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In this study, we report on what types of mathematical knowledge for teaching teachers (MKTT) mathematics teacher educators (MTEs) use and develop when they work together and reflect on their teaching in a Community of Practice while helping prospective primary teachers (PTs) generate their own mathematical knowledge for teaching in learning mathematics via problem solving. Two novice MTEs worked with an experienced MTE and reflected on the process of learning to teach via problem solving and supporting PTs in developing deep understandings of foundational mathematical ideas. Taking a position of inquiry as stance, we examined our experiences teaching mathematics content courses for PTs via problem solving. We found that all of the MTEs used and developed some MKTT through (a) understanding and deciding on the mathematical goals of both the individual lessons and the two-course sequence as a whole, (b) choosing and facilitating tasks, and (c) using questions to scaffold PTs learning and engage them in mathematical processes such as making conjectures, justifying their reasoning, and proving or disproving conjectures.

Teachers’ understanding of the concepts they teach affects the quality of instruction and students’ learning. This study used a sample of 303 teachers from across the USA to examine elementary school mathematics teachers’ knowledge of key concepts underlying fraction arithmetic. Teachers’ explanations were coded based on the accuracy of their explanations and the kinds of concepts and representations they used in their responses. The results showed that teachers’ understanding of fraction arithmetic was limited, especially for fraction division, yet a moderate relationship was found between teachers’ understanding of fraction addition and division. Furthermore, more experienced teachers seemed to have a deeper understanding of fraction arithmetic, whereas special education teachers had a substantially limited understanding.

Here, we systematically review research on teaching knowledge in the context of undergraduate STEM education, with particular attention to what this research reveals about knowledge that is important for evidence-based teaching. Evidence-based teaching can improve student outcomes in undergraduate STEM education. However, the enactment of promising evidence-based teaching strategies depends greatly on the instructor and potentially on the teaching knowledge they are able to deploy. The review includes an overview of prevalent teaching knowledge theory, including pedagogical content knowledge, mathematical knowledge for teaching, and pedagogical knowledge. We compare and contrast teaching knowledge theory and terminology across STEM disciplines in order to build bridges for researchers across disciplines. Our search for peer-reviewed investigations of teaching knowledge in undergraduate science, engineering and mathematics yielded 45 papers. We examined the theoretical frameworks used in each study and analyzed study approaches, comparing across disciplines. Importantly, we also synthesized findings from research conducted in the context of evidence-based teaching. Overall, teaching knowledge research is sparse and siloed by discipline, and we call for collaborative work and better bridge-building across STEM disciplines. Though disciplinary divergences are common in discipline-based education research, the effect is magnified in this research area because the theoretical frameworks are themselves siloed by discipline. Investigations of declarative knowledge were common, and we call for increased attention to knowledge used in the practice of teaching. Finally, there are not many studies examining teaching knowledge in the context of evidence-based teaching, but the existing work suggests that components of pedagogical content knowledge, pedagogical knowledge, and content knowledge influence the implementation of evidence-based teaching. We describe implications for future teaching knowledge research. We also call on those who develop and test evidence-based strategies and curriculum to consider, from the beginning, the teaching knowledge needed for effective implementation.

This paper sought to explore the Knowledge of Algebra for Teaching and the algebra difficulties of In-field and Out-of-field Junior High School mathematics teachers using the expanded KAT framework. The study employed the descriptive survey design and involved the participation of 374 mathematics teachers using an achievement test instrument. The study projected School Algebra Knowledge as the prevailing knowledge domain and revealed that the two categories of Junior High School mathematics teachers possess knowledge that falls below average for five (5) out of the seven (7) algebra knowledge domains and also for the overall Knowledge of Algebra for Teaching. However, In-field mathematics teachers showed higher knowledge as compared to Out-of-field mathematics teachers in six (6) out of the seven (7) algebra knowledge domains as well as the overall Knowledge of Algebra for Teaching. Also, the research revealed that mathematics teachers with 5 or more years of teaching experience have higher Knowledge of Algebra for Teaching as compared to those with below 5 years of teaching experience. The study finally identified eight major algebra difficulties among Junior High School mathematics teachers. The findings of the study have implications for teacher preparation, policy and practice.

Teacher education in Indonesia aims to prepare pre-service teachers in becoming professional teaching personnel. Two important knowledge need to be acquired are content knowledge (CK) and pedagogical content knowledge (PCK). However, it is still a challenge to measure the pre-service mathematics teachers' CK and PCK. Therefore, the first aim of the present study is to develop and validate a test instrument to measure CK and PCK of pre-service mathematics teachers on high school level mathematics. Second, the present study aims to measure the CK and PCK of pre-service mathematics teachers by using the developed instrument. The present study employed developmental research method and used students' empirical data in investigating the validity, objectivity, and reliability of the instrument. The results of the study and its implications for future research and practice then will be presented in the paper.

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

The purpose of this article is to contribute to the discussion of mathematics teacher knowledge, and the question of what makes it specialized. In the first part of the article, central orientations in conceptualizing mathematics teacher knowledge are identified and the more serious limitations of the grounds on which they stand are explicated. In the second part of the article, alternative views are offered to each of these orientations that direct attention to underexplored issues about what makes mathematics teacher knowledge specialized. Collectively, these alternative views suggest that specialization in mathematics teacher knowledge cannot be comprehensively accounted for by ‘what’ teachers know, but rather by ‘how’ teachers’ knowing comes into being. We conclude that it is not a kind of knowledge but a style of knowing that signifies specialization in mathematics teacher knowledge.

What mathematical knowledge is required for teaching has been researched by many, with Ball et al. (J Teach Educ 59(5):389–407, 2008 ) and their practice-based theory of mathematical knowledge for teaching (MKT) primary among them. However, what is required in terms of mathematical knowledge for teacher educator training has been researched much less. The present study builds on an emerging framework on mathematical knowledge for teaching teachers (MKTT) by Zopf (Mathematical knowledge for teaching teachers: the mathematical work of and knowledge entailed by teacher education, University of Michigan, Ann Arbor, 2010 ) augmented with more recent developments in the field. The empirical basis for the study stems from an implementation of a short course on history in mathematics education at the Danish School of Education, and in particular on two case studies involving three teacher educator students participating in that course. Based on analyses of their developments in terms of MKTT, we point to the potential of using history in both teacher and teacher educator training, and in particular historical source material. As part of the analyses, we also consider the teacher educator students’ development of disciplinary knowledge of mathematics, especially in relation to developing and making use of knowledge of the epistemology of mathematics and mathematical work. We assert that the cases we present here address Zopf’s call for case studies from different teacher education contexts as well as investigations of novice teacher educators, and further contribute to the developing theory of MKTT.

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.

This study examines the association between mathematical knowledge for teaching and instructional quality in a sample of first-year elementary school teachers. Ten teachers completed the mathematical knowledge for teaching (MKT) survey at the end of teacher preparation. Three mathematics lessons taught during their first year of teaching were videotaped and scored using the Mathematical Quality of Instruction. Findings replicate prior studies that were conducted with more experienced teachers. A strong, positive and statistically significant association was found between teacher knowledge and the mathematics is clear and not distorted dimension of instructional quality. In addition, associations of moderate strength were found between MKT and other dimensions of instructional quality centered on the mathematics taught in the lesson. Analyses also revealed individual differences among teachers and raised the question of what other factors might impact instructional quality. Three cases studies highlight the role of lesson design, mathematics tasks, and participation structures that support or inhibit instructional quality and the use of knowledge during teaching. Conclusions suggest that preparation and induction programs should include a focus on individual teachers’ mathematical knowledge for teaching, the development of a student-centered vision of mathematics instruction, and tailored support during the first year of teaching.