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1. Revealing prospective secondary teachers’ mathematical knowledge for teaching in teacher-created representations of practice

   Lischka, A. E.; Lai, Y.; Czap, L.; Strayer, J. (January 2021, The mathematician educator)

   To examine differences in dimensions of MKT observable in different creation mediums for teacher-created representations of practice, we analyzed 54 representations of practice (pairs of 27 video and 27 written) created by 27 PSMTs. We found 7 pairs where we observed higher MKT in the video than in the written representation of practice created by the same PSMT. We found 0 cases where we observed higher MKT in the written as compared to the video representation of practice. We examined these 7 pairs for qualities characterizing the pairs as a set. We use findings to problematize issues central to mathematics teacher education: how mathematics is conceptualized, and how representations of practice are used as assessment of dimensions of teachers’ MKT. 

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2. Mathematical Errors When Teaching: A Case of Secondary Mathematics Prospective Teachers’ Early Field Experiences

   Bieda, K.; Voogt, K. (January 2019, Proceedings of the 22nd Annual Conference on Research in Undergraduate Education)

   The construct of mathematical knowledge for teaching (MKT) has transformed research and practice regarding the mathematical preparation of future teachers. However, the measures used to assess MKT are largely written tasks, which may fail to adequately represent the nature of content knowledge as it is used in instructional decision making. This preliminary report shares initial findings into one measure of MKT in practice – mathematical errors made during planning and enactment of mathematics instruction. We analyzed lesson plans and classroom video from prospective secondary mathematics teachers (PSTs)’ supervised field experiences in college algebra course. We found that there tended be more errors related to understanding of functions (especially logarithmic), but relatively few errors happened overall during instruction. Of the errors made during planning, the majority of these errors were issues of mathematical precision. Implications for the mathematical preparation of secondary PSTs, as well as research on MKT in practice, are discussed. 

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3. Mathematics and science teacher educators' use of representations of practice: A mixed methods study

   https://doi.org/10.1111/ssm.12563  

   Austin, Christine K.; Kosko, Karl W.; Heisler, Jennifer L. (January 2023, School Science and Mathematics)

   Abstract This study sought to explore math and science teacher educators' use of various media to represent practice within methods courses. There is little understanding of why certain media is used over other representations and the rationale for these choices. Specifically, the study focused on the prevalence and familiarity of teacher educators with comics and animations, standard videos, and 360 videos. This mixed methods study utilized a survey and interviews to ascertain math and science teacher educators' level of familiarity and perceived usefulness of representations of practice. Results indicate that standard video is by far the most used representation of practice in methods classes with three themes explaining this finding: access to representations of practice, dimensions of representation, and pertinacity of using representations. Familiarity with representations of practice relates to teacher educators' perceptions of access thereby indicating a need for teacher educators to have better access to representations. Implications of this study include supporting current literature about the relationship between the level of familiarity and perceived usefulness in media along with the potential need for a central platform that houses these representations of practice resources for teacher educators. 

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4. Preservice Teachers’ Professional Noticing When Viewing Standard and 360 Video

   https://doi.org/10.1177/0022487120939544  

   Kosko, Karl W.; Ferdig, Richard E.; Zolfaghari, Maryam (May 2021, Journal of Teacher Education)

   Use of video as a representation of practice in teacher education is commonplace. The current study explored the use of a new format (360 video) in the context of preservice teachers’ professional noticing. Findings suggest that preservice teachers viewing 360 videos attended to more student actions than their peers viewing standard video. In addition, using a virtual reality headset to view the 360 videos led to different patterns in where preservice teachers looked in the recorded classroom, and to increased specificity of mathematics content from the scenario. Thus, findings and results support the use of 360 video in teacher education to facilitate teacher noticing. However, future research is needed to further explore this novel technology. 

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5. Noticing mathematics from multiple perspectives

   Kosko, K. W.; Ferdig, R. E.; Gandolfi, E. (January 2022, Psychology of Mathematics Education - North America)

   Olanoff, D.; Johnson, K.; Spitzer, S. (Ed.)

   A key aspect of professional noticing includes attending to students’ mathematics (Jacobs et al., 2010). Initially, preservice teachers (PSTs) may attend to non-mathematics specific aspects of a classroom before attending to children’s procedures and then, eventually their conceptual reasoning (Barnhart & van Es, 2015). Use of 360 videos has been observed to increase the likelihood that PSTs will attend to more mathematics-specific student actions. This is due to an increased perceptual capacity, or the capacity of a representation to convey what is perceivable in a scenario (Kosko et al., in press). A 360 camera records a classroom omnidirectionally, allowing PSTs viewing the video to look in any direction. Moreover, several 360 cameras can be used in a single room to allow the viewer to move from one point in the recorded classroom to another; defined by Zolfaghari et al., 2020 as multi-perspective 360 video. Although multiperspective 360 has tremendous potential for immersion and presence (Gandolfi et al., 2021), we have not located empirical research clarifying whether or how this may affect PSTs’ professional noticing. Rather, most published research focuses on the use of a single camera. Given the dearth of research, we explored PSTs’ viewing of and teacher noticing related to a six-camera multiperspective 360 video. We examined 22 early childhood PSTs’ viewing of a 4th grade class using pattern blocks to find an equivalent fraction to 3/4. Towards the end of the video, one student suggested 8/12 as an equivalent fraction, but a peer claimed it was 9/12. The teacher prompts the peer to “prove it” and a brief discussion ensues before the video ends. After viewing the video, PSTs’ written noticings were solicited and coded. In our initial analysis, we examined whether PSTs attended to students’ fraction reasoning. Although many PSTs attended to whether 8/12 or 9/12 was the correct answer, only 7 of 22 attended to students’ part-whole reasoning of the fractions. Next, we examined the variance in how frequently PSTs switched their camera perspective using the unalikeability statistic. Unalikeability (U2) is a nonparametric measure of variance, ranging from 0 to 1, for nominal variables (Kader & Perry, 2007). Participants scores ranged from 0 to 0.80 (Median=0.47). We then compared participants’ U2 statistics for whether they attended (or not) to students mathematical reasoning in their written noticing. Findings revealed no statistically significant difference (U=38.5, p=0.316). On average, PSTs used 2-3 camera perspectives, and there was no observable benefit to using a higher number of cameras. These findings suggest that multiple perspectives may be useful for some, but not all PSTs’. 

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