Reducing Abstraction: The Case of School Mathematics

There is a growing interest in the mathematics education community in the notion of abstraction and its significance in the learning of mathematics. Reducing abstraction is a theoretical framework that examines learners behavior in terms of coping with abstraction level. It refers to situations in which learners are unable to manipulate concepts presented in a given problem; therefore, they unconsciously reduce the level of abstraction of the concepts involved to make these concepts mentally accessible. This framework has been used for explaining students conception in different areas of undergraduate mathematics and computer science. This article extends the applicability scope of this framework from undergraduate mathematics to school mathematics. We draw on recently published research articles and exemplify how students behavior can be described in terms of various interpretations of reducing abstraction level.     

Confidence is Everything” – Perspectives of Teachers and Students on Learning Mathematics

This paper problematises confidence as a frequently used explanation for performance in the mathematics classroom. I report on an interview-based study of how some English Advanced level (16+) students who have chosen to study mathematics, and their teachers, speak about confidence with respect to the learning of mathematics. I outline what constitutes confident learners for these teachers and, differently, for their students and what the students feel teachers could do to improve the students confident state. I discuss the implications of this for the education of prospective teachers of mathematics.     

A Review of Solution Chemistry Studies: Insights into Students’ Conceptions

This study has reviewed the last two decades of student conception research in solution chemistry pertaining to aims, methods of exploring students conception, general knowledge claims, students conceptions and difficulties, and conceptual change studies. The aims of solution chemistry studies have been to assess students understanding level of solution chemistry and in some studies compare understanding based on age and year at school or college. The methods of exploring students conceptions consisted of interviews, paper and pencil surveys (open-ended questions and multiple-choice questions), free writing and drawings and the validity of these methods have been highlighted. The general knowledge claims synthesized in this study are students (a) attending to mechanical events, (b) preference for everyday language usage over chemical language, (c) confusing solution chemistry with non-related concepts, (d) lack of sub-microscopic explanation for macroscopic observation, (e) difficulty with visualizing and representing sub-microscopic ideas, (f) difficulty with symbolic representations, (g) inconsistent explanations, (h) development of student understanding with age, and (i) development of conservation reasoning with age. To incorporate students conceptions, conceptual change studies have used strategies such as worksheet, analogy, collaboratively working with a teacher, hypermedia, and group exploration. The results of conceptual change studies generally have had a positive impact enabling students to consider their ideas and develop plausible models of solution chemistry. For improvement of student learning in chemistry, this review of solution chemistry studies sheds light on teacher thinking and capacity building with respect to explicitly incorporating students conceptions into chemistry curriculum; practicing research-based strategies; forging links among types of chemical knowledge; collaborating for experimental teaching; and conducting further research.     

The Pseudo-Conceptual and the Pseudo-Analytical Thought Processes in Mathematics Learning

This paper suggests a theoretical framework to deal with some well known phenomena in mathematical behavior. Assuming that the notions conceptual and analytical are clear enough in the domain of mathematical thinking, the notions pseudo-conceptual and pseudo-analytical are proposed and explained. Examples from mathematics classrooms, mathematics exams, and homework assignments are analyzed and discussed within the proposed theoretical framework.The notions pseudo-conceptual and pseudo-analytical proposed in this paper, actually narrow the extension of the notion cognitive by restricting it to the domain of meaningful contexts. Analysis of meaningless behaviors, it is claimed, requires a different theoretical framework. The attempt to analyze meaningless behaviors in the same way as meaningful behaviors is called here the cognitive approach fallacy.     

Hierarchies in mathematics education

In mathematics education literature the term hierarchy is used in a number of ways. It is important that the mathematics educator consider the usefulness of the hierarchies presented by various researchers and theorists, in the light of their application to teaching. Current works on mathematical learning hierarchies are illustrated and in particular the work of the mathematics team of the research project Concepts in Secondary Mathematics and Science is examined.     

The Development of Authentic Assessments to Investigate Ninth Graders’ Scientific Literacy: In the Case of Scientific Cognition Concerning the Concepts of Chemistry and Physics

Scientific literacy and authenticity have gained a lot of attention in the past few decades worldwide. The goal of the study was to develop various authentic assessments to investigate students scientific literacy for corresponding to the new curriculum reform of Taiwan in 1997. In the process, whether ninth graders were able to apply school knowledge in real-life problems was also investigated. Over the course of our two-year study, we developed authentic assessments to investigate a stratified random sampling of 1,503 ninth graders levels of scientific literacy, including scientific cognition, process skills, application of science, habits of mind, nature of science, and attitude towards science. The purpose of this article is to discuss three different formats of authentic assessments: multiple-choice, open-ended, and hands-on test items, which we developed to investigate scientific cognition. To validate the three formats of authentic assessments, students performance on these three assessments were compared with the science section of Taiwans Academic Attainment Testing (STAAT), and the values of Pearson correlation coefficient were all at the significant level, ranging from 0.205 to 0.660 (p<0.01). We found that our three authentic assessments were better in evaluating students authentic abilities in science than standardized tests (such as STAAT). Further authentic assessments, particularly the hands-on activity, benefited low-achieving students. Concerning the common themes tested in the authentic assessments, students performed better in a multiple-choice test than an open-ended test on electricity and heat and temperature. In addition, two themes of chemical reactions and reactions of acid and base with indicators were performed best in a hands-on test than in the other two tests. In this article, we provide evidence that authentic assessments could be developed in different formats to investigate students scientific cognition as part of the national test. Of these formats, the multiple-choice, open-ended, and hands-on test items are all shown to be sensitive in their evaluation of students cognition in science.     

Child-methods in secondary mathematics

An earlier research project, the Concepts in Secondary Mathematics and Science (Mathematics) project, identified both a hierarchy of levels of understanding in different areas of secondary mathematics, and a number of particular errors which were made by significant proportions of the children tested. Preliminary consideration of these errors and the strategies which appear to have given rise to them suggests that the use of informal naive methods which are limited in their applicability is widespread even at fourth-year level. The suggestion is made that there may be two systems of mathematics coexisting in the secondary school classroom: the formal taught system, and a system of child-methods which are based upon a counting, adding-on or building-up approach, and by which children attempt to solve mathematical problems within a human-sense framework. The difficulties which some children appear to experience in mathematics is suggested to be due in part to these children's non-initiation into the formal taught system. The implication of such a view for teaching and research are indicated.     

Content and form in mathematics

The present article is based on the premise that mathematical activity exhibits an important duality. On the one hand pre-school children do mathematics spontaneously; the mathematics they do is often called informal arithmetic. On the other hand, the mathematics we teach is a cultural product developed by generations of mathematicians (Byers and Herscovics, 1977). Whatever premium school mathematics puts on individuality and original thought, it seems obvious that the mathematics done in school is neither spontaneous nor informal; the bulk of it is formal mathematics. We expect our pupils to do mathematics in a conventionally acceptable manner. In short, mathematics is both an individual and a social activity. We shall endeavour to show that duality in mathematical activity reflects a duality of mathematics as a discipline. The latter is a duality between content and form; in fact, it is the importance of form in this discipline that resulted in mathematics being classified as a formal science.     

HF: Working on mathematics education

First an impression will be given of the way Freudenthal worked on mathematics education during the period after WW II, until the early seventies. The sequence of events of that time form the leading principle of that period when important didactic concepts were being developed. In the following section the chronological order is set aside in an attempt to show how Freudenthal approached research, both in education and into mathematics education. In the next two sections two core themes are dealt with in detail: learning mathematics and developing mathematics education. Earlier didactic concepts are completed and further developed into key notions of a science of mathematics instruction.     

Attitudes to the Use of Alternative Assessment Methods in Mathematics: A Study with Secondary Mathematics Teachers in Sydney,Australia

Given issues related to differences in learner characteristics, effective sampling across the content domain, and recent emphases on assessing meaningfully contextualised abilities and higher-order cognitive processes, the traditional mathematics test arguably does not provide a valid measure of student ability. Consequently, there is a need to incorporate alternative methods of assessment that are able to effectively assess the range of students mathematical abilities. The present study investigated methods of assessment used by 60 mathematics teachers from 11 secondary schools in metropolitan Sydney, as well as their attitudes to a range of alternative assessment methods, together with reasons why they would or would not implement these. Results showed that teachers were satisfied with traditional tests as valid measures of student ability, particularly for senior school years. Teachers generally did not favour implementing alternative assessment methods, although those with the least years teaching experience reported more positive attitudes. A major concern raised by teachers about the use of alternative assessment methods related to their perceived subjectivity. Explanations for these findings are advanced for teachers who have varying lengths of teaching experience.     

Can Mathematics Education and History of Mathematics Coexist?

Despite the wide interest in combining mathematics education and the history of mathematics, there are grave and fundamental problems in this effort. The main difficulty is that while one wants to see historical topics in the classroom or an historical approach in teaching, the commitment to teach the modern mathematics and modern mathematical techniques necessary in thepure and applied sciences forces one either to trivialize history or to distortit. In particular, this commitment forces one to adopt a Whiggish approach to the history of mathematics. Two possible resolutions of the difficulty are (1) radical separation – putting the history of mathematics on a separate track from the ordinary course of instruction, and (2) radical accommodation – turning the study of mathematics into the study of mathematical texts.     

Curriculum matters: Creating a positive climate for diversity from the student perspective

The purpose of this paper is to identify the factors that predict students perceptions of their institutions success in achieving a positive climate for diversity. This study examines a sample of 544 students at a large, public, predominantly White Mid-Western institution. Results show that students perceptions of the institutions ability to achieve a positive climate for diversity is a reflection of students precollege interactions with diverse peers and the institutions ability to incorporate diversity-related issues into its curriculum. Results also indicate that these perceptions differ by race and gender. Implications for institutional researchers are discussed.     

Classroom Use of Multimedia-Supported Predict–Observe–Explain Tasks in a Social Constructivist Learning Environment

This paper focuses on the use of multimedia-based predict–observe–explain (POE) tasks to facilitate small group learning conversations. Although the tasks were given to pairs of students as a diagnostic tool to elicit their pre-instructional physics conceptions, they also provided a peer learning opportunity for students. The study adopted a social constructivist perspective to analyse and interpret the students conversations, focussing on students articulation and justification of their own science conceptions, clarification of and critical reflection on their partners views, and negotiation of new, shared meanings. Two senior science classes participated in this interpretive study. Data sources were mainly qualitative and included audio and video recordings of students small group discussions at the computer, interviews with selected students and their teachers, classroom observations, and student surveys. Findings indicate that the computer-based POE tasks supported students peer learning conversations, particularly during the prediction, reasoning and observation stages of the POE strategy. The increased level of student control of the POE tasks, combined with the multimedia nature of the program, initiated quality peer discussions. The findings have implications for authentic, technology-mediated learning in science.     

Applying the Philosophical Concept of Reduction to the Chemistry of Water: Implications for Chemical Education

Even though philosophical themes in science education have been advocated for several decades, little attention has been paid to how these themes can be contextualized in the teaching of a particular domain of science. The purpose of this paper is to provide an example theoretical framework for applying a philosophical theme, reduction, in chemistry education. The inclusion of the concept reduction in chemistry education not only elaborates on this important philosophical thesis in the specific context of chemistry but also it might offer some insight into how students interest in philosophical aspects of chemistry might be stimulated. The consideration of concepts such as chemical composition, molecular structure and bonding in chemistry of water provides an example for the application of reduction in chemistry education. The water example also promotes the reflection on the microscopic, macroscopic and symbolic features of substances, themes which are repeatedly covered in secondary as well as tertiary chemical education. Some implications include the need for restructuring the curriculum in higher education to be more inclusive of the controversial nature of quantum mechanical applications in chemistry.     

Connecting Teacher Beliefs to the Use of Children’s Literature in the Teaching of Mathematics

This article presents examples that illustrate how teachers use childrens literature in the teaching of mathematics. The examples are related to four curriculum ideologies that have influenced mathematics education in the USA for the last 75 years. It discusses why it is relevant to help teachers understand the ideological positions that influence their use of childrens literature during mathematics instruction, summarizes the four ideological positions, and presents results of a study of how teachers ideological positions relate to their use of childrens literature in the teaching of mathematics. The study examines two research questionsCan an instructional tool be developed that will highlight for teachers the different ways in which they and others use childrens literature to teach mathematics? and Can that instructional tool stimulate teacher discussion and reflection about their own beliefs and the ideological nature of the instructional environment in which they learned (as students) and teach (as teachers)? Study results indicate that both questions can be answered in the affirmative.     

Instructional Misconceptions in Acid-Base Equilibria: An Analysis from a History and Philosophy of Science Perspective

The implications of history and philosophy of chemistry are explored in the context of chemical models. Models and modeling provide the context through which epistemological aspects of chemistry can be promoted. In this work, the development of ideas and models about acids and bases (with emphasis on the Arrhenius, the Brønsted–Lowry, and the Lewis models) are presented. In addition, misconceptions (alternative and instructional ones) on acid-base (ionic) equilibria are examined from the history and philosophy of science perspective. The relation between the development of the models and students misconceptions are investigated. Finally, the hypothesis that history and philosophy could help educators anticipate students misconceptions is examined.     

Whitehead and a new look at teaching elementary science

If Whitehead is right, science teachers who try to increase student interest by making the science they teach more pure and by covering more material are going about their work in just the wrong way. Science, for purposes of precision in measurement, translates the dynamic world of feeling and force, of causal efficacy (for example, the San Francisco earthquake), into a static representation spatialized and given presentational immediacy (for example, the Richter scale). But notice that the Richter scale isn't very interesting (even as abstract art) apart from its connection, via symbolic reference, to the earthquake. Such reference is essential to give both a sense of reality and a feeling of interes to the subject, but it makes the science less pure, and it takes more time to cover the material. An example of teaching pure and impure formal logic is given as a case study.     

Parents of Traditionally Aged Community College Students: Communications and Choice

Five suburban community colleges in a metropolitan area collaborated on a survey to parents of high school graduates of the class of 2002 who attended the colleges in fall 2002. The study was undertaken to learn more about parents roles in their students selection of community colleges, to give insights and information about this important community college constituency, and to assist community colleges to shape more effective ways to foster positive and supportive opinions among parents of potential students. Parents want the community college to provide students with credits and grade point averages enabling transfer, overestimate their students academic skills as measured by course placement tests, engaged in a number of college-choice activities, and indicate that factors associated with lifestyle and money were major reasons for selecting the community college. Associations between parents social capital and perceptions of their students academic abilities were found for a number of dependent variables examined. The study also revealed important differences in how participating institutions record and calculate key measures such as credits earned and grade point averages.     

The American mixture of higher education in perspective: four dimensions

Several nations are currently considering privatization of parts of their higher education systems. This paper, mainly based on the American experience, examines privatizing public institutions as an alternative to establishing solely private institutions.Institutions are analyzed along four dimensions: (1) ownership (public or private); (2) control (external or internal); (3) financing (public or private funds); and (4) mechanisms for public financing (who controls fund distribution and how). There are varying mixtures along these four dimensions both within countries and around the world, with the American system exhibiting the widest range of combinations. Six categories are described, including four common in the U.S.: I. Independent private, where institutions are independent in ownership, in control, and in basic financing; II. Dependent private, independent in ownership and financing but dependent in control; III. Independent public, dependent in ownership but independent in control and substantially independent in financing; IV. Semi-independent public (state/guild type), dependent in ownership, mixed in control, and heavily dependent in financing (less common in the U.S., but typical of Italy and Latin America); V. Semi-independent public (state/trustee/guild type), where control is shared among state, academic guilds and lay boards of trustees but with mainly state-controlled financing; and VI. Dependent public, the model in the Communist nations.Kerr traces the historical path that led to the mixed American system and examines some of its positive consequences, which include institutional autonomy, diversity, and flexibility. Negative results include possible over-responsiveness to short-term pressures, as from the labor market or student preferences for courses of study, and from supporting business or industry.The author concludes that the American experience with privatized public institutions may serve as a model for those elsewhere who now seek greater institutional differentiation, autonomy, and flexibility within national systems of higher education.     

Idealisation and Galileo’s Pendulum Discoveries: Historical, Philosophical and Pedagogical Considerations

Galileos discovery of the properties of pendulum motion depended on his adoption of the novel methodology of idealisation. Galileos laws of pendulum motion could not be accepted until the empiricist methodological constraints placed on science by Aristotle, and by common sense, were overturned. As long as scientific claims were judged by how the world was immediately seen to behave, and as long as mathematics and physics were kept separate, then Galileos pendulum claims could not be substantiated; the evidence was against them. Proof of the laws required not just a new science, but a new way of doing science, a new way of handling evidence, a new methodology of science. This was Galileos method of idealisatioin. It was the foundation of the Galilean–Newtonian Paradigm which characterised the Scientific Revolution of the 17th century, and the subsequent centuries of modern science. As the pendulum was central to Galileos and Newtons physics, appreciating the role of idealisation in their work is an instructive way to learn about the nature of science.