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.     

The curriculum potential of Darwin's Theory of Evolution

Unity and diversity are best reconciled by Darwin's Theory of Evolution. Two major dilemmas related to the study of Evolution are described. The first, concerning pedagogy, can be solved by teaching the topic at an elementary level in the middle school so that the Theory may serve as an advanced oragnizer and returning with in-depth study in the last two years of high school. The second, concerning Faith, is a sensitive issue. Some successful approaches are discussed. The place of Darwin's Theory in the all-elective high school biology curriculum is described as well. The role of the Theory regarding explanations is especially highlighted.     

Education of the refugee in the United States

Conclusion This historical survey of the educational goals developed on behalf of the immigrant and refugee in America indicates that there has been a complete cycle. The first systematic educational efforts, as we have shown, were aimed at teaching the immigrant the English language so that he would be useful for his tasks in American industry. Then came the period which extended this educational goal by offering instruction in several aspects of acculturation (officially, assimilation, but a concept which has never been put into effect in the history of America's minorities). The post-war period saw the stress on the academic level on the cultural democracy concept; but this ideal has been more academic than practical, and World War II saw no definite evidences of the popular interest in any Americanization program. The recent waves of refugees, bringing in more or less well-educatedindividuals, saw a revival of educational efforts on their behalf; but they differ from the former endeavors in their emphasis on individual education and on instruction in English. The social aspects of acculturation — featuring, in general, the period between World War I and II — have been entirely ignored 1).     

Evaluating women's homemaking and volunteer work experiences for college credit

This project developed methods and materials to assist colleges in evaluating the competencies which women acquire from their nonmarket (volunteer work and homemaking) experiences. Competencies were identified through interviews and questionnaires of women's activities. Next an informal task analysis of each activity was conducted to develop a list of the competencies involved, the I Can lists. A twelve step counseling process, based on the use of the I Can lists, is described. The paper also discusses methods of dealing with faculty and administrative concerns about this type of experiential learning evaluation.The author wishes to thank Marlaine Lockheed and Abigal Harris for their assistance in this research.Paper presented at the conference on Continuing Education Programs for Women—Issues and Trends, Columbus, Ohio, May 12, 1978.     

On defining quality of education

In this paper quality of graduates is distinguished from quality of education, which is renamed efficiency of education. The first concept is defined in a quantitative manner, the second by way of a set of 66 statements concerning the efficiency of education derived from judgements about the effectiveness of particular teaching methods in the achievement of different types of teaching objectives.Paper presented at the conference on Efficiency of Teaching Methods in Higher Education, organized by the University Teaching Methods Unit of the University of London, 6–7 April, 1976.     

Different aspects in designing mathematics education: Three examples from the Freudenthal Institute

Three teaching designs are offered. They were developed for 12 to 16 year old pupils taking into account their personal differences in terms of gender (in Making Up the Cube), culture (in Mecca), knowledge, and use of calculators in The integrating calculator. The designs concern the representation of three-dimensional objects in the plane, directions and great circles on the sphere, and exploration of the integrating calculator.     

Beauty and Character: Values Beyond Beauty in Architecture

Formalist conceptions of aesthetic goals began to give way from the 17^th century on to rationalist accounts of experience. Beauty as described as essential by Renaissance thinkers was reconceived as just one possibility, with the sublime in Burke's account as another. In the early 20^th century Clive Bell again asserted a formalist position with his argument for significant form. The weakness of the formality position, and the need for a richer, more contextual account of aesthetic goals and experience is argued with reference to the idea of thick and thin concepts as applied by Bernard Williams to ethics, Wittgenstein's attention to the way in which language is used in a specific context, and Scruton's emphasis on an informed conception of the object of aesthetic attention. Beauty, in Alberti's procedural account if not in his version of the Platonic ideal, can be seen to continue to have critical value. The idea of character, however, offers a broader range of possibilities for experience and of goals for design.     

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.     

Print and reading: What do preschoolers know?

John Downing, an eminent reading researcher at Canada's University of Victoria, says that many children come to school in a state of cognitive confusion about the functions of reading and the terms we use when we start to teach them to read formally. Adults take for granted that young children know what they mean when they talk about a word, a letter, a sound, and other print-related terms. In his studies done in England and in Canada, Downing (1970; 1973–74) has found that many children, especially those who have not been read to regularly, simply don't understand the conventions of print and really don't understand what reading is all about.Joan T. Feeley is Professor of Reading and Language Arts at William Paterson College in Wayne, New Jersey.     

A Textbook Case of the Nature of Science: Laws and Theories in the Science of Biology

The two concepts law and theory are among the most important elements of the nature of science. They represent both the tools and products of science itself. Unfortunately, the variable meanings and use of these terms in general discourse and in other school disciplines results in much confusion with respect to their proper application in a science context. The project included the design of a six-part model definition for law and theory based on a review of the literature of the philosophy of science with special reference to biology. These model definitions were then compared with those provided in a range of U.S. secondary school biology textbooks. The majority of all current major U.S. secondary school biology texts were reviewed and analyzed with respect to how the concepts of law and theory were defined and applied, in an attempt to determine whether students and teachers using such texts would gain an accurate impression of these terms and the distinction between them. This study focuses on biology instruction since a life science course is completed as a graduation requirement by virtually all U.S. high school students and as such serves as a widely shared educational experience across the nation. The term law is rarely defined in any text but various laws such as those found in genetics are frequently included as examples. The term theory is frequently defined but with a wide range of completeness of the definitions. Only rarely are theories in biology included as examples.     

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.     

Reforming urban education: A review of recent reports and legislation

Following the National Commission on Excellence Report, A Nation at Risk, a number of additional reports were issued calling for major changes in American public education. Because advocates for poor and minority groups saw a class bias in the recommendations of these reports, there quickly followed a subset of reports arguing that the disadvantaged should not be overlooked in the education reform movement. Not surprisingly, these reports have received less attention than the mainstream ones. This article reviews recent urban education reform reports and legislation, paralleling them to determine whether the reformers have been able to influence policy and whether the reforms they advocate would be likely to benefit the disadvantaged.     

Three-dimensional art and sculpture

I am making it BIG (Evan). A secret slide with gold in there (Sydni). Look even a little car could fit in it (Joshua). This is a circle! Round, a tiger jumps through this hoop (Emily). Excitement runs high when children experiment with three-dimensional art projects using paper, boxes and styrofoam.Liz Seelhoff Byrum teaches four- and five-year-olds at a private school in New York City.     

Using problem solving teaching styles to prepare candidates for the CXC basic proficiency examinations

Two Grade 10 classes in an urban Jamaican High School were taught over a period of one academic year in two problem solving styles: an Explicit Style derived from Charles, and an Implicit Style derived from Isaacs. At the end of the academic year there was no significant difference in their performance on a problem solving test, or on the Problem Solving Profile of the Caribbean Examinations Council's Basic Proficiency papers. The two classes performed much better than the population who sat the Basic papers on the tasks measuring Recall and Algorithmic Thinking but only moderately better than the population on tasks measuring Problem Solving.The teaching project described in this paper was supported in part by grants from the Research and Publications Fund Committee of the University of the West Indies (Mona), and the Wolmers Trust, Kingston. The author wishes to thank the Registrar and the Pro-Registrar of the Caribbean Examinations Council for permission to use the CXC papers and data in this study. A modified version of this paper was presented at the 63rd Annual Conference of the National Council of Teachers of Mathematics held in San Antonio, Texas in 1985.     

Problems with problem-centred knowledge

The author completes theInterchange symposium on Referent-Centred and Problem-Centred Knowledge: Elements of an Educational Epistemology by responding to the commentaries of Paul Thagard, Stellan Ohlsson, and Francis Schrag.     

Visual processing during mathematical problem solving

This study investigated, in the context of mathematical problem solving by secondary school students, the nature of the visual schemata which Johnson (1987) hypothesises mediate between logical propositional structures and rich specific visual images. Four groups of grade 10 students were studied, representing all combinations of high and low operational ability in mathematics (equivalent to Johnson's logical propositional structures) and high and low vividness of visual imagery (corresponding to Johnson's rich images). The results suggested first, that success at problem solving was related to logical operational ability, but not to vividness of visual imagery. Second, a variety of visually based strategies were used during problem solving which differed in their level of generality and abstraction, and use of these strategies appeared related to either logical operational ability or vividness of visual imagery, depending on their level of abstraction. The results supported Presmeg's (1992b) continuum of abstraction of image schemata.Throughout the paper, the first High or Low denotes logical operational ability, and the second, vividness of visual imagery.     

Math activities for young children

Children use mathematical terminology throughout their daily play. While observing a group of children during play, one is likely to hear comments such as: He has more, These are my numbers, Jane has the biggest half, I'm bigger than you 'cause I'm five years old, Give me the round one, I gave each child one cupcake, and so on.Diane M. Kohl is Assistant Professor of Child and Family Development, The University of Georgia.     

Studying construction work: Connecting classroom centers

Johnny has one hammer, one hammer, one hammer; Johnny has one hammer, Johnny has two, sings a small group of children in the woodworking center as they pound small cuts of wood with hammer and nails. Hey, Ronnie, put that block here. We're making a chimney for our house, directs one youngster in the block center. We can move in when we're finished, he tells his two companions.Laverne Warner is Professor of Early Childhood Education within the Division of Teacher Education at Sam Houston State University in Huntsville, Texas. Melissa Sepulveda is a kindergarten teacher at Lowery Elementary School in the Cy-Fair Independent School District in Houston, Texas.