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.     

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.     

Computational Literacy and “The Big Picture” Concerning Computers in Mathematics Education

This article develops some ideas concerning the “big picture” of how using computers might fundamentally change learning, with an emphasis on mathematics (and, more generally, STEM education). I develop the big-picture model of computation as a new literacy in some detail and with concrete examples of sixth grade students learning the mathematics of motion. The principles that define computational literacy also serve as an analytical framework to examine competitive big pictures, and I use them to consider the plausibility, power, and limitations of other important contemporary trends in computationally centered education, notably computational thinking and coding as a social movement. While both of these trends have much to recommend them, my analysis uncovers some implausible assumptions and counterproductive elements of those trends. I close my essay with some more practical and action-oriented advice to mathematics educators on how best to orient to the long-term trajectory (big picture) of improving mathematics education with computation.     

Mathematics standards and curricula in the Netherlands

This paper addresses the question of what mathematics Dutch students should learn according to the standards as established by the Dutch Ministry of Education. The focus is on primary school and the foundation phase of secondary school. This means that the paper covers the range from kindergarten to grade 8 (4~14 years olds). Apart from giving an overview of the standards, we also discuss the standards' nature and history Furthermore, we look at textbooks and examination programs that in the Netherlands both have a key role in determining the intended mathematics curriculum. In addition to addressing the mathematical content, we also pay attention to the way mathematics is taught. The domain-specific education theory that forms the basis for the Dutch approach to teaching mathematics is called “Realistic Mathematics Education” Achievement scores of Dutch students from national and international tests complete this paper. These scores reveal what the standards bring us in terms of students' mathematical understanding. In addition to informing an international audience about the Dutch standards and curricula, we include some critical reflections on them.     

Mathematical creativity and giftedness: a commentary on and review of theory, new operational views, and ways forward

In this commentary we synthesize and critique three papers in this special issue of ZDM (Leikin and Lev; Kattou, Kontoyianni, Pitta-Pantazi, and Christou; Pitta-Pantazi, Sophocleous, and Christou). In particular we address the theory that bridges the constructs of “mathematical creativity” and “mathematical giftedness” by reviewing the related literature. Finally, we discuss the need for a reliable metric to assess problem difficulty and problem sequencing in instruments that purport to measure mathematical creativity, as well as the need to situate mathematics education research within an existing canon of work in mainstream psychology.     

The Mathematics Writer's Checklist: The Development of a Preliminary Assessment Tool for Writing in Mathematics

The use of writing as a pedagogical tool to help students learn mathematics is receiving increased attention at the college level ( Meier & Rishel, 1998 ), and the Principles and Standards for School Mathematics (NCTM, 2000) built a strong case for including writing in school mathematics, suggesting that writing enhances students' mathematical thinking. Yet, classroom experience indicates that not all students are able to write well about mathematics. This study examines the writing of a two groups of students in a college‐level calculus class in order to identify criteria that discriminate “;successful” vs. “;unsuccessful” writers in mathematics. Results indicate that “;successful” writers are more likely than “;unsuccessful” writers to use appropriate mathematical language, build a context for their writing, use a variety of examples for elaboration, include multiple modes of representation (algebraic, graphical, numeric) for their ideas, use appropriate mathematical notation, and address all topics specified in the assignment. These six criteria result in The Mathematics Writer's Checklist, and methods for its use as an instructional and assessment tool in the mathematics classroom are discussed.     

From how to why: A quest for the common mathematical meanings behind two different division algorithms

During their education cycle in mathematics, students are exposed to algorithms as early as primary school. Several studies show how students frequently learn to perform these algorithms without controlling the mathematical meanings behind them. On the other hand, several National Standards have highlighted the need to construct meanings in mathematics from the first cycle of education. In this paper we focus on division algorithms, investigating to what extent 6th graders can be guided to understanding the whys behind an algorithm, through the comparison of two different algorithms for integer division. Our results suggest, on the one hand, that “it could work!”, and on the other hand, that the exposure to different algorithms for the same mathematical operation seems particularly significant for bringing out the whys behind such algorithms, as well as for capturing the difference between a mathematical operation and algorithms for calculating the result of such an operation.     

Mathematics learning and tools from theoretical, historical and practical points of view: the productive notion of mathematics laboratories

In our research work, we have looked at the way in which artefacts become, for teachers as well as for students, instruments of their mathematical activity. The issues related to the use of tools and technologies in mathematical education are now widely considered. A look to history highlights the different ways in which the same questions have been studied at different times and in different places. This suggests that the contribution of artefacts to mathematics learning should be considered in terms of various contexts. Our “visits” to these contexts will be guided by the coordination of two main theoretical frameworks, the instrumental approach and the semiotic mediation approach from the perspective of mathematics laboratory. This journey through history and schooling represents a good occasion to address some questions: Are there “good” contexts in which to develop mathematical instruments? Are there “good” teaching practices which assist students’ instrumental geneses and construct mathematical meanings? How is it possible to promote such teaching practices? Some study cases are discussed.     

Applying Mathematical Concepts with Hands‐On,Food‐Based Science Curriculum

This article addresses the current state of the mathematics education system in the United States and provides a possible solution to the contributing issues. As a result of lower performance in primary mathematics, American students are not acquiring the necessary quantitative literacy skills to become successful adults. This study analyzed the impact of the Food, Math, and Science Teaching Enhancement Resource (FoodMASTER) Intermediate curriculum on fourth‐grade students' mathematics knowledge. The curriculum is a part of the FoodMASTER Initiative, which is a compilation of programs utilizing food, a familiar and necessary part of everyday life, as a tool to teach mathematics and science. Students exposed to the curriculum completed a 20‐item researcher‐developed mathematics knowledge exam (intervention n = 288; control n = 194). Overall, the results showed a significant increase in mathematics knowledge from pretest to posttest. These findings suggest that the food‐based science activities provided the students with the context in which to apply mathematical concepts to an everyday experience. Therefore, the FoodMASTER approach was successful at improving students' mathematics knowledge while building a foundation for becoming quantitatively literate adults.     

Shortcomings of Mathematics Education Reform in The Netherlands: A Paradigm Case?

This article offers a reflection on the findings of three PhD studies, in the domains of, respectively, subtraction under 100, fractions, and algebra, which independently of each other showed that Dutch students' proficiency fell short of what might be expected of reform in mathematics education aiming at conceptual understanding. In all three cases, the disappointing results appeared to be caused by a deviation from the original intentions of the reform, resulting from the textbooks' focus on individual tasks. It is suggested that this “task propensity”, together with a lack of attention for more advanced conceptual mathematical goals, constitutes a general barrier for mathematics education reform. This observation transcends the realm of textbooks, since more advanced conceptual mathematical understandings are underexposed as curriculum goals. It is argued that to foster successful reform, a conscious effort is needed to counteract task propensity and promote more advanced conceptual mathematical understandings as curriculum goals.     

Theories of Mathematics Education,Edited by Bharath Sriraman and Lyn English: Common Ground for Scholars and Scholars in the Making

Theories of mathematics education: Seeking new frontiers is the first book in a new Springer series, Advances in Mathematics Education. To some degree the book is based on a collection of previously published articles from special issues of ZDM—The International Journal on Mathematics Education (previously known as Zentralblatt für Didaktik der Mathematik). These articles, dealing with the role and use of theories in and about mathematics education, originally stem from various conferences and meetings such as PME and CERME. For this reason some of the articles in the book are already well known, a few may even be considered to be “modern classics” within theories of mathematics education such as Frank Lester's “On the theoretical, conceptual, and philosophical foundations for research in mathematics education.” What is new—and non-traditional—in the book, however, is its form of presentation and format, the main articles being accompanied by preludes and commentaries by established researchers and newcomers.     

The near-peer mathematical mentoring cycle: studying the impact of outreach on high school students’ attitudes toward mathematics

College students may be seen as near-peers to high school students and high school students are often able to see themselves in the college students who are but one step ahead. This nearness in maturity and educational level may place college students in a particularly powerful position when it comes to reaching out to high school students to promote higher education in math and science. In this study college students gave dynamic mathematics outreach presentations, MathShows, to minority and low-income high school students in a mid-sized public school district on the U.S. border with Mexico. The study investigated the impacts of this sort of outreach work on high school students’ attitudes towards mathematics using a mathematics attitudes survey. Results, obtained from N = 306 participants, showed statistically significant improvements in almost all components of mathematical attitudes, with less of an effect on the component of self-confidence in doing mathematics. Differences in impacts by specific student subgroups are all discussed.     

Areas,anti-derivatives,and adding up pieces: Definite integrals in pure mathematics and applied science contexts

Research in mathematics and science education reveals a disconnect for students as they attempt to apply their mathematical knowledge to science and engineering. With this conclusion in mind, this paper investigates a particular calculus topic that is used frequently in science and engineering: the definite integral. The results of this study demonstrate that certain conceptualizations of the definite integral, including the area under a curve and the values of an anti-derivative, are limited in their ability to help students make sense of contextualized integrals. In contrast, the Riemann sum-based “adding up pieces” conception of the definite integral (renamed in this paper as the “multiplicatively-based summation” conception) is helpful and useful in making sense of a variety of applied integral expressions and equations. Implications for curriculum and instruction are discussed.     

Possible Student Justification of Proofs

The National Council of Teachers of Mathematics calls for an increased emphasis on proof and reasoning in school mathematics curricula. Given such an emphasis, mathematics teachers must be prepared to structure curricular experiences so that students develop an appreciation for both the value of proof and for those strategies that will assist them in developing proving skills. Such an outcome is more likely when the teacher feels secure in his/her own understanding of the concept of “mathematical proof” and understands the ways in which students progress as they take on increasingly more complex mathematical justifications. In this article, a model of mathematical proof, based on Balacheff's Taxonomy of Mathematical Proof, outlining the levels through which students might progress as they develop proving skills is discussed. Specifically, examples of the various ways in which students operating at different levels of skill sophistication could approach three different mathematical proof tasks are presented. By considering proofs under this model, teachers are apt to gain a better understanding of each student's entry skill level and so effectively guide him/her toward successively more sophisticated skill development.     

Mathematikdidaktik: Die Erforschung theoretischen Wissens in sozialen Kontexten des Lernens und Lehrens

The problem of “defining” mathematics education as a proper scientific discipline has been discussed controversely for more than 20 years now. The paper tries to clarify some important aspects especially for answering the question of what makes mathematics education a specific scientific discipline and a field of research. With this aim in mind the following two dimensions are investigated: On the one hand, one has to be aware that mathematics is not “per se” the object of research in mathematics education, but that mathematical knowledge always has to be regarded as being “situated” within a social context. This means that mathematical knowledge only gains its specific epistemological meaning within a social context and that the development and understanding of mathematical knowledge is strongly influenced by the social context. On the other hand the specificity of the theory-practice-problem poses an essential demand on the scientific work in mathematics education.     

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.     

Proofs as bearers of mathematical knowledge

Yehuda Rav’s inspiring paper “Why do we prove theorems?” published in Philosophia Mathematica (1999, 7, pp. 5–41) has interesting implications for mathematics education. We examine Rav’s central ideas on proof—that proofs convey important elements of mathematics such as strategies and methods, that it is “proofs rather than theorems that are the bearers of mathematical knowledge”and thus that proofs should be the primary focus of mathematical interest—and then discuss their significance for mathematics education in general and for the teaching of proof in particular.     

Mathematics for All: Perspectives and Promising Practices

The Core-Plus Mathematics Project (CPMP, 1995) is one of four comprehensive curriculum development projects that, in 1992, were awarded 5-year grants by the National Science Foundation to design, evaluate, and disseminate innovative high school curricula that interpret and implement the recommendations of the National Council of Teachers of Mathematics (NCTM) Curriculum and Evaluation Standards for School Mathematics ( NCTM, 1989 ) and Professional Standards for Teaching Mathematics ( NCTM, 1991 ). This article describes CPMP perspectives on a new curriculum organization for high school mathematics, identifies implications of these perspectives for promoting access and equity for all students, and reports some of the supporting oral data from an ongoing formative evaluation of the curriculum.     

From the Written to the Enacted Curricula: The Intermediary Role of Middle School Mathematics Teachers in Shaping Students' Opportunity to Learn

In this paper is reported the extent of textbook use by 39 middle school mathematics teachers in six states, 17 utilizing a textbook series developed with funding from the National Science Foundation (NSF‐funded) and 22 using textbooks developed by commercial publishers (publisher‐generated). Results indicate that both sets of teachers placed significantly higher emphasis on Number and Operation, often at the expense of other content strands. Location of topics within a textbook represented an oversimplified explanation of what mathematics gets taught or omitted. Most teachers using an NSF‐funded curriculum taught content intended for students in a different (lower) grade, and both sets of teachers supplemented with skill‐building and “practice” worksheets. Implications for documenting teachers' “fidelity of implementation” ( National Research Council, 2004 ) are offered.     

'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/.