그래픽 계산기를 활용하는 수학 교수·학습에 관한 교사 연수 사례 보고

“Good luck on your first day as an assistant professor, Dr. Tanner! Have a great class!” On the wall above my desk, these words scream out from an otherwise encouraging note that is adorned with many exclamation points. This note has hung on my wall since my very first day as an Assistant Professor of Biology. As I was charging off to teach my first class, a senior faculty member who had been on my hiring committee slipped this note under my office door. In moments of pause years later, I still stare up at that note and breathe a sigh of relief that I had much more than luck to guide me on my first day as a college-level teacher. Although I continue to have much to learn—as all of us do no matter the number of years of teaching experience—I did arrive at the university with both formal and informal training in science education. I had had plenty of exposure to innovative pedagogical approaches, questioning strategies, and techniques for engaging diverse audiences in learning science. As a scientist educator, I had had the privilege of many years of collaboration with outstanding K–12 educators as well as a postdoctoral fellowship in science education. However, my training has been, to say the least, unconventional compared with that of my fellow junior faculty and unique in its preparation in regard to the teaching and learning of my discipline. It will not be news to anyone reading this article that university and college teaching is to a large extent a profession with no formal training. It’s startling but true that the majority of faculty members—and lecturers who often teach large numbers of students—have no formal training in the teaching and learning of their discipline. In fact, the hiring process in university science departments is structured primarily to evaluate a faculty candidate’s ability to be a productive researcher, with success measured in number of publications and magnitude of grant funds raised. Depending on the type of institution, for example, research university, state-level university, or liberal arts college, there may be a component of the faculty interview process that probes a candidate’s teaching ability, for example, requesting a statement of teaching philosophy and requiring the candidate to teach a sample lecture class. However, this sample lecture often screens for gross inadequacies, rather than looking for stellar innovations or pedagogical skills. This lack of formal, accredited training for university and college instructors stands in stark contrast to the requirements for a high school teacher who is charged with the education of students only a year junior to college freshmen. High school teachers in the United States must be credentialed as a secondary science teacher, demonstrate subject matter competency in every subject that they will be teaching, and must continually engage in professional development in the teaching and learning of their discipline throughout their career as a science teacher. With the 2002 federal No Child Left Behind legislation, the onus is upon each precollege science teacher to become “highly qualified” in terms of formal university-level training in science education. However, no such required professional training or measurable standards for teaching are required in institutions of higher education. Many policy documents have suggested standards of teaching practice in postsecondary science education (National Research Council, 1996, 1997; Siebert and McIntosh, 2001), but the extent of implementation of these ideals is unclear and has gone relatively unstudied, although national and regional accreditation boards do look at outcomes, asking colleges and universities to assess what their students have gained from four years of study at their institutions. Nonetheless, there is a striking reversal of accountability that happens when one crosses the precollege teaching to college-level teaching boundary (Table 1). During the K–12 school years, society expects K–12 teachers to be responsible for student learning. Salaries of teachers in many states are tied to student test scores, and poor student performance can potentially invoke penalties. At a college or university, several variables in the educational universe shift. Students are the ones responsible for learning. The evaluation and compensation of college-level teachers is not DOI: 10.1187/cbe.05–12–0132 Address correspondence to: Kimberly Tanner (kdtanner@sfsu.edu). CBE—Life Sciences Education Vol. 5, 1–6, Spring 2006

There are not enough faculty positions available for all of the current PhD students within the biomedical science field. Many trainees will leave research-intensive institutions and apply to faculty positions at academic institutions with a high emphasis on teaching responsibilities. The Professional Skills Training is a novel course offered by the APS for young trainees that provides useful formal training in teaching. This advancement toward the improvement in teaching skills in young trainees in science is critical and has the potential to make a significant impact in students9 lives and career paths.

This study explored the extent of utilization of differentiated teaching approach in achieving student engagement in learning mathematics by the secondary school mathematics teachers in SDO Urdaneta City, School Year 2024-2025. Employing an ex post facto one-shot case study research design, 112 mathematics teachers participated as respondents. Results revealed that most teachers were aged 21-30 (34.8%), predominantly female (55.4%), and married (64.3%), with 39.3% holding MA units and 33.0% having 6-10 years of teaching experience, 82.1 percent have been teaching in a barangay high school, and 66.9 percent have attended 6-20 trainings. The extent of utilization of differentiated teaching approach in achieving student engagement in learning mathematics was rated “Very Extensive,” with a grand weighted mean of 4.62. Significant differences and relationships in the extent of utilization of differentiated teaching approach in achieving student engagement in learning mathematics were observed based on civil status, teaching experience, and number of trainings attended. Recommendations include encouraging teachers to pursue advanced education, participate in relevant training, and collaborate with peers to refine differentiated strategies. Further studies were recommended to investigate other variables associated with the utilization of differentiated instruction in mathematics and these studies should be conducted under the auspices of accredited Graduate Institutions like the Urdaneta City University – Institute of Graduate and Advanced Studies (UCU – IGAS)

This study examined how secondary mathematics teachers understand and implement diagnostic assessments in their mathematics classes. To understand how mathematics teachers conceptualize diagnostic assessment and how they put it into practice, the study conducted in-depth interviews with 20 secondary school mathematics teachers using a phenomenography research approach and analyzed them thematically. The results showed that most mathematics teachers are unfamiliar with the concept of diagnostic assessment, misunderstand its role in the teaching and learning process, and refrain from using it because they don't know how to do so efficiently. Due to the overcrowding in classes and the widespread failure of national examinations, the few people who understood the concept admitted to not implementing it. The study recommends professional development for mathematics teachers to improve their conceptual understanding of diagnostic assessment as well as their capacity to integrate it in classrooms to improve the teaching and learning of mathematics and hence improve students' performance in the subject.

This JiM project set out to research the current state of understanding of mathematical justification by both Singapore secondary school students and Mathematics teachers across the different content strands in the Singapore secondary Mathematics curriculum. This timely and relevant project was started to complement the current goal of the Ministry of Education (MOE) on fostering reasoning in the learning of Mathematics in schools. While MOE focuses on the reasoning process, our study examined the outcomes of this reasoning process. For many years, mathematical reasoning and communication have been two key process skills in the Singapore Mathematics Framework under the Processes component. Thus, mathematical reasoning and justification are essential components of any mathematical activity that students should be familiar with. Yet from our personal communications with secondary school mathematics teachers between 2013 and 2016 and evidence from GCE O and N Level examiners' reports, we obtained the impression that many students, as well as mathematics teachers, struggled to navigate mathematical justification successfully. Mathematics teachers seemed to point students' difficulties in justification to several student-related factors. Other than those factors, we, however, speculated that there might be other potential obstacles, which contribute to students' poor performance in justification tasks. Therefore, this JiM project was undertaken to explore the quality of written justifications constructed by secondary school students and their mathematics teachers, with the aim of proposing a professional development (PD) programme that would allow the teachers to better engage their students in mathematical reasoning and justification more effectively, thereby improving the quality of written justifications provided by their students.

This study examines African American students’ interactions with the graphing calculator as a tool to support their mathematical learning. Two participants, chosen through purposive sampling, were the focus of this study. The participants were presented with a task-based interview, and the strategies they employed in solving these tasks with the help of a graphing calculator were examined. The participants’ strategies were analyzed using Goos, Galbraith, Renshaw, and Geiger’s (2003) framework of the varying modes of sophistication with which students work and interact with technology and the ways in which technology can mediate learning. These modes include technology as master, technology as servant, technology as partner, and technology as an extension of self. In solving the mathematical tasks, the participants’ modes of interacting with the graphing calculator technology were influenced by how and when they chose to use the graphing calculator. They were also influenced by their ability to reinforce their mathematical and technological knowledge mutually. It is this ability to develop an interactive relationship between mathematical and technological knowledge that shaped the participants’ fluency in solving the mathematical tasks.

An elementary mathematics teacher once argued that she and her students held four Rights of the Learner in the classroom: (1) the right to be confused; (2) the right to claim a mistake; (3) the right to speak, listen and be heard; and (4) the right to write, do, and represent only what makes sense. Written as an emerging framework to promote equity in the mathematics classroom through divergent formative assessment, the RotL assumes that students can take more explicit ownership of their learning, both in writing and in oral communication. Foregrounded in the literature, this paper discusses how the RotL can help children and teachers to embrace productive struggle and mistakes as valuable steps in the process of learning mathematics (and learning to teach mathematics). The paper also frames the RotL with divergent formative assessment as a tangible means of honoring students’ mathematical resources (e.g., native language, out-of-school knowledge and experiences) to help all students learn mathematics. The paper also presents the experiences of a mathematics teacher educator as she learned about and incorporated the RotL with her prospective elementary mathematics teachers in a university methods course. Implications for mathematics education and teacher education are discussed.

<span>This paper reports on a case study which examined factors leading to the adoption of graphics calculators (GCs) by secondary mathematics teachers in the state of New South Wales, Australia. In total, 587 teachers of the General Mathematics Course (Years 11 and 12) participated in the study. The median teachers' stage of adoption of GCs was found to be at the "Understanding and application of the process" level, the third lowest on a six point scale. The results also indicate that competence is the most important factor in explaining stages of adoption, training the second most important, followed by personal interest in GCs as well as faculty support. Teachers' gender, teaching experience, educational qualifications and the number of GCs in schools were not found to be statistically significant. Qualitative analysis of teachers' open ended comments also shed light on the nature of the explanatory variables. The findings are relevant to curriculum development policy and the design of professional development programs, and has implications for the introduction of other technologies.</span>

Background: In exploring the complexity of teaching, scholars have suggested that greater attention needs to be paid to the teachers' thoughts and actions and to how these affect quality teaching. Up to today, very few studies have investigated the relationship between teachers' beliefs and practices even though it has an important role to play in teaching. Whilst general research on primary teachers in physical education has been conducted, no research has yet specifically described trainee primary teachers' beliefs and practices about physical education. Purpose: The purpose of this study is to describe two trainee primary teachers' beliefs and practices in physical education during their student teaching. Participants and setting: Two trainee primary teachers, Marita and Anna, who attended a university in Cyprus, voluntarily participated in the study. Marita and Anna were in the fourth year of their programme during their student teaching course and they had already taken a compulsory methodology course on teaching elementary physical education and the course on early field experiences. Data collection: Data were collected through interviews (formal and informal), field observations, journals and document analyses (unit and lesson plans). Data analysis: Data from all sources were analysed inductively via individual-case and cross-case analysis. Data sources were analysed individually for each participant and then compared across data sources. Through multiple and careful examination of the data, the key links, properties, themes, and patterns in the data were identified in order to locate common categories. Based on this process, the data were organised and sorted into three major thematic clusters: subject matter, social sensitivity, and pedagogy. Within the thematic clusters there were several categories. Trustworthiness: To ensure the soundness of the data collection and analysis, four strategies were employed. First, triangulation was used to ensure that the findings were accurate. Second, data were scrutinised for negative cases that would dispute or undermine the interpretations. Third, two peer-debriefers were used to share developing interpretations and to challenge the researcher to support interpretations with data. Finally, member checks were conducted. Findings: The two trainee teachers believed that the major purpose of physical education is to provide students with opportunities to develop their psychomotor, cognitive and affective skills. Evidence from this study indicated that the participants not only articulated these views but also practised them in teaching. Both participants believed that physical education has the same status as the other subject areas in the school curriculum. The most important message the two trainee teachers wanted to convey to their students was that physical education is more than free play and similar to other subject areas in the curriculum. The two trainee teachers treated students of both genders equally and expected all students to participate in the learning activities. The way the participants in this study organised and delivered instruction and provided practice for students was a combination of teacher-mediated and student-mediated formats. Conclusions: Two major conclusions are drawn from the study. The first is that trainee teachers' beliefs play a significant role in designing and implementing meaningful teaching tasks that may affect student learning. This study suggests that teaching experiences can provide trainee teachers with powerful personal understanding of teaching practices. It is also clear that learning to teach physical education is a process which develops over time and it is affected by the trainee teachers' beliefs. A second conclusion from the study is that teacher education programmes can train trainee primary teachers to implement quality lessons in physical education. It seems that the methods course and teaching experiences forced the two trainee teachers to develop firm beliefs about good teaching practices. Reflective inquiry courses that provide trainees with a theoretical knowledge base about effective teaching in physical education and use reflective strategies that attempt to surface, challenge, and transform trainees' beliefs seem to be promising.

Este artigo é um ensaio que tem por objetivo problematizar e discutir o lugar da matemática na formação do futuro professor, em cursos de Licenciatura em Matemática. O ensaio é organizado em torno de duas questões básicas: de que matemática estamos falando, quando dizemos que o professor precisa saber bem matemática para ensiná-la? Que práticas formativas podem contribuir para que o futuro professor possa se apropriar dessa matemática fundamental para seu trabalho profissional? Para respondê-las, foram analisadas e problematizadas as diferentes práticas sociais do educador matemático, tentando perceber, nelas, indícios do tipo de saber matemático mobilizado e requerido. A seguir, foram trazidas algumas tentativas históricas de tratar o problema da formação matemática dos professores, tendo por base Felix Klein, Richard Courant e Bento de Jesus Caraça, e algumas pesquisas brasileiras e internacionais que têm como foco de estudo a formação matemática do professor. A partir desses estudos, o artigo destaca a existência, na formação inicial do professor de matemática, de uma quase tricotomia entre formação matemática, formação didático-pedagógica e prática profissional. Para romper com essa tradição tricotômica são sugeridas algumas mudanças em relação à prática e à pesquisa sobre formação de professores tais como: a formação do professor de matemática deve orientar-se pelas diferentes práticas sociais do educador matemático; adotar, na formação inicial, práticas e projetos nos quais os licenciandos possam integrar, fazendo contrastes, problematizações e investigações sobre as relações entre sua formação matemática na licenciatura, sua formação didático-pedagógica relacionada ao conteúdo, e a complexidade das práticas escolares.

<i>Rejoinder</i> : Misunderstanding the Problem of Out-of-Field Teaching

Background: This study aims to explore the effects of the availability of trained teachers on students’ academic performance in selected public schools in Mandera North Sub County, Mandera County. Methodology: The study will adopt a cross-sectional research design because it entails observing two study variables to define a corresponding statistical relationship between the variables. The study area will be public secondary schools in Mandera North Sub County; where the target population will be 9 schools with 2239 students, 104 teachers, and 9 principals. Yamane’s formula will be used to determine students’ sample size totaling 339, simple random sampling determine teachers’ sample size totaling 10, while purposive sampling will be used to identify 3 principals giving a total of 352 respondents. Questionnaires, interviews, and document analysis will be used to collect primary data. Results: Regarding the students enrolled in this study, 115 (38.3%) were females while 185 (61.7%) were males. All the teachers (100%) agreed that professional qualification can affect their output in class. There were all indications that the schools did not have enough trained teachers, and most schools had stayed for one year (11.1%), 2-3 years (33.4%), or more than 4 years (44.5%) without enough trained teachers. It was interesting that the principals agreed that qualified teachers are more effective in teaching, lack of trained teachers creates teaching overloads, teacher qualifications affect students’ academic output, and lack of trained teachers leads to delays in syllabus coverage Conclusion: There were only five to eight trained teachers in the schools surveyed. Out of the 9 teachers interviewed, only 5 attested to have received formal training in teaching. Recommendation: More trained teachers should be recruited to improve student’s performance in the Kenya Certificate of Secondary Education.

With the unveiling in the United States of the Common Core State Standards for Mathematics that value inquiry, cross-curricular connections, and the use of technology in instruction, school librarians may be best positioned to collaborate with mathematics teachers to instill these practices into mathematics teaching and learning. However, the mathematics curriculum is often perceived as challenging or even impossible for librarians to connect with. Consequently, such collaboration is often not pursued by either party. This paper examines the factors that shape or hinder collaboration between school librarians and mathematics teachers. Semi-structured interviews were conducted with mathematics teachers, school librarians, and school leaders in four middle schools in a highly resourced school district in the mid Atlantic region of the United States. We find several factors that influence the nature and opportunities for collaboration, including: perception; comfort with technology; institutional structures; testing pressure; and knowledge about mathematics teaching and learning. Our analysis reveals pivotal transformations that need to take place in order to strengthen the collaborative relationship between these two domains: (1) changes in the perceptions mathematics teachers and school librarians have of each other, and of various aspects of technology integration in mathematics teaching and learning and collaborative relationships; (2) enhanced structural support at the school district level; and (3) enhanced structural support at the school level to support collaboration. By examining the connections and contexts of these emerging themes from our study, we reveal the impact that school librarians can have in mathematics learning in middle schools.

Teaching Trainees to be Effective Mentors: A Needs-Based Assessment in Radiation Oncology

In the 1923 Report on the Reorganization of Mathematics in Secondary Education, it was pointed out that in some states the preparation of high school mathematics teachers was of such a low quality that the Committee judiciously refrained from giving the detailed data on those states. We have come a long way since then. Today, in general, secondary mathematics teachers are professionally and academically as well educated as teachers in any other field.