Peirce the logician

This paper traces the influence of the Boolean school, and more specifically of Peirce and his students, on the development of modern logic. In the 1890s it was Schröder's Algebra derLogik that represented the state of the art. This work mentions Frege, but the quantifier notation it adopts (a variant of the modern notation) is credited to Peirce and his students O. H. Mitchell and Christine Ladd-Franklin. This notation was widely adopted; both Zermelo and Löwenheim wrote famous papers in Peirce-Schröder notation. Even Whitehead (in 1908, in his Universal Algebra) fails to mention Frege, but cites the “suggestive papers” by Mitchell and Ladd-Franklin. (Russell credits Frege, with many things, but nowhere credits him with the quantifer; if the quantifiers in Principia were devised by Whitehead, they probably come from Peirce). The aim of this paper is not to detract from our appreciation of Frege's great work, but to emphasize that its influence came largely after 1900 (after Russell pointed out its significance). Although Frege discovered the quantifier in 1879 and Peirce's student Mitchell independently discovered it only in 1883, it was Mitchell's discovery (as modified and disseminated by Peirce) that made the quantifier part of logic. And neither Löwenheim's theorem nor Zermelo set-theory depended on Frege's work at all, but only on the work of the Boole-Peirce school.     

Revisiting decimal misconceptions from a new perspective: The significance of whole number bias in the Chinese culture

The aim of this study was to investigate Hong Kong Grade 4 students’ understanding of the decimal notation system including their knowledge of decimal quantities. This is a unique study because most previous studies were conducted in Western cultural settings; therefore we were interested to see whether Chinese students have the same kinds of misconceptions as Western students given the Chinese number naming system is relatively transparent and explicit. Three hundred and forty-one students participated in a written test on decimal numbers. Thirty-two students were interviewed to further explore their mathematical reasoning. In summary, the results indicated that many students had mastered reasonable knowledge of decimal notation and quantities, which may be attributed to the Chinese linguistic clarity of decimal numbers. More importantly, the results showed that some students’ construction of decimal concepts have been adversely affected by persistent misconceptions arising from whole number bias. Two kinds of whole number misconceptions, namely “-ths suffix error” and “reversed place value progression error”, were revealed in this study. This paper suggests that a framework theory approach to conceptual change may be an alternative approach to addressing students’ learning difficulties in decimals.     

Inequalities,assessment and computer algebra

The goal of this paper is to examine single variable real inequalities that arise as tutorial problems and to examine the extent to which current computer algebra systems (CAS) can (1) automatically solve such problems and (2) determine whether students’ own answers to such problems are correct. We review how inequalities arise in contemporary curricula. We consider the formal mathematical processes by which such inequalities are solved, and we consider the notation and syntax through which solutions are expressed. We review the extent to which current CAS can accurately solve these inequalities, and the form given to the solutions by the designers of this software. Finally, we discuss the functionality needed to deal with students’ answers, i.e. to establish equivalence (or otherwise) of expressions representing unions of intervals. We find that while contemporary CAS accurately solve inequalities there is a wide variety of notation used.     

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.     

Polynomial Approximation of Functions: Historical Perspective and New Tools

This paper examines the effect of applying symbolic computation and graphics to enhance students' ability to move from a visual interpretation of mathematical concepts to formal reasoning. The mathematics topics involved, Approximation and Interpolation, were taught according to their historical development, and the students tried to follow the thinking process of the creators of the theory. They used a Computer Algebra System to manipulate algebraic expressions and generate a wide variety of dynamic graphics; thus21st century technology was applied in order to “walk” with the students from the period of Euler in 1748 to the period of Runge in 1901. We describe some situations in which the combination of dynamic graphics,algorithms, and historical perspective enabled the students to improve their understanding of concepts such as limit, convergence, and the quality of approximation. This revised version was published online in July 2006 with corrections to the Cover Date.     

Explication as a lens for the formalization of mathematical theory through guided reinvention

Realistic Mathematics Education supports students’ formalization of their mathematical activity through guided reinvention. To operationalize “formalization” in a proof-oriented instructional context, I adapt Sjogren's (2010) claim that formal proof explicates (Carnap, 1950) informal proof. Explication means replacing unscientific or informal concepts with scientific ones. I use Carnap's criteria for successful explication – similarity, exactness, and fruitfulness – to demonstrate how the elements of mathematical theory – definitions, axioms, theorems, proofs – can each explicate their less formal correlates. This lens supports an express goal of the instructional project, which is to help students coordinate semantic (informal) and syntactic (formal) mathematical activity. I demonstrate the analytical value of the explication lens by applying it to examples of students’ mathematical activity drawn from a design experiment in undergraduate, neutral axiomatic geometry. I analyze the chains of meanings (Thompson, 2013) that emerged when formal elements were presented readymade alongside those emerging from guided reinvention.     

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.     

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.     

Helping Students Come to Grips With the Meaning of Division

Many years ago, Arons pointed out the incomprehension science students exhibit of the basic mathematical operations multiplication and division and the need to address the problem in physics classes to assure student understanding of the physical world. McDermott et al.'s Physics by Inquiry program does address this need directly and in detail (by defining two meanings for division). However, in the author's classes many students had relatively low scores (ranging from 60–80%) when trying to explain simple operations. Reported in this paper are ways to supplement the text that force students to address the actual meaning of division by stressing the relation between a “whole” and a “package,” and connect that meaning with previously learned operational definitions for area and volume.     

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.     

The interplay of empirical and deductive reasoning in proving “if” and “only if” statements in a Dynamic Geometry environment

This paper is situated within the ongoing enterprise to understand the interplay of students’ empirical and deductive reasoning while using Dynamic Geometry (DG) software. Our focus is on the relationships between students’ reasoning and their ways of constructing DG drawings in connection to directionality (i.e., “if” and “only if” directions) of geometry statements. We present a case study of a middle-school student engaged in discovering and justifying “if” and “only if” statements in the context of quadrilaterals. The activity took place in an online asynchronous forum supported by GeoGebra. We found that student's reasoning was associated with the logical structure of the statement. Particularly, the student deductively proved the “if” claims, but stayed on empirical grounds when exploring the “only if” claims. We explain, in terms of a hierarchy of dependencies and DG invariants, how the construction of DG drawings supported the exploration and deductive proof of the “if” claims but not of the “only if” claims.     

A learning trajectory in Kindergarten and first grade students’ thinking of variable and use of variable notation to represent indeterminate quantities

Given the importance of understanding and using indeterminate quantities in algebraic thinking, the development of learning trajectories about how Kindergarten and first grade students understand variable and use variable notation in the context of algebraic expressions is critical. Based on an empirically developed learning trajectory, we analyzed children’s responses at three different points in a classroom teaching experiment. Our purpose was to describe levels of thinking among 16 students (eight in Kindergarten and eight in first grade). Our results revealed qualitative changes in the thinking about indeterminate quantities of most student participants. As students progressed through the experiment, we found that they advanced from what we characterized as a “Pre-variable” Level to a “Letters as representing indeterminate quantities as varying unknowns; explicit operations on indeterminate quantities” Level. Learning trajectories such as that developed here hold promise for informing the design of interventions that support young children’s early algebraic thinking.     

A canonical representation of trivalent hamiltonian graphs

A canonical representation of trivalent hamiltonian graphs in the form of “span lists” had been proposed by J. Lederberg. It is here presented in a modified form due to H. S. M. Coxeter and the author, and therefore called “LCF notation.” This notation has the advantage of being more concise than Lederberg's original span lists whenever the graph has a hamiltonian circuit with rotational symmetry. It is also useful as a method for a systematic classification of trivalent hamiltonian graphs and allows one to define for such graphs two interesting properties, called, respectively, “antipalindromic” and “quasiantipalindromic.”.     

Mathematical modelling in engineering: an alternative way to teach Linear Algebra

Technological advances require that basic science courses for engineering, including Linear Algebra, emphasize the development of mathematical strengths associated with modelling and interpretation of results, which are not limited only to calculus abilities. Based on this consideration, we have proposed a project-based learning, giving a dynamic classroom approach in which students modelled real-world problems and turn gain a deeper knowledge of the Linear Algebra subject. Considering that most students are digital natives, we use the e-portfolio as a tool of communication between students and teachers, besides being a good place making the work visible. In this article, we present an overview of the design and implementation of a project-based learning for a Linear Algebra course taught during the 2014–2015 at the ‘ETSEIB'of Universitat Politècnica de Catalunya (UPC).     

Sounds and pictures: dynamism and dualism in Dynamic Geometry

In this paper, we examine and evaluate several new mathematical representations developed for The Geometer’s Sketchpad v5 (GSP5) from the perspective of their dynamic mathematical and pedagogic utility or expressibility. We claim the primary contributions of Dynamic Geometry’s principle of dynamism to the emerging concept of “Dynamic Mathematics” to be twofold: first, the powerful, temporalized representation of continuity and continuous change (dynamism’s mathematical aspect), and second, the sensory immediacy of direct interaction with mathematical representations (dynamism’s pedagogic aspect). Seen from this perspective, the growth of “Dynamic Mathematics software,” beyond the initial conception of first-generation planar geometry systems, represents a tremendous diversification and expansion of the mathematical domain of the dynamic principle’s applicability (for example, to dynamic statistics, graphing and 3D geometry). But at the same time, this expansion has come at the cost of a decrease in the immediacy of sensory interactions with mathematical representations, as in so-called dynamic graphing, wherein users modify a graph “at a distance” (through slider-based manipulation of the coefficients of its symbolic equation), or in solid geometry tools, in which users’ interactions with represented solids are mediated and distanced by the inevitably-2D communication interfaces of the computer mouse and screen. Thus we focus on this second aspect–sensory interaction with mathematical representations—in evaluating how novel dynamic representations in GSP5 affect mathematical modeling opportunities, student activity and engagement.     

Normalising geometrical figures: dynamic manipulation and construction of meanings for ratio and proportion

Enlarging-shrinking geometrical figures by 13 year-olds is studied during the implementation of proportional geometric tasks in the classroom. Students worked in groups of two using ‘Turtleworlds’, a piece of geometrical construction software which combines symbolic notation, through a programming language, with dynamic manipulation of geometrical objects by dragging on sliders representing variable values. In this paper we study the students’ normalising activity, as they use this kind of dynamic manipulation to modify ‘buggy’ geometrical figures while developing meanings for ratio and proportion. We describe students’ normative actions in terms of four distinct Dynamic Manipulation Schemes (Reconnaissance, Correlation, Testing, Verification). We discuss the potential of dragging for mathematical insight in this particular computational environment, as well as the purposeful nature of the task which sets up possibilities for students to appreciate the utility of proportional relationships.     

The role of symbols in mathematical communication: the case of the limit notation

Symbols play crucial roles in advanced mathematical thinking by providing flexibility and reducing cognitive load but they often have a dual nature since they can signify both processes and objects of mathematics. The limit notation reflects such duality and presents challenges for students. This study uses a discursive approach to explore how one instructor and his students think about the limit notation. The findings indicate that the instructor flexibly differentiated between the process and product aspects of limit when using the limit notation. Yet, the distinction remained implicit for the students, who mainly realised limit as a process when using the limit notation. The results of the study suggest that it is important for teachers to unpack the meanings inherent in symbols to enhance mathematical communication in the classrooms.     

Unpacking learner’s growth in geometric understanding when solving problems in a dynamic geometry environment: Coordinating two frames

The emergence of dynamic geometry environments challenges researchers in mathematics education to develop theories that capture learner’s growth in geometric understanding in this particular environment. This study coordinated the Pirie-Kieren theory and instrumental genesis to examine learner’s growth in geometric understanding when solving problems in a dynamic geometry environment. Data analysis suggested that coordinating the two theoretical approaches provided a productive means to capture the dynamic interaction between the growth in mathematical understanding and the formation/application of utilization scheme during a learner’s mathematical exploration with dynamic geometry software. The analysis of one episode on inscribing a square in a triangle was shared to illustrate this approach. This study contributes to the continuing conversation of “networking theories” in the mathematics education research community. By networking the two theoretical approaches, this paper presents a model for studying learner's growth in mathematical understanding in a dynamic learning environment while accounting for interaction with digital tools.     

Graphic calculators and connectivity software to be a community of mathematics practitioners

In a teaching experiment carried out at the secondary school level, we observe the students’ processes in modelling activities, where the use of graphic calculators and connectivity software gives a common working space in the class. The study shows results in continuity with others emerged in the previous ICMEs and some new ones, and offers an analysis of the novelty of the software in introducing new ways to support learning communities in the construction of mathematical meanings. The study is conducted in a semiotic-cultural framework that considers the introduction and the evolution of signs, such as words, gestures and interaction with technologies, to understand how students construct mathematical meanings, working as a community of practice. The novelty of the results consists in the presence of two technologies for students: the “private” graphic calculators and the “public” screen of the connectivity software. Signs for the construction of knowledge are mediated by both of them, but the second does it in a social way, strongly supporting the work of the learning community.