Various intervention programs for fostering at-risk students’ understanding of basic concepts (such as place value understanding or meanings of multiplication and division) have been developed and evaluated. However, little is known about the teacher expertise required to enact these intervention programs, and how this teacher expertise can be promoted. The article suggests a conceptual model for teacher expertise for fostering at-risk students’ understanding based on three recurrent jobs: (a) specify learning content (in basic concepts), (b) monitor students’ learning progress (in basic concepts) and (3) enhance students’ understanding (of basic concepts). Mastering these jobs with productive teaching practices involves four orientations in particular (conceptual rather than procedural orientation, diagnostic rather than syllabus-led orientation, communicative rather than individualistic orientation, and long-term rather than short-term orientation) as well as detailed pedagogical content categories for unpacking relevant knowledge elements. The paper reports on the professional development program Mastering Math which aims at promoting this expertise and its evaluation using a pre–post-design. For 95 participating teachers, the practices for specifying goals and monitoring and enhancing at-risk students’ understanding were captured in self-reports and situated in vignette-based activities for eliciting diagnostic judgments. Teachers’ development across different aspects of their expertise from the beginning and the end of the 1-year PD reveals the first quantitative evidence that the PD was effective in promoting growth of expertise. Whereas specifying and monitoring practices had substantially developed, the enhancement practices were hindered by a persistent short-term orientation.
Syntactic language features of word problems have often been identified by assessment studies and interview studies as potential obstacles in word problem solving. However, students' processes of noticing and interpreting these features have only rarely been investigated in depth because noticing is hard to investigate. Eye-tracking methodology has already been shown to be a suitable methodology for analyzing noticing other word problem features, but not yet for syntactic features. The longitudinal eye-tracking study presented here analyzes fifth graders' processes of noticing and interpreting of syntactic word problems before and after an intervention on word problem solving. Based on the eye-mind hypothesis, the study captures students' noticing through eyetracking, whereas interpreting can be captured by their solutions. The analysis of pre-intervention data reveals that long fixations cannot be identified with successful noticing and vice versa. But when comparing the net dwell time (i.e., the time that students look at the words indicating syntactic structures) before and after the intervention, an interesting decrease is revealed. This decrease occurs for both expected and less expected syntactic structures, regardless of the accuracy of mathematization. Meanwhile, analyzing the revisits of students indicates differences between items with more or less expected structures and correctly or incorrectly mathematized items. Methodologically, the article contributes to understanding which eye-tracking measures are relevant for capturing changes in students' processes of noticing syntactic language features: The revisits and scan paths reveal more insights than the net dwell time.
Teachers’ in-depth diagnostic thinking has been shown to be crucial for student-centered teaching as they need to perceive and interpret students’ understanding for well-informed decision-making on adaptive teaching practices. The paper presents a content-related approach to analyzing diagnostic thinking processes with respect to the mathematical knowledge elements that prospective teachers identify as students’ resources and obstacles. Prospective teachers’ challenge is that some relevant knowledge elements first have to be unpacked, because compact concepts (such as the place value concept) or procedures (such as for multi-digit multiplication) comprise several smaller knowledge elements (such as the positional property) that have to be made explicit for students to foster their learning processes adequately. Our study examines what knowledge elements prospective teachers perceive and interpret in a transcript vignettes on multi-digit multiplication (of decimal and natural numbers) and its underlying basic arithmetic concepts (place value understanding and meaning of multiplication) in written diagnostic judgments on students’ resources and obstacles (N = 196). A comparative design within the vignette is used to investigate how far the process of perceiving can be supported by thematic cues. The analysis reveals that those knowledge elements cued in the vignette by being already unpacked and explicitly addressed are perceived and interpreted more often (but with lower correctness) than those that are uncued and therefore have to be unpacked by the prospective teachers themselves. This confirms the need to prepare prospective teachers for unpacking mathematical concepts themselves.
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