فاعلیة تخطیط الدرس فی تنمیة المعرفة البیداغوجیة للمحتوى الریاضی لدى الطلبة / المعلمین

This study examined pre-service secondary mathematics teachers’ mathematical pedagogical content knowledge pertaining to the order of operations. The conceptual framework developed in this study has both theoretical and methodological significance and holds potential for further research on mathematical pedagogical content knowledge beyond the order of operations. The findings shed light on the participants' reasons for utilising the order of operations and identify misinterpretations they encountered. Furthermore, the study highlights the challenges faced by pre-service secondary mathematics teachers who lack mathematical pedagogical content knowledge when planning to teach the order of operations.

Abstract: This study examined how early childhood (EC) teachers’ mathematical content knowledge (MCK), mathematical pedagogical content knowledge (MPCK), mathematics anxiety, and mathematics enjoyment predicted children’s development in mathematics across two measurement occasions. Data on EC teachers ( n = 25) and children ( n = 236) was analyzed using multilevel structural equation modeling in order to account for the nested data structure. Children’s age was included as control variable. Results revealed that none of EC teachers’ variables predicted children’s development while MPCK and mathematics enjoyment predicted their initial achievement. EC teachers’ professional knowledge and emotions might be too distal to children’s outcome in order to directly predict children’s development. Instead, more proximal EC teacher variables (e. g., instructional quality) might be better suited. The present study offers valuable insights into the complex relationship between teacher variables and children’s mathematics development. Its methodological and content-related considerations make it a valuable reference for future research.

This study aims to identify the level of technological pedagogical content knowledge (TPCK) of mathematics trainee teachers at Universiti Pendidikan Sultan Idris (UPSI) and explore their teaching practices during practical training at school. The study was conducted in two phases using a mix-method research. In the first phase, a survey method using a questionnaire was carried out on 156 trainee teachers of Bachelor of Mathematics Education (AT14) and Bachelor of Science (Mathematics) with Education (AT48). The instrument used was a questionnaire that measures the level of content knowledge, pedagogy, technology and TPCK of mathematics. Data were analyzed using descriptive statistics, namely the mean. While in the second phase, the interview method involved four trainee teachers were performed. The instrument used was the semi-structured interview protocol to assess the trainee teacher's TPCK integration in their teaching practice. Data were analyzed using the content analysis. The findings showed that the level of knowledge of TPCK among trainee teachers was moderate with overall mean score of 3.60. This level did not show significant differences between the two programs with mean scores of 3.601 for the AT14 group and 3.603 for the AT48 group. However, there was a difference for gender classification such that the female trainees had mean score of 3.58 and male trainees with mean score of 3.72. Although students' TPCK level was moderate, the level of content knowledge (CK), technological knowledge (TK) and pedagogical knowledge (PK), showed a higher level with overall mean scores of 3.75, 3.87 and 3.84 respectively. The findings also showed that in terms of content knowledge, trainee teacher's learning mathematics background was good, but the knowledge of mathematics was limited in the curriculum, philosophy and application aspect. In terms of pedagogical content knowledge, all respondents tend to use lecture and discussion methods in teaching Trigonometry topic in almost the entire teaching and learning sessions. In terms of using technology in teaching and learning process, all respondents were more likely to use PowerPoint presentation as well as assisted by the use of a scientific calculator. The use of PowerPoint was limited in explaining the title and its content and the calculators were used during practice sessions. The four trainee teachers were still at the recognizing and acceptance levels of TPCK integration during their teaching practice. This study indicates that UPSI trainee teachers should be given exposure in the use of information and communication technology and trained to provide alternative teaching strategies in more creative ways if there are obstacles when integrating technology in teaching and learning process.

From existing research, it is known that early childhood teacher education pro-vides few opportunities to learn mathematical content or mathematics pedagogical con-tent knowledge. Furthermore, little is known about the extent to which teacher educators are qualified to teach early childhood mathematics within teacher education programs. This study investigates the frequency of opportunities to learn mathematical content knowledge and mathematics pedagogical content knowledge. Additionally, we investigate some characteristics of teacher educators, namely the sources they use for professional development with respect to early mathematics education. We report descriptive results as well as correlations. The results indicate that opportunities to learn mathematical con-tent knowledge and mathematics pedagogical content knowledge have increased over time. Moreover, the results indicate that teacher educator professional development for-mats that encourage professional exchange among teacher educators, experts in early mathematics education and in-service teachers may increase teacher educators’ compe-tence in early childhood mathematics and thereby the frequency with which this topic is addressed in teacher education.

The purpose of this paper is to examine trainee primary school teachers’ confidence in their mathematical content knowledge (MCK) and confidence to teach specific primary mathematics concepts (mathematics pedagogical content knowledge –MPCK) which was correlated to their actual MCK on specific tasks. For this correlational study survey and test data were collected from a cohort of 210 trainee teachers. It was found that confidence to do and to teach mathematics was reasonably strongly correlated with competence. Trainee teachers’ confidence varied greatly depending on the specific mathematics they were attempting. When presented with specific tasks, trainees were well aware of the link between personal numeracy levels and their potential to teach primary mathematics. A further finding was that the trainee teachers tended to over report their confidence. It is unknown if this is a cultural manifestation or a limitation of the scale. The data also add to the body of knowledge with respect to the MCK of about-to-graduate primary teachers.

Pedagogical content knowledge (PCK)—that is, knowledge necessary to make subject matter accessible to students—is considered to be a key component of teacher competence. Thus, how teachers develop PCK is an important issue for educational research and practice. Our study aimed at investigating the conditions of development of PCK, and especially at testing competing assumptions about the role of prior content knowledge (CK) and prior pedagogical knowledge (PK) for PCK development. We targeted three assumptions: (1) CK and PK amalgamate, (2) CK is a necessary condition and facilitates PCK development, and (3) CK is a sufficient condition for teachers’ PCK development. One hundred German pre-service elementary teachers participated in a randomized controlled trial. Participants’ prior knowledge was manipulated through five courses, constituting three experimental conditions and two controls. In this chapter, we report on the conceptualization of the treatments, and provide a detailed analysis of our knowledge measures. We further give an overview of the initial, preliminary results of the experiment. The findings of our study may have important implications for the discussion of how PCK can best be fostered in teacher education.

Cases can be used in a variety of ways with practicing teachers and teacher candidates to explore classroom practice and build knowledge needed for teaching. The richness of a case that portrays an authentic episode of teaching can afford teachers the opportunity to examine issues of mathematics content, pedagogical practices that support student learning, and the interactions between the two. In this chapter, I discuss a way of using cases in conjunction with the solving of a mathematical task that allows teachers to build these different facets of mathematical knowledge for teaching and strengthen the connections between them. Using records from a content-focused methods course for practicing teachers and teacher candidates, I examine how solving and discussing a mathematical task, then reading and analyzing a case of classroom practice featuring the same task, influenced teacher learning. This combination of exploring a mathematical task with the analysis of a related case afforded teachers the opportunity to integrate their mathematical, pedagogical, and pedagogical content knowledge in interesting ways.

ABSTRACTThis paper concludes the Special Issue (SI) ‘Probing the Amalgam: the relationship between science teachers’ content, pedagogical and pedagogical content knowledge’. We review the five papers (Sorge et al; Gess-Newsome et al; Kind; Pitjeng-Mosabala and Rollnick; and Liepertz and Bronowski) by discussing evidence these present regarding the relationships between content knowledge (CK), pedagogical knowledge (PK) and pedagogical content knowledge (PCK); the development of CK, PK and PCK in novice and experienced secondary science teachers and how CK, PK and/or PCK impact students’ learning. In conclusion, we draw these findings together in offering proposals for future research via reconsideration of Shulman’s amalgam. This includes post-hoc examination of a PCK model known as ‘the Consensus Model’ (Gess-Newsome, [2015]. A model of teacher professional knowledge and skill including PCK: Results of the thinking from the PCK Summit. In A. Berry, P. J. Friedrichsen, & J. Loughran (Eds.), Re-examining Pedagogical Content Knowledge in Science Education (pp. 28–42). New York, NY: Routledge; Neumann, Kind, & Harms [2018]. Probing the amalgam: The relationship between science teachers’ content, pedagogical and pedagogical content knowledge. International Journal of Science Education, 1–15) and presentation of a novel PCK structure based on evidence from the SI studies.

This study aims to assess and compare–at the national and regional sizes–mathematical knowledge for teaching of pre-service elementary mathematics teachers (PEMTs). Participants of the study were composed of 1,367 pre-service teachers selected among 21 universities in 12 regions of Turkey. Mathematical knowledge for teaching (MKT) test, which were developed by TEDS-M project were used to collect the data. The instruments were adapted into Turkish by the researchers. According to results, while there wasn’t much differentiation among the regions in terms of the MKT, pre-service teachers in the West Marmara, Mediterranean, and Northeast Anatolia regions are more successful than the other regions. PEMTs’ mathematics pedagogical content knowledge (MPCK) performance was close but somewhat better than their mathematics content knowledge (MCK) performance, in terms of both regions and universities. In the case of the MPCK test, the regions exhibited a more homogeneous picture, while in the case of MCK, the universities did so. Algebra content domain is where the PEMTs’ performance is lowest. When we compared the results obtained from this study with published international averages from TEDS-M study, Turkish PEMTs had appreciably higher performance on both MCK and MPCK components.

Family background, entry selectivity and opportunities to learn: What matters in primary teacher education? An international comparison of fifteen countries

German pre-service teacher education aims to foster the concurrent formation of content knowledge, pedagogical knowledge, and pedagogical content knowledge. Accordingly, the coordination, sequencing, and prioritization of learning opportunities for the three areas of professional knowledge represent vital issues of the organization of teacher education at German universities. In this context, reanalyzing selected data of a previous experiment, we examined effects of instruction on pedagogical content knowledge about fractions in sixth-grade mathematics on the formation of corresponding content knowledge and generic pedagogical knowledge. Totaling 59 pre-service elementary school teachers, three groups had received seven hours of intervention on either content knowledge, pedagogical knowledge, or pedagogical content knowledge. Analyses of video recordings, ratings of treatment quality, and tests of professional knowledge revealed strong internal validity of the data. We found small effects of instruction on pedagogical content knowledge on the formation of both content knowledge and pedagogical knowledge. However, contrary to our expectations, effects on pedagogical knowledge were more consistent and statistically robust than effects on content knowledge. For the development of pedagogical knowledge in teacher education, this finding highlights the potential of parallel or integrated coursework, in which pre-service teachers use specific instructional strategies and student conceptions as examples to derive generic principles of teaching and learning.

Mathematics and science are both essential to the field of agriculture; however, while science curriculum is currently integrated in many secondary school-based agricultural education classrooms nationwide, mathematics integration remains limited. The opportunity for students to engage in real world applications of mathematical content through school-based agricultural education programs exists, but if teachers do not possess the content knowledge necessary to teach mathematics, students are then left at a disadvantage for learning the content. The purpose of this study was to determine Arizona agriculture teachers’ perceived and actual mathematical content knowledge. The objectives of this study were to describe teachers’ perceived mathematical content knowledge, actual mathematical content knowledge, and the relationship between perceived and actual mathematical content knowledge. The content knowledge framework was utilized in determining teachers’ content knowledge for the subject area of mathematics. A quantitative analysis revealed school-based agricultural education teachers perceived their average mathematical ability as being at a moderate level, while their average actual mathematical ability was 44% on a mathematics content exam. The analysis also revealed a negative correlation between teachers’ perceived ability and years spent teaching and a positive correlation between teachers’ actual ability and years spent teaching. It is recommended that mathematics requirements at the teacher preparation level be reexamined. Additionally, professional development for Arizona school-based agricultural education teachers in various mathematics content is encouraged.

Although studies of teacher preparation programs have documented positive changes in mathematical knowledge for teaching with preservice teachers in mathematics content courses, this study focused on the impact of a mathematics methods course and follow-up student teaching assignment. The presumption was that preservice teachers would show growth in their mathematical knowledge during methods since the course was structured around active participation in mathematics, research-based pedagogy, and was concurrent with a two-day-per-week field experience in a local elementary school. Survey instruments utilized the computer adaptive test version of the Mathematical Knowledge for Teaching (MKT) measures from the Learning Mathematics for Teaching Project, and the Attitudes and Beliefs (towards mathematics) survey from the Mathematical Education of Elementary Teachers Project. A piecewise growth model analysis was conducted on data collected from 176 participants at 5 time-points (methods, 3 time-points; student teaching, 2 time-points) over a 9 month period. Although the participants' demographics were typical of U.S. undergraduate preservice teachers, findings suggest that initial low-level of mathematical knowledge, and a deep-rooted belief that there is only one way to solve mathematics problems, limited the impact of the methods and student teaching courses. The results from this study indicate that in (a) number sense, there was no significant change during methods (p = .392), but a significant decrease during student teaching (p < .001), and in (b) algebraic thinking, there was a significant decrease during methods (p < .001), but no significant change during student teaching (p = .653). Recommendations include that the minimum teacher preparation program entry requirements for mathematical knowledge be raised and that new teachers participate in continued professional development emphasizing both mathematical content knowledge and reform-based pedagogy to continue to peel away deep-rooted beliefs towards mathematics.

Although trigonometry is an important section in secondary school mathematics curricula, many teachers find it challenging to teach as a result of insufficient pedagogical content knowledge (PCK). Therefore, the aim of this article was to report on how PCK on trigonometry manifests itself in teachers' practice. This exploratory case study was underpinned by the mathematics knowledge for teaching (MKT) model of Hill et al. (2008). Twelve teachers were purposefully selected from six township schools. Qualitative data were collected through semi-structured one-on-one interviews, lesson plans, assessment tasks, and lesson observations; and analyzed using content analysis. The findings show that four elements of PCK on trigonometry manifest themselves in teachers’ practice, namely knowledge of subject matter, knowledge of teaching strategies, knowledge of students’ conceptions and knowledge of curriculum, but varied in levels of sufficiency. Awareness of these variances forms a useful basis for planning developmental opportunities that could address shortcomings in PCK on trigonometry in teachers’ practice. This study adds to few studies in PCK in mathematics by providing empirical evidence on how PCK on trigonometry manifests itself in teachers’ practice.

The aim of this study was to explore the influence of pedagogical content knowledge competencies of Ghana senior high school textile teachers on the modernisation of indigenous Ghanaian textile industry. A cross-sectional survey design was used to collect both qualitative and quantitative data. The study targeted senior high school textile teachers, senior high school textile graduates and heads of visual arts department. The sample size for the study was 243 (225 teachers, 12 textile graduates & 6 HoDs). Questionnaire, interview guide and observation were the research instruments employed for data collection. The result of H 01 indicated that the pedagogical content knowledge of textile teachers has influence on the skills textile graduates needed to modernise the indigenous textile industry. The findings of the study also signposted that the pedagogical content knowledge of the textile teachers cannot enable the textile students to acquire the requisite skills needed to modernise the indigenous textile industry. The findings of the study indicated most of the Ghana senior high schools lacked textile training facilities and resources needed to modernise the indigenous textile industry. The study recommended that ministry of education should periodically sponsor textile teachers to have industrial attachments in both local and international renowned textile industries to learn modern content knowledge of textile manufacturing. There should also be periodic capacity building workshops on vocational skills pedagogical knowledge to enable the textile teachers effectively transmit the content knowledge to their students effectively. Keywords: Content knowledge, pedagogical knowledge, pedagogical content knowledge. DOI : 10.7176/ADS/78-01 Publication date: November 30th 2019