1. UMass Dartmouth College of Arts & Sciences umassd.edu/cas Introduction and Purpose Methods Methods - ContinuedFindings - Continued Conclusions Acknowledgement We examine the ways in which middle school teachers understand and explain the relationship between similarity and proportions by building on the assumption that coherent and connected understandings of mathematics are necessary for teaching. Research Question: How do teachers make sense of the connections between similarity and proportions? We ground our work in the Knowledge in Pieces (KiP) theory.  Suggests that we accumulate our understandings of the world and store them as a variety of knowledge resources to be used in various in situ  Allows us to focus on knowledge resources that people have and the ways in which they use those resources rather than on discovering what knowledge individuals do not have. We rely on epistemic network analysis (ENA) as our analytical lens to analyze coherence.  Computational modeling technique derived from Social Network Analysis to analyze aspects of individuals’ cognition  A coding scheme is developed that captures important aspects of a participants’ understanding  Yields graphs showing the centrality of each of the knowledge resources over time  Uncovers links between codes by showing ideas that appear together in utterances  Provides a visual representation of those places in which multiple elements are interacting  At the end of the interview, we also asked a few open-ended questions to understand how participants think about and teach concepts related to proportions.  Considered responses from two of these questions:  When you think about similarity, do you think about it being related to proportional relationships?  Do you teach scale factor? If so, how do you define it for students?  A coding scheme was developed to capture the knowledge resources being invoked by the participants.  We then coded each utterance (turn of talk) using coding scheme to capture which knowledge resources were used. Pattern recognition for Relatively Low Score participants RLS participants used a wide variety of resources.  David (Figure 5) relied more on geometric resources  Bridgett (Figure 4) relied more on proportional and algebraic resources. In place of tiling, the RLS participants used pattern recognition combined with their other resources.  For Bridgett, pattern searching (which did not lead to finding a pattern) and pattern recognition both played a part in her reasoning.  David seemed to successfully identify patterns when used in conjunction with other knowledge resources. This suggests a more algebraic approach to approaching the tasks than the tiling approach seen in RHS participants. Many resources  Across all six of the participants, we saw many knowledge resources used.  We found no relationship between the number of co-occuring codes and the relative score on the LMT.  This suggests that the sheer number of co-occurring codes does not provide insight into the relative strength of a teachers’ knowledge.  There is something important about the ways in which the teacher has built connections between knowledge resources to allow flexible invocation of those resources in a variety of situations. Significance of the Study  There has long been a call for coherent teacher knowledge, yet little research has been conducted on what coherence might mean in measurable terms.  This study offers a first step in beginning to understand coherence and its potential for furthering our understanding of teacher knowledge.  Teachers use different resources to think about the relationships between similarity and proportions.  Teachers varied tremendously in their views about teaching the two concepts together.  More work is needed to understand the knowledge teachers use and the ways in which it is organized. This will lead to improvements in professional development and teacher education.  Six practicing middle school teachers of varying experience from urban or suburban districts in New England served as participants.  Each completed a written assessment (LMT-Proportional Reasoning) on proportional reasoning and participated in a clinical interview in which an iPad was used to present tasks designed to explore the relationship between similarity and proportions.  LMT scores were translated into a standardized score using equating tables provided by the assessment development team based on a national sample.  Both standard scores are reported as indicators of each participant’s relative strength as compared to the national sample of teachers.  Participants are grouped as relatively high scorers (RHS) and relatively low scorers (RLS).  Allows us a starting place for exploring trends in our sample with regards to their interpretations of the relationships between similarity and proportions. Figure 1a Figure 1b  Participants were provided with an interactive pair of “critters” in SketchPad Explorer.  We asked them a variety of questions about the growth of the critters and their relationship to either each other.  When both changed size such as in Figure 1a & 1b  Fixed critters, when only one changes sizes  Participants were allowed to interact as much or as little as they preferred with the interactive sketches. This research was supported by grant DRL-1054170 from the National Science Foundation. The views expressed here are those of the authors and do not necessarily reflect those of the NSF. ENA as a Tool for Exploring Teachers’ Understanding of Similarity and Proportion Kaput Center – School of Education Authors: Timothy Marum, Chandra Orrill, and James Burke Findings We determined the LMT scores for each participant. We then divided them into groups based on their relative scores. All participants scored statistically higher than the national average (a difference of 0.3 is generally accepted as significant.) We divided participants into 3 groups based on their relative scores. ParticipantForm AForm B Relatively High ScoreElla2.13962.0807 Alan1.89351.3626 Relatively Average ScoreMike1.66661.8151 Autumn0.84490.6417 Relatively Low ScoreDavid0.37660.6417 Bridgett0.21550.6417 Table 1. LMT Scaled Scores for Participants Tiling for Relatively High Scoring Participants One of the most striking difference between our RHS and RLS group was the use of tiling in comparing the relative sizes of the critters.  RHS participants relied on tiling the smaller critter into the larger one either by gesturing (Alan) or speaking about the relationship of one to the other (Ella).  Ella relied on more proportional ideas in discussing the tasks, Alan relied more on geometry.  Both used a number of resources with tiling being one of the key ones that differentiated them from other participants. Figure 2. Ella’s equiload showing the strong presence of tiling without gesturing. Figure 3. Alan’s equiload showing the strong presence of tiling with gesturing. Figure 4. Bridgett’s equiload showing pattern searching and pattern recognition. Figure 5. David’s equiload showing pattern recognition.