Search

Discusses the needs and the content of a philosophy of mathematics course. Provided are sample term paper topics, sample final exam items, and an annotated bibliography. (YP)

n a recent article, Inglis and Alcock (2012) contended that their data challenge the claim that when mathematicians validate proofs, they initially skim a proof to grasp its main idea before reading individual parts of the proof more carefully. This result is based on the fact that when mathematicians read proofs in their study, on average their initial reading of a proof took half as long as their total time spent reading that proof. Authors Keith Weber and Juan Pablo Mejía-Ramos present an analysis of Inglis and Alcock's data that suggests that mathematicians frequently used an initial skimming strategy when engaging in proof validation tasks. [See Inglis and Alcock's response to Weber and Mejía-Ramos ("Skimming: A Response to Weber and Mejia-Ramos") on pages 472-74. A download will include both articles.]

As a response to the attacks on ethnic studies in Arizona and the move to ban certain books, this essay presents theoretical and pedagogical reflections from two professors and addresses the ways teacher preparation programs can offer a resistance. Based on the authors' experience in teacher preparation programs, one in the humanities and the other in mathematics, they discuss fundamental concepts that undergird social change methodology from Gloria Anzaldua (la facultad and conocimiento) and from Isabel Gunning's work (World Traveling). Ultimately, our premise is that teachers of teachers can impact the curricula in significant ways that result in dismantling racism and in teaching that is focused on positive social change. We posit that the university classroom where future teachers are trained must address (1) Equity issues, (2) Cultural identity or cultural framing, and (3) Culturally relevant strategies and teaching, modeled by the university professor.

This is the author's response to Barbara and Derek Ball's comments to their three papers. Barbara and Derek believe that the "school run" infantilises children. The author disagrees. The school run does not infantilise children, but reflects officially sanctioned untruths and a systemic avoidance of issues of equity. The school run derives from fear and makes children victims. With regard to mathematics, the author disagrees with Barbara and Derek as to the means by which autonomy is achieved. Unlike Barbara and Derek, he thinks autonomy derives from positive conformity. He argues that the mathematical needs of children demand conformity to a coherent curricular and pedagogical tradition, rather than the encouragement of many. While many of Barbara and Derek's comments are focused on critiquing their proposal for greater regulation, the author thinks the real area of debate concerns the extent to which regulation implies a framework or a prescription. He is suggesting that intelligently-planned regulation would offer teachers affordances rather than constraints.

The integration of science, technology, engineering, and mathematics content (STEM) has become a mainstream topic within educational systems. This paper discusses the technology integration model for education and the factors to be considered when taking into account technology education as a focal point of integrated curricula. These factors are: (1) academic collaboration; (2) hands-on approaches; and (3) the use of creativity and problem solving. Considering that technology can be the driving force behind integration, the authors suggest cohorts of teachers from all academic areas, including technology education, work together to provide a comprehensive integrated curriculum, with technology leading the process and content. (Contains 1 figure and 1 table.)

In this essay, Vasileios Pantazis examines how two philosophers having different orientations acknowledge and study the phenomenon of the "encounter" ("Begegnung") and its fundamental importance to human life and education. On the one hand, Otto Friedrich Bollnow drew on existential philosophy and philosophical anthropology in his analysis of the encounter, while Alain Badiou, on the other hand, used psychoanalysis, mathematics, and Plato in exploring the concept. The approach Pantazis takes in this essay aims at fusing the concept of the encounter as developed by Bollnow with a philosophical view, specifically Badiou's understanding of the encounter in the context of his concept "event of truth." Through the "fusion of horizons," as Hans-Georg Gadamer put it, between these two views, Pantazis seeks to enrich the concept of the encounter and to draw out a renewed meaning for philosophical and educational theory. (Contains 29 footnotes.)

This response discusses the experiment reported in Krahmer et al.'s Letter to the Editor of "Cognitive Science". We observe that their results do not tell us whether the Incremental Algorithm is better or worse than its competitors, and we speculate about implications for reference in complex domains, and for learning from "normal" (i.e., non-semantically-balanced) corpora.

A report entitled "What we can learn from the English, mathematics and science curricula of high-performing jurisdictions" seems taken to suggest... "that imitating the content, pace and pedagogy of Singapore and Hong Kong in particular, would enable us to improve the mathematical performance of English children". This is surely "a gift" to the headline writers eager to undermine the work of educators, of classroom teachers and, in no small part, to belittle the performance of learners. But, learning in mathematics is a complex dynamic and any simplistic means to improve the status quo by policy makers should be subject to professional scrutiny. Here, the author scrutinises the facts and presents a forensic report on the reality of the situation. "Face value" can be seductive but, as ever, the demons can be slain by expert detailed analysis.

Cultural-historical activity theory--with historical roots in dialectical materialism and the social psychology to which it has given rise--has experienced exponential growth in its acceptance by scholars interested in understanding knowing and learning writ large. In education, this theory has constituted something like a well kept secret that is only in the process of gaining larger levels of acceptance. Mathematics educators are only beginning to realise the tremendous advantages that the theory provides over other theories. In this review essay, I articulate the theory as it may relate to the issues that concern mathematics education and educators with a particular focus on the way in which it addresses logical contradictions in existing theories.

The authors comment on the article by Morris and Hiebert in three ways. First, they add thoughts about why improvement efforts often focus on teachers, rather than teaching. Second, they offer evidence from U.S. lesson study research that focus on teaching can improve both students' learning and teachers' learning. Finally, they suggest that the instructional products and common assessments advocated by Hiebert and Morris are not sufficient, and that they need to be accompanied by practice-based, collegial learning in which teachers build shared knowledge and commitments for the hard work of improvement. Their research indicates that lesson study focuses on teaching, but improves teachers as well, increasing mathematical knowledge and beliefs that support instructional improvement, as well as improving student learning. (Contains 2 figures.)

For a long time, the body has had a bad rap among some big names in the Western philosophical tradition. Plato and Descartes come quickly to mind as particularly bad rappers. But it appears that the body is making a comeback in the cognitive and learning sciences. Long banished from the main stage by an idealized, inside-the-head information-processing worldview, the body is steadily being rediscovered in the work of thinking and learning. A reasonably active mind has been around since the beginning of the so-called cognitive revolution. That active mind is now being reintroduced to a sensing, feeling, "thinking" body in action. Mathematics has commonly been construed as among the forms of human activity most distant from the body's experiences and its understandings. This article discusses how the missing bodies of mathematical thinking and learning have been found. It draws its empirical examples of mathematical learning from classrooms.

The claim that faculty conduct research is one of the main justifications for the modern university. Supposedly, academe carries out important, cutting-edge inquiries in which society has an interest. In fact, states this author, research at American universities is becoming narrow and artificial, out of touch with social realities, and of interest mainly to other academics. This development the author calls scholasticism, after the medieval philosophers whom one remembers as being similarly ingrown. In recent decades, conservative critics of academe have focused on political correctness--the tendency of academics to privilege liberal perspectives to the exclusion of conservative ones. But scholasticism may pose a greater threat to the heart of university life, which is the intelligent construction of arguments. Rather than address important issues thoughtfully, today's academics are becoming pedants, mostly writing only for other specialists. Scholasticism is squeezing the intellectual life out of academe, and driving it toward other venues, including government and the blogosphere. In this article, the author describes the rise of scholasticism and asks about causes and cures: Why has obscurantism mushroomed in recent decades, and what can be done about it? He relies mostly on the trends he has seen in his own department and at New York University over more than thirty years. (Contains 35 footnotes.)

McLachlan (2011) and Vermunt (2011) each provided thoughtful replies to our original article (Steinley & Brusco, 2011). This response serves to incorporate some of their comments while simultaneously clarifying our position. We argue that greater caution against overparamaterization must be taken when assuming that clusters are highly elliptical in nature. Specifically, users of mixture model clustering techniques should be wary of overreliance on fit indices, and the importance of cross-validation is highlighted. Additionally, we note that "K"-means clustering is part of a larger family of discrete partitioning algorithms, many of which are designed to solve problems identical to those for which mixture modeling approaches are often touted. (Contains 3 footnotes.)

The classroom is a micro-culture where socio and socio-mathematical norms of interactions are fostered by an instructor who represents the mathematics community. Informed by the "emergent approach" developed by Cobb and Yackel (1996) that explains how mathematical growth takes place in the social context of classrooms, I explored the enactment of these norms in small groups' interactions that were not directly influenced by an instructor. I used audio recorded and later transcriptions of groups of undergraduate mathematics students' discussions on class tasks that were analysed using Polkinghorne's (1995) narrative analysis. Using excerpts from the discussions I argue that with a broadened view on socio-mathematical norms, and interpretations of interactions that focus on "learning in the moment", enactments of norms are seen in students' (a) contributions to discussions through initiating discussions, (b) taking responsibility for raising and offering answers, (c) acceptance of arguments that are efficient to the self. (Contains 1 figure and 2 tables.)

I argue that until ethnomathematical frameworks are explicitly used to investigate the specific mathematical practices in networks of powerful elites the term will continue to index that which is Other to "traditional" mathematics and run a risk of desubstantialization. I suggest that the field ally itself with disciplines and perspectives in which the imagination is central, such as mythopoetics, in order to address these risks. Further, I claim that ethnomathematics has been a mythopoetic endeavour involved in demythologizing and demystifying dominant discourses of mathematics and mathematics education and propose that there is need now for greater attention to mindful remythologizing.

Hillman's (2001) simple affirmation that "an idea of beauty is useful, functional, practical" is one this article attempts to pursue with teacher researchers in mind, based on the belief that to move from the "re"pression of beauty to its "ex"pression--or, at the very least, to its articulation--will enlighten rather than distract individuals. The author contends that, as a theoretical lens through which to observe and reflect upon practical issues, beauty has much to offer a range of research interests by no means restricted to learning in the arts. To illustrate the dynamic force of beauty, the author offers two examples. Although written by philosophers outside of the field of educational research, they nonetheless demonstrate how beauty can motivate an active spirit of inquiry. (Contains 1 figure and 2 notes.)

This article presents the authors' response to Asha K. Jitendra's Response to Xin and Zhang ("The Journal of Educational Research," 2009, Vol. 102, No. 6). The claim of "factual errors, including inappropriate quotes, misrepresentation of information, and inadequate scholarship in Xin and Zhang's text" are not justified. In this rejoinder, the authors address each of Jitendra's points. (Contains 1 figure.)

Contemporary argument increasingly relies on quantitative information and reasoning, yet our profession neglects to view these means of persuasion as central to rhetorical arts. Such omission ironically serves to privilege quantitative arguments as above "mere rhetoric." Changes are needed to our textbooks, writing assignments, and instructor development programs to broaden how both we and our students perceive rhetoric. (Contains 1 figure, 1 table and 7 notes.)

In this article, I make a case for the inputs that Martin Heidegger's theoretical perspective offers to current concerns about the nature of mathematics, its teaching and learning, and the problem of subjectivity. In particular, I consider Heidegger's notion of positive science and discuss both its applicability to mathematics and its importance to mathematics education. I argue that Heidegger's ontological position is consonant with some sociocultural approaches in mathematics education and that Heidegger's work can shed some light on the problem of knowing and being. Finally, I raise some questions concerning subjectivity and the link between language and mathematical objects.

In this commentary, the author discusses two recent replications (Santelices & Wilson, 2010; Scherbaum & Goldstein, 2008) of some of his earlier work on SAT items using the differential item functioning (DIF) statistic wherein he contrasted the test performance of African American examinees with White examinees (Freedle, 2003). In this commentary, it becomes clear how each of these two replications covers different aspects of the DIF statistic and addresses criticisms of his work. These replication studies find, in general, that whenever a significant ethnic correlation occurs between DIF and item difficulty, regardless of the item type, the African American group is found to perform differentially better on hard items than White examinees of matched ability and differentially worse than White examinees on easy items. As Scherbaum and Goldstein (2008) claim, this DIF correlation represents a counterintuitive finding, given the test bias found in previous studies (Freedle, 2003, p. 2; Gould, 1995). The author briefly advances a hypothesis explaining why this DIF correlation result is so counterintuitive. While Santelices and Wilson's (2010) findings for the verbal SAT items replicate the general pattern of DIF results the author reported (Freedle, 2003), there are some differences between these two studies when one focuses on particular item types. The author focuses on these differences because he believes they will put researchers on a path toward building a deeper theoretical understanding of the sources of test bias, at least at the individual item level. He also reviews the Scherbaum and Goldstein (2008) DIF replication study that used non-SAT content to address the assertion that his significant DIF findings (Freedle, 2003) represent a statistical artifact (see Wainer & Skorupski, 2005). Finally, he addresses some strategies for reducing mean test score differences between minority and White examinees. (Contains 2 notes.)

The article "Is Google Making Us Stupid?" in last Summer's "Atlantic Monthly," raised a number of provocative, and indeed worrisome, questions about computer usage and cognitive development. For instance, persons with considerable experience of reading for the sake of pleasure report that, after a couple of years using computers a great deal, they have experienced a loss of interest in pleasure-reading, even feeling impatient when written sources do not supply the information they seek quickly and conveniently. One suggestion is that these effects are somehow related to the mathematical basis of computing provided in the work of Alan Turing. Since, however, Turing's work is based on earlier work by Kurt Godel, the question may be referred to Godel's results and it is argued that the mathematical architecture underlying computer operation and applications in no way limits the creativity or even the cognitive activity of its users.

In this commentary, some remarks are offered on David Pimm, Mary Beisiegel, and Irene Meglis' article "Would the Real Lakatos Please Stand up." The commentary focuses on relatively recent developments in the philosophy of mathematics based on the work of Lakatos; on theory development in mathematics education; and offers critique on whether Lakatos' "Proofs and Refutations" (1976) can be directly implicated in mathematics education.

Research on cognitive modeling of information search and Web navigation emphasizes the importance of "information scent" (the relevance of semantic cues such as link labels and headings to a reader's goal; Pirolli & Card, 1999). This article shows that not only semantic but also structural knowledge is involved in navigating the Web (Juvina, 2006). This article argues for considering both semantic and structural knowledge in modelling Web navigation. A cognitive model is proposed that uses information scent to account for user's judgments of relevance (a semantic dimension) and "path adequacy" (the semantic similarity between a navigation path and a user's goal) to account for user's efficiency in traversing a Web structure (a structural dimension). Two empirical studies show that abilities to represent and manipulate spatial structures are complementary to semantic abilities in accounting for Web navigation performance. (Contains 2 figures and 3 tables.)

When causal inferences are to be synthesized across multiple studies, efforts to establish the magnitude of a causal effect should be balanced by an effort to evaluate the generalizability of the effect. The evaluation of generalizability depends on two factors that are given little attention in current syntheses: construct validity and external validity. The author concludes with six concrete suggestions for ways that future syntheses could be improved to make them more relevant for educational policy makers. (Contains 2 tables and 2 notes.)

The purpose of this article is to sketch a hypothetical descriptive framework of statistical knowledge for teaching. Because statistics is a discipline in its own right rather than a branch of mathematics, the knowledge needed to teach statistics is likely to differ from the knowledge needed to teach mathematics. Doing statistics involves many primarily nonmathematical activities, such as building meaning for data by examining the context and choosing appropriate study designs to answer questions of interest. Although there are differences between mathematics and statistics, the two disciplines do share common ground in that statistics utilizes mathematics. This connection suggests that existing research on mathematical knowledge for teaching can help inform research on statistical knowledge for teaching. (Contains 1 figure and 1 table.)

The notion of assumptions permeates school mathematics, but instruction tends to highlight this notion only in the advanced grades. In this article, I argue that it is important for even young children to develop a sense of the role of assumptions in proving, and I investigate what it might mean and look like for instruction to promote this goal. Toward this end, I study an episode from third grade that describes the first time that the students in the class were introduced in a deliberate and explicit way to the role of assumptions in proving. The central role of the mathematical task in the episode is identified, and features of mathematical tasks that can generate rich mathematical activity in the intersection of assumptions and proving are discussed. In addition, issues of the role of teachers in fostering productive interactions between students and mathematical tasks that have those features are considered. (Contains 4 footnotes and 3 figures.)

The author, director of the University of Chicago School Mathematics Project, tackles the following question: Should the United States have national standards with teeth, that is, a single set of standards tied to assessments and agreed to by the states? Proponents advance five main arguments for implementing such a standard. In his rebuttal, the author points out that a single set of national standards in mathematics assures neither high student performance nor a healthy economy, that mathematics curriculums are fairly prescribed and uniform across states, that a national curriculum won't resolve the equity issue or speed up change in schools, and that the politicization of education in the United States has resulted in a loss of expert guidance in mathematics at the national level. (Contains 4 endnotes.)

The series of articles presented in this journal issue reports a systematic approach to judging the validity of a test within a framework recommended by Kane (2001, 2004a). The authors' approach effectively integrates quantitative and qualitative tools, showing the role of psychometric models in validation, and emphasizes the importance of clearly explaining the theory underlying the development of the test. Indeed, validation of the test is commensurate with validation of the theory underlying its development. In this commentary, the author addresses the difficulties the authors encountered in connecting the empirical evidence with the theory, and suggests an alternative theory that might account for the difficulties. She proposes that teachers should have a deep understanding of the mathematics.

This article is reprinted from Mathematics Teacher, in honor of the 100th anniversary. It discusses the role of emotions in mathematics teaching and learning. (Contains 3 endnotes.)

Graphing predictions is especially important in classes where relationships between variables need to be explored and derived. In this article, the author describes how his students sketch the graphs of their predictions before they begin their investigations on two laboratory activities: Distance Versus Time Cart Race Lab and Resistance; and Ohm's Law Lab. Prior to collecting data, graphing a prediction helps students evaluate the relationship between their variables and establishes the foundation for the scientific concepts and mathematic principles underlying the investigation. After collecting and graphing data, it is easier for students to determine if they supported or rejected their prediction if they have a graph of their prediction in front of them. Perhaps most importantly, graphing their predictions forces students to analyze their thinking process prior to gathering data. It also provides a theoretical context for both the mathematic principles and scientific concepts underlying their investigation. This process leads students to a better understanding of both the nature of science and the nature of mathematics. (Contains 5 figures.)

Fumiko Samejima is best known for her pioneering work in polytomous response item response theory (IRT), yielding the eponymous model that has been used broadly for more than 30 years. In this interview, Samejima, on the verge of retiring from her faculty position at the University of Tennessee, discusses her life and career. She also describes how she first became interested in mathematics and the events that propelled her toward psychometrics.

This article focuses on the difficulties afflicting the system of higher education as a result of the decline in the overall level of student training and the "gap between the level of education of secondary school graduates and historically established requirements of higher education in Russia." The author has been teaching sociology majors in a number of the higher educational institutions of Moscow since 1989, the year in which higher sociological education was institutionalized in the country. With the decline in the overall training of the graduates of secondary school, the author examines secondary school graduates' knowledge in two disciplines that are of fundamental importance for future sociologists--history and mathematics. The author presents a wish list to the schools. The article further focuses on a number of the factors that account for secondary school graduates' lack of the knowledge and the ability to read and analyze texts that are necessary for success in institutions of higher learning. Drawing on his personal pedagogical experience, the author also examines a number of facts that reflect the statistical trend and presents conclusions. (Contains 5 notes.)

Over the years, research on students' epistemological beliefs has resulted in a growing common understanding but there are still some major points of discussion. Especially, the lack of consensus on the context-general and/or context-specific nature of epistemological beliefs deserves our attention. We argue that research in the field today is mainly characterized by a top-down approach that investigates students' domain-specific beliefs from a general epistemological perspective. Alternatively, we report on one of our studies as well as some other research that takes a bottom-up approach starting from students' domain-specific belief systems and analyzing their epistemic dimensions. Results of these studies point to the highly domain-specific nature of students' beliefs about knowledge and knowing. Therefore, a conceptual distinction between students' general epistemological beliefs and the epistemic dimensions of domain-related belief systems is recommended as a more appropriate way to address the context-general-context-specific discussion on epistemological beliefs.

Robinson interviews Howard Wainer, a scientist, who graduated from Rensselaer Polytechnic Institute in 1965 and Princeton in 1968 with degrees in mathematics and psychometrics, respectively. He taught at Temple University and The University of Chicago before moving to the Bureau of Social Science Research during the Carter Administration. Wainer also served as editor of the "Journal of Educational and Behavioral Statistics" from 2002 until 2004. Among other things, he talks about his experiences that led him to interests in science and mathematics. (Contains 3 notes.)

The purpose of this article is to contribute to the dialogue about the notion of advanced mathematical thinking by offering an alternative characterization for this idea, namely advancing mathematical activity. We use the term advancing (versus advanced) because we emphasize the progression and evolution of students' reasoning in relation to their previous activity. We also use the term activity, rather than thinking. This shift in language reflects our characterization of progression in mathematical thinking as acts of participation in a variety of different socially or culturally situated mathematical practices. For these practices, we emphasize the changing nature of students' mathematical activity and frame the process of progression in terms of multiple layers of horizontal and vertical mathematizing.

Students should be encouraged to focus on a big mathematical idea and to look for connections between problems and solution strategies. This unified view suggests that the students are developing computational fluency with fractions.

A few problems to dispel the notion that mathematics today admits no unsolved problem are presented. The vast area of the unknown and challenges that befit a twenty-first-century attack are discussed.

This essay investigates the rhetoric surrounding the appearance of the concept of the infinitesimal in the seventeenth-century Calculus of Sir Isaac Newton and Gottfried Wilhelm Leibniz. Although historians often have positioned rhetoric as a supplemental discipline, this essay shows that rhetoric is the "material" out of which a new and powerful mathematical system emerges. At the height of empiricism, the infinitesimal, thought by Newton and Leibniz to be evanescent or nascent, made available no recourse to empirical or geometric verification. Instead, the infinitesimal found its "substance" in the rhetorical arguments surrounding it, which ultimately precipitated an epistemic shift in scientific and mathematical practice.

In this article, the author presents several questions that are central to the debate over what should be taught in social studies classes; however, few academic subjects evoke as much passion as history. The debate comes when educators worry that social studies is vanishing from the curriculum, and with schools scrambling to meet requirements of the No Child Left Behind, history, geography, and civics are being pushed aside to spend more time on the subjects being tested--mainly reading and math.

In this commentary on Matthew Inglis' "Three Worlds and the Imaginary Sphere" (see EJ1106688), David Tall develops the theme that the building of theories is not an easy process. A theory in progress is a particularly delicate creation. Theories do not appear fully formed. There is a period of exploration and incubation that precedes the eventual formulation. In the case of the theory presented here (three worlds of mathematics), although it is moving towards a stable form, it is still in the process of being filled out and refined. Theories are built by reflecting on one's experiences and, "because we have different experiences, we naturally produce different theories or different aspects of a theory that can be made stronger by refinement." The development of the theory of "three worlds of mathematics" has gone beyond random specialization and on to systematic specialization in three areas: calculus (Tall, 2003), proof (Tall, 2002b) and vectors (Watson, Spyrou, and Tall, 2002). Tall presents the definition of the "three worlds of mathematics" as follows: The first grows out of our perceptions of the world and consists of our thinking about things that we perceive and sense, not only in the physical world, but in our own mental world of meaning. The second world is the world of symbols that we use for calculation and manipulation in arithmetic, algebra, calculus, and so on. The third world is based on properties, expressed in terms of formal definitions that are used as axioms to specify mathematical structures (such as "group," "field," "vector space," "topological space," and so on). In building a theory of different worlds of mathematics, Tall concludes, one cannot begin by stating definitions and proving theorems. There must be ideas that are tested out by trying out formulations to see if they make sense to others and to test the ideas in several different contexts (so far, calculus, vectors, and proof) to see if they have a useful practical meaning.

Discusses the issue of math problem solving and the concomitant concept of mental discipline/transfer of learning. Considers if there is any evidence to support the mental discipline theory. Discusses implications of the lack of mental discipline for the curriculum. Comments that "God gave humans two halves of the brain that are supposed to work together, not replace each other." (SG)

Many problem-solving activities include mathematical principles but students do not use them during the design and experimentation phases before creating a prototype or product. Restricting the amount and/or type of materials available to students will require them to calculate and requisition the materials needed. (JOW)

Considers the possibility of simplifying academic knowledge, and of taking its aesthetic dimension seriously, and the implications this may have for education. Suggests principles for simplifying knowledge and provides examples from mathematics and statistics to illustrate how ideas and methods can be simplified to assist with the problems of the expert, the layperson, and the student. (EV)

Describes some problems concerning mathematics instructors such as the effects of a less mathematically homogeneous entry of instructors. Outlines steps that are being taken to alleviate these problems. Introduces the Science and Engineering Foundation Studies (SEFI) Mathematics Working Group. (Author/YDS)

Addresses the question of how the concept of "bildung"--which encompasses the process of personal development and change throughout life as well as the goal of the formation of shaping or forming--can be used as a tool for exploring goals of teaching numeracy to adults who lack basic mathematical skills. (Author/VWL)

Argues that adults' relationship with mathematics has three components: cognitive, affective, and social. Focuses on the affective and social components and suggests that the differences between informal mathematics and formal school mathematics lead to the former not being recognized as mathematics. Believes this situation should be analyzed from three interrelated perspectives: the adult's mathematical skills; the adult's conception of mathematics; and the adults' self-perception in relation to mathematics. (Author/VWL)

Describes the ways in which a problem-based learning (PBL) approach to teaching precalculus can prepare students for calculus by helping students develop a conceptual understanding rather than learning through symbol manipulation and memorization. Discusses the exponential, trigonometric, polynomial, and rational function modules employed. (NB)

Can music in and of itself tell anything about the mind of the person who composes or performs it? This question is of general philosophical interest, but it takes on more than philosophical significance in contexts where there is reason to think that music may be the only significant point of contact between one human being and another. There are no doubt several contexts of which this is true, but the one this author concentrates on is that of the autistic. This essay constructs, in a preliminary way, a conceptual context for the evaluation of music therapy in the care and treatment of the mentally and emotionally impaired. It presents the implications of the author's argument for music therapy and art therapy more generally. The author's central purpose in this essay is to present a philosophically plausible account and interpretation of the phenomenon of the musical "idiot savant." (Contains 5 notes.)

Satisfactory reasons have not been found as to why mathematics is so effective in the sciences. Maharishi's Vedic Science explains the link between the subjective realm where mathematics is located and the objective world that science examines, and resolves this problem in a natural way. (Author/MM)

This article uses a self-organizing systems (SOS) framework to explain the complexity of learning disabilities (LD) and suggest that traditional classification schema may be inadequate. It explains principles of self-organizing systems and reviews research concerning reading and math disabilities and the roles of language, attention, working memory, and executive functioning as they relate to LD. Assessment practices are also addressed. (Contains references.) (Author/DB)