Mathematical Content Knowledge Research Articles

Student composition, equity, and mathematics learning outcomes during a time of educational reforms in Ethiopia

ABSTRACT This paper assesses the extent to which the implementation of a complex educational reform in Ethiopia was associated with changes in mathematics test scores for Grade 4 pupils from 2012–13 to 2018–19. During this period, while mathematics teachers’ educational qualifications and teacher content knowledge in mathematics improved, students in 2018–19 achieved, on average, lower test scores in mathematics than those in 2012–13. However, they also showed, on average, greater progress in mathematics test scores. Progress in mathematics test scores over the school year was associated with teacher content knowledge, but the magnitude of this association was larger in 2018–19 and was especially important for weaker students. The key to reconciling these paradoxical findings is to recognise that the reforms that underpin these improvements in school quality may well be working in multiple ways, changing the composition of the student body who attend school as well as increasing the amount of learning that takes place in the classroom, particularly among disadvantaged students.

To Praxis Test or Not? Mathematics Content Knowledge Tests for Teacher Licensure and Programmatic Considerations on Moving the Preparation Field Forward

ABSTRACT This study examined the relationships between recommended mathematics coursework for secondary mathematics teacher preparation, internal program measures, prior academic performance, and the Praxis-II testing requirement for initial teacher certification used in several states. Using structural equation modeling and path analysis, the research aimed to understand these relationships and their policy implications. Conducted over six years and holding the program measures and design constant for five cohorts, the study found strong support for specialized content knowledge courses, particularly for secondary mathematics teacher candidates who were not well-prepared in high school. Additionally, the findings suggest that the Praxis-II exam might not be necessary for all teacher candidates in mathematics, especially those who were well-prepared in high school and engaged in advanced mathematics coursework for teachers while meeting rigorous program requirements for teacher licensing. The study argues for reconsidering programmatic design in light of these findings while providing program faculty suggestions for program improvements. State Departments of Education and policy makers may consider the research methodology, validity of internal measures, and testing requirements for Praxis-II for future discussions.

Use of mathematical problems rooted in primary historical sources to reveal preservice teachers’ mathematical content knowledge

In this paper, the use of mathematical problems rooted in primary historical sources as a diagnostic tool for identifying learners’ understanding about area and volume is examined. The study follows preservice teachers in the context of a compulsory course in mathematics education while dealing with such problems embedding errors and challenging mistaken beliefs about the area of quadrilaterals and the volume of the cube. Although the initial aim was to involve the participants in inquiry-based activities, in the end, the history of mathematics served as a means to reveal the complete or partial understanding of the concepts making evident at the same time the participants’ misconceptions such as the overgeneralization of the rule ‘length times breadth’ and the confusion between area and perimeter concerning the area of the quadrilaterals task, and the illusion of linearity and the confusion between volume and surface area concerning the volume of the cube task.

Analysis of Mathematics Content Knowledge of Mathematics Teachers Based on Bloom's Taxonomy Revision

This study aims to assess mathematics teachers' understanding of Mathematics Content Knowledge based on the Revised Bloom Taxonomy. The method used is quantitative descriptive research with a survey approach. The respondents of this study were mathematics teachers who taught at SMP/MTs in Losari District, Brebes Regency. Data was collected using non-test instruments in the form of questionnaires and question tests. Data analysis involves presenting data, testing differences, and concluding. The results showed that teachers' understanding of Mathematics Content Knowledge in teaching and mastery of the material was categorized as strong. Based on Bloom's revised taxonomy, teachers have good mastery, especially in knowledge dimensions such as factual, conceptual, procedural, and metacognitive knowledge, which are categorized as high. There was no significant difference in the average Mathematics Content Knowledge score of teachers, either with a working period of less than 10 years, 10-15 years, or more than 15 years.

Mathematical Content Knowledge and Problem-Solving Skills of Preservice Teachers in the Era of Distance Learning

Mathematical Content Knowledge and Problem-Solving Skills of Preservice Teachers in the Era of Distance Learning

TPACK competence of mathematics education students in designing constructivist learning

The problem and the aim of the study. This study examines mathematics education students' TPACK (Technological Pedagogical Content Knowledge) and constructivist learning skills. The major issue is mathematical education students' ability to integrate technology, pedagogical expertise, and mathematical content knowledge in a constructivist learning setting. The study topics include how well students employ constructivist ideas in learning design, how technology supports mathematics learning, and what variables impact their TPACK abilities. By studying it more, this research may help enhance the mathematics education curriculum and teacher training to meet 21st-century education needs. Research methods. Qualitative case study research is used in this work. 23 students of the mathematics education program at Universitas Negeri Surabaya were selected. After observing students in constructivist mathematics learning design activities, in-depth interviews were conducted to understand better their views on TPACK and the constructivist approach, and their learning plans were analyzed. Students' TPACK abilities in constructivist learning will be assessed using thematic and triangulated data analysis. The research findings may help explain the problems and opportunities of constructing a mathematics education curriculum, particularly when merging TPACK and constructivist techniques. Results. The average TPACK competency of math education students is 78.4%, according to studies. According to the TPACK component analysis, students scored 85.2% on the Technology Knowledge Test (TK), 79.6% on the Pedagogical Knowledge Test (PK), and 75.8% on the Content Knowledge Test (CK). However, CK component proficiency varied, suggesting mathematical content knowledge should be improved. In-depth interviews showed that students with formal educational technology training or courses had higher TPACK ratings (M = 82.1%) than those without (M = 76.3%). These data show that formal education and training improve TPACK abilities. To optimize the constructivist method, the CK component must be improved. This research shows that technology integration and mathematics subject mastery are crucial to constructivist learning curricula and training programs for mathematics education students. Conclusion. The constructivist learning TPACK skills of mathematics education students are examined in this study. Students demonstrated technology integration, instructional competency, and mathematical topic knowledge with an average TPACK score of 78.4%. According to TPACK component analysis, pupils have good technical skills (TK) but varied mathematics topic knowledge. TPACK scores are higher for technology education students, demonstrating the role of formal learning in competency. These findings show that CK needs to include constructivism better. In conclusion, comprehensive training and mathematical content additions can improve students' TPACK skills, enabling constructivist technology usage. These insights may assist in improving mathematics curriculum and training.

The relationship between opportunities to learn in teacher education and Chinese preservice teachers’ professional competence

ABSTRACT Although research on the relationship between opportunities to learn (OTL) in teacher preparation and preservice teachers’ learning outcomes, including aspects of their knowledge, has significantly increased in recent years, the relationship between OTL and preservice secondary school teachers’ cognitive competence and, particularly, the situated aspect of competence, in Mainland China remain poorly understood. This paper examines this association based on a study involving 583 preservice teachers chosen from four preservice teacher education institutions. The results revealed that the preservice teachers received adequate OTL in various domains, such as tertiary-level mathematics and mathematics education pedagogy. However, mainly due to the influence of academically oriented teacher preparation models in Mainland China, there were quite weak overall connections between OTL and the four examined aspects of teachers’ professional competence (mathematics content knowledge, mathematics pedagogical content knowledge, a mathematics instruction-related perspective on noticing, and a general pedagogy-related classroom perspective on noticing). Comparatively speaking, relatively strong associations existed between OTL in teaching methods, school-based teaching practicums, and the situated aspect of competence (e.g. noticing). These findings show that, to a great degree, initial teacher preparation in Mainland China does not relate closely to preservice teachers’ professional competence.

Computational Thinking in Secondary Mathematics Education with GeoGebra: Insights from an Intervention in Calculus Lessons

Nowadays, mathematics teachers in K–12 strive to promote their students‘ mathematical knowledge and computational thinking (CT) skills. There is an increasing need for effective CT-embedded mathematics learning material and a better understanding of students’ views toward them. In this work, we present the results of a research study, which included the design of a six-lesson learning activity aimed at fostering 16- to 17-year-old secondary students’ CT skills in calculus lessons using the dynamic mathematics software GeoGebra. Our goal was to investigate how students experienced the CT-embedded calculus lessons with GeoGebra and what challenges they faced during their interaction with the learning material and software. We collected and analyzed data from students’ code in GeoGebra, workbooks, semi-structured interviews, and questionnaires. Our findings suggest that most students mastered using CT concepts in calculus activities to a satisfactory degree and could reason about their computational solutions using GeoGebra and the generated graphs. Students’ understanding of the mathematical content knowledge introduced was essential to complete the lesson series successfully and unnoticed gaps in prior knowledge emerged. Our study shows that students appreciate the CT-embedded calculus lessons and GeoGebra’s exploratory approach to mathematics problems when provided with appropriate support. We conclude that an integrated approach to mathematics education and CT is viable and can contribute not only to fostering CT but also to increasing interest in mathematics.

Seeing mathematical learning opportunities in preschool situations: the role of teacher disposition and type of picture book

Pre-school teachers’ competence to see and decide to use learning opportunities in informal play-based situations (i.e. situation-specific skills) is key for high-quality pre-school instruction. Current insights into pre-school teachers’ situation-specific skills and their association with underlying teacher disposition and instructional materials are limited. We aimed to complement existing findings by systematically analysing pre-service and in-service pre-school teachers’ situation-specific skills in association with their underlying disposition and the type of instructional material in the domain of early mathematics. We assessed 274 pre-service and in-service pre-school teachers’ mathematical content and pedagogical content knowledge (MCK, MPCK), beliefs and situation-specific skills, operationalised as the mathematical questions they formulate during simulated shared picture book reading activities. Logistic multilevel regression analyses revealed a complex interplay between pre-service and in-service pre-school teachers’ disposition, the type of instructional material and situation-specific skills, pointing to the need for future studies on the topic.

Effectiveness of the continuous professional development training on upper primary mathematics and science and elementary technology teachers’ Pedagogical Content Knowledge in Rwanda

This research aimed to determine the influence of Continuous Professional Development (CPD) training offered by Rwanda Quality Basic education for Capital Human Development Project (RQBEHCD) sub-component 1.2 on upper primary teachers’ Pedagogical Content Knowledge (PCK) in mathematics and science, and elementary technology (SET) subjects. The study adopted an explanatory sequential design to collect and analyze data. Quantitative data were collected from Mathematics and SET achievement tests while qualitative data were collected from semi-structured interview. Quantitative data were collected from 290 participants including 166 male and 124 female teachers. On the other side, the qualitative data were collected from 20 teachers including 12 males and 8 females selected randomly. Descriptive and inferential statistics (repeated measures Analysis of Variance were used to analyze quantitative data while interpretive and thematic analysis were used to analyze qualitative data. The results found high statistical significance in mean scores of mathematics teachers between pre and post-test, with a p < 0.001, medium learning gain of 0.37, and an effect size of 0.450. However, there were no significant differences found between male and female teachers (p > 0.05). Likewise, SET teacher's results showed a significant difference in mean score between pre and post-test with p < 0.01, learning gain of 0.20, and effect size of 0.432 but no significant difference was found between male and female teachers (p > 0.05). In addition, the qualitative results indicated that the CPD program enhanced engagement, interest, confidence, and self-efficacy in teaching Mathematics and Science. Thus, we recommended that the CPD program should be done nationwide to maintain teachers’ PCK in mathematics and science.

Fuzzy Delphi method for A-level mathematics technological pedagogical and content knowledge module

This study sought to identify the elements for designing and developing the A-level mathematics technological, pedagogical, and content knowledge (TPACK) teaching module. The aim of the module was to incorporate various pedagogies and technological tools for helping mathematics teachers to trigger, maintain, as well as sustain students’ interest in learning mathematics. This study used the Fuzzy Delphi method to determine the consensus on how A-level mathematics instruction should be designed. A semi-structured interview had first been conducted to develop a questionnaire. Then, a group of 16 experts was consulted together to discuss their ratings and reach a consensus. The group of experts included educational technology and subject matter experts. Data analysis results showed that the experts accepted all these elements through the expert consensus value above 75%, the threshold value (d)≤0.2, and the Fuzzy score (A)≥α-cut value of 0.5. This research provides implication to A-level mathematics teachers to systematically create an environment that is more conducive to learning and that will ultimately lead to an increase level of students’ interest towards learning.

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.

Academic Resilience Among Post Graduate Students in the Universities of Islamabad: A Comprehensive Assessment

Introduction: Mathematics plays a crucial role in the development of science and technology, and its importance is recognized at all educational levels. This study investigates the availability and utilization of mathematics facilities for the improvement and development of prospective teachers. Methodology: This quantitative study targeted students from four universities in Islamabad and Rawalpindi. A questionnaire was distributed to 300 respondents, with 175 completed responses collected through proportionate random sampling. The questionnaire was validated by experts and its reliability confirmed using Cronbach’s Alpha. Data analysis was conducted using the Statistical Package for Social Sciences (SPSS). Results/Findings: The findings revealed that most respondents strongly agreed that mathematical content knowledge significantly impacts the learning and understanding of prospective teachers. Additionally, the study highlighted the importance of practice in learning mathematics and emphasized the need for curriculum-based instruction that extends beyond exam preparation. Future Direction: The study recommends developing a framework to address identified challenges, enhance mathematical content, improve pedagogical approaches, and ensure the availability of up-to-date facilities for prospective teachers.

A STUDY TO INVESTIGATE THE CHALLENGES OF PROSPECTIVE TEACHER OF MATHEMATICS FACILITIES AT UNIVERSITY LEVEL(GRADUATE)

Introduction: Mathematics plays a crucial role in the development of science and technology, and its importance is recognized across all levels of education. This study aims to identify the challenges related to the availability and utilization of mathematics facilities for the improvement and development of prospective teachers. Methodology: The study employed a quantitative approach. The population consisted of students from four universities in Islamabad and Rawalpindi. A questionnaire was distributed to 300 respondents, with 175 valid responses collected through proportionate random sampling. The questionnaire was validated by experts, and its reliability was confirmed using Cronbach’s Alpha. Data analysis was conducted using the Statistical Package for Social Sciences (SPSS) Results/Findings: The findings indicated that most respondents agreed that mathematical content knowledge significantly impacts the learning and understanding of prospective teachers. Additionally, the participants strongly agreed that practical exercises play a vital role in learning mathematics, and there is a need for curriculum-based instruction that goes beyond exam preparation. Future Direction: The study recommends developing a framework to address identified challenges, enhance mathematical content, improve pedagogical approaches, and ensure the availability of up-to-date facilities for prospective teachers.

Prospective mathematics secondary teachers’ perceptions of their pedagogical content knowledge: an approach in the educational context of Zambia

ABSTRACT The purpose of the current study was to contribute to a better understanding of Zambian prospective mathematics secondary teachers’ pedagogical content knowledge of mathematics (M-PCK) as part of a larger project aimed at characterising their subject matter and pedagogical content knowledge (PCK) related to the function concept. K-means Cluster analysis was used to derive profiles based on M-PCK perceptions. Three profiles related to M-PCK perceptions of prospective teachers emerged from the clustering process namely (i) self-doubting (cluster 1), (ii) anti-M-PCK (cluster 2) and (iii) confident and enthusiastic (cluster 3). A One-way ANOVA test was performed to determine the existence of differences in prospective teachers’ perceptions between Clusters based on the five M-PCK sub-factors. The test revealed significant differences between Clusters of prospective teachers’ perception of all sub-factors. Prospective teachers in Clusters 1 and 2 showed low confidence in their M-PCK perceptions while those in Cluster 3 generally showed moderate confidence.

A promising approach to scaling up professional development: intelligent, interactive, virtual professional development with just-in-time feedback

AbstractThe scalability and accessibility of quality professional development (PD) is an ongoing concern in the teacher education community, yet little research has been conducted on potential solutions. We aimed to address this gap by developing an interactive, virtual PD program that uses intelligent tutoring systems and provides instant feedback to teachers. We then explored the role of this program in developing teachers’ content and pedagogical content knowledge of mathematics. We collected data from 60 teachers located across the USA and found that those who completed the program increased their content and pedagogical content knowledge.

Toward proficiency: Developing a multiplication mathematical content knowledge test for pre-service mathematics teachers in Indonesia and Türkiye

This study aimed to develop a specific Mathematical Content Knowledge (MCK) test focused on multiplication for pre-service mathematics teachers in Indonesia and Türkiye. No assessment tool exists to measure MCK related to multiplication among pre-service mathematics teachers. We used convenience sampling to obtain data from third- and fourth-year pre-service teachers in Indonesian and Turkish mathematics teacher education programs. The test items were administered to 423 Indonesian pre-service mathematics teachers and 413 Turkish pre-service mathematics teachers, and their responses were analyzed using factor analysis. The MCK test was found to be reliable, with 18 items that were grouped into four components: procedural knowledge of multiplication (C1), understanding of multiple representations (C2), conceptual knowledge of multiplication (C3), and anticipating students' thinking (C4). The study results suggest that a valid MCK test helps evaluate multiplication in these four components. Additionally, the study findings indicate that Indonesian pre-service teachers score higher in the C1 and C4 categories, while Turkish pre-service teachers score higher in the C2 and C3 categories.

Learners’ Misconceptions and Errors when Solving Inequalities

Learners’ Misconceptions and Errors when Solving Inequalities

Teaching measurement: The role of mathematics teachers' enacted PCK on gain in learner outcomes

Teaching is a challenging profession where teachers must create valuable learning opportunities to enhance learners’ conceptual understanding. Apart from mathematical content knowledge, teachers use their pedagogical content knowledge (PCK) that develops as they reflect on previously taught lessons and learner responses in assessments. International and national assessment studies showed that South African learners perform poorly in, among other topics, Measurement. Thus, we determined the gain in learner outcomes as revealed in a pre-and post-test on Measurement and studied one of the PCK domains, namely teachers’ enacted PCK as informed by the baseline assessment learner outcomes. The aim was to determine how teachers’ enacted PCK relate to learner outcomes. Underpinned by a social constructivist paradigm, the study used a mixed-method research approach. Data were gathered from a pre-and post-test written by 124 Grade 9 learners taught by two experienced mathematics teachers in a city school in South Africa. Findings revealed that although some improvements are evident after the topic has been taught, the test was still experienced as difficult by almost all the learners. However, from the observations, there is little evidence that the experienced teachers extensively used the baseline assessment outcomes to inform their teaching.

Effect of Mathematical Content Knowledge on Mathematical Knowledge for Teaching

Effect of Mathematical Content Knowledge on Mathematical Knowledge for Teaching