An investigation into whether student use of graphics calculators matches their teacher's expectations

This research examines students’ use of graphics calculators and investigates the extent to which the students’ use meets their teachers aim when using graphics calculators in the classroom. The teacher's use of her graphics calculator was analysed over a week using Key Record software. The teacher was questioned about her aims and expectations for the students when using a graphics calculator. As a result an interview schedule for students was constructed in order to determine whether the teacher's aims had been met. It was found that in general all of the teachers’ aims were met to some extent by most of the students.     

New challenges in the 2011 revised middle school curriculum of South Korea: mathematical process and mathematical attitude

The ‘future-oriented middle school mathematics curriculum focused on creativity and personality’ was revised in August of 2011 with the aim of nurturing students’ mathematical creativity and sound personalities. The curriculum emphasizes: contextual learning from which students can grasp mathematical concepts and make connections with their everyday lives; manipulation activities through which students may attain an intuitive idea of what they are learning and enhance their creativity; and reasoning to justify mathematical results based on their knowledge and experience. Since students will not be able to engage in the intended mathematical process with the study-load imposed by the current curriculum, the newly revised curriculum modifies or deletes some parts of the contents that have been traditionally taught mechanically. This paper provides a detailed overview of the main points of the revised curriculum.     

On the use of history of mathematics: an introduction to Galileo's study of free fall motion

In this paper, we report on an experimental activity for discussing the concepts of speed, instantaneous speed and acceleration, generally introduced in first year university courses of calculus or physics. Rather than developing the ideas of calculus and using them to explain these basic concepts for the study of motion, we led 82 first year university students through Galileo's experiments designed to investigate the motion of falling bodies, and his geometrical explanation of his results, via simple dynamic geometric applets designed with GeoGebra. Our goal was to enhance the students’ development of mathematical thinking. Through a scholarship of teaching and learning study design, we captured data from students before, during and after the activity. Findings suggest that the historical development presented to the students helped to show the growth and evolution of the ideas and made visible authentic ways of thinking mathematically. Importantly, the activity prompted students to question and rethink what they knew about speed and acceleration, and also to appreciate the novel concepts of instantaneous speed and acceleration at which Galileo arrived.     

Objectifying the adjacent and opposite angles: a cultural historical analysis

The angle topic is central to the development of geometric knowledge. Two of the basic concepts associated with this topic are the adjacent and opposite angles. It is the goal of the present study to analyze, based on the cultural historical semiotics framework, how high-achieving seventh grade students objectify the adjacent and opposite angles’ concepts. We videoed the learning of a group of three high-achieving students who used technology, specifically GeoGebra, to explore geometric relations related to the adjacent and opposite angles’ concepts. To analyze students’ objectification of these concepts, we used the categories of objectification of knowledge (attention and awareness) and the categories of generalization (factual, contextual and symbolic), developed by Radford. The research results indicate that teacher's and students’ verbal and visual signs, together with the software dynamic tools, mediated the students’ objectification of the adjacent and opposite angles’ concepts. Specifically, eye and gestures perceiving were part of the semiosis cycles in which the participating students were engaged and which related to the mathematical signs that signified the adjacent and the opposite angles. Moreover, the teacher's suggestions/requests/questions included/suggested semiotic signs/tools, including verbal signs that helped the students pay attention, be aware of and objectify the adjacent and opposite angles’ concepts.     

Teachers’ views on dynamically linked multiple representations, pedagogical practices and students’ understanding of mathematics using TI-Nspire in Scottish secondary schools

Do teachers find that the use of dynamically linked multiple representations enhances their students’ relational understanding of the mathematics involved in their lessons and what evidence do they provide to support their findings? Throughout session 2008–2009, this empirical research project involved six Scottish secondary schools, two mathematics teachers from each school and students from different ages and stages. Teachers used TI-Nspire PC software and students the TI-Nspire handheld technology. This technology is specifically designed to allow dynamically linked multiple representations of mathematical concepts such that pupils can observe links between cause and effect in different representations such as dynamic geometry, graphs, lists and spreadsheets. The teachers were convinced that the use of multiple representations of mathematical concepts enhanced their students’ relational understanding of these concepts, provided evidence to support their argument and described changes in their classroom pedagogy.     

SVG+Ajax+R: a new framework for WebGIS

This paper first proposes a method of establishing a Web-based system that can visualize statistical data that are accompanied by geographical information and analyze it interactively using dynamic graphics. In terms of the graphics format, our proposed system uses XML-based 2D vector graphics, known as Scalable Vector Graphics (SVG). To install an enhanced interactive function, we adopted a technique of server–client asynchronous communication using JavaScript, called Asynchronous JavaScript and XML (Ajax) and R, which perform statistical analysis on the server side. This enables Web developers to construct a lightweight system including statistical computing rapidly. Furthermore, many users get possible to utilize such data effectively and efficiently anywhere anytime. The latter half of this paper introduces the WebGIS realized by this framework. We then discuss the possibility and advantages of applying this new method to the dynamic graphics proposed previously.     

An Exploratory Study of College Students' Views of Mathematical Thinking in a Historical Approach Calculus Course

Heuristic training alone is not enough for developing one's mathematical thinking. One missing component is a mathematical point of view. This study reports findings regarding outcomes of a historical approach calculus course to foster Taiwanese college students' views of mathematical thinking. This study consisted of 3 stages. During the initial phase, 44 engineering majors' views on mathematical thinking were tabulated by an open-ended questionnaire, and 9 randomly selected students were invited to participate in follow-up interviews. Students then received an 18-week historical approach calculus course in which mathematical concepts were problematized to challenge their intuition-based empirical beliefs about doing mathematics. Near the end of the semester, all participants answered the identical questionnaire, and we interviewed the same students to pinpoint any shifts in their views on mathematical thinking. We found that participants were more likely to value logical sense, creativity, and imagination in doing mathematics. Further, students were leaning toward a conservative attitude toward certainty of mathematical knowledge. Participants' focus seemingly shifted from mathematics as a product to mathematics as a process.     

An Exploratory Study of College Students' Views of Mathematical Thinking in a Historical Approach Calculus Course

Heuristic training alone is not enough for developing one's mathematical thinking. One missing component is a mathematical point of view. This study reports findings regarding outcomes of a historical approach calculus course to foster Taiwanese college students' views of mathematical thinking. This study consisted of 3 stages. During the initial phase, 44 engineering majors' views on mathematical thinking were tabulated by an open-ended questionnaire, and 9 randomly selected students were invited to participate in follow-up interviews. Students then received an 18-week historical approach calculus course in which mathematical concepts were problematized to challenge their intuition-based empirical beliefs about doing mathematics. Near the end of the semester, all participants answered the identical questionnaire, and we interviewed the same students to pinpoint any shifts in their views on mathematical thinking. We found that participants were more likely to value logical sense, creativity, and imagination in doing mathematics. Further, students were leaning toward a conservative attitude toward certainty of mathematical knowledge. Participants' focus seemingly shifted from mathematics as a product to mathematics as a process.     

Where do Fractions Encounter their Equivalents? – Can this Encounter Take Place in Elementary-School?

The concept of equivalence class plays a significant role in the structure of Rational Numbers. Piaget taught that in order to help elementary school children develop mathematical concepts, concrete objects and concrete reflection-enhancing-activities are needed. The “Shemesh” software was specially designed for learning equivalence-classes of fractions. The software offers concrete representations of such classes, as well as activities which cannot be constructed without a computer. In a discrete Cartesian system students construct points on the grid and learn to identify each such point as a fraction-numeral (a denominator-numerator pair). The children then learn to construct sets of such points, all of which are located on a line through the origin point. They learn to identify the line with the set of its constituent equivalent fractions. Subsequently, they investigate other phenomena and constructions in such systems, developing these constructions into additional fraction concepts. These concrete constructions can be used in solving traditional fraction problems as well as in broadening the scope of fraction meaning. Fifth-graders who used “Shemesh” in their learning activities were clinically interviewed several months after the learning sessions ended. These interviews revealed evidence indicating initial actual development of the desired mathematical concepts. This revised version was published online in July 2006 with corrections to the Cover Date.     

Computer Use and Visualization in Students' Construction of Reflection and Rotation Concepts

The effects of a dynamic instructional environment (based on use of The Geometer's Sketchpad, N. Jackiw, 1991 , in a computer lab) and visualization on eighth-grade students' (N= 241) construction of the concepts of reflection and rotation were investigated. Also investigated were the effects of the environment on students' two- and three-dimensional visualization. After controlling for initial differences, it was concluded that students experiencing the dynamic environment significantly outperformed students experiencing a traditional environment on content measures of the concepts of reflection and rotation, as well as on measures of two-dimensional visualization. The students' environment did not significantly affect their three-dimensional visualization.     

An investigation of 6th graders’ solutions of Cartesian product problems and representation of these problems using arrays

Two hour-long interviews were conducted with each of 14 sixth-grade students. The purpose of the interviews was to investigate how students solved combinatorics problems, and represented their solutions as arrays. This paper reports on 11 of these students who represented a balanced mix of students operating with two of three multiplicative concepts that have been identified in prior research (Hackenberg, 2007, 2010; Hackenberg & Tillema, 2009). One finding of the study was that students operating with different multiplicative concepts established and structured pairs differently. A second finding is that these different ways of operating had implications for how students produced and used arrays. Overall, the findings contribute to models of students’ reasoning that outline the psychological operations that students use to constitute product of measures problems (Vergnaud, 1983). Product of measures problems are a kind of multiplicative problem that has unique mathematical properties, but researchers have not yet identified specific psychological operations that students use when solving these problems that differ from their solution of other kinds of multiplicative problems (cf. Battista, 2007).     

Improving Algebra Preparation: Implications From Research on Student Misconceptions and Difficulties

Through historical and contemporary research, educators have identified widespread misconceptions and difficulties faced by students in learning algebra. Many of these universal issues stem from content addressed long before students take their first algebra course. Yet elementary and middle school teachers may not understand how the subtleties of the arithmetic content they teach can dramatically, and sometimes negatively, impact their students' ability to transition to algebra. The purpose of this article is to bring awareness of some common algebra misconceptions, and suggestions on how they can be averted, to those who are teaching students the early mathematical concepts they will build upon when learning formal algebra. Published literature discussing misconceptions will be presented for four prerequisite concepts, related to symbolic representation: bracket usage, equality, operational symbols, and letter usage. Each section will conclude with research‐based practical applications and suggestions for preventing such misconceptions. The literature discussed in this article makes a case for elementary and middle school teachers to have a deeper and more flexible understanding of the mathematics they teach, so they can recognize how the structure of algebra can and should be exposed while teaching arithmetic.     

An Atom is Known by the Company it Keeps: A Constructionist Learning Environment for Materials Science Using Agent-Based Modeling

This article reports on “MaterialSim”, an undergraduate-level computational materials science set of constructionist activities which we have developed and tested in classrooms. We investigate: (a) the cognition of students engaging in scientific inquiry through interacting with simulations; (b) the effects of students programming simulations as opposed to only interacting with ready-made simulations; (c) the characteristics, advantages, and trajectories of scientific content knowledge that is articulated in epistemic forms and representational infrastructures unique to computational materials science, and (d) the principles which govern the design of computational agent-based learning environments in general and for materials science in particular. Data sources for the evaluation of these studies include classroom observations, interviews with students, videotaped sessions of model-building, questionnaires, and analysis of artifacts. Results suggest that by becoming ‘model builders,’ students develop deeper understanding of core concepts in materials science, and learn how to better identify unifying principles and behaviors within the content matter.     

Algorithmic contexts and learning potentiality: a case study of students’ understanding of calculus

The study explores the nature of students’ conceptual understanding of calculus. Twenty students of engineering were asked to reflect in writing on the meaning of the concepts of limit and integral. A sub-sample of four students was selected for subsequent interviews, which explored in detail the students’ understandings of the two concepts. Intentional analysis of the students’ written and oral accounts revealed that the students were expressing their understanding of limit and integral within an algorithmic context, in which the very ‘operations’ of these concepts were seen as crucial. The students also displayed great confidence in their ability to deal with these concepts. Implications for the development of a conceptual understanding of calculus are discussed, and it is argued that developing understanding within an algorithmic context can be seen as a stepping stone towards a more complete conceptual understanding of calculus.     

Developing Student Understanding: Contextualizing Calculus Concepts

This study looked at the practice of one high school teacher who provided students with concrete examples from their physics class to give them a contextually rich environment in which to explore the abstractions of calculus. Students discovered connections between the physics concepts of position, velocity, and acceleration and the calculus concepts of function, derivative, and antiderivative. The qualitative study sought to describe several critical aspects of understanding: students' ability to explain concepts and procedures, to apply concepts in a physics context, and to explore their own learning. It included 32 seniors at a large, urban, comprehensive, religious school in a midwestern stale. Samples of student work and reflections were collected by the teacher, as well as by students in individual portfolios. The teacher kept a reflective journal. This study suggests that making connections between calculus and physics can yield deep understandings of semantic as well as procedural knowledge.     

Dynamic mathematics and the blending of knowledge structures in the calculus

This paper considers the role of dynamic aspects of mathematics specifically focusing on the calculus, including computer software that responds to physical action to produce dynamic visual effects. The development builds from dynamic human embodiment, uses arithmetic calculations in computer software to calculate ‘good enough’ values of required quantities and algebraic manipulation to develop precise symbolic values. The approach is based on a developmental framework blending human embodiment, with the symbolism of arithmetic and algebra leading to the formalism of real numbers and limits. It builds from dynamic actions on embodied objects to see the effect of those actions as a new embodiment that needs to be calculated accurately and symbolised precisely. The framework relates the growth of meaning in history to the mental conceptions of today’s students, focusing on the relationship between potentially infinite processes and their consequent embodiment as mental concepts. It broadens the strategy of process-object encapsulation by blending embodiment and symbolism.     

The fractional knowledge and algebraic reasoning of students with the first multiplicative concept

To understand relationships between students’ quantitative reasoning with fractions and their algebraic reasoning, a clinical interview study was conducted with 18 middle and high school students. Six students with each of three different multiplicative concepts participated. This paper reports on the fractional knowledge and algebraic reasoning of six students with the most basic multiplicative concept. The fractional knowledge of these students was found to be consistent with prior research, in that the students had constructed partitioning and iteration operations but not disembedding operations, and that the students conceived of fractions as parts within wholes. The students’ iterating operations facilitated their work on algebra problems, but the lack of disembedding operations was a significant constraint in writing algebraic equations and expressions, as well as in generalizing relationships. Implications for teaching these students are discussed.     

Russell and His Sources for Non-Classical Logics

My purpose here is purely historical. It is not an attempt to resolve the question as to whether Russell did or did not countenance nonclassical logics, and if so, which nonclassical logics, and still less to demonstrate whether he himself contributed, in any manner, to the development of nonclassical logic. Rather, I want merely to explore and insofar as possible document, whether, and to what extent, if any, Russell interacted with the various, either the various candidates or their, ideas that Dejnožka and others have proposed as potentially influential in Russell’s intellectual reactions to nonclassical logic or to the philosophical concepts that might contribute to his reactions to nonclassical logics.     

Teaching and learning of inclusive and transitive properties among quadrilaterals by deductive reasoning with the aid of SmartBoard

Learning to identify Euclidean figures is an essential content of many elementary school geometry curricula. Students often learn to distinguish among quadrilaterals, for example, by categorizing their geometric properties according to two attributes, namely the length of the edges and the size of the interior angles. But knowing how to differentiate them based on their geometric properties does not necessarily help students to develop the abstract concepts of the inclusive and transitive properties among the quadrilaterals. With the aid of dynamic geometry multimedia software in SmartBoard (SB), a kind of digital whiteboard (DWB), we enhanced the teaching and learning effectiveness by the effect of “animation-on-demand” in classrooms. This is basically a dual delivery of geometric concepts by texts, narrations and words accompanied by pictures, illustrations and animations. The preliminary results of our study on 9-year-old students’ performance in tests given after three such lessons show that those students could differentiate with reasons why a square is a rhombus (inclusion) as well as a parallelogram (transitivity).     

A modularized tablet-based approach to preparation for remedial mathematics

Basic arithmetic forms the foundation of the math courses that students will face in their undergraduate careers. It is therefore crucial that students have a solid understanding of these fundamental concepts. At an open-access university offering both two-year and four-year degrees, incoming freshmen who were identified as lacking in basic arithmetic skills were engaged in an experimental technology-enhanced workshop designed to provide them with a deeper understanding of arithmetic prior to their initial remedial coursework. Customized online content was created specifically for this experiment, and the first implementation (n = 27) yielded statistically significant improvement, not only from pre-test to post-test, but also in the subsequent remedial course. This paper also analyses the accuracy of students’ self-assessment from pre-test to post-test, as well as student attitudes about this experimental approach.