Publication Type: Abstract
Abstract: Often teachers use pedagogical methods that focus and elaborate on the peripheral issues of a topic. Such emphasis is mostly directed towards the very specific applications of concepts to certain phenomena. Scattering the material in this way delivers a message to the students that what they are leaning is specific only to the course they are in. They fail to apply solutions to problems that they may later face in another course or in their professional life because of their inability to perceive the underlying basic concepts in what is at hand, despite the fact that they were previously exposed to these concepts. However, their exposure was centered on an application of a concept with very little deep understanding of the concept itself.

In order to facilitate students’ learning, it is much more effective to start with the concept as a bull’s eye of a multi-concentric circle system and then expand outward towards applications and adaptations of the concepts to different areas of relevance to learning the concept. The expansion is in the form of concentric circles whereby the specificity of the applications increases as the circles expand outwards.

This pedagogical approach constantly reminds the students of the very basic concept involved, ie, the bull’s eye. The poster will illustrate such a modality using the ionization character of weak acids and bases as the concept and expanding from there to applications such as absorption, distribution, metabolism, and excretion of drugs that are weak acids and bases or salts thereof. Other applications are buffers, precipitation as pH changes, IV admixtures, and pharmaceutical products design and development, among others. Finally, this modality nurtures problem-solving and critical thinking skills as the learner is directed to reexamine the assumptions made as he/she tries to delve into the concept upon which an application is based.

Publication Type: Session Presentation
Abstract: The paper discusses middle voice in Mandarin Chinese from two research perspectives, semantic and syntax. It contains two major purposes. The first purpose is to explain the syntactic constraints and verbal semantic conditions. Second, this paper will demonstrate a proper pedagogical grammar of middle voice construction for a Chinese teaching curriculum. Based on our analysis of the middle voice construction and the concerns of frequency, structural complexity and crossligulistic analysis, we provide a teaching sequence of middle voice construction.

Publication Type: Conference Paper/Unpublished Manuscript
Abstract: For the last two decades, the issue of academic responsibility has been a common concern of both mainstream and critical scholars of international relations (IR). Most of the discussions have framed the problematic in terms of the relationship between theory and practice, and as such, have focused more on the content of the knowledge produced in IR than on the process of knowledge production. Missing from the debate, specifically, is a systematic examination of the social relations of knowledge production.Starting from the claim that IR scholarship has historically been, and continues to be complicit in producing and sustaining imperialist/colonial power relations, this paper argues for the need to critically reflect on the conditions of (im)possibility for the theorization and practice of global politics created through the obscuring of the socio-political relations of knowledge production. Informed by a feminist project of (de)colonizing knowledge production in IR, this paper challenges the colonial compartmentalization of being and knowing by re-locating IR pedagogical practices in the socio-political process of knowledge production. By drawing connections between the ?intimate? site of identity production within the IR classroom and the broader political power dynamics of imperialism, this paper shows how IR pedagogical practices participate in the creation and naturalization of gendered, racialized, classed, and sexualized categories and subjects. Informed by an understanding of Empire-building as involving the production and deployment of not only militaries and capital, but also of subjectivities, the paper concludes by suggesting that those of us who are committed to a project of resistance and opposition to imperializing/colonializing discourses and practices need to reflect on the (im)possibilities created by our pedagogical practices.

Publication Type: Conference Paper/Unpublished Manuscript
Review Method: Peer Reviewed
Abstract: Exploring Middle School Teachers’ Pedagogical Content Knowledge of Fractions and Decimals

This study is a part of a larger project that aims at developing and sustaining a professional development program that will support middle school mathematics teachers throughout their career stimulating their intellectual growth and improving their content knowledge and pedagogical skills. An important premise of the approach this project is taking is that the program should evolve based on local situations and local needs and any activities that we intend to design for achieving the goals of the program should be tailored by the specificities of the local middle school. Hence, as a first phase of this project we engaged in a study that focused on identifying what are the areas of teachers’ knowledge that need improvement and, respectively, what content should the professional development program deliver to teachers in order to have an impact on students’ achievement. This paper reports on the findings of this study, and more particularly it focuses on describing the main difficulties that 6 grade teachers experience in teaching the notions of fractions and decimals.
Rationale
Since the end of the 1980’s there has been a strong effort to strengthen and reform the mathematics education in all levels and to improve the preK-12 instructional force (NCTM, 2000; National Research Council, 2001; National Commission on Mathematics and Science Teaching for the 21st Century, 2000). It is widely recognized that high quality teacher preparation both in content and pedagogy is critical for the improvement of student outcomes in mathematics. Numerous studies show that many teachers, especially at the middle school level do not have sufficient content knowledge or adequate background for teaching mathematics. In many schools, middle school teachers with a generalist background, K-8 certification are assigned to teach science and mathematics exclusively (NCR, 2001). Further, teachers need assistance in developing challenging and updated curricula and instructional materials to bring the teaching and learning of mathematics to high standards. Many teachers have been driven to develop their own curriculum materials because the available textbooks were not sufficient to meet the goals of reasoning, communicating, connecting, and problem solving in mathematics; nor were the text materials sufficient to develop an understanding of important concepts, skills, and procedures in mathematics (Lappan, 1998).
Current research in teacher development suggests that teachers should also incorporate content-appropriate methods of teaching that promote students conceptual understanding in mathematics. Several studies (Steinberg, 1985; Carlsen, 1988; Brown & Borko, 1992; Manouchehri, 1997; Ball, 1998; Ball &Bass, 2000) have concluded that content knowledge and understanding of the methods of inquiry in mathematics is at the core of effective teaching and learning. For instance, it has been found (Carlsen, 1988; Ball & Bass, 2000) that teachers with deep conceptual understanding of mathematics not only know more content but also use their content knowledge more effectively in their classrooms. In addition, the research on teachers’ preparation in mathematics (Shulman, 1986, Ball & Bass, 2000) suggests that teachers should possess knowledge that integrates content and pedagogy, called pedagogical content knowledge. In mathematics, this kind of knowledge may include useful representations, unifying concepts, clarifying examples and counter examples, helpful analogies, and important relationships and connections among concepts (Grouws & Shultz, 1996). Thus, pedagogical content knowledge is essential for planning and executing lessons that facilitate effective students learning.
Following the above mentioned recommendations of national organizations and the suggestions of recent research in mathematics education, we centered our professional development efforts at improving teachers’ pedagogical content knowledge. In order to develop effective means (workshops, study groups, summer institutes) for improving teachers’ instructional strategies and students achievement in mathematics, we in engaged in a study of the current status of teachers’ pedagogical content knowledge as reflected in their everyday classroom practices.
Methodology
The research took place in a rural middle school in the northeast, over the course of a year, as a part of efforts for establishing a professional development program for middle school teachers. In this paper we focus on three sixth-grade teachers in the process of teaching the notions of fractions and decimals. We made regular classroom observations and we had weekly meetings with the teachers discussing teaching strategies, students’ learning, and curriculum issues. During the classroom observations and the discussions with the teachers we took detailed notes and wrote memos which subsequently were analyzed and briefly summarized.
Results
In summary, we found out that all of the three teachers were trying to employ innovative approaches for instruction incorporating problem solving, classroom discourse and hands-on activities. However, their instructional activities were designed and carried out as goals in themselves and did not lead to conceptual understanding of mathematical ideas and the connections among them. In our weekly discussions the teachers shared their concerns and asked for assistance about various aspects of their mathematical preparation and teaching practices. Their main concerns were that they experience difficulties in organizing and sequencing the mathematical topics in a way that will allow them to present the mathematical ideas in a coherent and connected way; they had difficulties in identifying the conceptual prerequisite necessary for the introduction of new concepts and connecting it to other previously studied mathematical concepts; they had difficulties in explaining mathematical ideas and engaging the students in the process of exploring them.
For instance, some of the difficulties experienced by the teachers in the context of fractions and decimals were related to:
1. Learning theories. Teachers failed to develop the operational conception (Sfard, 1991) of fraction as cutting on and taking parts of the whole; instead they introduced directly the symbolic form of a fraction and illustrated various fractions with manipulatives and pictures. Later the teachers had difficulties in relating fractions to division of whole numbers.
2. Connecting mathematical ideas. Teachers had difficulties in connecting previously studied prime factorization of numbers with simplifying fractions and finding common denominator for comparing and adding fractions. They introduced new rather complicated algorithms for finding greatest common divisor and least common multiple.
3. Employing hands on activities and problem solving. In introducing the operations with fractions, manipulatives and problem situations were used mainly for illustration of readily given rules rather then as means for exploration and discovery.
4. Sequencing the topics. One of the teachers, contrary to the textbook suggestion, introduced percentages before introducing the notions of ratio and proportions and, thus caught by surprise, experienced difficulties teaching. On the other hand, all of the teachers intuitively felt that the suggested by the textbook sequencing of topics, first decimals then fractions, is inappropriate and reversed the order of introducing the concepts. However, they had difficulties in developing their own curriculum materials and lessons to make the transition from fractions to decimals smooth and they failed to make the connections between fractions and decimals explicit. The main difficulties were coming from teachers’ reluctance to engage in multi-step operations with symbols. As a result, many of the students did not develop deep conceptual understanding of decimals and some common misconceptions surfaced.
Conclusions
The results of our study suggest that the activities of the professional development program addressing teachers’ pedagogical content knowledge should focus on enhancing teachers’ ability to connect mathematical ideas and, particularly, on their skills to work with symbols when properties of operations with fractions and decimals can be meaningfully derived from prior notions and conceptually connected with them. An important finding of our study is that teachers need further instruction and opportunities for exploration on how to use effectively hands-on activities and manipulatives in a classroom instruction. More specifically, teachers need help in making the transition from using hands-on activities and manipulatives just as means for illustration of directly introduced mathematical concepts to utilizing these tools as means for exploration and discovery leading to students’ deep conceptual understanding of mathematics.

References
Lappan, G. (1998). Texts and teachers: Keys to improved mathematics learning. http://www.nctm.org/news/pastpresident/1998-07_lappan.htm
Ball, D.L. (1998). Unlearning to teach mathematics. For the Learning of Mathematics 8(1): 40-48.
Ball, D.L. & Bass, H. (2000). Interweaving content and pedagogy in teaching and
learning to teach: Knowing and using mathematics. In Jo Boaler (Ed.), Multiple
Perspectives on Teaching and Learning (pp.83-104). Westport, CT: Ablex Publishing.
Before is too late. (2000). A report to the nation from the National Committee on Mathematics and Science Teaching for the 21st Century. http://www.ed.gov/americacounts/glenn/report.pdf
Brown & Borko, (1992)
Carlsen, W. (1988). The effect of science teacher subject matter knowledge on teacher questioning and classroom discourse. Unpublished doctoral dissertation. Stanford University, CA.
Grouws, D.A., and Schultz, K.A. (1996). In Sikula, J. (ed.), Handbook of Research on Teacher Education, 2nd ed. New York: Macmillan.
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National Research Council (2001). Educating teachers of science, mathematics, and technology: New practices for the new millennium. Committee on science and teacher preparation. http://www.nap.edu/books/0309070333/html/
NRC (1999). How People Learn: Brain, Mind, Experience, and School. Bransford, John D., Brown, Ann L., and Cocking, Rodney R. (eds.). Washington, DC: National Academy Press.
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Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher 15,4-14.
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Publication Type: Research Report
Abstract: This study delineates an effective method for assessing elementary students’ understanding of a core representational format, the number line. Confronting students with irregular transformations (e.g., non-equally spaced tickmarks) provides opportunities to elicit students’ implicit understandings. Students have opportunities to grapple with, and eventually make explicit, hidden conventions of this representation, providing an assessment of student understanding and difficulty.