The State of Readiness of Initial Level Preservice Middle Grades Science and Mathematics Teachers and Its Implications on Teacher Education Programs

The purpose of this study was to document through interview and videotaped data the current state of readiness of 10 preservice middle grade teachers, regarding their ability to plan, implement, and reflect on an integrated mathematics and science lesson. The results showed that only one student was successful in implementing a lesson that compared favorably to national standards. This student's lesson plan contained minimal pedagogical considerations and consisted primarily of notes emphasizing fine detail of distinction about the content of the lesson using her own examples. The lesson plan and post-lesson-plan interview data of the remaining students indicated an adherence to algorithmic learning, rote memorization, and procedural knowledge. There were numerous content errors in the plans, and these students orally described a lack of self-confidence in their ability to teach this lesson successfully. The most successful student demonstrated her competence in meeting standards of pedagogical content knowledge and was most successful in analyzing her own teaching. The results showed that most subjects of this study needed extensive training in content and pedagogy and in synthesizing these in a way consistent with modern learning theory.     

Teaching and learning of mathematics: what really matters to teachers and students?

The learner’s perspective study, motivated by a strong belief that the characterization of the practices of mathematics classrooms must attend to learner practice with at least the same priority as that accorded to teacher practice, is a comprehensive study that adopts a complementary accounts methodology to negotiate meanings in classrooms. In Singapore, three mathematics teachers recognized for their locally defined ‘teaching competence’ participated in the study. The comprehensive sets of data from the three classrooms have been used to explore several premises related to the teaching and learning of mathematics. In this paper the student interview data and the teacher interview data were examined to ascertain what do students attach importance to and what do teachers attach importance to in a mathematics lesson? The findings of the student interview data showed that they attached importance to several sub-aspects of the three main aspects, i.e., exposition, seatwork and review and feedback of their teachers’ pedagogical practices. The findings of the teacher interview data showed that they attached importance to student’s self assessment, teacher’s demonstration of procedures, review of prior knowledge and close monitoring of their student’s progress in learning and detailed feedback of their work. It was also found that teachers and students did attach importance to some common lesson events.     

The Relationship Between Lesson Study and Self‐Efficacy

This article addresses a gap in the literature by developing a theory that bridges lesson study and self‐efficacy. Since self‐efficacy has been linked to student achievement, the theory is important as an explanatory mechanism linking lesson study to student achievement. The theory was developed using grounded theory based on primary source data primarily from planning and debriefing meetings. The theory demonstrates that lesson study is linked to self efficacy through a content continuum which implies that issues from pedagogical content knowledge to deeper conceptual matters can be addressed. It also implies that constraints arising in lesson study are most frequently limitations of pedagogical content knowledge which can be mitigated by a community of practice that provides appropriate resources and subject expertise.     

Exploring a structure for mathematics lessons that initiate learning by activating cognition on challenging tasks

While there is widespread agreement on the importance of incorporating problem solving and reasoning into mathematics classrooms, there is limited specific advice on how this can best happen. This is a report of an aspect of a project that is examining the opportunities and constraints in initiating learning by posing challenging mathematics tasks intended to prompt problem solving and reasoning to students, not only to activate their thinking but also to develop an orientation to persistence. Data were sought from teachers and students in middle primary classes (students aged 8–10 years) via online surveys. One lesson focusing on the concept of equivalence is described in detail although mention is made of other lessons. The research questions focused on the teachers’ reactions to the lesson structure and the specifics of the implementation in a particular school. The results indicate that student learning is facilitated by the particular lesson structure. This article reports on the implementation of this lesson structure and also on the finding that students’ responses to the lessons can be used to inform subsequent learning experiences.     

Distinguishing between mathematics classrooms in Australia, China, Japan, Korea and the USA through the lens of the distribution of responsibility for knowledge generation: public oral interactivity and mathematical orality

The research reported in this paper examined spoken mathematics in particular well-taught classrooms in Australia, China (both Shanghai and Hong Kong), Japan, Korea and the USA from the perspective of the distribution of responsibility for knowledge generation in order to identify similarities and differences in classroom practice and the implicit pedagogical principles that underlie those practices. The methodology of the Learner’s Perspective Study documented the voicing of mathematical ideas in public discussion and in teacher–student conversations and the relative priority accorded by different teachers to student oral contributions to classroom activity. Significant differences were identified among the classrooms studied, challenging simplistic characterisations of ‘the Asian classroom’ as enacting a single pedagogy, and suggesting that, irrespective of cultural similarities, local pedagogies reflect very different assumptions about learning and instruction. We have employed spoken mathematical terms as a form of surrogate variable, possibly indicative of the location of the agency for knowledge generation in the various classrooms studied (but also of interest in itself). The analysis distinguished one classroom from another on the basis of “public oral interactivity” (the number of utterances in whole class and teacher–student interactions in each lesson) and “mathematical orality” (the frequency of occurrence of key mathematical terms in each lesson). Classrooms characterized by high public oral interactivity were not necessarily sites of high mathematical orality. In particular, the results suggest that one characteristic that might be identified with a national norm of practice could be the level of mathematical orality: relatively high mathematical orality characterising the mathematics classes in Shanghai with some consistency, while lessons studied in Seoul and Hong Kong consistently involved much less frequent spoken mathematical terms. The relative contributions of teacher and students to this spoken mathematics provided an indication of how the responsibility for knowledge generation was shared between teacher and student in those classrooms. Specific analysis of the patterns of interaction by which key mathematical terms were introduced or solicited revealed significant differences. It is suggested that the empirical investigation of mathematical orality and its likely connection to the distribution of the responsibility for knowledge generation and to student learning ourcomes are central to the development of any theory of mathematics instruction and learning.     

Student teachers’ types of probing questions in inquiry-based mathematics teaching with and without GeoGebra

Previous studies have produced several typologies of teacher questions in mathematics. Probing questions that ask students to explain are often included in the types of questions. However, only rare studies have created subtypes for probing questions or investigated how questioning differs depending on whether technology is used or not. The aims of this study are to elaborate on different ways of asking students to give explanations in inquiry-based mathematics teaching and to investigate whether questioning in GeoGebra lessons differs from questioning in other lessons. Data was collected by video recording 29 Finnish mathematics student teachers’ lessons in secondary and upper secondary schools. The lesson videos were coded for the student teachers’ probing questions. After this, categories for the types of probing questions were created, which is elaborated in this paper. It was found that the student teachers who used GeoGebra emphasized conceptual probing questions during the explore phase of a lesson slightly more than the other student teachers.     

A Longitudinal Study of Elementary Pre‐service Teachers' Mathematics Beliefs and Content Knowledge

This study investigated the mathematics beliefs and content knowledge of 103 elementary pre‐service teachers in a developmental teacher preparation program that included a two course mathematics methods sequence. Pre‐service teachers' pedagogical beliefs became more cognitively‐oriented during the teacher preparation program with these changes occurring during the two methods courses. Pedagogical beliefs remained stable during student teaching. The pre‐service teachers also significantly increased their personal efficacy for teaching mathematics throughout the program with these shifts occurring across both methods courses and into student teaching. Pedagogical beliefs and teaching efficacy beliefs were not related at the beginning of the program, but, in general, were positively related throughout the program. In addition, the pre‐service teachers' pedagogical beliefs were positively related to their specialized content knowledge for teaching mathematics at the end of the program.     

Sensitivity to complexity—an important prerequisite of problem solving mathematics teaching

Teaching is deciding and acting in a complex system. If a teacher attempts to fulfil demands to teach mathematics with a stronger problem solving orientation, it becomes even more complex. This complexity must not be reduced arbitrarily. Instead, a sufficient degree of sensitivity is necessary to competently and flexibly deal with emerging demands on the teacher. In this article I provide an introduction to the concept of sensitivity to complexity of mathematics teaching and report on specific realistic and interactive diagnostic instruments. A particular focus is placed on a diagnostic interview about decision-making situations which could occur in a mathematics lesson. A first pilot study with student teachers from different German universities—briefly outlined in the last part of this article—suggests its suitability for gaining important indications of the agent's degree of sensitivity to complexity of problem solving mathematics teaching.     

Searching for good mathematics instruction at primary school level valued in Taiwan

In this article, we aim to provide a glimpse of what is counted as good mathematics instruction from Taiwanese perspectives and of various approaches developed and used for achieving high-quality mathematics instruction. The characteristics of good mathematics instruction from Taiwanese perspectives were first collected and discussed from three types of information sources. Although the number of characteristics of good mathematics instruction may vary from one source to another, they can be generally organized in three phases including lesson design before instruction, classroom instruction during the lesson and activities after lesson. In addition to the general overview of mathematics classroom instruction valued in Taiwan, we also analyzed 92 lessons from six experienced teachers whose instructional practices were generally valued in local schools and counties. We identified and discussed the characteristics of their instructional practices in three themes: features of problems and their uses in classroom instruction, aspects of problem–solution discussion and reporting, and the discussion of solution methods. To identify and promote high-quality mathematics instruction, various approaches have been developed and used in Taiwan including the development and use of new textbooks and teachers’ guides, teaching contests, master teacher training program, and teacher professional development programs.     

Including Curriculum Focus in Mathematics Professional Development for Middle‐School Mathematics Teachers

This paper examines professional development workshops focused on Connected Math, a particular curriculum utilized or being considered by the middle‐school mathematics teachers involved in the study. The hope was that as teachers better understood the curriculum used in their classrooms, i.e., Connected Math, they would simultaneously deepen their own understanding of the corresponding mathematics content. By focusing on the curriculum materials and the student thought process, teachers would be better able to recognize and examine common student misunderstandings of mathematical content and develop pedagogically sound practices, thus improving their own pedagogical content knowledge. Pre‐ and post‐mathematics content knowledge assessments indicated that engaging middle‐school teachers in the curriculum materials using pedagogy that can be used with their middle‐school students not only solidified teachers' familiarity with such strategies, but also contributed to their understanding of the mathematics content.     

A Preservice Secondary Teacher's Moves to Protect Her View of Herself as a Mathematics Expert

This paper discusses the experience of a preservice secondary mathematics teacher during lesson study. Although the preservice teacher was a strong undergraduate mathematics student, she used compensation “moves” to deflect attention away from her insecurities about her conceptual understanding of secondary mathematics. She feared being labeled as “dumb” and redirected conversations in order to protect her identity as a knower of mathematics. This paper investigates the culture in which preservice teachers develop confidence in their personal mathematics knowledge and how that confidence may influence behavior.     

Using Culturally Responsive Stories in Mathematics: Responses From the Target Audience

This study examined how Black students responded to the utilization of culturally responsive stories in their mathematics class. All students in the two classes participated in mathematics lessons that began with an African American story (culturally responsive to this population), followed by mathematical discussion and concluded with solving problems that correlated to the story. The researcher observed and recorded responses by students during each part of these lessons with protocols. Students independently reflected weekly by answering five questions to share their perspective on the African American stories. The teacher reflected on each lesson as well, describing thoughts on how these students responded to the story in each lesson. This paper examines the analyzed data from the target audience: Black students. Results revealed that Black students responded to the use of African American stories with high self‐rated levels of engagement and enjoyment and that the stories helped them think about mathematics to varying degrees. Since students who are engaged and are thinking about mathematics are more likely to achieve mathematical understanding, the researcher concludes that this strategy should continue to be tested in diverse classrooms with an emphasis on student reflection to determine if the outcomes are transferable and generalizable.     

Conceptions of Turkish mathematics teachers about the effectiveness of classroom teaching

This study investigated how Turkish mathematics teachers evaluate the effectiveness of classroom teaching in terms of improving students’ mathematical proficiency. To this purpose, teachers were asked to evaluate a mathematics lesson as presented them in a vignette. By means of cluster analysis, the participants’ evaluations of the lesson were described in five thematic dimensions, which could be further assembled into two overriding categories: students’ understanding of the subject, and teachers’ classroom practices. The overall aim of the current paper is to propose a preliminary model of the framework that Turkish mathematics teachers use to evaluate a mathematics lesson.     

Conditions and decisions of urban elementary teachers regarding instruction of STEM curriculum

The study was situated in a National Science Foundation supported Math Science Partnership between a private university and an urban school district. This study sought to understand the decision‐making process of elementary teachers as they implement an integrated science, technology, engineering, and mathematics (STEM) curriculum in their classrooms and the interactions that occur between the teachers and curriculum during that process. This qualitative study utilized a comparative case study approach to understanding the decision‐making process of three elementary teachers enacting the same lesson. Analysis of the interactions revealed that the teachers' perceptions of student ability, their pedagogical design capacity, and time were influences that impacted implementation. These findings have implications for STEM‐focused professional development of elementary teachers.     

Improper Use of Physics‐Related Context in High School Mathematics Problems: Implications for Learning and Teaching

This article discusses potential pedagogical difficulties arising from using physics‐related contexts in high‐school mathematics problems. It is suggested that such problems should not require any external knowledge of physics by the students; meanwhile, the problems should not contradict the physics knowledge that the students may already have. Several examples of recently published textbook problems are presented along with the discussion of the context pitfalls. A simple technique (SCAN list) is proposed for evaluating the physics‐related mathematics problems.     

Going beyond the basics: creating and inspiring problem solvers

This paper offers a general discussion of a complex student project used in a first-semester, first-year mathematics course that goes beyond the basics taught in the class and inspires creative problem solving. The project requires the student to model the transition of vehicles among regions of a vehicle rental company. A penalty cost is introduced when the regional inventory drops below an established threshold. The project allows the company to move vehicles by rail among regions to reduce or alleviate the penalty cost. In this phase of the project, student teams attempt to minimize the total cost to the company (penalty cost plus transportation costs), thus searching for an ‘optimal’ solution. The project allows the students to use technology to numerically develop an approximate solution to a problem that is easily understood, but whose analytical solution goes well beyond the scope of the course.     

Japanese and American teachers' evaluations of mathematics lessons: A new technique for exploring beliefs

In our study, we use a novel technique to explore the beliefs of Japanese and American elementary school teachers. Four American and four Japanese teachers watched a mathematics lesson—videotaped in either Nagano, Japan or Chicago, Illinois—and commented on the lesson's strengths and weaknesses. The major pedagogical issues that differentiated the teachers' comments were: what students should do during a lesson, how instructors should use language, how instructors should pace lessons and address ability differences, and how instructional materials should be used. The specific beliefs of the American and Japanese teachers in this study mapped easily onto common instructional practices in elementary school mathematics classes in the United States and Japan. We conclude that, at least for the teachers in this sample, beliefs are linked to practices and they may help to tie teachers to their culturally preferred method of mathematics instruction.     

Inventing Creativity: An Exploration of the Pedagogy of Ingenuity in Science Classrooms

Concerns with the ability of U.S. classrooms to develop learners who will become the next generation of innovators, particularly given the present climate of standardized testing, warrants a closer look at creativity in science classrooms. The present study explored these concerns associated with teachers' classroom practice by addressing the following research question: What pedagogical factors, and related teacher conceptions, are potentially related to the demonstration of creativity among science students? Seventeen middle‐level, high school, and introductory‐level college science teachers from a variety of school contexts participated in the study. A questionnaire developed for this study, interviews, and classroom observations were used in order to explore potential areas of relatedness between pedagogical factors and manifestations of student creativity in science. Five categories ultimately emerged and described potential areas in which teachers would have to explicitly plan for creativity. These areas could inform the pedagogical considerations that teachers would have to make within their lesson plans and activities in order to support its manifestation among students. These provide a starting point for science teachers and science teacher educators to consider how to develop supportive environments for student creative thinking.     

Case-based pedagogy for prospective teachers to learn how to teach elementary mathematics in Korea

Cases have been used in mathematics teacher education with increasing prominence. Yet, there is little research that confirms cases as pedagogical tools to improve prospective teacher expertise, specifically in Asian contexts. This article illustrates how a specific case-based pedagogy was developed and implemented in Korea to increase prospective elementary teacher expertise in terms of paying attention to the mathematics-specific features of a lesson. The results showed that the participant teachers’ analytic foci moved from general to substantive features of a mathematics lesson. This tendency was evident when they reflected on their own teaching and was confirmed by their self-assessment. Given this, issues and suggestions in teacher education programs to promote teacher expertise in terms of mathematics-specific analysis ability are discussed.