Teachers' and Students' Cognitional Knowledge for Classroom Teaching and Learning

The study explored the influence of teachers’ mathematical knowledge for teaching on students’ achievement, retention and attitude towards mathematics in secondary schools in Nigeria. The study was guided by three research questions and hypotheses. The study employed correlational survey research design. The population of the study comprised 3.5 million secondary school students and 100,000 mathematics teachers from the six geopolitical zones of Nigeria.The sample size for the study consisted of 500 secondary school students and 100 mathematics teachers, selected proportionally from each geopolitical zone.The primary research instrument for this study was a structured questionnaire designed to collect data on teachers’ MKT and students’ achievement, retention, and attitude towards mathematics. The content and construct validity of the instrument was established. The instrument had a reliability value of 0.76 determined using Cronbach alpha. Data obtained with the aid of the instrument were analysed using coefficient of determination and Pearson correlation. The results revealed significant positive relationship between the components of mathematical knowledge for teaching (MKT) and students’ achievement, retention and attitude towards mathematics. The study concluded that teachers’ mathematical knowledge for teaching (MKT)—which encompasses subject matter content knowledge (SMCK), pedagogical content knowledge (PCK), knowledge of learners’ learning difficulties (KLD), and understanding of learners’ misconceptions (ULM)—plays a significant role in shaping students’ achievement, retention, and attitude towards mathematics. It was recommended among others that school administrators should provide regular, targeted training programs to improve teachers’ subject matter content knowledge, pedagogical content knowledge, and ability to address students’ misconceptions, thereby enhancing student achievement, retention, and attitudes towards mathematics.

Introduction: Although there is ambiguity about the elements of teacher knowledge, all researchers accept that being able to anticipate what errors can be made, the reasons for and the strategies to overcome these errors, in short, the knowledge of students is important for student achievement. In this study, knowledge of students refers to being aware of students’ possible errors and underlying reasons for these errors and knowing how to overcome these errors. Based on this consideration, the purpose of this study is to investigate pre-service elementary teachers’ knowledge of students on the subtraction topic. Methods: Considering the purpose, the data were collected from 118 pre-service elementary teachers who were enrolled in a four-year Elementary Teacher Education program via a task-based questionnaire related to the topic of subtraction and semi-structured interviews following the questionnaire. The task-based questionnaire included three completed incorrect subtraction tasks and was prepared considering the related literature and the elementary school mathematics curriculum of Turkey. Each task in the questionnaire contained a different type of error. The pre-service elementary teachers’ answers to the task-based questionnaire were categorized as correct, partially correct, wrong, or no answer by means of categorical analysis. The pre-service elementary teachers who gave correct and partially correct answers to the tasks were asked to participate in the second part of the study to learn their possible strategies to overcome the errors made in the tasks. Results: As a result of the quantitative and qualitative analysis of the pre-service elementary teachers’ responses, it was found that their knowledge of students for subtraction is limited. Specifically, although the pre-service elementary teachers were partially able to identify the errors in the first and second task, they were not able to identify the error in the third task. Furthermore, they were better able to determine a more commonly occurring subtraction error compared to the uncommon ones. The pre-service elementary teachers could not identify the underlying reasons that led the students to the errors. They could not explain what conceptual knowledge related to the topic of subtraction the student lacked that resulted in the errors. As the pre-service teachers did not attain these reasons, they were not able to provide strategies to overcome these errors different than restating the rules or procedures to overcome students’ errors. Discussion: Only being able to identify the errors is not enough to make the instruction effective. Teachers also need to know and provide a rationale for why the errors happen and how to overcome them (Even & Tirosh, 1995). Contrary to this statement, the pre-service elementary teachers in this study could not attempt to understand the students’ thinking or could not explain the reasons behind students’ errors with the notion of subtraction. Son (2013) emphasizes that pre-service teachers tend to explain the reasons for students’ errors as procedural. Similar to this emphasis, the pre-service teachers identified the errors as resulting from not applying procedures carefully or not enough knowing algorithms to find correct solutions. However, without addressing the reasons or focusing on procedures does not promote students’ understanding of the related topic (An, Kulm, & Wu, 2004). The pre-service teachers’ difficulties in identifying the possible conceptual reasons may result from deficits in knowledge of students (Fennema & Franke, 1992; Ball et al., 2008). Limitations: The findings were limited with the responses of the pre-service elementary teachers participating in this study. Moreover, the pre-service elementary teachers’ knowledge of students was investigated within the scope of subtraction. Conclusions: Although the pre-service elementary teachers could identify students’ errors on subtraction, they had difficulty in identifying the reasons and suggesting strategies to overcome the errors. Therefore, it can be concluded that the pre-service teachers’ knowledge of students for the topic of subtraction is limited. An implication of this is that teacher educators need to include error-analysis tasks to help pre-service teachers be aware of the importance of their knowledge of students.

Possession of prior knowledge has been shown to critically influence students' learning in major courses. Yet the topic remains unexplored in the realm of general education in higher education. The study aims at investigating whether students lacking relevant high school knowledge background could benefit in the two courses, one inclined to science and one inclined to the humanities, in a general education program at a university in Hong Kong. Interactive learning has been intensively practiced in the program to minimize the unfavorable impact brought up by a lack of prior knowledge among students. Data from an entry-exit survey show that a lack of prior knowledge does not limit students' learning in outcomes related to generic cognitive skills and subject matter content in both courses. An interactive learning approach is suggested to have mediated the problem of the lack of prior knowledge in students' subsequent learning in the two general education courses.

An important part of the mathematics syllabuses at the secondary school level in most countries is the concept of function. However, secondary school students often experience difficulties with this concept. These difficulties are well-known in the research literature. The study applies the mathematical knowledge for teaching (MKT) framework, including the category knowledge of content and students (KCS). Teachers’ ability to anticipate students’ difficulties is one aspect of KCS. The aim of this study is to investigate secondary mathematics student teachers’ KCS regarding the concept of function. Ten mathematics student teachers participating in a Supplementary Teacher Education Program answered a questionnaire about fictive secondary school students’ various difficulties with the concept of function. Follow-up interviews were conducted with four of the respondents. Compared to the findings of previous research on students’ difficulties with the concept of function, the respondents in the study sometimes provide reasonable suggestions about the sources of students’ difficulties. Some of the respondents demonstrate an aspect of KCS when they suggest that students can reason that a function must be defined by one algebraic expression only, and that students only know about continuous functions. However, no respondent suggests that one source of students’ difficulties with a constant function with an implicit domain is the missing domain. In addition, some respondents take for granted that students can interpret the algebraic representation of a piecewise-defined function and translate it into a graph.

هدف اصلی این پژوهش، بررسی و مقایسه دیدگاه‌های دو صاحب نظر در زمینه دانش پایه تدریس و دانش معلمان، لی شولمن و گری فنسترماخر می‌باشد. شولمن اشکال دانش تدریس را شامل انواع دانش گزاره‌ای (موضوعی)، دانش موردی و دانش راهبردی می‌داند و دانش پایه تدریس و دانش معلمان را شامل طبقه‌های مختلف دانش محتوایی، دانش محتوایی- تربیتی، دانش عمومی تربیتی، دانش برنامه درسی، دانش درباره یادگیرندگان و خصوصیات آنها، دانش درباره موقعیت تربیتی و دانش درباره هدف‌ها، مقاصد و ارزش‌های تربیتی و زمینه‌های تاریخی و فلسفی معرفی می‌کند. فنسترماخر، اشکال دانش تدریس را به دو دسته کلی، دانش رسمی تدریس و دانش عملی تدریس تقسیم می‌کند. فنسترماخر عناصر تدریس خوب را شامل فعالیت‌های منطقی، فعالیت‌های روان‌شناختی و فعالیت‌های اخلاقی می‌داند. هر دو دیدگاه از اندیشه‌های شواب تأثیر پذیرفته اند و هر دو دانش تدریس و دانش معلمان را هم از جهت عملی و هم از جهت نظری و موضوعی مورد مطالعه قرار دادند. در این میان، فنسترماخر تاکید بیشتری روی عمل معلمان و دانش عملی آنها دارد.

In the previous Chapters, the knowledge implementation was defined in the context of both human learning and machine learning. The knowledge implementation is focused on learning of the knowledge and skills by machine (SUS) and is concerned with two main aspects of human learning: learning of the visual knowledge in the context of the categorical structure of the learned categories of the visual objects and learning of the knowledge that is connected with understanding of the content of the text. As it was described in Chapter 3, SUS operates in two main modes, learning and understanding mode. SUS ability to understand depends on the effectiveness of learning process and learning of the new knowledge depends on SUS ability to understand.

The study aimed to investigate the effect of a training program based on the principles of conceptual understanding in enhancing the pedagogical knowledge of mathematics teachers. The study sample consisted of 34 male and female in-service mathematics teachers, who teach the middle grades (5-8). To achieve the objectives of the study, a quasi-experimental approach was used, with a pre-post design for two groups. In order to collect data, a pedagogical knowledge test was used for the content of numbers and algebra contained in the middle grades curriculum. The results showed a significant effect of the training program in enhancing the pedagogical knowledge of mathematics teachers, with two dimensions (knowledge of teaching mathematics: approaches and strategies, knowledge of students’ thinking). The results also showed an improvement in performance levels on the test in general among the teachers of the experimental group, as the percentage of teachers in the level of skilled performance increased from 0% on the pre-test to 53% on the post-test. In light of the results, the study recommended to conduct more research through employing more principles of conceptual understanding to enhance pedagogical knowledge among broader samples of mathematics teachers and different mathematics content.

The research intends to study the application of NVC of English majors’ classroom learning and teaching based on a case study in Sichuan International Studies University. Three approaches are involved in this paper: classroom observation, interview and questionnaires. The author does a classroom observation for two weeks in three English majors’ classes. Follow-up interviews are then conducted with the students and the teachers. 100 English majors and 20 English teachers complete the questionnaires. The study aims to make the university English teachers and English majors realize the fact that NVC takes a great part in English majors’ classroom teaching and that both teachers and students have to be equipped with sufficient knowledge of NVC so as to improve the teaching and learning efficiency.

Integrating Technology into Instructional Practices Focusing on Teacher Knowledge

This study focuses on a particular type of knowledge which is referred to throughout as tacit knowledge and has been identified as a personal form of knowledge, which plays a key role in the practice and professional development of teachers. In the context of initial teacher education tacit knowledge is important to consider as what student teachers are learning and how they are learning it is of current, pressing concern to initial teacher education in England. The role of and access to tacit knowledge in student teacher learning were investigated using a case study approach employing stimulated recall interviews wherein mentors were asked to reflect upon their teaching and explain their teaching actions. Stimulated recall interviews involved student teachers taking on the role of interviewer. Thematic analysis of data revealed that student teachers probed key teaching episodes and encouraged mentors to reveal complex decision making which may otherwise have remained undisclosed. Main findings show that both student teachers and experienced teachers see tacit knowledge as useful in the learning to teach process. Interviews reveal that observable teaching actions had complex underlying reasoning and this was elicited via objective-led dialogue between student teacher and mentor, post-teaching. Student teachers were able to independently select teaching events to examine further, and were able to understand their mentor’s actions and judgements more deeply as a result of stimulated recall interviewing. The implications arising out of this study include a need for change to observation and debriefing practices in school, with student teachers and mentors becoming more aware of the value and purpose of tacit knowledge. The benefits of student teachers being instrumental in initiating and asking questions which can elicit experienced teachers’ tacit knowledge are clear; and findings show the potential for student teachers to become more pivotal in their own learning. Where stimulated recall interviews were used mentors were encouraged to reveal embedded reasoning underpinning their classroom practice.

The present qualitative study investigated if school building conditions have a relationship with student achievement and behavior in twenty Lebanese Armenian schools applying the 9th grade official program. The study relied on self-report measures, as the principals of the schools completed a 22-item and 4-question surveys (adapted from the 1993 study of C. Cash), as well as providing academic and behavior scores of their 9th grade students (n = 361) for the academic year 2009-2010. Also, the principals answered five open-ended questions on their opinion of the relationship of the school building condition and several aspects of the building with student achievement and behavior. A qualitative analysis of the obtained data showed that there is no relationship between school building conditions and student achievement and behavior. Moreover, the scores for behavior were high across the schools, and the achievement scores were low, in general, for all three kinds of school buildings: substandard, standard and above standard, regardless of the school building condition.

The current education system not doing anything for understanding students or helping them in problem solving skills. It provides nothing but only a usual information package to those who pass through the gateways. It is better to diversifying the teaching strategies rather than changing the curriculum for solving this issue. The Theatre Activities which can be used in classrooms are extremely useful in this matter. The theatre has accepted a new approach for creating a bond between classroom learning and drama instruments. The major goal of this attempt is to modify the knowledge, attitude and behaviour of students. The People who have acquired sufficient training are considered as prospective teachers hence, they must be familiar with teaching strategies like Theatre Activities. It is a study conducted among prospective teachers for their opinion about the effectiveness of Theatre Activities. It includes the opinion of prospective teachers about the effectiveness and possibilities of Theatre Activities. The findings of the study reveals that most of the prospective teachers have good awareness about the learning strategies using Theatre Activities. This study reveals that it is very effective to use of Theatre Activities in the classroom for developing good relationship, leadership qualities and multiple intelligence among the students and usage of Theatre Activities are very effective for diversity in classroom and to sustain the learning interest among students.

As a comparative analysis of teacher preparation in its sociohistorical contexts, this study examines the official educational aims and curricula of 49 mathematics teacher preparation programs in South Korea and the United States, where substantial differences have been observed in both student achievement and teacher knowledge. Overall, the findings from this study suggest that transnational commonalities and national differences exist simultaneously in social expectations for teacher knowledge. The authors argue that attending to both culturally contextualized and semantically decontextualized dimensions helps us have a more balanced comparative perspective from which we can better assess current conditions of teacher education. Constructive international dialogue can be facilitated by such a balanced perspective that may further enrich teacher education without ignoring either profound differences in sociohistorical contexts or important commonalities in epistemic models of teacher education across countries.

In this chapter, we discuss comparative knowledge expectations for mathematics teachers in their respective sociohistorical contexts. Specifically, this study examines the official educational aims and curricula of 49 mathematics teacher preparation programs in South Korea and the United States, where substantial differences have been observed in both student achievement and teacher knowledge. Overall, our findings suggest that transnational commonalities and national differences simultaneously affect social expectations for teacher knowledge. We argue that attending to both culturally contextualized and semantically decontextualized dimensions offers a more balanced comparative perspective from which we can better evaluate the current status of teacher education. Constructive international dialogue can be facilitated by such a balanced perspective, and may further enrich teacher education without ignoring either profound differences in sociohistorical contexts or important commonalities in epistemic models of teacher education across countries.

NP26 Healthy Schoolhouse 2.0: Results From Year One