Mathematics-for-Teaching: an Ongoing Investigation of the Mathematics that Teachers (Need to) Know

In this article we offer a theoretical discussion of teachers' mathematics-for-teaching, using complexity science as a framework for interpretation. We illustrate the discussion with some teachers' interactions around mathematics that arose in the context of an in-service session. We use the events from that session to illustrate four intertwining aspects of teachers' mathematics-for-teaching. We label these aspects “mathematical objects,” “curriculum structures,” “classroom collectivity,” and “subjective understanding”. Drawing on complexity science, we argue that these phenomena are nested in one another and that they obey similar dynamics, albeit on very different time scales. We conjecture (1) that a particular fluency with these four aspects is important for mathematics teaching and (2) that these aspects might serve as appropriate emphases for courses in mathematics intended for teachers.     

Ready...Fire...Aim! Toward meaningful technology standards for educators and students

Conclusion I applaud ISTE, AASL, AECT, and the other organizations involved for tackling the “messy work” of developing standards for the use of technology and information resources in schools. And, at the same time, I call for a “second generation” of standards that define realistic expectations for teachers based on the subjects and levels they are called upon to teach. I propose that professional organizations from each subject work with ISTE and AECT to complete this huge task, and I propose that we consider as a “next step” the creation of a set of on-line learning experiences through which teachers can gain the identified skills and knowledge by using the very technologies we’re hoping they’ll embrace in their own teaching. There’s an old saying, “If you don’t know where you’re going, any road will do.” As far as educational technologies are concerned, this is also true. For many, the goal seems to have been simply to “get more computers into the schools,” without much thought about purpose. To return to Phil Schlechty’s metaphor, It’s generally been a brief and misguided “Ready” stage (occupied with questions like “How many do we need?” “What type?” “Where?” and “How shall we connect them?”), followed by “Fire!” (the acquisition and installation of equipment). What we need is: “Ready” (the creation of appropriate teams of people who will combine their insights to plan for the district)... “Aim” (a series of discussions about what technologies can accomplish for schools and the students they serve)... “Fire” (acquisition, installation, and professional development according to plan)... “Aim” (an assessment of how well the technologies and related programs met the intended goals, and a new planning effort designed to close the gap)... “Fire” (acquisition and implementation designed to eliminate the gap)... “Aim” (another gap assessment)... “Fire” (another attempt to close gaps)..., And so on.     

Audio-visual education in urban school systems

Summary impressions In some ways, this report shows progress in audio-visual education. The work is being headed up by more widespread adoption of central coordination, more classroom teachers are prepared to use audio-visual materials in their classes, and there is more wide-spread use of the materials. On the other hand, there are some urgent needs that become so evident from these data. “More time” is the need, “more central coordination” is the need, “more adapted classrooms” is the need, and separately and behind all these is the need for “better support.” This article is a digest of a report given at the DAVI national convention in Los Angeles in April. The address was entitled “A First Look at Findings of the NEA Survey of the Status of Audio-Visual Education in City and County School Systems.” Victor Hornbostel is assistant director of the Research Division of the National Education Association and supervisor of the survey which is reported.     

“Is This a Boy or a Girl?”: Rethinking Sex-Role Representation in Caldecott Medal-Winning Picturebooks, 1938–2011

A number of previous studies have addressed gender role-stereotyping in Caldecott Award-winning picturebooks. Building upon the extensive scholarship examining representations of females in Caldecott books, this current study offers a critical investigation of how gender is represented in Caldecott Medal-winning literature from 1938 to 2011 by exploring the ways in which “femininity” and “masculinity,” biological sex, and gender are constructed in these texts. The investigators briefly address author and illustrator gender and the representations of males and females as characters or images in pictures and text before departing from previous scholarship to offer a rereading of books that feature “ungendered” leading characters, those that are not identified in the text as being either “male” or “female” and are therefore open to the interpretation of individual readers. By resisting cultural cues and normative constructions of gender and biological sex, these ungendered depictions extend the range of possible ways in which readers may see themselves or those in their lives represented in Caldecott Medal-winning picturebooks.     

Differential Patterns of Change and Stability in Student Learning Outcomes in Holland’s Academic Environments: The Role of Environmental Consistency

The findings of this study show wide variation in the learning patterns of college students in the academic environments of Holland’s theory and, more importantly, that such variability differs based on the level of “consistency” or “inconsistency” of the environments. Differences in the learning patterns of students in “consistent” academic environments tend to be more in alignment with the premises of Holland’s theory than those of students in “inconsistent” environments. Implications of these findings for future research using Holland’s theory to understand longitudinal patterns of change and stability in the attitudes, interests, and abilities of college students are discussed, and attention is devoted to policy development and practical implications for academic advisors, career counselors, campus leaders, and governmental and accrediting officials.     

The “System of Chymists” and the “Newtonian dream” in Greek-speaking Communities in the 17th–18th Centuries

The acceptance of new chemical ideas, before the Chemical Revolution of Lavoisier, in Greek-speaking communities in the 17th and 18th centuries did not create a discourse of chemical philosophy, as it did in Europe, but rather a “philosophy” of chemistry as it was formed through the evolution of didactic traditions of Chemistry. This “philosophical” chemistry was not based on the existence of any academic institutions, it was focused on the ontology of principles and forces governing the analysis/synthesis of matter and formulated two didactic traditions. The one, named “the system of chymists”, close to the Boylean/Cartesian tradition, accepted, contrary to Aristotelianism, the five “chymical” principles and also the analytical ideal, but the “chymical” principles were not under a conceptual and experimental investigation, as they were in Europe. Also, a crucial issue for this tradition remained the “mechanical” principles which were under the influence of the metaphysical nature of the Aristotelian principles. The other, close to the Boylean/Newtonian tradition, was the integrated presentation of the Newtonian “dream”, which maintained a discursive attitude with reference to the “chemical attractions”–“chemical affinities” and actualised the mathematical atomism of Boscovich, according to which the elementary texture of matter could be causally explained within this complex architecture of mathematical “punkta”. In this tradition also coexisted, in a discursive synthesis, the “chemical element” of Lavoisier and the arguments of the new theory and its opposition to the phlogiston theory, but the “chemical affinities” were under the realm of the “physical element” as “metaphysical point”.     

Hutchins’s University of Utopia: Institutional Independence,Academic Freedom,and Radical Restructuring

In a crisis-plagued world looking to higher education for knowledge, wisdom, and solutions, higher education itself is stumbling. Its transformational thinking has frozen up like an overstressed computer program; and we need, in effect, to “push the reset button.” In 1953, the renowned and controversial president of the University of Chicago, Robert M. Hutchins, authored a refreshing and provocative work, The University of Utopia, containing ideas that still challenge today’s paradigms. He argued for institutional independence over “accountability,” “outcomes,” and “stakeholders.” He indicted educational evils he called “industrialization,” “specialization,” “philosophical diversity,” and “social and political conformity” and suggested ways to defeat them. Although his 56-year-old thoughts on reconceptualizing the multiversity are not a panacea, they could help higher education make a fresh start. This essay reintroduces the modern reader to Hutchins’s iconoclastic and stimulating ideas in the hope of restarting the stalled agenda for educational reform.     

The study on the core concepts of contemporary sociology of education and its theoretical construction

Within the sphere of contemporary social sciences, the terms “modernity,” “post-modernity” and “globalization” have penetrated, as the core concepts, into various fields of social sciences in a logical way. In constituting the concept of “modernity,” sociology of education develops the educational theory, as sociological theory does, into a “grand narrative” and “foundationalist” theory; the contribution of post-modernity is pluralism and self-examination in an attempt to transcend modernity. Globalization, a kind of expanded modernity, makes education sociologists broaden their perspective from single and traditional “nationality,” “society” and “nation” to an international society and even to the global society, which has broken through the research paradigm of modernity with ethnocentrism, anthropocentrism and Euro-centrism. These changes have required urgent constitution of the conceptions and theoretical frameworks of sociology of education to be applied on a global level. __________ Translated from Peking University Education Review, 2006 (1)     

Knowledge of derivational suffixes in students with language learning disabilities

Knowledge of derivational suffix meanings was investigated in 10- to 12-year-old students with language learning disabilities and individually matched chronological- and language-age peers. Students produced derived forms from nonce bases in an elicitation task and selected derived forms in a forced choice task. For instance, students produced a word to label the baby of an invented animal called a TEM (TEMlet, TEMette, TEMkin, etc.) and chose an appropriate label from four options (TEMlet, TEMkin, TEMship, TEMhood). Twenty-eight suffixes (e.g., less, y, ize) conveying seven derivational meanings (e.g., “without X,” “approximately X,” “to make X”) were studied. All groups showed higher accuracy on the forced choice task than on the elicitation task. However, elicitation task accuracy of students with language learning disabilities fell substantially below that of typically achieving students. Suffixes for “agent X,” “character of X,” and “to make X” were produced in the elicitation task with higher accuracy than suffixes for “approximately X,” “diminutive X,” and “state of X.” In both tasks, suffix use was associated with productivity (i.e., regularity of suffix attachment). All groups chose highly productive suffixes (e.g., TEMlet) over less productive suffixes (e.g., TEMkin) to convey each meaning.     

The use of examinations for revealing and distinguishing between students' misconceptions,misunderstandings and “no conceptions” in college chemistry

Students at all ages hold a wide variety of scientifically faulty knowledge structures called “misconceptions”. As far as misconceptions in chemistry are concerned, college science students are no exception. Systematic administration to freshman biology majors of specially-designed mid-term and term higher-order cognitive skills (HOCS)-oriented examinations within the courses “General and Inorganic Chemistry” and “Introduction to Modern Organic Chemistry” proved these examinations to be very effective in revealing and distinguishing between students'misconceptions, misunderstandings, and“no conceptions”. Several of these have never been mentioned before in the relevant research literature. Accordingly, reflective teaching strategies to overcome this “misconceptions problem” and affect meaningfully subsequent learning have been explored and implemented within our longitudinal effort to develop students' HOCS. The study results combined with accumulated experience indicate that properly designed HOCS-oriented examinations may be very effective for revealing, but notper se for overcoming, students' misconceptions. However, within HOCS-oriented chemistry teaching, the assessment of students by such examinations is very useful particularly for providing data for remediation purposes via appropriate modification of the teaching strategies. Eventually, this leads to gains in students' HOCS which is in line with the overall goal of the current reform in science education.     

Workplace correlates and scholarly performance of clinical pharmacy faculty

The workplace, especially the academic department or division, provides an intriguing research site for study of correlates of scholarly performance. The objective of this study was to estimate a correlate model of three-year scholarly performance of clinical faculty in colleges of pharmacy. Two-hundred and ninety-six clinical faculty in pharmacy responded to an author-developed questionnaire. The scholarly performance items factored into three dimensions: “Refereed Research,”, “Grants/Books Research,” and “Nonresearch Scholarship.” A fourth single-item variable, “Contracts,” was also used as a dependent dimension. Eight correlates, including two related to the departmental workplace, emerged as factors significantly explaining the variation in scholarly performance. The two most significant were “Off-Campus Conversations” and “Department: Percent Time in Research.” These results as well as others were contrasted with prior correlate studies and suggestions were made for future additional study.     

An Experiment in Teaching Ratio and Proportion

This paper summarizes our analysis of the complexity of ratio problems at Grades 6 and 7, and reports a two-year experiment related to the teaching and learning of rational numbers and proportionality in these grades. Two classes were followed and observed. Part of the teaching material was common to both classes, mainly the objectives and the corpus of ratio problems in a physical context. But in one class, here called “Partial-experiment”, the learning environment was exclusively a paper-pencil one and the teacher followed his usual method in designing and conducting teaching sequences. In the other class, here called “Full-experiment”, the teaching was based on a framework, emerging from our analysis of complexity of ratio problems, involving precise guidelines and a specific computer environment. Using a pre-test and a post-test, we observed clear progress in both classes compared to a sample of “standard” pupils. Our comparative pupil-oriented study indicates more complete improvement in the “full-experiment” class, i.e., a better acquisition of fractions and their use for solving usual proportionality problems. The average pupil’s progress is greater in the “full experiment”, with the pupils who were initially high- or low-level attainers benefiting the most from the “full-experiment”.     

Affection and cognition in transaction and the mapping of cultural space

Summary In this article, an attempt has been made to treat affection and cognition “transactionally” and to report work underway in the development of a rationale for mapping “sign components” of “cultural space.” The development of statistical procedures and applications remains to be undertaken.     

No Country for young people? Anxieties in Australian society and education

Australia is often portrayed as a place in which young people are in jeopardy. In 19th century literature and art, the recurring motif of a child lost in “the bush” became an increasingly significant dimension of European settlers’ experiences of Australia, whereas during the latter half of the 20th century analogous cultural narratives shifted towards urban environments and the plight of young people endangered by their parents’ generation. In contemporary popular culture, Australia’s cities and suburbs are places where children are aborted, abandoned, exploited, murdered or never conceived. Like the bush-lost children before them, these “at risk” young people symbolise adult fears of self, society, and the future. My concern is that the most common public policy response to these persistent fears and insecurities is to retreat to a politics of complexity reduction. Many politicians and public opinion leaders see teachers and schools as being in the vanguard of people and institutions dedicated to Australian children’s educational ruin, and simplistically seek to “protect” them with blunt instruments such as back-to-phonics literacy and a national curriculum. I argue that Australia’s young people are much more seriously endangered by the symbolic violence of those who position them as docile receptors of whatever schools and teachers serve up to them, and who treat them as passive screens upon which to project their own anxieties.     

How Children Regulate their Own Collaborative Learning

In this article we analyze the dialogic learning of one pair of students in order to investigate how these students cope with a collaborative learning situation in the classroom. Our aim is to substantiate the claims that not only are young students (8 year olds) capable of solving mathematical problems collaboratively, but that they also take an active role in regulating their collaborative learning activities. More specifically, our claim is that children appear to apply constructs of “mathematical level raising”, “social interaction” and “division of time” to steer their own collaborative learning and that they are rather successful in balancing these three aspects. The analysis is exploratory, but this new perspective on collaborative learning is relevant theoretically and consequential for classroom practice.     

Facing multiculturalism’s challenges in Korean education and society

Multicultural policy in South Korea faces variants of challenges endemic to multiculturalism. These challenges are “dilemmas of difference,” “variable terms of inclusion,” and “legitimacy.” In Korea, these challenges arise in a setting in which ethnic diversity is of relatively recent origin, an ideology of ethnic homogeneity is prevalent, and official multicultural policy is limited in its reach to those who are designated as “multicultural families,” that is families in which one spouse is Korean and the other an immigrant, usually the wife, and their offspring. The exclusion of migrant workers and their families from Korea’s multicultural framework poses a core contradiction in Korean multicultural policies. This contradiction must be resolved if multiculturalism in education and other spheres is to promote equality and provide a foundation for national integration on terms that are equitable to the diverse constituents of Korean society. Inevitably, this will require a redefinition of what it means to “be Korean.” Even if multicultural policies fall short in their immediate effects on those toward whom they are directed, multiculturalism represents a significant shift in the discourse of Korean identity and will be terrain on which the status of diverse groups in Korea will be contested.     

Hypervideo

Methods for incorporating video in hypermedia programs are discussed in this article. Hypermedia is defined, and linear and interactive video are contrasted. It is argued that hypermedia presents unique problems in knowledge representation affecting video production, and that techniques are required differing from those for linear video and other forms of interactive video. Suggestions are made for giving interactive video the “look and feel” of hypermedia that primarily involve condensing information and accelerating presentation pace. They are based on the authors' experiences evaluating interactive media and creating “compressed” video programs. Existing media and visual ability research is re-interpreted to support these interactive, “hypervideo” methods.     

Mathématiques de la vie quotidienne au Burkina Faso: une analyse de la pratique sociale de comptage et de vente de mangues

Mathematics teaching in Burkina Faso is faced with major challenges (high illiteracy rates, students’ difficulties, and high failure rates in mathematics, which is a central topic in the curriculum). As evidenced in many of these studies, mathematics is reputed to be tough, inaccessible, and far from what students live daily. Students here look as though they are living in two seemingly distant worlds, school and everyday life. In order to better understand these difficulties and to contribute in the long run to a more adapted teaching of mathematics, we tried to document and elicit the “mathematical resources” mobilized in various daily life social practices. In this paper, we focus on one of them, the counting and selling of mangoes by unschooled peasants. An ethnographic approach draws on the observation of the situated activity of counting and selling mangoes (during harvesting) and on “eliciting interviews” of the involved actors. The analysis of results highlights a richness of structuring resources mobilized and distributed through this practice, related to what Lave (1988) call “the experienced lived-in-world” and “constitutive order.” The mathematical resources take the form of “knowledge in action” and “theorems in action” (Vergnaud, Rech Didact Math 10(23):133–170, 1990), embedded in the social, economic, and even cultural structures of actors.     

Help! I'm selecting children's books

Publishing houses, like the “mill” in the old folk tale “Why the Sea is Salty”, churn out books unceasingly day after day. There are over 2,000 children's books published each year and these books are of varying quality.