Pub Date : 2020-12-23DOI: 10.51272/PMENA.42.2020-161
Erin Pomponio, S. Greenstein, Denish Akuom
Operating from an enactivist theory of cognition, this work seeks to understand the emergent nature of mathematical activity mediated by manipulatives. Enactivism takes a biological approach and theorizes that perception consists in perceptually guided action enabled by cognitive structures that emerge from recurrent sensorimotor patterns (Proulx, 2013; Varela, Rosch, & Thompson, 1992). These processes are non-linear, unfolding, and ongoing events where meanings emerge and transform in interactions, not inside of minds and bodies (Malafouris, 2013; Proulx, 2019). Further, one’s way of knowing is driven by an evolutionary imperative to act in an adequate, fitting, and harmonious way with one’s environment (Maheux & Proulx, 2015). This search for harmony leads to a structural coupling between the individual and their environment with the individual’s history of recursive interactions playing a crucial role in structurally determining this course of evolution (Proulx, 2013). This work seeks to elucidate the nature of emergent mathematical activity mediated by manipulatives by addressing the question, “What role might manipulatives play in the emergent processes of sense making?” To do so, we analyzed the activity of “Dolly” and “Lyle” as they aimed to make sense of the flip-and-multiply algorithm for fraction division in a problem-solving interview using a manipulative Dolly created for engagement with fraction concepts. The data comes from a larger study that is exploring how an open-ended and iterative design experience centered in Making (Halverson & Sheridan, 2014) might inform prospective mathematics teachers’ (PMTs’) pedagogy. We took a revelatory case study approach to analyze and transcribe the video data (Yin, 2014), and focused our analysis on the particular interactions aiming to coordinate meanings of fraction division in the manipulative and in the algorithm that presumably substantiates those meanings (Malafouris, 2013). Our analysis illuminates the role manipulatives can play in establishing a notion of sense making that is grounded in embodied understandings. For example, although Dolly and Lyle arrived at the correct answer with the manipulative early in their problem solving, they were dissatisfied because it did not seem to fit with the answer they derived from the algorithm. Eventually, this dissonance gave way as they established harmony between the two, thereby revealing the compelling power that embodied tool use can have for altering a space of possible actions and consequently on sense-making activity. Our analysis also reveals what might be problematic about a pedagogical practice where a procedure is adequate and sense making is not the criteria for fit. The enactment of the algorithm was disrupted through use of a tool, ultimately leading to an authentic understanding of what it means to do fraction division. These findings further substantiate extant arguments for engaging mathematics learners in embodied, tool-mediated
{"title":"Making sense of senseless things: an enactivist analysis of harmony and dissonance in problem solving","authors":"Erin Pomponio, S. Greenstein, Denish Akuom","doi":"10.51272/PMENA.42.2020-161","DOIUrl":"https://doi.org/10.51272/PMENA.42.2020-161","url":null,"abstract":"Operating from an enactivist theory of cognition, this work seeks to understand the emergent nature of mathematical activity mediated by manipulatives. Enactivism takes a biological approach and theorizes that perception consists in perceptually guided action enabled by cognitive structures that emerge from recurrent sensorimotor patterns (Proulx, 2013; Varela, Rosch, & Thompson, 1992). These processes are non-linear, unfolding, and ongoing events where meanings emerge and transform in interactions, not inside of minds and bodies (Malafouris, 2013; Proulx, 2019). Further, one’s way of knowing is driven by an evolutionary imperative to act in an adequate, fitting, and harmonious way with one’s environment (Maheux & Proulx, 2015). This search for harmony leads to a structural coupling between the individual and their environment with the individual’s history of recursive interactions playing a crucial role in structurally determining this course of evolution (Proulx, 2013). This work seeks to elucidate the nature of emergent mathematical activity mediated by manipulatives by addressing the question, “What role might manipulatives play in the emergent processes of sense making?” To do so, we analyzed the activity of “Dolly” and “Lyle” as they aimed to make sense of the flip-and-multiply algorithm for fraction division in a problem-solving interview using a manipulative Dolly created for engagement with fraction concepts. The data comes from a larger study that is exploring how an open-ended and iterative design experience centered in Making (Halverson & Sheridan, 2014) might inform prospective mathematics teachers’ (PMTs’) pedagogy. We took a revelatory case study approach to analyze and transcribe the video data (Yin, 2014), and focused our analysis on the particular interactions aiming to coordinate meanings of fraction division in the manipulative and in the algorithm that presumably substantiates those meanings (Malafouris, 2013). Our analysis illuminates the role manipulatives can play in establishing a notion of sense making that is grounded in embodied understandings. For example, although Dolly and Lyle arrived at the correct answer with the manipulative early in their problem solving, they were dissatisfied because it did not seem to fit with the answer they derived from the algorithm. Eventually, this dissonance gave way as they established harmony between the two, thereby revealing the compelling power that embodied tool use can have for altering a space of possible actions and consequently on sense-making activity. Our analysis also reveals what might be problematic about a pedagogical practice where a procedure is adequate and sense making is not the criteria for fit. The enactment of the algorithm was disrupted through use of a tool, ultimately leading to an authentic understanding of what it means to do fraction division. These findings further substantiate extant arguments for engaging mathematics learners in embodied, tool-mediated ","PeriodicalId":68089,"journal":{"name":"数学教学通讯","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90499107","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-23DOI: 10.51272/PMENA.42.2020-218
S. Prough
How past research considers the influence of parents on their children’s mathematical understanding does not often focus on the experience and intent of the parents themselves. In a crosscase analysis, I address how two mothers’ experience shapes their mathematical positioning and the resulting interactions with their children in the subject. The attention on parents in mathematics takes on particular gendered roles, which is not often considered in research. There are more complex factors that shape how parents see and interact with mathematics. This study begins to show what alternative possibilities exist for engagement by U.S. parents in the content that still positively support children’s early learning.
{"title":"Mother-centered understanding of mathematical interactions with children: Pursuing positive intent","authors":"S. Prough","doi":"10.51272/PMENA.42.2020-218","DOIUrl":"https://doi.org/10.51272/PMENA.42.2020-218","url":null,"abstract":"How past research considers the influence of parents on their children’s mathematical understanding does not often focus on the experience and intent of the parents themselves. In a crosscase analysis, I address how two mothers’ experience shapes their mathematical positioning and the resulting interactions with their children in the subject. The attention on parents in mathematics takes on particular gendered roles, which is not often considered in research. There are more complex factors that shape how parents see and interact with mathematics. This study begins to show what alternative possibilities exist for engagement by U.S. parents in the content that still positively support children’s early learning.","PeriodicalId":68089,"journal":{"name":"数学教学通讯","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89182729","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-23DOI: 10.51272/PMENA.42.2020-36
M. Tremblay, Mireille Saboya
Transforming the teaching practice involves the adoption of new methods that must engage teachers in reflecting on their practice (CSE, 2004; Day, 1999). The roles of the researcher are multiplying. His quest to understand the teaching/learning activity is mirrored by a concern to examine the teaching profession from more than just a normative standpoint, thereby contributing to the reflection on actions and participating in the decisions that will guide their interventions (Bednarz, 2009). Influenced by historical and cultural theories (Radford, 2011; Engestrom, 1999) and more specifically by the concept of contradiction, our communication illustrates our way of “grasping” the awareness process and the coming into being of generality layers linked to the development of algebraic thinking occurring in the various classrooms of the teachers we are supporting and acting as trainers with them.
{"title":"A study of contradictions as a lever for continuing education: A case involving algebraic generalization activities / Étude de contradictions comme levier de formation continue: un exemple autour d’activités de généralisation algébrique","authors":"M. Tremblay, Mireille Saboya","doi":"10.51272/PMENA.42.2020-36","DOIUrl":"https://doi.org/10.51272/PMENA.42.2020-36","url":null,"abstract":"Transforming the teaching practice involves the adoption of new methods that must engage teachers in reflecting on their practice (CSE, 2004; Day, 1999). The roles of the researcher are multiplying. His quest to understand the teaching/learning activity is mirrored by a concern to examine the teaching profession from more than just a normative standpoint, thereby contributing to the reflection on actions and participating in the decisions that will guide their interventions (Bednarz, 2009). Influenced by historical and cultural theories (Radford, 2011; Engestrom, 1999) and more specifically by the concept of contradiction, our communication illustrates our way of “grasping” the awareness process and the coming into being of generality layers linked to the development of algebraic thinking occurring in the various classrooms of the teachers we are supporting and acting as trainers with them.","PeriodicalId":68089,"journal":{"name":"数学教学通讯","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86699505","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-23DOI: 10.51272/PMENA.42.2020-204
Brigitte Johana Sánchez-Robayo, J. Wilkins
Brigitte Johana Sánchez Robayo Virginia Tech brigsa1@vt.edu Jesse L. M. Wilkins Virginia Tech wilkins@vt.edu This study investigated the intuitive knowledge of conditional probability in one four-year-old child. Six clinical interviews were video-recorded for analyzing transcripts and expressions from the child. Findings from this research suggest that the child has a pre-operational intuition about change in sample space in non-replacement situations. He also seems to have an intuitive understanding of independent events. Furthermore, although his judgments are mainly subjective, he does use quantitative justifications, although, inconsistently.
{"title":"Conditional probability in early childhood: a case study","authors":"Brigitte Johana Sánchez-Robayo, J. Wilkins","doi":"10.51272/PMENA.42.2020-204","DOIUrl":"https://doi.org/10.51272/PMENA.42.2020-204","url":null,"abstract":"Brigitte Johana Sánchez Robayo Virginia Tech brigsa1@vt.edu Jesse L. M. Wilkins Virginia Tech wilkins@vt.edu This study investigated the intuitive knowledge of conditional probability in one four-year-old child. Six clinical interviews were video-recorded for analyzing transcripts and expressions from the child. Findings from this research suggest that the child has a pre-operational intuition about change in sample space in non-replacement situations. He also seems to have an intuitive understanding of independent events. Furthermore, although his judgments are mainly subjective, he does use quantitative justifications, although, inconsistently.","PeriodicalId":68089,"journal":{"name":"数学教学通讯","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85275019","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-23DOI: 10.51272/PMENA.42.2020-368
B. Pak, Brent Jackson, Jillian M. Cavanna
{"title":"Beginning teachers’ equitable and ambitious number talks","authors":"B. Pak, Brent Jackson, Jillian M. Cavanna","doi":"10.51272/PMENA.42.2020-368","DOIUrl":"https://doi.org/10.51272/PMENA.42.2020-368","url":null,"abstract":"","PeriodicalId":68089,"journal":{"name":"数学教学通讯","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89920601","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-23DOI: 10.51272/PMENA.42.2020-14
Fabio Milner, José Ramón Cordo Rodríguez
{"title":"Working Group: Didactic contrasts between Calculus and Analysis / Grupo de trabajo: Contrastes didácticos entre cálculo y análisis","authors":"Fabio Milner, José Ramón Cordo Rodríguez","doi":"10.51272/PMENA.42.2020-14","DOIUrl":"https://doi.org/10.51272/PMENA.42.2020-14","url":null,"abstract":"","PeriodicalId":68089,"journal":{"name":"数学教学通讯","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84850079","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-23DOI: 10.51272/PMENA.42.2020-246
F. Arbaugh, Raymond LaRochelle, Seonmi Do, Azaria Cunningham, Kevin Voogt, M. Cirillo, Kristen N. Bieda
We examined the coaching practices of three mathematics teacher educators as they engaged in realtime coaching with secondary mathematics preservice teachers. Situated in a novel early field experience and under close supervision, preservice teachers instructed undergraduate students in an introductory mathematics course; teacher educators coached in real time during these teaching episodes. Forty-four preservice teachers participated in this study, resulting in a data corpus of 44 videos of their teaching. Findings indicate that direct coaching was used more than indirect coaching, and pacing was the most prevalent focus of direct coaching.
{"title":"Real-time coaching with secondary preservice teachers: The practices of mathematics teacher educators","authors":"F. Arbaugh, Raymond LaRochelle, Seonmi Do, Azaria Cunningham, Kevin Voogt, M. Cirillo, Kristen N. Bieda","doi":"10.51272/PMENA.42.2020-246","DOIUrl":"https://doi.org/10.51272/PMENA.42.2020-246","url":null,"abstract":"We examined the coaching practices of three mathematics teacher educators as they engaged in realtime coaching with secondary mathematics preservice teachers. Situated in a novel early field experience and under close supervision, preservice teachers instructed undergraduate students in an introductory mathematics course; teacher educators coached in real time during these teaching episodes. Forty-four preservice teachers participated in this study, resulting in a data corpus of 44 videos of their teaching. Findings indicate that direct coaching was used more than indirect coaching, and pacing was the most prevalent focus of direct coaching.","PeriodicalId":68089,"journal":{"name":"数学教学通讯","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87211668","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-23DOI: 10.51272/PMENA.42.2020-301
Jennifer D. Cribbs, Jianna Davenport, Lisa C. Duffin, M. M. Day
{"title":"Contributing factors to secondary mathematics teachers’ professional identity","authors":"Jennifer D. Cribbs, Jianna Davenport, Lisa C. Duffin, M. M. Day","doi":"10.51272/PMENA.42.2020-301","DOIUrl":"https://doi.org/10.51272/PMENA.42.2020-301","url":null,"abstract":"","PeriodicalId":68089,"journal":{"name":"数学教学通讯","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75678833","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-23DOI: 10.51272/PMENA.42.2020-159
A. Ozturk
{"title":"Exploring high school students’ validation methods in the mathematical modeling process","authors":"A. Ozturk","doi":"10.51272/PMENA.42.2020-159","DOIUrl":"https://doi.org/10.51272/PMENA.42.2020-159","url":null,"abstract":"","PeriodicalId":68089,"journal":{"name":"数学教学通讯","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72654484","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-23DOI: 10.51272/PMENA.42.2020-295
N. Ehrenfeld, K. S. McGugan, Samantha A. Marshall, Brette Garner
{"title":"Reconciling local contexts and external conceptual resources in mathematics teachers’ collaborative sensemaking","authors":"N. Ehrenfeld, K. S. McGugan, Samantha A. Marshall, Brette Garner","doi":"10.51272/PMENA.42.2020-295","DOIUrl":"https://doi.org/10.51272/PMENA.42.2020-295","url":null,"abstract":"","PeriodicalId":68089,"journal":{"name":"数学教学通讯","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73873884","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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