Elementary Preservice Teachers' Conceptions of and Reflections on Student Autonomous Problem-Solving and Mathematical Practices
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The findings of this dissertation study are from a single-site, qualitative, case study involving undergraduate elementary preservice teacher participants from a two-week summer mathematics camp for elementary and middle school-aged students. The purpose of this study was to investigate three elementary preservice teachers’ conceptions of and reflections on student autonomous problem-solving and the mathematical practices of justification, mathematical language, mathematical troubles, perseverance, and visual representations. This study presents the elementary preservice teachers’ conceptions, student interactions, and their reflections from the two-week camp. Additionally, I present how the preservice teachers’ conceptions, interactions, and reflections align with each other and with what research tells us by using an analytic framework based on a corpus of literature that would accurately capture what the participants were saying. This study aided in answering researchers call to understand elementary preservice teachers’ mathematical conceptions (Bransford, Brown, & Cocking, 1999; Conference Board of the Mathematical Sciences, 2001; Thanheiser, Browning, Edson, Kastberg, & Lo, 2013).
This study utilized data from written surveys, survey interviews, clinical interviews, interaction video observations, and stimulated recall interviews, all collected within the two weeks. The theoretical perspective of this study, which aligns with the setting’s theoretical orientation, operationalized a social constructivist perspective of collaborative learning and teaching mathematics (Vygotsky, 1978, 1986). Additionally, the analytic framework used in the analysis of this data was based on literature using and building on the camp’s governing precepts, which formed the camp’s foundational approach to teaching and learning mathematics. Three individual case reports, as well as a cross-analysis, are presented. The findings of this study indicate that:
(1) All three elementary preservice teachers supported the elementary-aged students in was that surpassed their conceptions of the given practices. Moreover, the elementary preservice teachers were aware of the times in which their enactments did not match their conceptions or pointed out instances where their enactments supported the students in ways they did not conceptualize. For instance, the elementary preservice teachers’ all conceptualized perseverance as something they could foster through questioning the students to focus on their strategies or to try other methods. However, the elementary preservice teachers also reflected on other ways of fostering perseverance that were not mentioned in their conceptions, such as changing the participation format to include group work or would praise the students for their effort or progress in problem-solving.
(2) The elementary preservice teachers tended to focus their enactments through supporting students’ justifications. Thus, all their conceptions linked back to supporting justification; however, many of the links between the other practices were missing in the elementary preservice teachers’ conceptions. Moreover, the elementary preservice teachers’ views of justification did not match those of the mathematics education research community in that they did not differentiate between reasoning about a process and justifying why the process is true.
(3) The elementary preservice teachers reflected on the difficulties of supporting a student’s mathematical trouble, particularly with troubles that required more than one repair cycle (Ingram, 2012). Moreover, these difficulties occurred most often in more complex interactions that involved the use of visual representations, troubles with mathematical language, and troubles with justifications of the students’ work. This complex interaction involved multiple mathematical practices, consequently causing the elementary preservice teachers difficulty supporting the students’ autonomous problem-solving.
(4) The elementary preservice teachers attributed several lived experiences to their conceptions and enactments of support for the mathematical practices. Most of these lived experiences involved some type of decomposition, representation, or approximation of practice (Grossman, Compton, Igra, Ronfeldt, Shahan, & Williamson, 2009a) centered around teaching; such as the camp experience, talking with teachers or their peers, the university coursework, or working with students. However, other lived experiences such as babysitting, working as a retail clothing salesperson, or playing volleyball were also mentioned as having an impact on their teaching.
Lastly, I include (i) a discussion of these key findings, and more, situated within the context of existing literature, (ii) implications for the strengthening and development of university coursework for elementary teacher preparation courses, and (iii) future research recommendations based on the finding from this study.