Mathematics Pedagogical Content Knowledge Research Articles

What role does professional noticing play? Examining connections with affect and mathematical knowledge for teaching among preservice teachers

This study examined the intersections of preservice elementary teachers’ (PSET) professional noticing (PN) of children’s mathematical thinking, two mathematical knowledge for teaching (MKT) domains: mathematics content knowledge (MCK) and mathematics pedagogical content knowledge (MPCK), and two affective domains: attitudes and dispositions toward teaching mathematics. An instructional module focused on PN of children’s early algebraic thinking, as defined by Jacobs et al. (J Res Math Educ 41: 169–202, 2010) to include three components: attending, interpreting, and deciding, was implemented with 170 PSETs. The PSETs, who participated in the instructional module, the implementation group, showed significant positive growth in attending, interpreting, and attitudes toward mathematics. There was no significant change in dispositions toward teaching mathematics and a decrease was observed in PSET deciding and MKT. A comparison group of 126 PSETs enrolled in mathematics methods courses did not participate in the instructional module. PSETs in the comparison group showed a decrease across all measures including attending, interpreting, deciding, attitudes, dispositions, and MKT. Results showed statistically significant connections within and between some of the constructs; however, the limitations of the study call for further investigation.

The relationship between preschool teachers pedagogical content knowledge in mathematics, childrens' math ability and liking

This study attempts to identify preschool teachers' pedagogical content knowledge levels in mathematics and to examine its effect on preschool children's mathematics ability and their liking of mathematics. The study utilised the relational survey model, one of the general survey models. The working group was composed of 600 children aged 54-66 months old and a total of 150 teachers. The study employed three data collection tools: with the 'Preschool Teachers' Pedagogical Content Knowledge Scale regarding Mathematics' and the 'Mathematics Liking Scale for Children' developed by the researchers; 'Early Mathematics Ability Test' to determine children's mathematics ability. The data were analysed using IBM's SPSS 22 statistical package. The correlation analysis results suggested that there were no significant relationships between teachers' pedagogical content knowledge in mathematics, children's mathematics ability, and their liking of mathematics, while a positive and significant relationship was found between children's mathematics ability and their liking of mathematics.

Exploration of pre-service teachers’ pedagogical content knowledge in mathematics learning in senior high school based on gender and academic skills

In Indonesia, pre-service mathematics teacher education is strictly supervised so that Pedagogical Content Knowledge (PCK) becomes an important aspect to build the quality of learning. This study aimed to explore pre-service mathematics teachers’ PCK based on gender and academic skills. To obtain rich and in-depth data, a qualitative approach was used. A total of 70 subjects aged between 19 – 21 years old participated in this study. There were two subjects selected based on their academic skills and gender. Using a grounded theory approach, we conducted a preliminary analysis, open coding, axial coding to obtain the three PCK components, namely Knowledge of Subject Matter (KSM), Knowledge of Pedagogy (KP), and Knowledge of Student (KS). Research findings revealed that the pre-service teachers’ pedagogical content knowledge in terms of knowledge of subject matter was categorized as good in mathematics learning. As for their knowledge of pedagogy, the male subjects presented the concepts by employing the expository strategy, the female subjects with high skills used the guided discovery, and the female subjects with average skill also employed the strategy of expository. In the aspect of knowledge of students, the subjects with average skills overcame students’ misconception by explaining the procedures and using the strategy of asking, but the subjects with high academic skills did not only implement the two previous strategies but also used their reasoning behind every procedure of problem-solving that they carried out. These findings can be used as recommendations for the development of mathematics learning.

Mentoring and preparing primary school mathematics teachers

This article reports on the mentoring of foundation phase teachers and their development of disciplinary-specific knowledge, particularly how the practicum provides a space for mathematics content knowledge (MCK) and mathematics pedagogical content knowledge (MPCK) for student teachers, given the generalist knowledge base of many primary school teacher mentors. Data was collected through semi-structured interviews with two Grade 2 mentors and focus group interviews with third-year student teachers in an urban university setting. A two-step process of analysis comprising open-ended coding, followed by a form of deductive coding, within a framework derived from literature on the development of MCK and MPCK, lead to two main findings. Firstly, mentors might have hindered student teachers’ development of MCK and MPCK, thus impacting on the building of comprehensive conceptual models of good mathematics teaching practice. Secondly, the efficacy of the university-led training is questioned. We therefore recommend that greater emphasis be placed on the development of subject-specific mentoring practices, especially in scarce-skills areas such as mathematics, and argue for a set of standards for mathematics-specific mentoring practices in South African primary school teacher education.

Predictive Validity of Entry Level Mathematical on Mathematical Knowledge of Pre-service Teachers for Teaching Basic School mathematics in Ghana

Introduction: Knowledge about teaching mathematics remains a contentious issue in the preparation of pre-service teachers in Ghana. This study explored relationships among entry-level mathematical knowledge (ELMK), mathematics content knowledge (MCK) and mathematics pedagogical content knowledge (MPCK) scores of pre-service teachers in colleges of education (CoEs) in Ghana. Methods: The cross-sectional survey design was adopted for this study. A total of 998 pre-service teachers from ten (10) public colleges of education were sampled via multi-stage sampling technique. Structured questionnaire and unstructured observation schedule were used for the survey. College mathematics test results were used as secondary data. The data were analysed using descriptive statistics (frequency count, percentage) and inferential statistics (Pearson’s correlation, paired-samples t-test and regression analysis) with the aid of Statistical Package for Social Sciences software version 22. Results: The majority of pre-service teachers in Ghanaian colleges of education had average but relatively weak grades in ELMCK. ELMCK did not significantly predict pre-service teachers’ MCK statistically (p ≥ 0.05) although a statistically significant positive correlation existed between the variables. Although a negative correlation existed between ELMCK and PMCK, ELMCK significantly predicted pre-service teachers’ PMCK statistically (p ≤ 0.05) but the effect size was statistically negligible. Conclusions: This study concluded that, CoEs in Ghana admit pre-service teachers with average but weak grades in ELMK. Therefore, Ghanaian CoEs should structure and introduce PMCK alongside MCK for pre-service teachers. Mathematics tutors in CoEs should organize remedial lessons for pre-service teachers to scale-up their MCK and PMCK that adequately equip them with MKT. Keywords: Entry-level mathematical knowledge (ELMK), mathematics content knowledge (MCK), mathematics pedagogical content knowledge (MPCK), mathematical knowledge for teaching (MKT), pre-service teachers.

Unpacking Mathematics Pedagogical Content Knowledge for Elementary Number Theory: The Case of Arithmetic Word Problems

“Number” is an important learning dimension in primary mathematics education. It covers a large proportion of mathematical topics in the primary mathematics curriculum, and teachers use most of their class time to teach fundamental number concepts and basic arithmetic operations. This paper focuses on the nature of mathematics pedagogical content knowledge (MPCK) concerning arithmetic word problems. The aim of this qualitative research was to investigate how well the future primary school teachers in Hong Kong had been prepared to teach mathematical application problems for third and sixth graders. Nineteen pre-service teachers who majored in both mathematics and primary education were interviewed using two sets of scenario-based questions. The results revealed that innovative approaches were suggested for teaching third graders while the strategies suggested for teaching sixth graders were mostly based on a profound understanding of mathematical content knowledge. Many participants demonstrated sound knowledge about the sixth grader’s mathematical misconception, but most of them were unable to precisely indicate the third grader’s error in presenting a complete solution for a typical mathematics word problem. A deep understanding of elementary number theory seems to be a precondition for developing pre-service teachers’ MPCK in teaching arithmetic word problems.

A Comparative Study on the Constructive Beliefs and Mathematical Attitudes about Mathematics Pedagogical Content Knowledge of Korean and Chinese Early Childhood Teachers

A Comparative Study on the Constructive Beliefs and Mathematical Attitudes about Mathematics Pedagogical Content Knowledge of Korean and Chinese Early Childhood Teachers

Differences in teacher education programmes and their outcomes across Didaktik and curriculum traditions

ABSTRACT Teacher education is of vital importance for what teachers are capable to do for their pupils, but little is known about student teachers’ pedagogical knowledge. The Didaktik and the curriculum traditions are two main education approaches underpinning formal schooling and teacher education programmes (TEPs) in the Western world. The main difference between the two traditions lies in the content and objectives of teacher education, which are either theoretical or action-oriented. Two questions are addressed quantitatively: How do teacher education programmes and their outcomes vary across Didaktik and curriculum traditions? How do opportunities to learn and beliefs about teaching methods affect mathematical content knowledge (MCK) and mathematical pedagogical content knowledge (MPCK) scores? Empirical data from the Teacher Education and Development Study in Mathematics (TEDS-M) are used, with samples from Norway, Germany, Switzerland, and the US. The study offers alternative explanations for variations of TEPs’ outcomes within the Western world.

Pre-service Secondary Teachers’ Mathematical Pedagogical Content Knowledge Self-concept related to their Content Knowledge of Functions and Students

Pre-service teachers’ beliefs, attitudes, and values about mathematics have a significant influence on their self-concept about mathematics as a subject and determine how confident they are to teach it. The purpose of this study was to examine Zambian pre-service secondary mathematics teachers’ pedagogical content knowledge self-concept about the function concept in relation to their knowledge of content and students. Data was collected from 150 pre-service teachers using a sequential approach in two phases. The first, quantitative phase, involved 150 pre-service teachers who responded to a functions survey and a mathematical pedagogical content knowledge survey. The second, qualitative phase, involved two pre-service teachers who were purposively selected from phase one to respond to vignettes and interviews for an in-depth understanding of their knowledge. Results of the study revealed that pre-service teachers’ level of their pedagogical content knowledge self-concept was low. They would not be confident enough to teach the function concept in secondary school. Results further revealed that their knowledge of content and students was weak. A weak correlative relationship between pre-service teachers’ KMLS and KM was revealed whereas a moderate correlative relationship of their KL and KC was revealed. It was further revealed that there was no significant correlative relationship between KTS and their knowledge of the function concept Thus, pre-service teachers needed to improve before leaving university for them to effectively teach secondary school concepts.

Relationship between pre-service mathematics teachers\u2019 knowledge, beliefs and instructional practices in China

In this paper we examine the relationship between teachers’ knowledge, beliefs and instructional practices based on a study with 495 Chinese pre-service mathematics teachers. The results indicate that Chinese pre-service mathematics teachers tend to hold mixed beliefs about the nature of mathematics, and a constructivist view about mathematics teaching and learning, and that they are inclined to report that their teaching is inquiry-oriented. Mathematical content knowledge (MCK) and mathematics pedagogical content knowledge (MPCK) were found not to correlate with the teachers’ self-reported instructional practice, in contrast to pre-service mathematics teachers’ beliefs, which showed a stronger association with their self-reported inquiry-oriented instructional practice. Moreover, pre-service teachers’ dynamic beliefs about the nature of mathematics, and constructivist beliefs about mathematics teaching and learning, acted as mediators between pre-service mathematics teachers’ MCK, MPCK and instructional practice respectively.

Investigating subject matter knowledge and pedagogical content knowledge in mathematics with the Concept Cartoons method

The article focuses on an educational tool called Concept Cartoons and its possible use in professional preparation of future primary school teachers. In particular, it presents the method of how Concept Cartoons can be employed as a tool for diagnosing subject matter knowledge and pedagogical content knowledge in mathematics. The first part of the contribution introduces the concept of teachers' knowledge that is used in the article, and the Concept Cartoon tool. It describes the original Concept Cartoons method for classroom use at primary and secondary school levels that was established by Keogh and Naylor, and then it introduces our work on diagnostic Concept Cartoons method, including a commented summary of our recent research on qualitative diagnosis of subject matter knowledge and pedagogical content knowledge in mathematics. The second part of the contribution introduces another step in the methodology, a mixed approach to the issue that enables to enrich the qualitative results with quantitative characteristics. The mixed method is illustrated through a small empirical study that shows how exactly the quantitative enrichment might be provided.

Development and validation of a survey instrument for measuring pre-service teachers’ pedagogical content knowledge

ABSTRACT In Australia, one of the key findings of the Teacher Education Ministerial Advisory Group was that not all graduating pre-service teachers possess adequate pedagogical content knowledge (PCK) to teach effectively. The concern is that higher education providers working with pre-service teachers are using pedagogical practices and assessments which are not informed by research. Due to its multifaceted nature, PCK is difficult to measure. In the case of mathematics PCK, some researchers have used multiple-choice questions while others use constructed-response formats depending on the specific attribute of PCK being measured. In either case, researchers need to use instruments that are appropriate to each task. This paper reports on the construction and reliability of a survey instrument used in a larger study to measure pre-service teachers’ mathematics PCK. Using four stages of survey development, in addition to inter-rater agreement measures and Rasch modelling techniques, data analyzes found high levels of validity and reliability for the Mathematics Pedagogical Content Knowledge Instrument (MPCKI).

Intelligence and knowledge: the relationship between preschool teachers’ cognitive dispositions in the field of mathematics

A large research gap exists between psychological research about intelligence on the one hand and educational research about competence on the other hand. Both constructs are only rarely examined within one study. From an educational psychology point of view and for construct validation of competence tests, this is necessary though to gain more insight into the relation of the domain-specific constructs. The present article examined the relation between two domain-specific facets of preschool teachers’ competence, mathematical content knowledge (MCK) and mathematics pedagogical content knowledge (MPCK), to intelligence by hypothesizing a nested-factor model. Three hundred and fifty-three preservice preschool teachers were tested on their MCK and MPCK and took the screening version of I‑S‑T 2000 R covering verbal, numerical, and figural intelligence. The data revealed, as expected, a strong impact of intelligence on the ability to solve the domain-specific test items. Nevertheless, MCK and MPCK had a significant additional impact on this ability. The initially strong relation of the two domain-specific constructs could fully be explained by the higher-order g factor which supported construct validity of the assessments furthermore. The discussion focusses on implications for measurement issues and future research besides practical suggestions.

Early Childhood Teachers' Pedagogical Content Knowledge in Mathematics: A Research Report from China

The focus on Pedagogical Content Knowledge (PCK) of pre-school teachers is a breakthrough point to promote teacher's professional development and quality of early childhood education. Based on an investigation of PCK of 161 pre-school teachers from Shanghai, this study described the level of PCK for early mathematics (PCK-EM) of pre-school teachers and examined the influence of their background on the level of PCK-EM. Further, the study put forward implications and suggestions on improving and promoting early mathematics education.

예비유아교사의 수학교과교육학지식과 수학교수효능감 간의 관계 내에서 수학정체성의 매개효과

본 연구는 예비유아교사의 수학교과교육학지식, 수학정체성, 수학교수효능감 간의 관계를 밝히고 수학정체성이 수학교과교육학지식과 수학교수효능감 간의 관계에서 가지는 매개효과를 알아보는데 목적을 두었다. 조사에 응한 예비유아교사 대상의 391부의 설문자료를 최종분석에 활용하여 매개회귀분석을 실시하였다. 연구결과, 첫째, 수학교과교육학지식의 하위영역은 학습자에 대한 지식, 교과내용에 대한 지식, 교육과정에 대한 지식, 교수학습방법에 대한 지식, 과정기술 지식 순으로 높게 나타났다. 수학교수효능감은 수학교수결과기대효능감, 개인적 수학교수효능감의 순으로 높게 나타났고, 수학정체성은 수학적 자기효능감, 수학에 대한 흥미, 자신 또는 타인에 의한 인정의 순으로 높게 나타났다. 둘째, 예비유아교사의 수학교과교육학지식과 수학교 수효능감의 모든 하위요인 간 상관, 예비유아교사의 수학정체성과 수학교수효능감의 모든 하위요인 간 상관, 예비유아교사의 수학교과교육학지식과 수학정체성의 모든 하위요인 간 상관은 통계적으로 유의미한 정적 상관이 있는 것으로 나타났다. 셋째, 예비유아교사의 수학교과교육학지식과 수학교수효능감 간의 관계에서 수학정체성은 부분매개효과를 지니는 것으로 나타났다.

Mathematikangst bei angehenden frühpädagogischen Fachkräften: Bedeutsam für professionelles Wissen und Wahrnehmung von mathematischen Inhalten im Kita-Alltag?

The skill to perceive math-related situations during play-based activities in preschool can be conceptualized as an inherent part of early childhood teachers’ professional competence. It is an important aspect for fostering children’s mathematical competencies and should be based on mathematics content knowledge as well as mathematics pedagogical content knowledge. How emotions affect the perception of mathematics-related situations is still unknown. Mathematics anxiety can be seen as a relevant achievement emotion of (pre-service) early childhood teachers. The current study investigates the effects of mathematics anxiety, mathematics content knowledge and mathematical pedagogical content knowledge on early childhood teacher students’ perception of math-related situations. Therefore, n = 354 early childhood teacher students were surveyed with well-established questionnaires and tested with paper-pencil-based as well as video-based assessments on their mathematics anxiety, knowledge and the perception of math-related situations. Results indicate significant indirect effects of mathematics anxiety on math-related perception, mediated by knowledge. Our findings are discussed with respect to early childhood teachers’ training in Germany.

Mathematical pedagogical content knowledge of early childhood teachers: a standardized situation-related measurement approach

Measurement instruments of early childhood teachers’ mathematical pedagogical content knowledge (MPCK) have to consider the special characteristics of early childhood teaching. Early childhood teaching includes some planned activities but in contrast to learning in school, it is often motivated and generated by situations which unfold spontaneously in play and everyday activities. Therefore, and to respect that early childhood teachers as experts in their field may fail to explain their knowledge explicitly but indeed show implicit knowledge in their daily work, a measurement approach related to concrete situations seems to be suitable. While there are qualitative MPCK interview-instruments following this approach, there is still a lack of standardized instruments suitable for large-scale studies. As a first proposal to approach this gap, we developed a paper–pencil test with multiple-choice items measuring MPCK, and respecting early childhood teachers’ implicit knowledge. The test is based on descriptions of four typical kindergarten situations. Linked to each situation, we developed a set of items measuring professional knowledge concerning mathematical abilities of children and two multiple-choice items measuring professional knowledge of adaptive mathematical learning activities. The 39 items were piloted with a sample (N = 149) of pre- and in-service early childhood teachers. The results show validity evidence related to the test content, response processes, the internal structure of the test, as well as convergent evidence. Therefore, the test provides a suitable first approach to measure early childhood teachers’ MPCK, while addressing their implicit knowledge.

Helping Preservice Secondary Mathematics Teachers Construct Pedagogical Content Knowledge

This paper intends to shed light on understanding of Mathematical Pedagogical Content Knowledge (MPCK) from an empirical angle. Examples are utilized to explicitly demonstrate how a class activity in a capstone course of preservice secondary mathematics teachers (PSMTs) could be structured for PSMTs to build MPCK.

The effects of kindergarten teacher’s background variables and DAP beliefs on their mathematics pedagogical content knowledge

The effects of kindergarten teacher’s background variables and DAP beliefs on their mathematics pedagogical content knowledge

Mathematical pedagogical content knowledge in Early Childhood Education: tales from the ‘great unknown’ in teacher education in Portugal

ABSTRACTThe study aims to explore the specificity of Mathematics pedagogical content knowledge in Early Childhood Education Pedagogy in Initial Teacher Education. It addresses the issue by characterizing student teachers’ perspectives and by analyzing student teachers’ knowledge mobilized in a situation of planning for teaching. The answers to a task developed by students in an Initial Teacher Education program are analyzed in terms of the mathematical knowledge and pedagogical options presented. The results contribute to the discussion in terms of (un)balance between teacher-initiated and child-led activities. The discussion deepens the importance to assert specific/particular ways of teaching in Early Childhood Education, contrasting with the more restricted view of only adult-led moments being teaching. Strong content knowledge and pedagogical content knowledge are valued because of their relevance both at the level of adults’ knowledge needed to support children initiatives and plan curricular/didactic activities and at the level of the knowledge children interact within their daily environment and routine.