How well prepared are the teachers of tomorrow? An examination of prospective mathematics teachers' pedagogical content knowledge

There is a growing emphasis in the teaching profession on pedagogical content knowledge (PCK) as an important knowledge component. The study reported in this article investigates Turkish prospective mathematics teachers’ mathematics teaching knowledge in the numbers content domain. A series of 10 open-ended scenario-type questions were adopted to challenge 83 prospective mathematics teachers’ knowledge of the learner and presentation of content in the context of PCK. The participants’ responses were analysed by means of rubrics and scoring guides developed by the researchers. The results showed that many of the future teachers performed well in determining what misconceptions students might express in the given scenarios. However, a majority of the participants performed poorly on presentation of content in terms of instructional strategies. In line with these results, the authors offer some suggestions for teacher training programmes.     

Analyzing connections between teacher and student topic-specific knowledge of lower secondary mathematics

The interpretive cross-case study focused on the examination of connections between teacher and student topic-specific knowledge of lower secondary mathematics. Two teachers were selected for the study using non-probability purposive sampling technique. Teachers completed the Teacher Content Knowledge Survey before teaching a topic on the division of fractions. The survey consisted of multiple-choice items measuring teachers’ knowledge of facts and procedures, knowledge of concepts and connections, and knowledge of models and generalizations. Teachers were also interviewed on the topic of fraction division using questions addressing their content and pedagogical content knowledge. After teaching the topic on the division of fractions, two groups of 6th-grade students of the participating teachers were tested using similar items measuring students’ topic-specific knowledge at the level of procedures, concepts, and generalizations. The cross-case examination using meaning coding and linguistic analysis revealed topic-specific connections between teacher and student knowledge of fraction division. Results of the study suggest that students’ knowledge could be associated with the teacher knowledge in the context of topic-specific teaching and learning of mathematics at the lower secondary school.     

Construction of Teacher Knowledge in Context: Preparing Elementary Teachers to Teach Mathematics and Science

The purpose of this study was to further the understanding of how preservice teachers construct teacher knowledge and pedagogical content knowledge of elementary mathematics and science in a school‐based setting and the extent of knowledge construction. Evidence of knowledge construction (its acquisition, its dimensions, and the social context) was collected through the use of a qualitative methodology. The methods course was content‐specific with instruction in elementary mathematics and science. Learning experiences were based on national standards with a constructivist instructional approach and immediate access to field experiences. Analysis and synthesis of data revealed an extensive acquisition of teacher knowledge and pedagogical content knowledge. Learning venues were discovered to be the conduits of learning in a situated learning context. As in this study, content‐specific, school‐based experiences may afford preservice teachers greater opportunities to focus on content and instructional strategies at deeper levels; to address anxieties typically associated with the teaching of elementary mathematics and science; and to become more confident and competent teachers. Gains in positive attitudes and confidence in teaching mathematics and science were identified as direct results of this experience.     

Establishing the link between the graph of a parametric curve and the derivatives of its component functions

ABSTRACT

This study aimed to explore the effects of an instructional intervention in which GeoGebra and the think-pair-share method were used to teach the relationship between the graph of a parametric curve and the derivatives of its component functions. The participants in the study were 19 prospective mathematics teachers. To assess the understanding of the participants regarding the content of the instruction, two comparable tests were administered as a pre- and post-test. In order to determine whether there was a difference between the students’ performance on the two tests, a paired samples t-test was conducted on the test scores; the results revealed a significant difference in favour of the post-test. Thus, it was concluded that the adopted teaching method, which included the use of GeoGebra, in delivering the focused content had a positive impact on the students’ understanding.     

What Is a Fraction? Developing Fraction Understanding in Prospective Elementary Teachers

Classroom teachers need a well‐developed deep understanding of fractions and pedagogic practices so they can provide meaningful experiences for students to explore and construct ideas about fractions. This study sought to examine prospective elementary teachers' understandings of fraction by focusing specifically on their use of fractions meanings and interpretations. Results indicated that prospective elementary teachers bring with them to their final methods course a limited understanding of fractions and that experiences in methods courses resulted only in minor improvement of those limited understandings. The limited part‐whole understanding of fractions that prospective elementary teachers entered the course with was resilient. The implications of this study suggest a need for prospective elementary teachers to continue to develop their conceptual understanding of fractions and for changes to the content and instructional strategies of mathematics content courses designed for prospective elementary teachers.     

The missing link in teachers’ knowledge about common fractions division

The current study explored the difficulties teachers encounter when teaching common fractions division, focusing on teachers’ knowledge concerning this issue. Nine teachers who study towards a M.Ed. degree in mathematics education demonstrated the algorithms they apply in order to solve fractions division problems, described how they teach the subject, and attempted to explain a student's mistake, in understanding a word problem involving dividing by fraction. The findings indicate there is a missing link in the teachers’ pedagogical capability, stemming from insufficient content knowledge. They presented different solution algorithms and reported using constructivist teaching methods, yet the methods they described couldn't lead a student to understand the logic behind the algorithm they teach (invert-and-multiply – multiplication by an inverse number, in accordance with the requirements of the curriculum). Furthermore, the participating teachers did not possess specialized mathematics content knowledge (SCK) and knowledge of content and students (KCS), enabling them to identify the source of a student's misconception.     

Building teachers’ expertise in understanding, assessing and developing children’s mathematical thinking: the power of task-based, one-to-one assessment interviews

In this paper, we outline the benefits to teachers’ expertise of the use of research-based, one-to-one assessment interviews in mathematics. Drawing upon our research and professional development work with teachers and students in primary and middle years in Australia and the research of others, we argue that the use of the interviews builds teacher expertise through enhancing teachers’ knowledge of individual and group understanding of mathematics, and also provides an understanding of typical learning paths in various mathematical domains. The use of such interviews also provides a model for teachers’ interactions and discussions with children, building both their pedagogical content knowledge and their subject matter knowledge.     

Including Curriculum Focus in Mathematics Professional Development for Middle‐School Mathematics Teachers

This paper examines professional development workshops focused on Connected Math, a particular curriculum utilized or being considered by the middle‐school mathematics teachers involved in the study. The hope was that as teachers better understood the curriculum used in their classrooms, i.e., Connected Math, they would simultaneously deepen their own understanding of the corresponding mathematics content. By focusing on the curriculum materials and the student thought process, teachers would be better able to recognize and examine common student misunderstandings of mathematical content and develop pedagogically sound practices, thus improving their own pedagogical content knowledge. Pre‐ and post‐mathematics content knowledge assessments indicated that engaging middle‐school teachers in the curriculum materials using pedagogy that can be used with their middle‐school students not only solidified teachers' familiarity with such strategies, but also contributed to their understanding of the mathematics content.     

On the content-dependence of prospective teachers’ knowledge: a case of exemplifying definitions

In this article, we demonstrate that prospective teachers’ content knowledge related to defining mathematical concepts is dependent on content area. We use the example of generation (a research tool we developed in a previous study) to investigate prospective teachers’ knowledge. We asked prospective secondary mathematics teachers to provide multiple examples of definitions of concepts from different areas of mathematics. We examined teacher-generated examples of concept definition and analysed individual and collective example spaces, focusing on their correctness and richness. We demonstrate differences in prospective teachers’ knowledge associated with defining mathematical concepts in geometry, algebra and calculus.     

Reflections on the promise and complexity of mathematics coaching

If students are to develop mathematical proficiency, then mathematics teaching must both change and improve. In an effort to provide site-based professional development addressing the mathematical content and pedagogical demands that teachers encounter in reality of public schooling, many school districts are turning to elementary mathematics coaches. Knowledgeable coaches can have a significant positive impact on teachers, yet this study documents substantial variance in the amount of coaching delivered and in the nature of activity that coaches undertake within schools. Coaches are frequently responsive to the needs of individual teachers. If this support is primarily marked by shared teaching or provision of instructional materials, it may not transform either instruction or teacher knowledge. Similarly if coaches assume duties that primarily address an administrator’s needs, they will have less time to enhance a school’s mathematics program. Coaches need to engage teachers in fundamental dialogue about mathematical content, mathematical learning, and student understanding. It may be that this dialogue and the effectiveness of a coach’s work with individual teachers would benefit from a coach’s concurrent work with grade-level teams. When a coach leads a grade-level team through discussion of targeted goals and approaches, the coach may facilitate individual teacher learning while building collective learning. When coupled with the support of a principal, this partnership may foster instructional change across a school.     

Teachers’ mathematical problem-solving knowledge: In what way is it constructed during teachers’ collaborative work?

This research aims to analyze the type of mathematics problem-solving knowledge for teaching used when working collaboratively in a Lesson Study (LS) process and examine how dialogic interactions contribute to knowledge construction. Five meetings during one LS cycle of a group of eight Swiss primary teachers were video recorded, transcribed and coded with the help of qualitative data analysis software. This analysis is conducted by crossing theoretical frameworks from two different fields in education, namely mathematics education and dialogic analysis. The mixed-method uses quantitative analysis with Markov chains and cross-tables, as well as qualitative analysis at micro-, meso- and macro-levels. This research suggests that participants collectively use their mathematics and their problem-solving content knowledge to focus on pedagogical problem-solving knowledge, that they navigate between different knowledge levels and that the roles of teachers and facilitators are differentiated but are also coequal.     

Secondary mathematics teachers’ pedagogical content knowledge and content knowledge: validation of the COACTIV constructs

Research interest in the professional knowledge of mathematics teachers has grown considerably in recent years. In the COACTIV project, tests of secondary mathematics teachers’ pedagogical content knowledge (PCK) and content knowledge (CK) were developed and implemented in a sample of teachers whose classes participated in the PISA 2003/04 longitudinal assessment in Germany. The present article investigates the validity of the COACTIV constructs of PCK and CK. To this end, the COACTIV tests of PCK and CK were administered to various “contrast populations,” namely, candidate mathematics teachers, mathematics students, teachers of biology and chemistry, and advanced school students. The hypotheses for each population’s performance in the PCK and CK tests were formulated and empirically tested. In addition, the article compares the COACTIV approach with related conceptualizations and findings of two other research groups.     

The challenges facing initial teacher education: Irish prospective elementary teachers’ mathematics subject matter knowledge

Given the acknowledged relationship between teachers’ knowledge, their teaching and pupil learning, teachers’ mathematics subject matter knowledge (MSMK) has received increased attention internationally. As children's early mathematics experiences have been recognized as a critical stage, elementary teachers’ MSMK has become a focal point among researchers and policy makers alike. International research findings have uncovered that in many cases, there is a mismatch between what is perceived to be an appropriate MSMK for teaching elementary mathematics and that demonstrated by many qualified and prospective elementary teachers. Following repeated incidences of weak MSMK during interactions with prospective elementary teachers in one Irish College of Education (provider of initial teacher education programme for elementary teachers), this study sought to examine and address the issue purposefully through two cycles of action research. This article focuses on the data collected prospective teachers’ MSMK in the initial stage (reconnaissance) of these cycles, i.e. pre-test findings. While considerable differences were evident among the pre-test population, the findings suggest that prior to the intervention stage many participating prospective teachers; regardless of previous mathematics achievements or the level of mathematics study; demonstrate weaknesses and gaps in their ‘common’ MSMK. Particular difficulties were evident in relation to pre-test items requiring knowledge of rational numbers, conceptual understanding or problem solving. These findings highlight the inadequacy of previous mathematics achievements and indeed minimum entry requirements as predictors of MSMK for teaching. As well as its contribution at a local and national level, the findings provide an Irish perspective on this international issue.     

Connecting the learning of advanced mathematics with the teaching of secondary mathematics: Inverse functions,domain restrictions,and the arcsine function

Prospective secondary mathematics teachers are typically required to take advanced university mathematics courses. However, many prospective teachers see little value in completing these courses. In this paper, we present the instantiation of an innovative model that we have previously developed on how to teach advanced mathematics to prospective teachers in a way that informs their future pedagogy. We illustrate this model with a particular module in real analysis in which theorems about continuity, injectivity, and monotonicity are used to inform teachers’ instruction on inverse trigonometric functions and solving trigonometric equations. We report data from a design research study illustrating how our activities helped prospective teachers develop a more productive understanding of inverse functions. We then present pre-test/post-test data illustrating that the prospective teachers were better able to respond to pedagogical situations around these concepts that they might encounter.     

Mathematics teachers’ views about the limited utility of real analysis: A transport model hypothesis

In the United States and elsewhere, prospective teachers of secondary mathematics are usually required to complete numerous advanced mathematics courses before obtaining certification. However, several research studies suggest that teachers’ experiences in these advanced mathematics courses have little influence on their pedagogical practice and efficacy. To understand this phenomenon, we presented 14 secondary mathematics teachers with four statements and proofs in real analysis that related to secondary content and asked the participants to discuss whether these proofs could inform their teaching of secondary mathematics. In analyzing participants’ remarks, we propose that many teachers view the utility of real analysis in secondary school mathematics teaching using a transport model, where the perceived importance of a real analysis explanation is dependent upon the teacher’s ability to transport that explanation directly into their instruction in a secondary mathematics classroom. Consequently, their perceived value of a real analysis course in their teacher preparation is inherently limited. We discuss implications of the transport model on secondary mathematics teacher education.     

Standards for professionalization of mathematics teachers: policy, curricula, and national teacher employment test in Korea

International comparative studies such as the Trend in International Mathematics and Science Study (TIMSS) and the OECD Programme for International Student Assessment (PISA) indicate that Korean students have consistently performed well. In addition, a recent study titled Mathematics Teaching in the 21st Century (MT21) compared prospective teachers’ knowledge and beliefs about teaching and learning in six participant countries, reporting that Korean prospective secondary mathematics teachers were better prepared than those in other countries. In this context, this study has examined the curricula for mathematics teacher education and teacher employment tests in order to investigate the social expectation for teacher professionalization in Korea, particularly focusing on teacher knowledge. The analysis shows that while elementary mathematics teacher education emphasizes pedagogical knowledge, the secondary mathematics education curricula are highly content knowledge oriented. However, the secondary mathematics teacher education includes various aspects of pedagogical content knowledge in its curricula and teacher employment test. This research also identifies the discourse concerning mathematics instruction for diversity and equity and the emphasis of reflective practice as the significant development of the current Korean teacher education.     

The relationship between preservice elementary teachers’ solutions to a pattern generalization problem and difficulties they anticipate in teaching it

To explore the relationship between elementary preservice teachers’ (PTs’) solutions to a pattern generalization problem and the difficulties they expected to encounter when teaching the same problem to students, we administered a task-based questionnaire to 154 participants at a large Southwestern university in the US. Employing inductive content analysis, we identified possible links between PTs' solutions and their anticipated difficulties. PTs who solved the problem using figurative reasoning tended to anticipate difficulties related to pedagogical moves to support students’ mathematical understanding. In contrast, PTs who solved the problem using algebraic formulations were likely to anticipate difficulties related to teaching algebraic knowledge and supporting procedural fluency. Also, only PTs who solved the problem using figurative reasoning anticipated difficulties associated with eliciting and evaluating student thinking, whereas PTs who used formulas to solve the problem expected difficulties related to their own self-efficacy and confidence. We discuss three implications for mathematics teacher education.     

The impact of integrated STEM modeling on elementary preservice teachers’ self‐efficacy for integrated STEM instruction: A co‐teaching approach

The purpose of the current study was to evaluate the impact of co‐taught integrated STEM methods instruction on preservice elementary teachers’ self‐efficacy for teaching science and mathematics within an integrated STEM framework. Two instructional methods courses (Elementary Mathematics Methods and Elementary Science Methods) were redesigned to include STEM integration components, including STEM model lessons co‐taught by a mathematics and science educator, as well as a special education colleague. Quantitative data were gathered at three time points in the semester (beginning, middle, and end) from 55 preservice teachers examining teacher self‐efficacy for integrated STEM teaching. Qualitative data were gathered from a purposeful sample of seven preservice teachers to further understand preservice teachers’ perceptions on delivering integrated STEM instruction in an elementary setting. Quantitative results showed a significant increase in teacher self‐efficacy across all three time points. Item‐level analysis revealed that self‐efficacy for tasks involving engineering and assessment (both formative and summative) were low across time points, while self‐efficacy for tasks involving technology and flexibility were consistently high. Qualitative results revealed that the preservice teachers did not feel adequately prepared by university‐level science and mathematics courses, in terms of content knowledge and integration of science and mathematics for elementary students.     

A case study of pedagogy of mathematics support tutors without a background in mathematics education

This study investigates the pedagogical skills and knowledge of three tertiary-level mathematics support tutors in a large group classroom setting. This is achieved through the use of video analysis and a theoretical framework comprising Rowland's Knowledge Quartet and general pedagogical knowledge. The study reports on the findings in relation to these tutors’ provision of mathematics support to first and second year undergraduate engineering students and second year undergraduate science students. It was found that tutors are lacking in various pedagogical skills which are needed for high-quality learning amongst service mathematics students (e.g. engineering/science/technology students), a demographic which have low levels of mathematics upon entering university. Tutors teach their support classes in a very fast didactic way with minimal opportunities for students to ask questions or to attempt problems. It was also found that this teaching method is even more so exaggerated in mandatory departmental mathematics tutorials that students take as part of their mathematics studies at tertiary level. The implications of the findings on mathematics tutor training at tertiary level are also discussed.     

Characteristics of good mathematics teaching in Singapore grade 8 classrooms: a juxtaposition of teachers’ practice and students’ perception

This paper examines the instructional approaches of three competent grade 8 mathematics teachers. It also examines their students’ perception of the lessons they taught as well as characteristics of good lessons. The findings of teachers’ practice and students’ perception are juxtaposed to elicit characteristics of good teaching in Singapore grade 8 classrooms. With limitation, the findings of the paper suggests that good mathematics teaching in Singapore schools centres around building understanding and is teacher-centred but student focused. Some characteristic features of good lessons are that their instructional cycles have specific instructional objectives such that subsequent cycles incrementally build on the knowledge. The examples used in such lessons are carefully selected and vary in complexity from low to high. Teachers actively monitor their student’s understanding during seatwork, by moving from desk to desk guiding those with difficulties and selecting appropriate student work for subsequent whole-class review and discussion. Finally, during such lessons teachers reinforce their students’ understanding of knowledge expounded during whole-class demonstration by detailed review of student work done in class or as homework.