Rhetoric,Reality, and Possibilities: Interdisciplinary Teaching and Secondary Mathematics

The National Council of Teachers of Mathematics has proposed a broad core mathematics curriculum for all high school students. One emphasis in that core is on “mathematical connections” both among mathematical topics and between mathematics and other disciplines of study. It is suggested that mathematics should become a more integrated part of all students' high school education. In this article, working definitions for the terms curriculum, interdisciplinary, and integrated and a model of three categories of curriculum design based on the work of Harold Alberty are developed. This article then examines how a “connected” mathematics core curriculum might be situated within the different categories of curriculum organization. Examples from research on interdisciplinary education in high schools are presented. Issues arising from this study suggest the need for a greater emphasis on building and using models of curriculum integration both to frame and to give impetus to the work being done by teachers and administrators.     

'Model your genes the mathematical way'--a mathematical biology workshop for secondary school teachers

This article describes a mathematical biology workshop givento secondary school teachers of the Danville area in Virginia,USA. The goal of the workshop was to enable teams of teacherswith biology and mathematics expertise to incorporate lessonplans in mathematical modelling into the curriculum. The biologicalfocus of the activities is the lactose operon in Escherichiacoli, one of the first known intracellular regulatory networks.The modelling approach utilizes Boolean networks and tools fromdiscrete mathematics for model simulation and analysis. Theworkshop structure simulated the team science approach commonin today's practice in computational molecular biology and thusrepresents a social case study in collaborative research. Theworkshop provided all the necessary background in molecularbiology and discrete mathematics required to complete the project.The activities developed in the workshop show students the valueof mathematical modelling in understanding biochemical networkmechanisms and dynamics. The use of Boolean networks, ratherthan the more common systems of differential equations, makesthe material accessible to students with a minimal mathematicalbackground. High school students can be exposed to the excitement of mathematicalbiology from both the biological and mathematical point of view.Through the development of instructional modules, high schoolbiology and mathematics courses can be joined without havingto restructure the curriculum for either subject. The relevanceof an early introduction to mathematical biology allows studentsnot only to learn curriculum material in a innovative setting,but also creates an awareness of new educational and careeropportunities that are arising from the interconnections betweenbiological and mathematical sciences. The materials used in this workshop are available at a websitecreated by the directors: http://polymath.vbi.vt.edu/mathbio2006/.     

Developing mathematical modelling skills: The role of CAS

Mathematical modelling as one component of problem solving is an important part of the mathematics curriculum and problem solving skills are often the most quoted generic skills that should be developed as an outcome of a programme of mathematics in school, college and university. Often there is a tension between mathematics seen at all levels as ‘a body of knowledge’ to be delivered at all costs and mathematics seen as a set of critical thinking and questioning skills. In this era of powerful software on hand-held and computer technologies there is an opportunity to review the procedures and rules that form the ‘body of knowledge’ that have been the central focus of the mathematics curriculum for over one hundred years. With technology we can spend less time on the traditional skills and create time for problem solving skills. We propose that mathematics software in general and CAS in particular provides opportunities for students to focus on the formulation and interpretation phases of the mathematical modelling process. Exploring the effect of parameters in a mathematical model is an important skill in mathematics and students often have difficulties in identifying the different role of variables and parameters This is an important part of validating a mathematical model formulated to describe, a real world situation. We illustrate how learning these skills can be enhanced by presenting and analysing the solution of two optimisation problems.     

Mimicry of proofs with computers: the case of Linear Algebra

In common teaching practice the habit of connecting mathematics classroom activities with reality is still substantially delegated to wor(l)d problems. During recent decades, a growing body of empirical research has documented that the practice of word problem solving in school mathematics does not match this idea of mathematical modelling and mathematization. If we wish to construct ‘real problems arising from real experiences of the child’ following the spirit of these new suggestions, we have to make changes. On the one hand we have to replace the type of activity in which we delegate the process of creating an interplay between reality and mathematics by substituting the word problems with an activity of realistic mathematical modelling, i.e. of both real-world based and quantitatively constrained sense-making; and, on the other hand, to ask for a change in teacher beliefs; furthermore, a directed effort to change the classroom socio-math norms will be needed. This paper discusses some classroom activities that takes these factors into account.     

Mathematical modelling at secondary school: the MACSI-Clongowes Wood College experience

In Ireland, to encourage the study of STEM (science, technology, engineering and mathematics) subjects and particularly mathematics, the Mathematics Applications Consortium for Science and Industry (MACSI) and Clongowes Wood College (County Kildare, Ireland) organized a mathematical modelling workshop for senior cycle secondary school students. Participants developed simple mathematical models for everyday life problems with an open-ended answer. The format and content of the workshop are described and feedback from both students and participating teachers is provided. For nearly all participants, this workshop was an enjoyable experience which showed mathematics and other STEM components in a very positive way.     

Problem solving heuristics,affect, and discrete mathematics

It has been suggested that activities in discrete mathematics allow a kind of new beginning for students and teachers. Students who have been “turned off” by traditional school mathematics, and teachers who have long ago routinized their instruction, can find in the domain of discrete mathematics opportunities for mathematical discovery and interesting, nonroutine problem solving. Sometimes formerly low-achieving students demonstrate mathematical abilities their teachers did not know they had. To take maximum advantage of these possibilities, it is important to know what kinds of thinking during problem solving can be naturally evoked by discrete mathematical situations—so that in developing a curriculum, the objectives can include pathways to desired mathematical reasoning processes. This article discusses some of these ways of thinking, with special attention to the idea of “modeling the general on the particular.” Some comments are also offered about students' possible affective pathways and structures.     

Mental Computation of Students in a Reform-Based Mathematics Curriculum

Traditional school instruction in mathematics has generally produced students who are poor at mental computation and exhibit a weak sense of number and mathematical operations. In this study, fifth graders who had been in a reform-based mathematics curriculum since kindergarten were given a whole-class test on mental computation problems. Baseline data with students in traditional mathematics curricula were used as a comparison. The students in this reform-based mathematics curriculum performed much higher than the comparison group on all but one problem, and on most problems, this difference was substantial. Additionally, a student preference survey indicated that students in the reform curriculum were more likely to consider the calculator as an option than were the baseline group. They were also more able to recognize problems that did not lend themselves to mental computation. Individual interviews indicated that experiences in the primary grades with “invented” algorithms and discussing alternative solutions led to a better ability to compute mentally and a stronger number sense.     

Ideal and reality: Cross-curriculum work in school mathematics in South Africa

Within various school mathematics dispensations in South Africa the intention for cross-curriculum work is expressed in the official documents describing the intended school mathematics curriculum. This paper traces this expressed intention from 1962 to the present. The view is adopted that textbook authors are the major interpreters of the intended curriculum and therefore the manifestations of the cross-curricular ideal in school textbooks for the various periods are described and commented on. Using the manifestation of the cross-curricular ideal in the South African situation as backdrop, the paper concludes by suggesting ways to deal with three issues that seemingly mitigate against the realisation of this ideal. It is argued that the applications of and modelling in mathematics should be treated as a distinet separate section in the school mathematics curriculum, that mathematics activities, should be designed so that learners with various levels of mathematical sophistication and expertise can deal with both the embedded context and the mathematics and that problems which use context only as a disguise for “pure” mathematics should not be summarily dismissed and written off as useless for the realisation of cross-curricular goals.     

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.     

Literacy, Matheracy, and Technocracy: A Trivium for Today

This article focuses on the relations between mathematics and mathematics education on the one hand and human behavior, societal models, and power on the other. Based on a critical analysis of school systems and of mathematical thinking, its history and its sociopolitical implications, anew concept of curriculum is suggested, organized in 3 strands: literacy, matheracy, and technoracy. This new concept sees education and scholarship as pursuing a major, comprehensive goal of building up a new civilization that rejects arrogance, inequity, and bigotry. Because the development of mathematics has been intertwined with all forms of human behavior in the history of human- kind, it is relevant to discuss mathematics and mathematics education with this major goal in mind.     

Integration of Science and Mathematics: A Theoretical Model

Interest in interdisciplinary, integrated curriculum development continues to increase. However, teachers, who have been given primary responsibility for developing these materials, are often working with little guidance. At present there exists no clear definition of the meaning of integration of mathematics and science. A continuum model of integration is proposed as a useful tool for curriculum developers as they create new integrated mathematics and science curricula or adapt commercially prepared materials. On the continuum, activities range from mathematics or science involving no integration to those activities including balanced mathematics and science concepts. Several examples are given to illustrate the utility of the continuum model for analyzing integrated curricula. The continuum model is intended to be used by curriculum developers to clarify the relationship between the mathematics and science activities and concepts and to guide the modification of lessons.     

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.     

Mathematics teachers’ conceptions about modelling activities and its reflection on their beliefs about mathematics

The current study examines whether the engagement of mathematics teachers in modelling activities and subsequent changes in their conceptions about these activities affect their beliefs about mathematics. The sample comprised 52 mathematics teachers working in small groups in four modelling activities. The data were collected from teachers' Reports about features of each activity, interviews and questionnaires on teachers' beliefs about mathematics. The findings indicated changes in teachers' conceptions about the modelling activities. Most teachers referred to the first activity as a mathematical problem but emphasized only the mathematical notions or the mathematical operations in the modelling process; changes in their conceptions were gradual. Most of the teachers referred to the fourth activity as a mathematical problem and emphasized features of the whole modelling process. The results of the interviews indicated that changes in the teachers' conceptions can be attributed to structure of the activities, group discussions, solution paths and elicited models. These changes about modelling activities were reflected in teachers' beliefs about mathematics. The quantitative findings indicated that the teachers developed more constructive beliefs about mathematics after engagement in the modelling activities and that the difference was significant, however there was no significant difference regarding changes in their traditional beliefs.     

Supporting Middle School Teachers’ Implementation of STEM Design Challenges

We describe and analyze a professional development (PD) model that involved a partnership among science, mathematics and education university faculty, science and mathematics coordinators, and middle school administrators, teachers, and students. The overarching project goal involved the implementation of interdisciplinary STEM Design Challenges (DCs). The PD model targeted: (a) increasing teachers’ content and pedagogical content knowledge in mathematics and science; (b) helping teachers integrate STEM practices into their lessons; and (c) addressing teachers’ beliefs about engaging underperforming students in challenging problems. A unique aspect involved low‐achieving students and their teachers learning alongside each other as they co‐participated in STEM design challenges for one week in the summer. Our analysis focused on what teachers came to value about STEM DCs, and the challenges in and affordances for implementing DCs. Two significant areas of value for the teachers were students’ use of scientific, mathematical, and engineering practices and motivation, engagement, and empowerment by all learners. Challenges associated with pedagogy, curriculum, and the traditional structures of the schools were identified. Finally, there were four key affordances: (a) opportunities to construct a vision of STEM education; (b) motivation to implement DCs; (c) ambitious pedagogical tools; and, (d) ongoing support for planning and implementation. This article features a Research to Practice Companion Article . Please click on the supporting information link below to access.     

The Impact of Enacted Mathematics Curriculum Models on Prospective Elementary Teachers' Course Perceptions and Beliefs

This paper communicates the impact of prospective teachers' learning of mathematics using novel curriculum materials in an innovative classroom setting. Two sections of a mathematics content course for prospective elementary teachers used different text materials and instructional approaches. The primary mathematical authorities were the instructor and text in the textbook section and the prospective teachers in the curriculum materials section. After one semester, teachers in the curriculum materials section (n= 34) placed significantly more importance on classroom group work and discussions, less on instructor lecture and explanation, and less on textbooks having practice problems, examples, and explanations. They valued student exploration over practice. In the textbook section (n= 19), there was little change in the teachers' beliefs, in which practice was valued over exploration. These results highlight the positive impact of experiences with innovative curriculum materials on prospective elementary teachers' beliefs about mathematics instruction.     

Problem Solving: Teachers' Perceptions,Content Area Models,and Interdisciplinary Connections

Features of common problem-solving models in mathematics and science, as well as those found in business and industry today, are discussed. Commonalties in the models are used to advance a case for interdisciplinary or integrated instruction in mathematics, science and technology. The Integrated Mathematics, Science and Technology (IMaST) program's problem-solving model is presented as an example of a curriculum project that draws upon the commonalties in the problem-solving models as a basis for a seventh grade integrated curriculum.     

Game and decision theory in mathematics education: epistemological, cognitive and didactical perspectives

In the 1950s, game and decision theoretic modeling emerged—based on applications in the social sciences—both as a domain of mathematics and interdisciplinary fields. Mathematics educators, such as Hans Georg Steiner, utilized game theoretical modeling to demonstrate processes of mathematization of real world situations that required only elementary intuitive understanding of sets and operations. When dealing with n-person games or voting bodies, even students of the 11th and 12th grade became involved in what Steiner called the evolution of mathematics from situations, building of mathematical models of given realities, mathematization, local organization and axiomatization. Thus, the students could participate in processes of epistemological evolutions in the small scale. This paper introduces and discusses the epistemological, cognitive and didactical aspects of the process and the roles these activities can play in the learning and understanding of mathematics and mathematical modeling. It is suggested that a project oriented study of game and decision theory can develop situational literacy, which can be of interest for both mathematics education and general education.     

Expanding context and domain: A cross-curricular activity in mathematics and physics

This article is based on my 15 years of experience as a teacher of mathematics and physics in the Danish Gymnasium (high school), and it gives an example of an interdisciplinary course between mathematics and physics. The course is centered around the concept of exponential functions. The starting point is that concepts are rooted in practice and gain their meaning through application, and the concept of a function is regarded as a tool for modelling real-world situations. It is the intention to teach a course that emphasizes factors that promote transfer of the concept and use of the various representations of the concept, to make it more practical and meaningful for the students. It is concluded that a coordinated cross-curricular activity between mathematics and physics, by offering a great variety of domain relations and context settings, has a great potential for creating a learning environment where the students, through applicational and modelling activities, are engaged actively in constructing and using knowledge.     

Modelling in Mathematics Classrooms: reflections on past developments and the future

This paper describes the development of mathematical modelling as an element in school mathematics curricula and assessments. After an account of what has been achieved over the last forty years, illustrated by the experiences of two mathematician-modellers who were involved, I discuss the implications for the future—for what remains to be done to enable modelling to make its essential contribution to the «functional mathematics», the mathematical literacy, of future citizens and professionals. What changes in curriculum are likely to be needed? What do we know about achieving these changes, and what more do we need to know? What resources will be needed? How far have they already been developed? How can mathematics teachers be enabled to handle this challenge which, scandalously, is new to most of them? These are the overall questions addressed. The lessons from past experience on the challenges of large-scale of implementation of profound changes, such as teaching modelling in school mathematics, are discussed. Though there are major obstacles still to overcome, the situation is encouraging.     

Developing the roots of modelling conceptions: ‘mathematical modelling is the life of the world’

A study conducted with 25 Year 6 primary school students investigated the potential for a short classroom intervention to begin the development of a Modelling conception of mathematics on the way to developing a sense of mathematics as a way of thinking about life. The study documents the developmental roots of the cognitive activity, actions and conceptions of both modelling and mathematics that these beginners to modelling displayed. Understanding the conceptions of mathematics that students might hold or be developing and how these can be influenced in early schooling are essential ingredients in any plans for introducing modelling seriously into primary school classrooms. The majority of the students (22/25) were identified as displaying a developing conception of modelling as a way of problem handling. The three other students displayed the developmental roots of a way of understanding the world conception of modelling. These three students also displayed a Modelling conception of mathematics with one showing indications of developing towards a Life conception of mathematics.