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The purpose of this paper is to offer three alternatives to patterns or visualization used to justify division of fraction "algorithm" invert and multiply". The three main approaches are historical, similar denominators and algebraic, that teachers could use to justify the standard algorithm of division of fraction. The historical approach uses the Babylonia definition of division as the dividend multiplied by the reciprocal of the divisor. The similar denominators approach converts dissimilar denominators to similar ones and proceeds as division of whole numbers. The algebraic approach uses the property of division of numbers, along with a little algebraic manipulation to show how the "invert and multiply" algorithm can be justified. The pedagogical merits of each approach are discussed. The paper concludes that difficulties with fractions, especially, division of fractions can be traced to students' and teachers' personal experience or image. In particular, there is confusion between fractions as mathematical objects and the mathematical operations of fractions.

Efficacy and its effects upon student achievement need nourishment. For a student to believe that he/she can and therefore does achieve needs fostering continuously within each student in the curriculum. With the tremendous emphasis being placed on reading instruction and student achievement by legislators and governors of individual states, efficacy skills need to be promoted so that each learner feels confident in high attainment. This paper discusses how teachers can help students develop feelings of self efficacy, finding that they need to develop a curriculum which stresses the success of the learner within the framework of a challenging program of reading. The paper also considers several curriculum areas besides reading, including social studies, mathematics, and science. It notes that each curriculum area has its own contributions to make in developing the feeling of efficacy in the individual. (NKA)

A person can communicate more accurately by possessing a rich vocabulary, be it in listening, speaking, reading, or writing. Then too, students need a well developed vocabulary which is integrated across the entire curriculum. This paper focuses on the plethora of vocabulary terms in diverse curriculum areas, including social studies, science, mathematics, and language arts. The paper discusses disciplines in each curriculum area in turn and suggests strategies to develop a strong and relevant vocabulary knowledge for each one. It states that the teacher needs to provide a variety of learning opportunities to assist student vocabulary development achievement, and that these learning opportunities should do the following: they should be engaging and involving; they should increase student interest in achieving instructional objectives; they should develop student purpose for learning; they should encourage learning motivation; they should provide for individual learning styles and intelligences; they should help students to become lifelong learners; and they should assist students to evaluate their own progress. (Contains 10 references.) (NKA)

This paper proposed the question "How do we teach logical thinking and sophisticated mathematics to unsophisticated college students?" One answer among many is through the writing of computer programs. The writing of computer algorithms is mathematical problem solving and logic in disguise and it may attract students who would otherwise stop taking mathematics courses after their required sequence is finished. In college classrooms in the United States, there is often an over-involvement with the calculation aspect of mathematics, especially in today's technical environment. The emphasis should fall on the teachers' developing of logic in students. Just like mathematical algorithms, computer algorithms however simple, employ logical steps which will result in the desired conclusion. Mathematics teachers should take advantage of the inumerable opportunities, even in a beginner's computer programming course, to play with algorithms that may aid students in the development of logical ways to approach mathematical problems. (MA)

This paper highlights an attempt by two grade 8 teachers, Bulelwa and Kevin, to draw in the everyday in the teaching of mathematics. Though located in different South African contexts and settings, both teachers tend to enable their learners' access to mathematics by rendering the everyday inauthentic. I argue that inauthenticating the everyday is an unavoidable strategy by which the everyday considerations are silenced and not necessarily teachers' lack of empowerment, as it is sometimes claimed. (Contains 1 figure.) [For complete proceedings, see ED496851.]

In this paper we put forward a theoretical position that, in cognitive terms, a differentiation should be made between a correspondence and a function. Important in understanding this difference is the role of an assignation rule; the correspondence acts as a way to identify a rule in context, whilst the function accommodates the rule in a more formal framework providing a secure base for argumentation. This perspective is used to interpret some students' behavior in a task where the identification of a particular relationship is crucial for its solution. [For complete proceedings, see ED489632.]

The presentation of physics content knowledge in conjunction with the lack of examples in solving problems leads high school students to struggle with mathematical problem solving in physics courses. This paper explains the importance and necessity of having mathematical courses for all high school students. (YDS)

Since nearly 10% of the students in the U.S. who receive doctorates in the mathematical sciences begin their undergraduate studies in two-year institutions, it is clear that these schools are a significant part of the mathematics educational pipeline. Yet, minority students enrolled in two-year colleges are one-fifth as likely to earn a bachelor's degree as those who start out in four-year schools. Given that approximately 50% of minorities in college are enrolled in two-year institutions, the role of these colleges in increasing minority participation in mathematics-related fields cannot be overstated. Mathematics teachers and professors, and those charged with facilitating the learning process, are the major change agents for improving the delivery of mathematics for minorities, as well as for all students. Proven change strategies include: (1) supporting a paradigmatic shift which asserts that all students can and must learn mathematics, and that minorities can succeed in mathematics-based fields; (2) setting up articulation and collaborative programs that make transfer from two- to four-year institutions smoother; (3) encouraging the best students to go into teaching; (4) intensifying minority recruitment; (5) promoting mathematics within minority communities by highlighting the successes of two-year college students in these communities; (6) restructuring remedial courses to incorporate cooperative learning, peer tutoring, and computer-assisted instruction; (7) setting numerical targets for minority student transfer; (8) establishing partnerships with industry; (9) seeking financial and human resources from government; and (10) promoting the teaching and learning function in mathematics. (MAB)

Using literature as a natural catalyst in an interdisciplinary approach, teachers can effectively bring the basics of reading, writing, and mathematics together to provide a learning atmosphere that promotes risk-taking as a natural and necessary part of learning. Integrating these areas throughout the curriculum enables students to develop a sense of purpose in content area classes. Such a sense of purpose will also help students recognize the connections between what is learned in school and what is used in real life. For students to realize a practical application of reading, writing, and mathematics, materials must be interesting and relevant. (MG)

In this update the National Governors Association presents the testimony of Honorable Bill Ritter, Jr., as submitted to the U.S. House Education and Labor Committee. Ritter speaks about the Common Core State Standards Initiative, a joint project by the National Governors Association (NGA) and Council of Chief State School Officers (CCSSO) to develop common core standards in English language arts and mathematics by February 2010. He believes that this initiative has a high probability for success. He further notes how governors recognize that the adoption of a strong set of academic standards is just an initial step toward upgrading state education systems. States have both the authority and the responsibility to provide students with a high-quality education, and many states are already deeply engaged in efforts to raise standards, advance teaching quality, and improve low-performing schools. State adoption standards, state leadership for education improvement, and federal government support are also discussed. Ritter concludes on the topic of federal government support, that moving forward, the reauthorization of the Elementary and Secondary Education Act will be a critical opportunity to rethink and evolve the new federal-state partnership and capitalize on the power of state-led innovations to improve education. Much work remains, and governors stand ready to work with the committee on this important reauthorization.

The relationship between mathematicians and researchers in mathematics education has often been fragile. Yet it is crucial. We conducted a series of themed Focus Group interviews with mathematicians from six UK universities. Pre-distributed samples of mathematical problems, typical written student responses, observation protocols, interview transcripts and outlines of relevant bibliography were used to trigger an exploration of pedagogical issues. Here we elaborate the theme "the relationship, and its potential, between mathematicians and researchers in mathematics education" that emerged from the data analysis. We do so by presenting the participants views on this relationship in terms of: obstacles, desired characteristics and potential benefits. [For complete proceedings, see ED489538.]

The High Schools That Work (HSTW) program is geared to help public high schools meet high standards for many more students by building educational programs around college-preparatory-level academic studies and challenging career concentrations. The program is based on the belief that almost all students can learn advanced-level academic and technical concepts when they are nurtured in an environment of high expectations and support. Based on the results of the 1998 HSTW Assessment of more than 34,000 students at 770 HSTW sites in 22 states, this report answers 3 questions: (1) What progress is being made in getting more career-bound students to meet the HSTW performance goals? (2) Which school and classroom experiences are associated with higher achievement? (3) How can states move toward getting 85 percent of career-bound students to meet the HSTW performance goals? If states' political and educational leaders share the Southern Regional Education Board's belief that rigorous vocational studies can motivate more students to complete an upgraded academic core, and if these leaders accept the lessons described in this report, they need to help more schools and students benefit from these experiences. This effort includes integrating some of the essential practices from the HSTW initiative into state accountability systems and broadening state assessments to link quality vocational studies and challenging academic studies. (DFR)

During the 1994 fall semester, an instructor taught an English 101 section at Central Arizona College-Superstition Mountain Campus that used readings from Graham Flegg's "Numbers: Their History and Meaning" as the basis for 3 of the assigned readings. Only 3 of the 5 assigned essays were based on math--as opposed to all of them--for several related reasons: (1) the degree of difficulty of such assignments for beginning composition students; (2) the belief that not all of the composition students would improve as writers if they felt completely alienated by the essay subject matter (an all-math class); and (3) the sense that retention in the class would suffer if students were not allowed to write an equal number of essays dealing with more "language-friendly" subject areas (history, culture, sociology). Although the students were never completely comfortable with the idea of writing about math and the plaintive request for "regular English essays" was heard frequently, the outcome was generally successful. The class came to realize that written composition can be an effective tool in areas other than literature-based or personal narrative pieces of writing. However, the intellectual difficulty of the assignments may have impeded the students' ability to improve their writing processes and grammatical/syntactical skills. (Contains student responses to the math agenda at various points in the semester, sample writing assignments, and a student writing sample.) (TB)

In New Delhi (India), the Central Institute of Educational Technology in collaboration with Computer Maintenance Corporation and Kentucky Educational Television launched the Classroom 2000+ project to demonstrate the use of interactive technology for distance learning and to experiment with its development in various learning situations. Six schools in Indian cities were linked with the central studio. Sixty-five students participated, responding on computer keypads to lessons in physics and mathematics. Students responded favorably to the lessons and method of presentation. Most (almost 84%) thought they had adequate time to answer teacher questions, and almost 60% thought that lesson pacing was adequate. Instructional technologists and experts expressed both approval and some concerns about implementation. Educational technology may answer the questions of equity and access regarding education, particularly in remote areas of India. (SLD)

Much of what teachers know about how to teach and how to conceptualize specific academic content is learned from their undergraduate professors. In 1990, California State University, Fullerton, received a grant from the Fund for the Improvement of Postsecondary Education (FIPSE), Project Teach, to improve the quality of instruction which prospective elementary school teachers experience in undergraduate academic coursework. Project Teach has engaged junior and senior faculty from three schools at the University (Human Development and Community Service, Humanities and Social Science, and Natural Science and Mathematics) in a discussion of effective teaching and of the relationship of their own teaching to the preparation of teachers. In this monograph, participants representing such diverse fields as child development, elementary education, health, physical education and recreation, psychology, English, political science, linguistics, anthropology, mathematics, physics, chemistry, and biology explain their views of the teaching process through the use of metaphors and describe teaching strategies found to be effective in their own classes. (LL)

Teacher educators have begun to explore the value of teaching cases in the preparation of teachers. The 10 cases presented in this casebook were developed by teachers to reflect on the problems of nurturing young women's talent in mathematics and science and are based on teachers' personal and professional experiences. The casebook describes classroom dilemmas in which gender issues surface in both central and subtle ways. Each case is preceeded by a short abstract and followed by an epilogue and a list of questions to consider. In these cases, teachers tell their own stories and reflect on dilemmas they faced and choices they made. For example, a teacher dedicated to the development of mathematical talent in young women describes how she nurtured a mathematically gifted young girl whose interests suddenly switched at adolescence from mathematics to males; or how an elementary enrichment teacher with a strong mathematics background established an accelerated mathematics class for gifted math students only to have her efforts derailed by school politics. Following an introduction which discusses case development, purposes of teaching cases, and studying and teaching a case, the document is organized into two parts: (1) Professional Problems of Female Teachers; and (2) Puzzles Young Women Pose for Teachers. (LL)

Quantitative literacy, the ability to discriminate between good and bad data, the disposition to use quantitative information to think through complex problems are capacities that educators across fields should be helping students develop. Burke leads the reader to this conclusion through an extended example of income distribution, clarified and made powerful by visual representation. Noting that the content of thought and the depth of understanding are dependent on the tools brought to a task, the writer advocates that the ability to think in terms of quantitative data, in terms of tables and graphs, is indispensable for understanding the modern world and should be part of what is taught to all students, not just those in selected courses or selected majors. We write, after all, says Burke, to figure out what we think. Students must be taught to write so that they will learn how to think.

Asera assesses the challenges that community colleges face in educating students in basic skills. For a significant group of college students the seemingly automatic skills of literacy and numeracy have become opaque, creating a particular challenge for community colleges, 98 percent of which offer at least one remedial reading, writing or mathematics course. Originally such programs were designed to reacquaint returning adults with skills that had become rusty over time; what was needed was a refresher where they could relearn things they had previously learned in high school. Today, these courses are more likely to be populated by students recently out of high school, indicating that they never mastered these essential skills of English and math. Many of these students have had years of negative experiences with school and need courses in which they can, in effect, more successfully learn the content and learn to be students. Over the years, the jargon for such courses has changed: from remedial, to basic skills, developmental education, and pre-collegiate education. Although terminology has changed, teaching, writes Asera, has not. Too often, that is, basic skills courses are taught through drill and memorization of rules, lacking intellectual vitality and engagement. A long-term solution to the problem of under-preparation and student failure must be systemic, addressing alignment of curriculum and assessment across educational sectors. Noting that even as a long-term solution is required, the pre-collegiate classroom needs immediate attention, the writer advocates a different way to think about teaching "basic skills", one that depends on remembering what is actually entailed in successful reading, writing and problem solving, and making the complexity of those processes visible for students so that they can develop strategies for improvement by creating learning environments where students learn about themselves as learners and develop strategies for success.When campus leadership, including administrators, faculty, tutors, counselors, institutional researchers and student peers, join to make pre-collegiate education a campus-wide priority, the ideal of college access for all, Asera's understanding of the mission of community colleges, will evolve from pipedream to pipeline for learning and success.

The first two parts of this paper deal with the potential and reality of mathematics in education. In the third section, some recommendations are made for narrowing the gap which seems to exist between the potential and the reality. Among these fifteen recommendations are: (1) the public should be educated about the nature of mathematics; (2) steps must be taken to improve the attitude of mathematics teachers; (3) there should be a curriculum emphasis on the dynamic process of mathematizing rather than on the static product of this process which is mathematical fact; (4) we must stress the historical roots and contemporary applications of mathematics; and (5) the mathematical community must give educational considerations a higher priority than is now the case. (MK)

This essay explores how the application of the decision sciences in the interdisciplinary training, research, and development activities of model graduate professional schools of management, urban and public affairs, business, government, and regional planning might be linked to current efforts to improve leadership development and training programs in vocational education. Three major topics are treated by addressing three corresponding sets of questions. The first set of questions provides a framework for describing the mathematical and decision sciences, their contemporary influences in graduate professional education, and the issue of whether a common core of mathematical science skills exists that all practitioners should possess. The second group of questions offers a guide for examining the influences of the decision sciences in graduate professional schools traditionally associated with the managerial and social sciences. Discussion focuses on how various professions and specific institutions are approaching the development of mathematical methods and techniques as practical tools for use in everyday situations. The final group of questions explores opportunities to link the decision sciences with leadership development and training efforts in vocational education. Seven appendixes and four inserts, amounting to over one-half of the essay, are papers, articles, or supplemental material on the topic. (YLB)