Mathematical Knowledge For Teaching Research Articles

Toward a Better Understanding of the Nature of Pedagogical Content Knowledge for Fractions: The Role of Experience

ABSTRACT Mathematical knowledge for teaching (MKT) is a professional knowledge crucial for effective mathematics instruction, encompassing both content knowledge (CK) and pedagogical content knowledge (PCK). The purpose of this study is to investigate the hierarchical structure of PCK for fractions by seeking further evidence for the PCK-Fractions assessment. The PCK-Fractions measure is designed to examine the effectiveness of teacher education program and field experiences in facilitating preservice teachers’ (PSTs) PCK for children’s fraction reasoning in grades 4–6. Emplying165 undergraduate students, the research investigates how different levels of coursework and field experience influence PCK-Fractions scores. Results suggest that both completing coursework in a teacher education program and having grades 4–6 field experience each led to higher PCK-Fractions scores. Additionally, this study provides further evidence supporting the validity of the PCK-Fractions measure as a reliable indicator of PSTs’ proficiency in teaching fractions. Implications for mathematics teacher education and educational research are discussed.

The Effect of Training Program Based on Instructional Practices According to the (NCTM) Standards on Improving the Pedagogical Knowledge among Mathematics Teachers in Jordan

The current study aims at revealing the effect of a training program based on teaching practices according to NCTM standards in improving the pedagogical knowledge of mathematics teachers. The study followed the semi- experimental method. To achieve the aims of the research, a training program that is based on the standards of NCTM was designed. The program consists of eight packages which are (Establish mathematics goals to focus learning, implement tasks that promote reasoning and problem solving, Use and connect mathematical representations, facilitate meaningful mathematical discourse, pose purposeful questions, build procedural FLuency from conceptual understanding, support productive struggle in learning mathematics, Elicit and use evidence of student thinking). After confirming the validity of the tool, a sample of (20) male and female teachers from Deir Ala Directorate of Education in the academic year 2020-2021.To reveal the effectiveness of the program, a test in pedagogical knowledge was prepared. The test consists of six respects which are identification of mathematics content, the pedagogical knowledge of the mathematical content and knowing students' thinking. The test validity and reliability were checked. The results of the study showed that there were statistically significant differences between measuring averages in the post application of the test. These results were pro- sample in all dimensions and in the total degree. Depending on these results, the study asserted the necessity for the training department in the Jordanian Ministry of Education to benefit from the suggested and used program in the study to focus more on the teaching practices.

Analysis of STEM Knowledge of Pre-Service Science and Mathematics Teacher

This research is motivated by the importance of STEM knowledge for pre-service science and mathematics teachers in the 21st century. This research aims to analyze the STEM knowledge of pre-service science and mathematics teachers. The research method was carried out by distributing questionnaires with STEM knowledge indicators to the sample. The sample in this research was 86. They were 59 pre-service science teachers and 27 pre-service mathematics teachers, with different years of education, namely the 1st, 2nd, 3rd, and 4th year of Universitas Negeri Semarang. The results of this research show that the average understanding of pre-service science and mathematics teachers regarding to the STEM-based learning in the definition aspect is 4.11, objectives are 3.94, benefits are 3.94, aspects are 3.69, components are 3.71, characteristics is 3.75, and implementation is 3.86. This proves that pre-service science and mathematics teachers' understanding of the definition of STEM-based learning is very good, while pre-service science and mathematics teachers' understanding of the objectives, benefits, aspects, components, characteristics, and implementation of STEM-based learning is good.

Preservice teachers' mathematical knowledge for teaching: Focus on lesson planning and reflection

AbstractMathematics teacher educators have suggested preservice mathematics teachers' (PMTs') practices provide evidence of their mathematical knowledge for teaching (MKT). PMTs engage in approximations of practice to rehearse teaching. To explore connections between MKT and PMTs’ practices, we operationalized six MKT subdomains in terms of approximations of practice. We used that operationalization to investigate, which subdomains were evidenced in PMTs’ lesson plans and their lesson plan reflections. Eleven secondary PMTs participated in our study. Using a qualitative content analysis method, we found Knowledge of Content and Teaching most evidenced, suggesting that the PMTs demonstrated MKT when they described mathematical tasks and associated instructional activities. Also, the PMTs demonstrated Knowledge of Content and Students when they described how their students would engage with proposed mathematical tasks. However, the PMTs struggled to plan for activities by building on students’ potential mathematical thinking and reasoning. Only limited instances of Horizon Content Knowledge suggest that the PMTs had limited opportunities to demonstrate how their advanced mathematical knowledge connected with their mathematics teaching in the approximations of practice provided. We propose alternative forms of approximations of practice to optimize PMTs’ opportunities to demonstrate and develop MKT.

A model for problem creation: implications for teacher training

AbstractThe invention of problems is a fundamental competence that enhances the didactic-mathematical knowledge of mathematics teachers and therefore should be an objective in teacher training plans. In this paper, we revise different proposals for categorizing problem-creation activities and propose a theoretical model for problem posing that, based on the assumptions of the Onto-Semiotic Approach, considers both the elements that characterize a problem and a categorization of different types of problem-posing tasks. In addition, the model proposes a description of the mathematical processes that occur during the sequence of actions carried out when a new problem is created. The model is illustrated by its application to analyze the practices developed by pre-service teachers in three problem-posing tasks aimed at specific didactic-mathematical purposes (mobilizing certain mathematical knowledge or reasoning, contributing to achieving learning goals, or addressing students’ difficulties). We conclude discussing the potential of our model to analyze the mathematical processes involved in problem creation from the perspective of teacher education.

Relationship between mathematical pedagogical content knowledge, beliefs, and motivation of elementary school teachers

Pedagogical content knowledge (PCK) is one form of teachers’ professional knowledge in subject teaching, and teachers’ rich PCK enables effective instruction and improves students’ academic performance. However, there has been limited research on the relationships of individual difference characteristics of teachers to PCK among in-service elementary school teachers. Therefore, in addition to the demographic variables (gender and years of teaching experience) and psychological variables (beliefs about teaching and learning and teacher efficacy) examined in previous studies, this study attempted to clarify whether motivation for teaching is related to PCK. We conducted a web survey of in-service elementary school teachers in Japan (n = 267). The results showed that the traditional beliefs that students are to be controlled by their teachers and indifference, which describes a state of lack of motivation to prepare for class, were negatively associated with two elements of mathematical PCK (knowledge of learners and knowledge of instruction). Furthermore, multiple regression analysis revealed that traditional beliefs about teaching and learning were negatively associated with the knowledge of learners and indifference to subject instruction with knowledge of instruction. This suggests that teachers’ motivation for teaching is related to PCK, in addition to the variables that have been previously examined.

Specialized Knowledge of Prospective Mathematics Teachers on the Concept of Mathematical Proof

[Objetivo] En este artículo, se presentan los resultados de una investigación cualitativa de carácter descriptivo que tiene como objetivo caracterizar el conocimiento especializado de los profesores de matemáticas en formación inicial de la Universidad Nacional en Costa Rica (UNA), sobre el concepto de la demostración matemática, mediante el modelo Mathematics Teacher’s Specialized Knowledge (MTSK). [Metodología] La investigación se posiciona en el paradigma interpretativo y tiene un enfoque cualitativo. Se aplicó un cuestionario, durante el primer semestre del 2021, a 42 profesores de matemáticas en formación inicial de cuarto y quinto año de la carrera Bachillerato y Licenciatura en Enseñanza de la Matemática de la Universidad Nacional de Costa Rica. Para examinar la información, se empleó el análisis de contenido y se hicieron agrupaciones de las respuestas, con el propósito de generar ideas centrales sobre el concepto de demostración. [Resultados] De los resultados se desprenden cuatro ideas centrales sobre lo que es una demostración matemática para los sujetos de la investigación, cercanas a aspectos formales lógico-sintácticos y matemáticos (ALSM) o aspectos informales semánticos (AIS). Se encontraron evidencias de las cinco funciones de De Villiers (1993) para la demostración y surgieron nuevas funciones relacionadas con esta en las matemáticas y en las matemáticas escolares. [Conclusiones] Los resultados brindan insumos a formadores de profesores de matemáticas e investigadores, en la revisión y análisis de programas de formación docente, y contribuyen en la búsqueda de nuevas áreas de investigación relacionadas con el tema.

From “learning to variate” to “variate for learning”: Teachers learning through collaborative, iterative context-based mathematical problem posing

Problem posing (PP) has been found to contribute to teachers’ mathematical pedagogical knowledge. However, little is known about what and how teachers learn when engaged in continuous iterative PP. We use the variation theory of learning to conceptualize what and how teachers learn during iterative PP, illustrating these processes via a case study. The main argument is that what teachers learn from engaging in iterative PP are different task variables we refer to as “dimensions of possible variation.” Awareness of these dimensions allows teachers to skillfully generate new problems or re-formulate previously posed ones to achieve desired pedagogical goals. We show how, during a collaborative design process with the PD coordinator, a teacher-designer became aware of some new-to-her dimensions and developed corresponding techniques for diversifying tasks. These awarenesses were still evident in an interview six months after the end of the PD. Recommendations for teacher educators are suggested.

Conceptions of teaching and justice as pivotal to mathematics teacher educators’ thinking about mathematical knowledge for teaching

Conceptions of teaching and justice as pivotal to mathematics teacher educators’ thinking about mathematical knowledge for teaching

Normative Evaluation Scale of Pedagogical Content Knowledge with Emphasis on Proving knowledge of Secondary School Mathematics Teachers

Normative Evaluation Scale of Pedagogical Content Knowledge with Emphasis on Proving knowledge of Secondary School Mathematics Teachers

Rwandan teacher training college’s mathematics teachers’ pedagogical content knowledge for teaching: assessment toward competency-based curriculum

IntroductionTeachers’ knowledge and positive attitudes significantly impact educational settings. This study aimed to assess teachers’ mathematical knowledge for teaching (MKT) and their pedagogical content knowledge (PCK) attitudes.MethodologyIt was conducted on all 23 mathematics teachers from Rwandan teacher training colleges (TTCs). Employing a survey design, standardized performance and attitude tests were utilized, employing instruments known for their standardization, validity, and reliability.ResultsThe findings informed that teachers possess a low MKT but a high PCK attitude. Thus, they performed the MKT test poorly as the average score was below 50%. However, teachers were found to have better content knowledge than other MKT constructs and performed better in pattern function and algebra than in other areas of mathematics. Teachers demonstrated a highly positive attitude toward integrating PCK in teaching mathematics.DiscussionSuch a positive attitude was inferred from the pedagogical training offered by implementing a competency-based curriculum. Therefore, teachers need to improve their MKT performance, and more research is needed on other teachers apart from those in TTCs.

Pre-appropriation of games about the first-degree equation proposed in textbooks

This article results from a master’s thesis developed in the Postgraduate Program in Science Teaching, whose objective was to analyze the pre-appropriation of games for teaching first-degree equations by mathematics teachers, proposed in textbooks approved by the National Program of the Textbook. The research was guided by the model of appropriation of a new resource, structured in three phases (pre-appropriation, original appropriation, and reappropriation), and the reflective methodology was used based on the Documental Approach to Didactics. This article is limited to analyzing the pre-appropriation phase of the games. Two mathematics teachers from the final years of the elementary school in the municipal network of Recife, Pernambuco, participated in the research. The results highlight the influence of mathematical knowledge for teaching in the instrumentalization process of the games and the different types of planned instrumental orchestrations, in particular, for the use of the equivalent equations game in the classroom.

The effect of inquiry based technology integration on conceptual and procedural geometry knowledge of preservice mathematics teachers

The effect of inquiry based technology integration on conceptual and procedural geometry knowledge of preservice mathematics teachers

Relationships between elementary mathematics specialist certification, knowledge, beliefs, and classroom learning environments

AbstractWe report on the differences in mathematics learning environments in classes taught by certified Elementary Math Specialists (EMSs) (n = 28) and their peers (n = 33) as determined by observations of instruction. We used path analysis to examine how variables such as mathematical knowledge for teaching, beliefs, and background characteristics were related to the learning environment. We used the Classroom Learning Environment Measure (CLEM) observation protocol, which attends to aspects of mathematics lessons such as opportunities for students to justify their reasoning and attend to mathematical concepts. Our analysis revealed that learning environments incorporating such elements were significantly more prevalent in classes taught by EMSs, and that there were two paths indicating mediation effects on the relationship between EMS status and learning environment. One path was related to teachers’ beliefs about the primacy of computation in learning mathematics; the other path was related to teachers’ mathematical knowledge for teaching and their beliefs about the extent to which mathematical knowledge is constructed by the learner. We share implications for EMS programs and recommendations for future research on the impact of EMSs in elementary schools.

Linear algebra in engineering: a study of specialized knowledge of Chilean and Brazilian teachers

Abstract The objective of this paper is to establish comparisons between the Linear Algebra (LA) approach practiced in Engineering courses offered by Chilean and Brazilian Higher Education Institutions and, for that, teachers from both countries were interviewed. From a methodological point of view, the research consists of two case studies. For the analysis of the interviews, we used the precepts of Content Analysis. The data were qualitatively analyzed considering the MTSK theoretical model. In this analysis, we sought to identify what specialized knowledge these teachers have regarding AL and about its teaching and learning. The results have shown that, in relation to the Mathematical Knowledge (MK) domain, which refers to content knowledge, the Knowledge of Topics subdomain (KoT), linked to the way that teachers know the topics they teach, is the most present among interviewees. In the Pedagogical Content Knowledge (PCK) domain, covering specific pedagogical knowledge linked to Mathematics, the Knowledge of Mathematics Teaching (KMT), regarding strategies, teaching theories, resources and materials, and Knowledge of Mathematics Learning Standards (KMLS) subdomains, concerning the curricular specifications and the moments in which the student must or can learn certain content and with what level of depth prevail.

Keyakinan dan mathematics pedagogical content knowledge calon guru matematika: fokus pada domain bilangan

The purpose of this study was to identify and describe the beliefs and mathematics pedagogical content knowledge (MPCK) of pre-service mathematics teachers. This research is a type of quantitative research with a descriptive design. This research involved 29 final year mathematics pre-service teacher at a college in Southeast Sulawesi, Indonesia. The instruments used in this study were a confidence questionnaire and the MPCK test based on the TEDS-M indicator. The data analysis technique used in this research is descriptive analysis technique. The results showed that the pre-service teacher's belief in mathematics and teaching mathematics was in the high category. Meanwhile, the MPCK of pre-service teachers is in the less category. This shows that pre-service teachers still have difficulties in planning appropriate activities, appropriate methods, predicting student responses, evaluating student mathematical solutions, identifying student misconceptions and representing mathematical concepts or procedures. Future research may consider expanding the focus on the predictive relationship between belief and MPCK pre-service teachers.

Mathematical Knowledge for Teaching in the Initial Education of Special Education Teachers

Mathematical Knowledge for Teaching in the Initial Education of Special Education Teachers

Primary Pre-service Teachers’ Knowledge of the Concepts of Mean and Median

The development of statistical literacy has become a key outcome of mathematics curricula in various countries. The purpose of the study was to explore primary pre-service teachers’ (PSTs’) knowledge of aspects of measures of central tendency, which is a central concept in statistical literacy. The 183 PSTs’ written responses to four tasks were analysed in terms of demands elucidated by the Mathematical Knowledge for Teaching (MKT) framework. The results showed that almost all the participants were able to carry out simple computations of the mean and median and were generally able to identify learner errors. However, they found it much harder to solve problems that required unpacking of the procedures and comparison of different measures of central tendency. They also struggled with providing suitable feedback based on learner errors. This study shows that primary school PSTs need more support in developing these MKT skills with respect to the concepts of the mean and median.

Mathematical knowledge for teaching and mathematics didactic knowledge: a comparative study

AbstractThis paper compares and contrasts two approaches that are widely used in the English- and German-speaking discourse on mathematics teacher knowledge: ‘mathematical knowledge for teaching’ and ‘mathematics didactic knowledge’. It is proposed that these constructs are based on distinct theoretical and conceptual positions and origins. Mathematical knowledge for teaching is viewed as a utilitarian-pragmatic approach rooted in English-speaking traditions as it focuses on its use in teaching and represents a practice-based conceptualization of knowledge domains required for mathematics teaching. Mathematics didactic knowledge, on the other hand, is considered normative-descriptive as it is formulated based on didactic principles and broader theoretical perspectives, providing a theory-driven conceptualization of knowledge domains rooted in traditions of German-speaking didactics of mathematics. The paper further highlights similarities and differences in these two constructs through an examination of two central knowledge domains: specialized content knowledge (part of mathematical knowledge for teaching) and subject matter didactic knowledge (part of mathematics didactic knowledge).

Mathematical knowledge for teaching as a didactic praxeology

It is widely acknowledged that teachers' work has both practical and theoretical dimensions, both praxis and logos, and that teacher education should address both dimensions. However, as argued in this article, existing discourses for describing the mathematical knowledge of teachers and what it takes to become competent in it are not always clear to what extent these discourses address knowledge about the practices of mathematics teaching or theories that provide arguments for the practices. Specifically, I argue for the coordination of the framework of mathematical knowledge for teaching (MKT) with the analytical distinction of logos and praxis combined in teaching practices. Moreover, I have backed this argument with examples of the proposed categories of the adjusted framework from the existing body of work in mathematics education. Each category of the proposed framework, whether concerning a teaching method or a theoretical consideration, was indeed exemplified by some concept, approach, or perspective from the field of mathematics education. The framework, I believe, would prove useful in engaging mathematics teacher education discursively, for instance, in analyzing teacher education materials to determine the extent to which they offer mere methods, theoretical arguments, or a combination of both.