The role of the graphic calculator in mediating graphing activity

The PIGMI (Portable Information Technologies for supporting Graphical Mathematics Investigations) Project ¹ investigated the role of portable technologies in facilitating development of students' graphing skills and concepts. This paper examines the impact of a recent shift towards calculating and computing tools as increasingly accessible, everyday technologies on the nature of learning in a traditionally difficult curriculum area. The paper focuses on the use of graphic calculators by undergraduates taking an innovative new mathematics course at the Open University. A questionnaire survey of both students and tutors was employed to investigate perceptions of the graphic calculator and the features which facilitate graphing and linking between representations. Key features included visualization of functions, immediate feedback and rapid graph plotting. A follow-up observational case study of a pair of students illustrated how the calculator can shape mathematical activity, serving a catalytic, facilitating and checking role. The features of technology-based activities which can structure and support collaborative problem solving were also examined. In sum, the graphic calculator technology acted as a critical mediator in both the students' collaboration and in their problem solving. The pedagogic implications of using portables are considered, including the tension between using and over-using portables to support mathematical activity.     

Positive undecidable numberings in the Ershov hierarchy

A sufficient condition is given under which an infinite computable family of Σ^-1 _a -sets has computable positive but undecidable numberings, where a is a notation for a nonzero computable ordinal. This extends a theorem proved for finite levels of the Ershov hierarchy in [1]. As a consequence, it is stated that the family of all Σ^-1 _a -sets has a computable positive undecidable numbering. In addition, for every ordinal notation a > 1, an infinite family of Σ^-1 _a -sets is constructed which possesses a computable positive numbering but has no computable Friedberg numberings. This answers the question of whether such families exist at any—finite or infinite—level of the Ershov hierarchy, which was originally raised by Badaev and Goncharov only for the finite levels bigger than 1.     

Exploring Mathematics in Imaginative Places: Rethinking What Counts as Meaningful Contexts for Learning Mathematics

This paper explores what happens when students engage with mathematical tasks that make no attempt to be connected with students' everyday life experiences. The investigation draws on the work of educators who call for a broader view of what might count as real and relevant contexts for studying mathematics. It investigates students' experiences with two imaginative tasks and reports on the students' intellectual and emotional engagement. This engagement is examined and described in terms of the character and quality of the class and group discussions generated. Findings suggest that students can indeed engage productively with mathematics when it is explored in imaginative settings and that such contexts can help students support and sustain their engagement with the mathematics in the task.     

Reducing abstraction: The case of constructing an operation table for a group

This paper describes students' mental processes while constructing an operation table for a group. More specifically, undergraduate students' approaches are analyzed as the students fill in an operation table for four elements—a, b, c, and d—in such a way that it represents a group of order four. The data are analyzed from the perspective of reducing abstraction, which aims to explain students' conceptions of abstract algebra concepts. From this perspective, most students' responses and conceptions can be attributed to their tendency to work on a lower level of abstraction than the level on which concepts are introduced in class.     

Exploring the state of science stereotypes: Systematic review and meta-analysis of the Draw-A-Scientist Checklist

This systematic review and meta-analysis examined recent articles that have used the 1995 Draw-A-Scientist Checklist (DAST-C). This study was focused on the current state of students' stereotypes of scientists and the appropriateness of the DAST-C as a tool to assess these perceptions. Articles included in the review were published between 2003 and 2018, resulting in n = 30 studies. Mean results across studies are presented to describe current stereotypes of scientists, and the current format of the DAST-C is evaluated. Findings suggest that students' perceptions of scientists have largely remained consistent across time: scientists are still perceived as Caucasian, middle-aged or elderly males who wear lab coats and work indoors. However, while the DAST-C is a generally appropriate measure to assess students' perceptions of scientists, recommended revisions to the DAST-C could assist in capturing more modern scientist stereotypes and culturally bound perceptions of scientists.     

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.     

Efficacy of Different Concrete Models for Teaching the Part-Whole Construct for Fractions

The effectiveness of different concrete and pictorial models on students' understanding of the part-whole construct for fractions was investigated. Using interview data from fourth and fifth grade students from three different districts that adopted the Mathematics Trailblazers series, authors identified strengths and limitations of models used. Pattern blocks had limited value in aiding students' construction of mental images for the part-whole model as well as limited value in building meaning for adding and subtracting fractions. A paper fraction chart based on a paper folding model supported students' ability to order fractions with same numerators but was less useful in helping students on estimation tasks. The dot paper model and chips did not support fifth grade students' initial understanding of the algorithm.     

High School Students' Goals for Working Together in Mathematics Class: Mediating the Practical Rationality of Studenting

In this article I explore high school students' perspectives on working together in a mathematics class in which they spent a significant amount of time solving problems in small groups. The data included viewing session interviews with eight students in the class, where each student watched video clips of their own participation, explaining and justifying their behaviors. Analysis of data involved an investigation of students' goals for working together, which were found to vary along multiple dimensions. The dimensions that emerged from these data were mathematical versus nonmathematical goals, individual versus group goals, and personal versus normative goals. I present cases of four individual students to illustrate these dimensions. Such goals are important for illuminating how students' practical rationality is mediated by their personal goals for working together; additionally, these goal dimensions can be used as tools for considering challenges involved with using small group collaboration in high school classes where students' goals may be diverse.     

Logical form,mathematical practice,and Frege's Begriffsschrift

For over a century we have been reading Frege's Begriffsschrift notation as a variant of standard notation. But Frege's notation can also be read differently, in a way enabling us to understand how reasoning in Begriffsschrift is at once continuous with and a significant advance beyond earlier mathematical practices of reasoning within systems of signs. It is this second reading that I outline here, beginning with two preliminary claims. First, I show that one does not reason in specially devised systems of signs of mathematics as one reasons in natural language; the signs are not abbreviations of words. Then I argue that even given a system of signs within which to reason in mathematics, there are two ways one can read expressions involving those signs, either mathematically or mechanically. These two lessons are then applied to a reading of Frege's proof of Theorem 133 in Part III of his 1879 logic, a proof that Frege claims is at once strictly deductive and ampliative, a real extension of our knowledge. In closing, I clarify what this might mean, and how it might be possible.     

A Modeling Perspective on the Teaching and Learning of Mathematical Problem Solving

This study analyzed the processes used by students when engaged in modeling activities and examined how students' abilities to solve modeling problems changed over time. Two student populations, one experimental and one control group, participated in the study. To examine students' modeling processes, the experimental group participated in an intervention program consisting of a sequence of six modeling activities. To examine students' modeling abilities, the experimental and control groups completed a modeling abilities test on three occasions. Results showed that students' models improved as they worked through the sequence of problem activities and also revealed a number of factors, such as students' grade, experiences with modeling activities, and modeling abilities that influenced their modeling processes. The study proposes a three-dimensional theoretical model for examining students' modeling behavior, with ubsequent implications for the teaching and learning of mathematical problem solving.     

Diagrams and math notation in e-learning: growing pains of a new generation

Current e-learning environments are ill-suited to college mathematics. Instructors/students struggle to post diagrams and math notation. A new generation of math-friendly e-learning tools, including WebEQ, bundled with Blackboard 6, and NetTutor's Whiteboard, address these problems. This paper compares these two systems using criteria for ideal math-friendly e-learning systems. NetTutor's Whiteboard is, apparently, the only system allowing two-way communication of both diagrams and math notation between instructor and students. This paper also summarizes a case study of two community college mathematics courses (calculus and algebra) using NetTutor over two semesters. Pilot studies, interviews and experimental problems revealed that NetTutor's Whiteboard is effective for 2-way communication of diagrams and math notation in college courses. Learning difficult concepts was comparable to face-to-face courses.     

A Partnership Approach to Improving Student Attitudes About Sharks and Scientists

This article describes the methods and impact of a student–teacher–scientist research partnership on student attitudes. The partnership objective was to teach students about the diverse roles of sharks in the marine environment while personally connecting students with scientific study. Students (N = 229) participated in lessons about shark biology and helped the partnering scientist design experimental protocols and analyze data. A self‐selected subset of students also volunteered (n = 82) for a field component working with live hammerhead sharks (Sphyrna lewinii). Student surveys before and after the partnership suggested that negative attitudes about sharks are due largely to lack of exposure, and direct attention to students' stereotypes about sharks resulted in significant attitude improvement. Change in students' attitudes toward scientists, however, was minimal. Students' negative views of scientists did decline significantly, but their overall views of scientists were relatively positive to begin with. Also of interest was the students' unremitting association of scientists with specialized equipment and the students' lack of personal connection to scientific ways of examining the world, suggesting that partnerships may be more effective at personally connecting students with scientific process if they explicitly incorporate activities designed to improve students' view of themselves as scientists.     

New directions in differential equations: A framework for interpreting students' understandings and difficulties

The purpose of this paper is to offer a framework for interpreting students' understandings of and difficulties with mathematical ideas central to new directions in differential equations. These new directions seek to guide students into a more interpretive mode of thinking and to enhance their ability to graphically and numerically analyze differential equations. The framework reported here is the result of investigating in depth six students' understandings through a series of task-based individual interviews and classroom observations. The two major themes of the framework, the function-as-solution dilemma theme and students' intuitions and images theme, extend previous research on student cognition at the secondary and collegiate level to the domain of differential equations and reflect the increased recognition of situating analyses of student learning within students' learning environment. For new areas of interest such as differential equations, mapping out students' understandings of important mathematical ideas can be an important part of curricular and instructional design that seeks to refine and build on students' ways of thinking.     

Testing significance of a mean vector—A possible alternative to Hotelling'sT 2

In problems involving multivariate measurements experimental considerations often indicate grouping of variables into subsets ordered according to their importance. In such situations, the problems such as comparison of two mean vectors and profile analysis may be treated by Hotelling'sT ²-test adapted along the lines of the step-wise procedure of J. Roy [10], or the well known test for additional information due to Rao [9]. In this paper we study a modification of the step-wise procedure obtained by combining the component tests. The exact Bahadur slopes of resulting procedures are computed and it is shown that the procedure based upon Fisher's combination method is asymptotically equivalent to Hotelling'sT ². A Monte Carlo study suggests that even in small samples the power functions of the new method and Hotelling'sT ²-test are practically equivalent. Research sponsored by the Air Force Office of Scientific Research, Air Force Systems Command, USAF under Grant No. AFOSR-77-3360. The United States Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation hereon.     

Are You Convinced? Middle‐Grade Students' Evaluations of Mathematical Arguments

Students learn norms of proving by observing teachers generating proofs, engaging in proving, and generalizing features of proofs deemed convincing by an authority, such as a textbook. Students at all grade levels have difficulties generating valid proof; however, little research exists on students' understandings about what makes a mathematical argument convincing prior to more formal instruction in methods of proof. This study investigated middle‐school students' (ages 12–14) evaluations of arguments for a statement in number theory. Students evaluated both an empirical and a general argument in an interview setting. The results show that students tend to prefer empirical arguments because examples enhance an argument's power to show that the statement is true. However, interview responses also reveal that a significant number of students find arguments to be most convincing when examples are supported with an explanation that “tells why” the statement is true. The analysis also examined the alignment of students' reasons for choosing arguments as more convincing along with the strategies they employ to make arguments more convincing. Overall, the findings show middle‐school students' conceptions about what makes arguments convincing are more sophisticated than their performance in generating arguments suggests.     

Challenging assumptions of notational transparency: the case of vectors in engineering mathematics

ABSTRACT

The notation for vector analysis has a contentious nineteenth century history, with many different notations describing the same or similar concepts competing for use. While the twentieth century has seen a great deal of unification in vector analysis notation, variation still remains. In this paper, the two primary notations used for expressing the components of a vector are discussed in historical and current context. Popular mathematical texts use the two notations as if they are transparent and interchangeable. In this research project, engineering students’ proficiency at vector analysis was assessed and the data were analyzed using the Rasch measurement method. Results indicate that the students found items expressed in unit vector notation more difficult than those expressed in parenthesis notation. The expert experience of notation as transparent and unproblematically symbolic of underlying processes independent of notation is shown to contrast with the student experience where the less familiar notation is experienced as harder to work with.     

Taiwanese Gifted Students' Views of Nature of Science

This study examined the conceptions of nature of science (NOS) possessed by a group of gifted seventh‐grade students from Taiwan. The students were engaged in a 1‐week science camp with emphasis on scientific inquiry and NOS. A Chinese version of a NOS questionnaire was developed, specifically addressing the context of Chinese culture, to assess students' views on the development of scientific knowledge. Pretest results indicated that the majority of participants had a basic understanding of the tentative, subjective, empirical, and socially and culturally embedded aspects of NOS. Some conflicting views and misconceptions held by the participants are discussed. There were no significant changes in students' views of NOS after instruction, possibly due to time limitations and a ceiling effect. The relationship between students' cultural values and development of NOS conceptions and the impact of NOS knowledge on students' science learning are worth further investigation.     

Teachers' ways of listening and responding to students' emerging mathematical models

In this paper, I present the results of a case study of the practices of four experienced secondary teachers as they engaged their students in the initial development of mathematical models for exponential growth. The study focuses on two related aspects of their practices: (a) when, how and to what extent they saw and interpreted students' ways of thinking about exponential functions and (b) how they responded to the students' thinking in their classroom practice. Through an analysis of the teachers' actions in the classroom, I describe the teachers' developing knowledge when using modeling tasks with secondary students. The analysis suggests that there is considerable variation in the approaches that teachers take in listening to and responding to students' emerging mathematical models. Having a well-developed schema for how students might approach the task enabled one teacher to press students to express, evaluate, and revise their emerging models of exponential growth. Implications for the knowledge needed to teach mathematics through modeling are discussed.     

NAMING A COLUMN ON A SPREADSHEET

Spreadsheets use a meaningful algebra-like notation which, research suggests, can support pupils in developing an understanding of variables. This paper discusses the activity of Year 8 pupils who were taught to name a column on a spreadsheet, and who were asked to reflect upon their activity in a stimulated recall interview. More specifically, it considers the pupils' understanding of notation, such as 'A2' and 'm', which they used when constructing spreadsheet formulae. It is suggested that experience of naming columns may help pupils to develop a clearer sense of the notation as a variable, and to make links between their spreadsheet activity and use of standard algebraic notation [1].