions or solution methods are not immediately obvious, and require the application of concepts from different areas of mathematics, and, in some instances, knowledge from other content areas, investigations are powerful vehicles for helping students achieve expertise in solving problems. The nature of investigations and their educational benefits are described. Three types of investigations, whimsical, real, and mathematical, are defined and illustrated. For each investigation, the mathematical content and problem-solving strategies are identified, and a solution method is presented. The responsibilities of the teacher, before, during and after an investigation are described.

Problem Formulation: Insights from Student Generated Questions

This article describes an exploration of one facet of problem formulation. Prospective elementary and secondary teachers were presented with a mathematical situation consisting of data and conditions, but no questions were presented. They were asked to create questions that could be answered using the given information. In the spirit of exploration, this paper analyzes the student‐formulated questions from two perspectives—the nature of the information contained within the questions posed and the responses which those questions elicited. A careful analysis of questions posed by students will enable education professionals to understand and promote better problem posing in mathematics classrooms.

Laminar Fluid Transients in Conduits of Arbitrary Cross Section

Conduit cross sections of nonelementary shape are the rule in biological fluid systems. Because of mathematical and experimental difficulties, however, relatively little is known about the dynamics of flow in these vessels, particularly under time-varying conditions, although knowledge of the fluid motion therein in terms of properties such as driving head, velocity, and wall shear stress is clearly essential for proper diagnosis of a flow problem. This paper presents an analytical method for solving problems of unsteady laminar flow in long ducts of arbitrary cross section. The implied assumption of parallel flow renders the mathematical situation linear, but the proposed method is far from conventional because it allows consideration of almost any cross-sectional configuration conceivable. This novel approach is made possible by joining a steady-state superposition method for creating new conduits from known basic shapes with a regular perturbation scheme found suitable for dealing with fluid transients...

Lagois connections — a counterpart to Galois connections

In this paper we define a Lagois connection, which is a generalization of a special type of Galois connection. We begin by introducing two examples of Lagois connections. We then recall the definition of Galois connection and some of its properties; next we define Lagois connection, establish some of its properties, and compare these with properties of Galois connections; and then we (further) develop examples of Lagois connections. Via these examples it is shown that, as is the case of Galois connections, there is a plethora of Lagois connections. Also it is shown that several fundamental situations in computer science and mathematics that cannot be interpreted in terms of Galois connections naturally fit into the theory of Lagois connections.

Teaching Mathematics With Technology: Using the Video Camera in Mathematical Problem Solving

In the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989), technology, including the use of video, is discussed as a way to develop mathematical thinking and reasoning abilities, promote problem solving, and apply mathematics in the classroom. Effective use of the video camera and instructional television may help students construct meaning in mathematical situations. Video technology brings in the outside world and adds excitement to the classroom by contributing a visual element that is often lacking in class discussions.

Mathematical Iteration through Computer Programming

Mathematical iteration is a process for generating a sequence in which one or more initial terms are given and each subsequent term is determined from its predecessors in the same way. An equation that describes the relationship between a term and its predecessors is called a recurrence relation. Arithmetic and geometric sequences, common topics in high school algebra courses, are examples of iterative processes. Arithmetic sequences are generated iteratively from an initial term, a1, a common difference, d, and a recurrence relation, an+1, = an+ d. Geometric sequences are generated from an initial term, a1 a common ratio, r, and a recurrence relation, an+1 =an •r. Mathematical iteration is used in many other mathematical situations and algorithms, such as the Fibonacci sequence, the Euclidean algorithm, and Newton's method for solving equations.

Children's invented symbol systems: Exploring parallels between music and mathematics.

This is a short report of an exploratory study linking children's invented mathematical notations and music notations. Ten 8-year-old children participated in the study. Prior research in music notations has indicated that children have a variety of symbols at their disposal, and their choice of symbols depends on the entity they are trying to capture. This was also the case for notations of tessellations. Although there were similarities found between the use of symbols in the two domains, a consideration of the intended reader did not appear to be a contributing factor in designing mathematical notations. This research note describes the first of four phases of the research comparing children's invented notations for mathematics and music. The overall aims of the research are to explore the authentic connections between music and mathematics in terms of notations, as well as form and transformations, and to adapt methodologies used in researching children's music cognition to address parallel situations in mathematics. There is much to learn from the research on music notations in advancing our understanding of mathematics notations. A diverse body of research indicates that children use a rich and varied repertoire of symbols and strategies in notating rhythm sequences and melodies. They notate with pictures or icons, letters and words, numbers, lines and squiggles and crosses, music symbols, and color to create engaging systems of recording the music they (Bamberger, 1982; Davidson & Colley, 1986; Davidson & Scripp, 1988; Davidson, Scripp, & Welsh, 1988; Smith, 1989; Upitis, 1987, 1990a) and compose (Borstad, 1989; Upitis, 1990b, 1992). In an examination of the research on children's music notations, three other features consistently emerge. The first is that children are more likely to engage in exploring and using music notations when they are involved in making something that they value as real—perhaps composing a piece that others will in a class concert (Upitis, 1992). Second, research on music notations indicates the importance of the intended or imagined reader. Children are less likely to use music symbols if they know that the person who will be reading their notation is not fluent in standard notation (Upitis, 1990a, 1992). Finally, research indicates that when children truly embrace an idea, they make it a part of their world, using the idea to see the everyday phenomena in new ways (Hodgkin, 1985). Thus, children who become musicians and composers may hear the music in the pattern on a sweater or in a mosaic (Upitis, 1990b).