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Variance and invariance are two powerful mathematical ideas to support geometrical and spatial thinking, yet there is limited research about teachers' knowledge of variance and invariance. In this paper, we examined how high school teachers deal with the task of looking for invariant properties in a dynamic geometry environment (DGE) setting. Specifically, we investigated if they even attend to invariant properties; what invariant properties they discern and discuss; and how DGE can support such discernment. Our analysis found that teachers tend to discern and discuss invariant properties mainly when they were probed to consider invariance. We also found four categories of invariant properties that seem to be important for a robust and rich understanding of geometric objects in the context of invariance and DGE. The use of DGE allowed teachers to see and interact with invariant properties, thus suggesting that accessing geometry dynamically may have structural affordances especially when exploring invariance. Teachers were able to enact different DGE movements to discern and discuss invariant properties, as well as to reason with and about them. We also saw that teachers' backgrounds and past experiences can play an important role in their descriptions of invariant properties. Possible future research directions and implications to teacher education are discussed. [ABSTRACT FROM AUTHOR]

Teaching mathematical modeling is a demanding task. Thus, fostering teachers' competencies in this regard is an essential component of teacher education. Recent conceptualizations of teachers' competencies include situation-specific skills based on the concept of noticing, which is of particular interest for the spontaneous reactions needed when teaching mathematical modeling. The study described in this paper aims to analyze the development of a video-based instrument for measuring teachers' noticing competencies within a mathematical modeling context and obtain evidence for the validity of the instrument. Three kinds of validity are examined in three different studies: content validity, elemental validity and construct validity. Indicators for content validity could be found through different expert ratings and implementation with the target group, where participants were able to perceive all relevant aspects. The qualitative analysis of participants' reasoning, which is consistent with the coded level, indicates elemental validity. Moreover, the results of the confirmatory factor analysis suggest construct validity with one overall factor of noticing competence within a mathematical modeling context. Taken together, these studies imply a satisfactory validity of the video-based instrument. [ABSTRACT FROM AUTHOR]

Learning to name and notice students' mathematical strengths is a challenging process requiring time and multiple iterations of practice for prospective teachers (PTs) to adopt. Mathematics teacher educators (MTEs) can approximate and decompose the complex practice of naming and noticing students' mathematical strengths so PTs learn to teach mathematics while emphasizing what students know and can do. This study uses two tools MTEs can use to support PTs as they learn to name and notice students' mathematical strengths: A LessonSketch experience, a digital platform with comic-based storyboards showing children engaged in a mathematics task, and a strengths-based sentence frame. Our study presents the findings from the 111 noticing statements from 18 PTs as they engaged in the LessonSketch digital experience and practiced making noticing statements about what children know about mathematics. The study found that after a sentence-frame intervention, the PTs are more likely to use strengths-based language and more likely to identify mathematical evidence in their noticing statements. Uncommitted language (statements that do not align with a strength- or deficit-based coding scheme), suggests a fruitful, yet complex space for supporting more PTs as they learn to name and notice students' mathematical strengths. The paper concludes with implications for future research in teacher education. [ABSTRACT FROM AUTHOR]

The development of mathematical reasoning skills has increasingly been of focus for the teaching and learning of mathematics. This research utilizes a teaching simulation using the methodology of scriptwriting, in which prospective teachers are asked to complete a script of a dialogue from a classroom simulation involving fraction multiplication and division with justification, assisting fictional students to work through their difficulties and helping them to justify their reasoning. Such tasks allow for the examination of the prospective teacher moves to support student reasoning through their imagined action and choice of words. Scripts from forty-one prospective primary teachers were examined for the study, and five clusters based on the type of teacher move for supporting student reasoning were found. Overall, the prospective teachers emphasized the elicitation and facilitation of students' ideas. The cluster analysis, however, provided a nuanced examination of the cohort's teacher moves. While cluster one saw the highest incident of eliciting teacher moves, albeit only in the low potential category, clusters two and three mostly used facilitating teacher moves, but varied in their use of high and low potential moves. Cluster four concentrated moves on facilitating, eliciting, and responding to student reasoning. Cluster five employed teacher moves from all main categories, with some instances of high potential moves in all categories except extending student reasoning, which can better support reasoning. The prospective mathematics teachers' scripts and the five clusters that were found during analysis are discussed with implications for future teacher education and the support of building mathematical reasoning. [ABSTRACT FROM AUTHOR]

Definitions are an integral aspect of mathematics. In particular, they form the backbone of deductive reasoning and facilitate precision in mathematical communication. However, when multiple non-equivalent definitions for the same term exist, their ability to serve these purposes can be called into question. While ambiguity can be productive, the lack of an agreed upon definition can make the truth value of certain mathematical statements unclear. In this study, we presented pre-service and in-service mathematics teachers with three mathematical claims in which definitional ambiguity was a consideration; then, we asked them to determine the correctness of each claim, justify their positions, and anticipate what reasoning might lead to a possible counterargument. Based on their responses, we identified several factors that influenced the participants' approaches in each situation as well as cross-cutting themes that describe their considerations. Finally, we consider the pedagogical implications of employing such a task in teacher preparation programs and in-service teacher education. [ABSTRACT FROM AUTHOR]

Tertiary mathematics has a central place in teacher education, yet in recent years there is growing evidence that realizing its potential affordances in secondary mathematics teaching is far from trivial. Research suggests that utilizing tertiary mathematics in secondary teaching requires interweaving it with knowledge for teaching secondary mathematics. Little is known about the underlying processes, which are often tacit and highly personal. In this article we analyze affordances of tertiary mathematics for teaching secondary probability. A group of mathematicians and experienced secondary teachers jointly inquired into the mathematics that could be addressed in school when discussing a popular probability game – the River Crossing game (henceforth "the game"). This context was chosen as an extreme case, in the sense that the mathematics underlying the game is so nuanced and complex that applying tertiary knowledge to mathematize and understand it is generally not feasible for secondary teachers. Thus, it is not clear how tertiary mathematics can inform teachers about using the game in class. Our analysis shows how the conflicting perspectives of teachers and mathematicians on what mathematics students may learn by playing the game initially hindered utilization of tertiary mathematics. Nevertheless, rapprochement was achieved, highlighting four different trajectories for interweaving knowledge of tertiary mathematics with knowledge for teaching secondary mathematics towards using the game in ingenious ways that respect both mathematical and pedagogical concerns. Our findings suggest that tertiary mathematics may have affordances for secondary mathematics teaching even in situations where teachers lack tertiary-level understanding of the underlying subject-matter. [ABSTRACT FROM AUTHOR]

This study examined elementary and secondary prospective teachers' (PTs') abilities to analyze a classroom lesson in order to make claims about student thinking around specific mathematical learning goals based on relevant and revealing evidence. Previous research suggests PTs have some skills in analyzing evidence but apply them inconsistently. Our goal was to describe in more detail the strengths and weaknesses in PTs' ability to analyze evidence of student thinking. Results indicate that PTs can make some appropriate claims about student learning in a lesson transcript, but more often make overly broad and general claims. PTs were able to support their claims with specific student work but often used poorly aligned evidence. PTs also often explicitly recognized the shortcomings of evidence from the lesson transcript, but then relied on that evidence to make claims about student thinking. Finally, PTs' background, such as number of teacher education courses completed, does not appear to strongly influence their ability to make claims and support them with evidence, though secondary PTs were more likely to recognize the limitations of evidence than elementary PTs. These results have implications for teacher educators, pointing to the importance of designing interventions to help PTs look beyond the most visible and salient features of a lesson when analyzing student thinking. [ABSTRACT FROM AUTHOR]

Within a global context of more standardization of teacher preparation, programs in the USA are increasingly using standardized teacher performance assessments such as edTPA to evaluate prospective teachers' instructional effectiveness and readiness for their own classroom. This study was conducted in a state recently requiring successful completion of edTPA for initial teacher certification. Phenomenological methodology was used to examine the lived experiences of six elementary prospective teachers completing Elementary Education edTPA: Assessing Students' Mathematics Learning (i.e., Math Task 4) during student teaching. Data were collected via 24 individual, phenomenological interviews at specific phases during the student teaching semester. Whole-Part-Whole analysis revealed the prospective teachers found themselves in a state of elusiveness as they engaged in Math Task 4. Three themes provide the structure for elusiveness, including juggling Math Task 4 in student teaching classrooms, from relying on experts to standing alone, and navigating edTPA processes. Understanding prospective teachers' lived experiences with Math Task 4 provides insights into both needed supports for their successful completion of the task and some of the issues with a high stakes teacher performance assessment embedded in the student teaching semester. [ABSTRACT FROM AUTHOR]

In a previous study, we validated the fractions schemes and operations trajectory (Norton and Wilkins 2012; Steffe in J Math Behav 20(3): 267–307, 2002; Steffe and Olive Children's fractional knowledge. Springer, New York, 2010; Wilkins and Norton 2011) with PreK-8 prospective teachers and explored PreK-8 prospective teachers' understanding of fractions based on fraction schemes and operations (Lovin et al. 2016). In that study, we found that most of the prospective teachers had constructed the lower-level fraction schemes and operations, but less than half had constructed the more sophisticated ones. In the current study, we explored ways of evoking the more sophisticated fraction schemes and operations, in particular, by implementing simple instructional changes to a mathematics course for PreK-8 prospective teachers. One change focused on fraction language, specifically encouraging the use of iterative fraction language (e.g., 3/5 is three equal-size pieces, each of which is 1/5 of the whole). Another change focused on intentional number choice, specifically emphasizing improper fractions. After focused fraction instruction that included these instructional changes, we found significant increases in PreK-8 prospective teachers' interiorization of coordinating three levels of units as well as in PreK-8 prospective teachers' construction of the reversible partitive fraction scheme and the iterative fraction scheme. We conclude with implications for practice and research in mathematics courses intended for PreK-8 prospective teachers. [ABSTRACT FROM AUTHOR]

In Australia, there is increasing scrutiny of teacher education processes in learning to teach primary mathematics. In this study, mixed methods, including linear regression, are used to examine the entry mathematical content knowledge of third-year Bachelor of Education Pre-service Students and to relate this to their graduating level of content and a measure of pedagogical content knowledge based upon capacity to describe student errors and provide learning support. The data indicate low levels of mathematical content knowledge at the beginning of the course, prompting questioning of the focus of earlier mathematics curriculum courses. Over the life of the study, there was improvement in some domains, which is to be expected where knowledge of mathematics was an intended outcome. Mathematical content knowledge at the start, but particularly at the end of the study, was highly predictive of expressions of mathematical pedagogical content knowledge. The stronger predictive value of mathematical content knowledge at the end of the study for mathematical pedagogical content knowledge suggests merit in developing the two aspects of teacher knowledge in tandem, rather than in different courses. The relevance of the data to teacher preparation in the institution and more broadly is discussed. [ABSTRACT FROM AUTHOR]