Bridging second-grade children's thinking and mathematical recording

The focus of this article is how what students do and say to solve a problem may be recorded to mirror the student's actions or thoughts rather than portraying a common algorithm that is not connected to the student's thinking. It illustrates the multiple strategies used by a second-grade class that has solved a problem with three addends and how the teacher tries to faithfully map their thinking into the system of mathematical notation. Emphasis is placed on the step-by-step linking of action, thought, and symbol that must occur. The AFT Thinking Mathematics program that the teacher is using is briefly described as well as how the teacher developed number sense and a culture in which student thinking is respected. The article stresses that teachers need professional development experiences that help them understand how children best learn mathematics to be able to effectively address the needs of a class of heterogeneous learners and open doors to greater achievement.     

Assessing and understanding U.S. and Chinese students' mathematical thinking

If the main goal of educational research and refinement of instructional program is to improve students' learning, it is necessary to assess students' emerging understandings and to see how they arise. The purpose of this paper is to address issues related to assessments of students' mathematical thinking in cross-national studies and then to discuss the lessons we may learn from these studies to assess and improve students' learning. In particular, the issues related to assessing U.S. and Chinese students' mathematical thinking were discussed. Then, this paper discussed the findings from two studies examining the impact of early algebra learning and teachers' beliefs on U.S. and Chinese students' mathematical thinking. Lastly, the issues related to interpreting and understanding the differences between U.S. and Chinese students' thinking were discussed.     

Technology-active mathematical modelling

Problems in mathematical modelling and data analysis are discussed from a constructivist perspective. This approach provides students with realistic opportunities to connect mathematics to significant social and environmental problems while incorporating recent advances made possible by today's mathematically powerful calculators. Also included are methods for enhancing students' abilities to shift among a wide range of representations using the modelling capabilities in graphing utilities. Consideration is further given to the changes that technology imposes on the classroom culture, including changes in students' attitudes about modelling techniques and difficulties in locating appropriate problems. The article concludes by discussing the integration of teaching and assessment with mathematical modelling.     

Teaching mathematical modelling in microeconomics

Three modelling excercises from microeconomics are presented, both as examples of a methodology for teaching economics to mathematically inclined students, and as a vehicle for teaching modelling as a general skill. The first exercise, a price adjustment model for perfect competition, is given in detail within a formulation/solution/generalization/validation scheme with suggestions for an interactive teaching strategy. The results of giving the exercise to a second‐year BSc class in Mathematics and Computing are reported. Outlines of two further exercises relating to oligopoly and related models are given, and an Appendix provides a series of graduated problems for the first of these exercises.     

Shifts of mathematical thinking in adolescence

This theoretical paper relates key features of the mathematics adolescents are expected to learn in school to other aspects of adolescent development. Difficulties in mathematical learning at that age include changes in perspective and in the actions that are mathematically productive. Commonly-recommended methods of trying to engage adolescents in mathematics do not necessarily enable students to shift to new perceptions and new ways of constructing mathematical understandings, yet the shifts students need to make are in accord with other aspects of adolescent development.     

Applied mathematical modelling in marine science

The essential elements of a general interdisciplinary marine model are reviewed; state variables are identified, evolution equations are discussed and major parameters are indicated. The methods of reducing the size of the model by aggregation in state space (reduction of scope) or averaging in physical space (reduction of support) are revised with particular emphasis on the associated parameterization of the non-linear effects. Examples of application are given in illustration and comparison of the models' predictions with observations indicates the degree of accuracy and the limits of viability of present state marine modelling.     

On examples in mathematical thinking and learning

The importance of examples and exemplification in mathematical thinking, learning and teaching, is well recognized by mathematicians, epistemologists and mathematics educators. In the collection of papers on these topics presented in this issue, we aim to contribute to the debate on this theme, proposing original studies carried out from different approaches and perspectives, and linked to other relevant topics within mathematics education.     

Methodological conceptualization of mathematical modelling

The experience of the author and colleges, as mathematicians working in interdisciplinary groups, have shown the necessity to make the process of mathematical modelling more precise and to establish its different phases. In this way, the specific role of the mathematician in working teams can be better understood by the other members of the team and his or her specific capabilities can be used more efficiently. The proposed structuration of the mathematical modelling process is resumed in a following diagram, especially when computational schemes are the desired result (see Figure 1).

The discussion tends to delineate a concept of modelling from a standpoint where the difference between mathematics as a language and mathematics as a science, having its own dynamic and semantics, plays a fundamental role.     

Approximate mathematical modelling of motions

This paper discusses different approaches to constructing approximate mathematical models. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 7, pp. 5–9, 2005.     

Introducing mathematical modelling to undergraduates

Some motivation for, and thoughts behind, the development of a course in mathematical modelling are described. The structure of the course and some of the problems of teaching and assessment are reviewed. The course has the characteristics that it is introductory and discipline‐independent.     

An examination for mathematical modelling

First year mathematics degree students at Leicester Polytechnic attend a course in mathematical modelling. The aim is to introduce the students to mathematical modelling concepts and to model development. Work is set and marked during the course and this forms a vital part of the students' assessment. In addition to this, however, the students are assessed by means of a three hour examination at the end of the year. This examination is significantly different from the normal ‘five out of eight’ type. The philosophy and organization of the examination are discussed in this paper. An example of a particular examination, that for June 1986, is included as an appendix to illustrate the points made in the discussion.     

Promoting interdisciplinarity through mathematical modelling

This article presents one approach to addressing the important issue of interdisciplinarity in the primary school mathematics curriculum, namely, through realistic mathematical modelling problems. Such problems draw upon other disciplines for their contexts and data. The article initially considers the nature of modelling with complex systems and discusses how such experiences differ from existing problem-solving activities in the primary mathematics curriculum. Principles for designing interdisciplinary modelling problems are then addressed, with reference to two mathematical modelling problems—one based in the scientific domain and the other in the literary domain. Examples of the models children have created in solving these problems follow. A reflection on the differences in the diversity and sophistication of these models raises issues regarding the design of interdisciplinary modelling problems. The article concludes with suggested opportunities for generating multidisciplinary projects within the regular mathematics curriculum.