The Influence of Different Representations on Solving Concentration Problems at Elementary School

This study investigated the students’ learning process of the concept of concentration at the elementary school level in Taiwan. The influence of different representational types on the process of proportional reasoning was also explored. The participants included nineteen third-grade and eighteen fifth-grade students. Eye-tracking technology was used in conducting the experiment. The materials were adapted from Noelting’s (1980a) “orange juice test” experiment. All problems on concentration included three stages (the intuitive, the concrete operational, and the formal operational), and each problem was displayed in iconic and symbolic representations. The data were collected through eye-tracking technology and post-test interviews. The results showed that the representational types influenced students’ solving of concentration problems. Furthermore, the data on eye movement indicated that students used different strategies or rules to solve concentration problems at the different stages of the problems with different representational types. This study is intended to contribute to the understanding of elementary school students’ problem-solving strategies and the usability of eye-tracking technology in related studies.     

Using diagrams as tools for the solution of non-routine mathematical problems

The Mathematics education community has long recognized the importance of diagrams in the solution of mathematical problems. Particularly, it is stated that diagrams facilitate the solution of mathematical problems because they represent problems’ structure and information (Novick & Hurley, 2001; Diezmann, 2005). Novick and Hurley were the first to introduce three well-defined types of diagrams, that is, network, hierarchy, and matrix, which represent different problematic situations. In the present study, we investigated the effects of these types of diagrams in non-routine mathematical problem solving by contrasting students’ abilities to solve problems with and without the presence of diagrams. Structural equation modeling affirmed the existence of two first-order factors indicating the differential effects of the problems’ representation, i.e., text with diagrams and without diagrams, and a second-order factor representing general non-routine problem solving ability in mathematics. Implicative analysis showed the influence of the presence of diagrams in the problems’ hierarchical ordering. Furthermore, results provided support for other studies (e.g. Diezman & English, 2001) which documented some students’ difficulties to use diagrams efficiently for the solution of problems. We discuss the findings and provide suggestions for the efficient use of diagrams in the problem solving situation.     

Natural language as a tool for analyzing the proving process: the case of plane geometry proof

In the field of human cognition, language plays a special role that is connected directly to thinking and mental development (e.g., Vygotsky, 1938). Thanks to “verbal thought”, language allows humans to go beyond the limits of immediately perceived information, to form concepts and solve complex problems (Luria, 1975). So, it appears language can be studied as a cognitive process (Chomsky, 1975). In this investigation, I study language as a means for making the cognitive process explicit. In particular, I analyze the role of the verbalization produced by pairs of students solving a plane geometry problem. The basic idea of my research is that, during the resolution process of a plane geometry problem, natural language can play roles beyond that of communication: Natural language can be seen as a tool for supporting students’ cognitive processes (Robotti, 2008), and, at the same time, it can also be seen as a researchers’ tool which allows us to shed light on the evolution of students’ cognitive processes. With regard to language as researchers’ tool, I show how natural language (in our case, students’ verbalization during resolution of a plane geometry problem) can be used by the researcher to make explicit, to study, and to describe the development of the students’ cognitive processes during the resolution process. To this end, I present a model I have developed that allows us to identify, in students’ verbalization, different phases of their cognitive processes.     

Re-situation challenges for international students ‘becoming’ researchers

This paper presents data generated during a semester-long programme to support international students from countries in Melanesia and Asia embarking on masters research in education in a New Zealand university. All were scholarship recipients. The researcher-and facilitator-of the programme, was interested in documenting and understanding the nature of the students’ experience as they planned and wrote research proposals. The process of developing a research proposal, as one of the early stages of ‘becoming’ a researcher, highlighted a number of challenges for the six case study students. The challenges are viewed from a transition or ‘resituation’ perspective (Eraut in Stud Contin Educ 26(2): 247–74, 2004, 2008) rather than an adjustment one. A resituation perspective assumes that students brought with them “personal expertise, practical wisdom and tacit knowledge” (Eraut 2008, p. 42) which needed to be reconciled with what was demanded of them by different aspects of the research planning process. The resituation challenges experienced by the students included situating a perceived problem or issue in the research literature; reconciling personal research goals with the limitations of one’s own agency as a researcher; integrating new learning with research goals; and reconciling the new role or identity as a researcher with the previous role as colleague or community member. The paper presents a case for providing a context for postgraduate students in which explicit recognition of what they bring to the research task, and acknowledgement of the resituation challenges can take place.     

An examination of middle school students’ representation practices in mathematical problem solving through the lens of expert work: towards an organizing scheme

Representation is viewed as central to mathematical problem solving. Yet, it is becoming obvious that students are having difficulty negotiating the various forms and functions of representations. This article examines the functions that representation has in students’ mathematical problem solving and how that compares to its function in the problem solving of experts and broadly in mathematics. Overall, this work highlights the close connections between the work of experts and students, showing how students use representations in ways that are inherently similar to those of experts. Both experts and students use representations as tools towards the understanding, exploration, recording, and monitoring of problem solving. In social contexts, experts and students use representations for the presentation of their work but also the negotiation and co-construction of shared understandings. However, this research also highlights where students’ work departs from experts’ representational practices, hence, providing some directions for pedagogy and further work.     

On negotiating White science: a call for cultural relevance and critical reflexivity

This article is a response to Randy Yerrick and Joseph Johnson’s article “Negotiating White Science in Rural Black America: A Case for Navigating the Landscape of Teacher Knowledge Domains”. They write about research conducted by Yerrick in which videos of his teaching practice as a White educator in a predominately Black rural classroom were examined. Their analysis is framed through Shulman’s (1986) work on “domains of teacher knowledge” and Ladson-Billings’ (1999) critical race theory (CRT). Although we appreciate a framework that attends to issues of power, such as CRT, we see a heavier emphasis on Shulman’s work in their analysis. We argue that a culturally relevant pedagogy (CRP) framework has the potential to provide a more nuanced analysis of what occurred in Yerrick’s classroom from a critical lens. Thus we examine Yerrick and Johnson’s work through the five main CRP components (as defined by Brown-Jeffy and Cooper 2011) and ultimately argue that science educators who want to promote equity in their classrooms should engage in continuous critical reflexivity, aid students in claiming voice, and encourage students to become not only producers of scientific knowledge but also users and critics of such knowledge.     

Making the most of the mosaic: facilitating post-school transitions to higher education of disadvantaged students

Research studies of post-school education and training conducted in Australia and internationally have revealed a mosaic of students’ education and employment experiences, with a multiplicity of nonlinear pathways. These tend to be more fragmentary for disadvantaged students, especially those of low socio-economic background, rural students, and mature aged students seeking a ‘second chance’ education. Challenges faced by students in their transitions to higher education are made more complex because of the intersection of vertical stratification created by institutional and sectoral status hierarchies and segmentation, especially relating to ‘academic’ and ‘vocational’ education and training, and the horizontal stratification of regional, rural and remote locations in which students live. If we are to achieve the equity goals set by the Bradley Review (Bradley et al., Review of Australian Higher Education Final Report, 2008) we need to acknowledge and work with the complex realities of disadvantaged students’ situations, starting at the school level. Interrelated factors at the individual, community and institutional level which continue to inhibit student take-up of higher education places are discussed in the context of discursive constructions of ‘disadvantage’ and ‘choice’ in late modernity. Research highlights the need to facilitate students’ post-school transitions by developing student resilience, institutional responsiveness and policy reflexivity through transformative education.     

Developing a Reflective Practitioner Through the Connection Between Educational Research and Reflective Practices

This paper outlines the connection between qualitative research methods in education and teacher reflective practices as they relate to Valli’s (Reflective teacher education: cases and critiques. State University of New York Press, Albany, 1992; Peabody J Educ 72(1): 67–88, 1997) model of reflection. Using the authors’ own experiences in performing and guiding educational research, and existing research in the field of teacher education pertaining to reflective practitioners, explicit connections are made between the two paradigms. These connections illustrate the importance of integrating authentic research experiences into the teacher education curriculum outside the context of methods courses, much like models established in the sciences.     

Justification and Persuasion about Cloning: Arguments in Hwang’s Paper and Journalistic Reported Versions

We examine the argumentative structure of Hwang et al.’s (2004) paper about human somatic cell nuclear transfer (SCNT, or ‘therapeutic cloning’), contrasted with four Journalistic Reported Versions (JRV) of it, and with students’ summaries of one JRV. As the evaluation of evidence is one of the critical features of argumentation (Jiménez-Aleixandre 2008), the analysis focuses on the use of evidence, drawing from instruments to analyze written argumentation (Kelly et al. 2008) and from studies about the structure of empirical research reports (Swales 2001). The objectives are: 1) To examine the use of evidence and the argumentative structure of Hwang et al.’s Science, 303: 1669–1674 (2004) original paper in terms of the criteria: a) pertinence of the evidence presented to the claims; b) sufficiency of the evidence for the purpose of supporting the claims; and c) coordination of the evidence across epistemic levels. 2) To explore how the structure of Hwang’s paper translates into the JRV and into university students’ perceptions about the evidence supporting the claims. The argumentative structure of Hwang’s paper is such that its apparently ostensible main claim about NT constitutes a justification for a second claim about its therapeutic applications, for which no evidence is offered. However, this second claim receives prominent treatment in the JRV and in the students’ summaries. Implications for promoting critical reading in the classroom are discussed.     

FROM “EXPLORING THE MIDDLE ZONE” TO “CONSTRUCTING A BRIDGE”: EXPERIMENTING IN THE SPIRAL BIANSHI MATHEMATICS CURRICULUM

The spiral bianshi curriculum, an improvement on bianshi teaching developed by Gu (2000) and in line with Marton’s theory of variation (Marton & Booth, 1997), was tried out in a primary school in Hong Kong. This improved theoretical framework for the spiral bianshi curriculum comprises four types of bianshi problems—the inductive bianshi, the broadening bianshi, the deepening bianshi, and the applicative bianshi. Based on this framework, the research team developed a set of teaching materials on the three topics of division of fraction, speed, and volume. The materials were tried out in 21 Primary 6 classes (a total of 686 students) in a school. The effect was compared with a reference group using standard textbook materials in Hong Kong. A series of instruments, pre-tests, and post-tests were administered to gauge the effects on students’ performance in solving routine and non-routine problems, as well as the affective outcomes including self-concept, attitude towards learning mathematics, approaches to learning, and conceptions of mathematics. The intervention effects of the experimental design were examined by hierarchical regression analysis. The research reveals that students using spiral bianshi teaching materials performed significantly better than their counterparts using standard textbook materials. However, no significant differences were identified among affective learning outcome variables despite the positive results on cognitive learning outcomes. The findings indicate that spiral bianshi curriculum has high potential in enhancing students’ learning effectiveness. However, further studies are needed to map its strengths in detail.     

Pedagogy for Equity: Teaching in a Hispanic-Serving Institution

Three female tenure-track faculty members at a Hispanic-Serving Institution explored how their cultural backgrounds inform their pedagogical approaches toward equity. They drew upon Mills’s (1959) and Collins’s (1993) frameworks to examine how their personal biographies, local social contexts, and broader systemic institutions affect their teaching processes for diverse students. These teaching processes include limiting assumptions about students, encouraging students to consider their own personal biographies in relation to the social world, welcoming students’ multiple modes of expression, serving as role models, and challenging inequities in schooling. They conclude with recommendations for enhancing inclusivity in student learning and faculty development.     

UNDERSTANDING STUDENTS�� OUT-OF-SCHOOL MATHEMATICS AND SCIENCE PRACTICE

This study examined standard 6 and 8 (Standards 6 and 8 are the sixth and eighth years, respectively, of primary level schooling in Kenya.) students’ perceptions of how they use mathematics and science outside the classroom in an attempt to learn more about students’ everyday mathematics and science practice. The knowledge of students’ everyday mathematics and science practice may assist teachers in helping students be more powerful mathematically and scientifically both in doing mathematics and science in school and out of school. Thirty-six students at an urban school and a rural school in Kenya were interviewed before and after keeping a log for a week where they recorded their everyday mathematics and science usage. Through the interviews and log sheets, we found that the mathematics that these students perceived they used outside the classroom could be classified as 1 of the 6 activities that Bishop (Educ Stud Math 19:179–191, 1988) has called the 6 fundamental mathematical activities and was also connected to their perception of whether they learned mathematics outside school. Five categories of students’ perceptions of their out-of-school science usage emerged from the data, and we found that 4 of our codes coincided with 2 activities identified by Lederman & Lederman (Sci Child 43(2):53, 2005) as part of the nature of science and 2 of Bishop’s categories. We found that the science these students perceived that they used was connected to their views of what science is.     

White Women Preparing to Teach in Urban Schools: Looking for Similarity and Finding Difference

Research points to particular problems in the experiences of White teachers teaching students of color (Cochran-Smith et al., 2004). Despite good intentions, teaching students of diverse backgrounds and experiences can be challenging for teachers who are unfamiliar with their students’ backgrounds and communities. The purpose of this paper is to describe the development of notions about “good urban teaching” for three women in a preservice teacher preparation program. Reporting on two years of data, we show how the three women negotiated their beliefs and identities in light of program demands and classroom realities. The lack of synchronicity within the women’s experiences highlights that the traditional (white, female, middle class) students in preservice teacher education programs are not homogeneous. The significance of this difference is highlighted through the concept of heterogeneity. We define heterogeneity as the differences that exist among traditional students in preservice teacher preparation programs. Our research suggests that heterogeneity is complicit in the progress or lack of progress of preservice teachers developing professional identities. This paper was originally presented at the Annual Meeting of the American Education Research Association April 7–11, 2006 San Francisco, CA An erratum to this article can be found at     

Development research of a teachers’ educational performance support system: the practices of design, development, and evaluation

This study adopted design and development research methodology (Richey & Klein, Design and development research: Methods, strategies, and issues, 2007) to systematically investigate the process of applying instructional design principles, human–computer interaction, and software engineering to a performance support system (PSS) for behavior management in a classroom. The purpose was to examine how a proposed instructional design framework based on Ausubel’s (The psychology of meaningful verbal learning, 1963) advance organizer theoretical approach could be used to address inherent problems of technically driven PSSs. Development data were collected from a six-phase participatory rapid prototyping process using both qualitative and quantitative methods. Findings indicated that (a) the advance organizer concept combined with a matrix design metaphor provided an effective way to illustrate conceptual connections and relations among PSS modules and their elaborated information, (b) the system served as a reference to support participants for exchanging ideas with other teachers as well as with parents of students, and (c) the rapid prototyping process established parameters that helped the project team maintain a focus on users and collect data useful for advancing to a higher phase of system development. Potential drawbacks pertaining to the proposed design strategies and their possible corrective actions are also reported and discussed.     

Danish and Finnish PISA results in a comparative, qualitative perspective: How can the stable and distinct differences between the Danish and Finnish PISA results be explained?

The research project presented in this article was designed to provide a better understanding of the stable and significant differences in the PISA results between two otherwise very similar Nordic welfare states, Denmark and Finland. In the PISA studies, Finnish students repeatedly achieve the highest Nordic (and partly worldwide) scores in e.g. reading, science and math, while Danish students score lower. Even though Denmark has one of the world’s most expensive educational systems, the OECD ranks the Finnish school system as the world’s best both in terms of quality and equity (OECD 2004). The basic research question is why these differences continue to persist. The case study methodology was mainly inspired by Kirsti Klette’s classroom research (Klette 2003) which involves both interviews and observations. Thus, the overall design could be labeled mixed methods (Johnson and Onwuegbuzie Educational Researcher, 33(7): 14-26, 2004). Five regular public schools in each country were sampled for the comparative classroom studies. The basic theoretical approaches follow Hundeide’s theory of pedagogical communication and relations (2003) and Csikszentmihalyi’s flow-theory (1992). Both this study and statistical studies (S?rensen 2008) show that the difference in the Danish and Finnish PISA results mainly consists in the relatively better score by the lowest scoring 25% of the Finnish pupils compared to the lowest scoring Danish quartile. The results of this study point to a number of possible classroom related reasons why the Finnish school system can produce a better outcome for the lowest scoring quartile of pupils. These reasons are presented and discussed in the article. The study underlines the need to focus more on good classroom management in Denmark—and recommends further international, comparative research in order better to understand the huge differences shown in large scale international programmes such as PISA, PIRLS and TIMMS. The study also reveal the need for more knowledge about inclusive classroom practices, the use of teacher assistants and free, healthy school meals for all pupils.     

Assessing primary school student-teachers’ pedagogic implementations in child sexual abuse protection education

Teacher-educators need ways of assessing the adequacy of university curriculum and the extent to which student-teachers meet learning objectives. One potentially useful tool is Anderson and Krathwohl’s (Addison Wesley Longman, New York, 2001) theoretical framework, which can be applied to assess student-teachers’ knowledge types and their cognitive processes in critical pre-service curriculum areas such as child sexual abuse and personal safety. This study aims to illustrate the use of Anderson and Krathwohl’s framework in assessing student-teachers’ learning of pedagogies in relation to child sexual abuse and personal safety. Participants were a final year cohort of 122 Bachelor of Education (Primary School) students at an Australian university. Student-teachers’ essays regarding the pedagogical practices that they would implement in teaching about child sexual abuse and personal safety were content analysed using Anderson and Krathwohl’s framework. Pedagogies identified by the student-teachers were unevenly distributed across the cells within the theoretical framework. Well-represented pedagogies tended to reflect mid-level cognitive processes (those of understand, apply, and analyse), and low to mid-level knowledge types (those of factual, conceptual, and procedural). Under-represented were pedagogies reflecting higher-level cognition such as creating (in all four knowledge types) and evaluation (of factual and conceptual knowledge). The findings provide a basis for assessing the adequacy of current university teacher-education curriculum structures. Student-teachers’ understandings of under-used theoretical and pedagogical strategies can be identified as a guide to enhance their cognitive processes and knowledge dimensions. This study illustrates a method that has applicability as a diagnostic and assessment tool across a wide range of pre-service teacher education curriculum areas.     

Books and Good Stuff: A Strategy for Building School to Home Literacy Connections

Engaging young children in literacy activities at home is one way for families to augment and enrich the home literacy setting and to participate in their child’s education at an early age (St. Pierre et al. in Dev Psychol 41(6): 953–970, 2005). Burgess et al. (Read Res Quart 4(4): 408–426, 2002) suggested that the resources families have at their disposal, the quality of literacy role models provided by parents, and the types of literacy and language activities in which parents and children engage, are all related to young children’s developing literacy and language abilities. Other studies demonstrated that even modest literacy-promoting interventions can significantly enhance a young child’s early literacy environment by increasing the frequency of parent–child book-sharing activities (Weitzman et al. in Pediatrics 113(5):1248–1253, 2004). Dever (J Early Educ Fam Rev 8(4):17–28, 2001) and Dever and Burtis (Early Child Dev Care 172(4):359–370, 2002) emphasize the use of family literacy bags for early childhood development. Developing and sharing take-home literacy bags is an exciting literacy-promoting activity that may be shared with children and families to provide support for emergent literacy. This article explores the development of the BAGS (Books and Good Stuff) take-home literacy kits and provides suggestions for content, construction, implementation, and evaluation. Sixteen current books are reviewed and recommended by theme.     

ENACTED TYPES OF ALGEBRAIC ACTIVITY IN DIFFERENT CLASSES TAUGHT BY THE SAME TEACHER

The aim of this study was to examine how teachers enact the same written algebra curriculum materials in different classes. The study addresses this issue by comparing the types of algebraic activity (Kieran, 2004) enacted in two 7th grade classes taught by the same teacher, using the same textbook. Data sources include lesson observations and an interview with the teacher. The findings show that students in the two classes were offered somewhat different algebraic experiences. At one school, more emphasis was placed on global/meta-level activities (activities that are not exclusive to algebra and suggest general mathematical processes), whereas at the other school, more emphasis was placed on transformational activities (“rule-based” algebraic activities). Analysis of the sources of the differences related to the ways in which the teacher used and enacted the curriculum materials in the two classes revealed that these were linked to the teacher’s attempts to be attentive to the students in the class and to the nature of the students’ work.