Pub Date : 2020-12-23DOI: 10.51272/PMENA.42.2020-256
Y. Ko, Dalton Edgecomb, Nathan Kooi
Developing communally agreed-on criteria for proof in a mathematics classroom has been found to empower pre-service secondary mathematics teachers’ (PSMTs’) learning of proof. To date, we do not know how creating class-based criteria for proof throughout a semester-long course with a focus on secondary school mathematics can promote PSMTs’ understanding of proof. In this paper, we reported PSMTs’ evolution of what constitutes proof by comparing their initial and revised classbased criteria for proof and investigating their videotaped lessons and video transcripts. Results indicated that PSMTs perceived mathematical values and norms of what counts as proof in their mathematics classroom community as the semester progressed.
{"title":"Pre-service secondary mathematics teachers’ evolution of communally agreed-on criteria for proof","authors":"Y. Ko, Dalton Edgecomb, Nathan Kooi","doi":"10.51272/PMENA.42.2020-256","DOIUrl":"https://doi.org/10.51272/PMENA.42.2020-256","url":null,"abstract":"Developing communally agreed-on criteria for proof in a mathematics classroom has been found to empower pre-service secondary mathematics teachers’ (PSMTs’) learning of proof. To date, we do not know how creating class-based criteria for proof throughout a semester-long course with a focus on secondary school mathematics can promote PSMTs’ understanding of proof. In this paper, we reported PSMTs’ evolution of what constitutes proof by comparing their initial and revised classbased criteria for proof and investigating their videotaped lessons and video transcripts. Results indicated that PSMTs perceived mathematical values and norms of what counts as proof in their mathematics classroom community as the semester progressed.","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":"73442837","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}
{"title":"Teachers’ problem posing in paper-and-pencil and GeoGebra / Planteamiento de problemas por profesores en papel-y-lápiz y GeoGebra","authors":"Lucero Hernández-Cruz, César Martínez Hernández, Rodolfo Rangel Alcántar, Norma Angélica Barón Ramírez","doi":"10.51272/PMENA.42.2020-377","DOIUrl":"https://doi.org/10.51272/PMENA.42.2020-377","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":"73034594","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.42.2020-126
Doris Jeannotte, S. Sampson, S. Dufour
{"title":"Elementary teachers’ discourse about mathematical reasoning","authors":"Doris Jeannotte, S. Sampson, S. Dufour","doi":"10.51272/PMENA.42.2020-126","DOIUrl":"https://doi.org/10.51272/PMENA.42.2020-126","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":"72394426","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.42.2020-46
Adam Scharfenberger, Leah M. Frazee
Usiskin (1999) described four conceptions of algebra: Algebra as Generalized Arithmetic, Algebra as a Study of Procedures for Solving Certain Kinds of Problems, Algebra as the Study of Relationships among Quantities, and Algebra as the Study of Structures. The Algebra as the Study of Relationships among Quantities conception relates to the NCTM (2000) Algebra Standard expectation that students “understand patterns, relations, and functions” (p. 37). Algebraic thinking “includes being able to think about functions and how they work, and to think about the impact that a system’s structure has on calculations” (Driscoll, 1999, p. 1). Analyzing students’ algebraic thinking with patterning tasks in two variables allows researchers to understand how students think about functions, how they work, and how the representation provided in the question impacts student thinking about the structure of the problem. In this study, one elementary student solved patterning problems in two variables with different representations during a task-based interview (Goldin, 2000). Preliminary findings suggest that this student used different reasoning strategies when given pattern problems in two different representations. On a task consisting of a visual pattern of figures growing in an arithmetic sequence, the student visualized how the growth occurred in each successive figure. The student used the rate of growth to compute the size of the figure at future iterations. In the context of this task, the evidence suggests that the student was thinking covariationally (Confrey & Smith, 1994) about the relationship between the increase in figure size and increase in figure number. When presented with a task showing a linear relationship between values in an input-output table of numbers, the student was asked to determine the output value when the input value was 38. Upon receiving this question, the student intensely looked at the problem before stating: Oh, I see it now. Okay, so I see if you multiply this by – each number [points at all the numbers in the left input column] by two and add 1, that’s the number on this side [points at all the numbers in the right output column]. So take 15 for example. 15 times 2 is 30, plus 1 is 31 and that is in the out. [15 and 31 correspond to each other in the table. 15 being in the input column and 31 being in the output column]. The student used this mapping between the numbers in the input column and the output column to determine 38 corresponds to 77. In this context, the student used a correspondence approach (Confrey & Smith, 1994) to determine the output when the input was 38. In conclusion, both tasks contained the same structure as linear functions. However, the student thought differently about how the functions “worked” when given a visual pattern of growth as opposed to when given an input-output table. This student showed the capacity to reason through covariation and correspondence while the context of the problem may ha
{"title":"Elementary Algebraic thinking with patterns in two variables","authors":"Adam Scharfenberger, Leah M. Frazee","doi":"10.51272/PMENA.42.2020-46","DOIUrl":"https://doi.org/10.51272/PMENA.42.2020-46","url":null,"abstract":"Usiskin (1999) described four conceptions of algebra: Algebra as Generalized Arithmetic, Algebra as a Study of Procedures for Solving Certain Kinds of Problems, Algebra as the Study of Relationships among Quantities, and Algebra as the Study of Structures. The Algebra as the Study of Relationships among Quantities conception relates to the NCTM (2000) Algebra Standard expectation that students “understand patterns, relations, and functions” (p. 37). Algebraic thinking “includes being able to think about functions and how they work, and to think about the impact that a system’s structure has on calculations” (Driscoll, 1999, p. 1). Analyzing students’ algebraic thinking with patterning tasks in two variables allows researchers to understand how students think about functions, how they work, and how the representation provided in the question impacts student thinking about the structure of the problem. In this study, one elementary student solved patterning problems in two variables with different representations during a task-based interview (Goldin, 2000). Preliminary findings suggest that this student used different reasoning strategies when given pattern problems in two different representations. On a task consisting of a visual pattern of figures growing in an arithmetic sequence, the student visualized how the growth occurred in each successive figure. The student used the rate of growth to compute the size of the figure at future iterations. In the context of this task, the evidence suggests that the student was thinking covariationally (Confrey & Smith, 1994) about the relationship between the increase in figure size and increase in figure number. When presented with a task showing a linear relationship between values in an input-output table of numbers, the student was asked to determine the output value when the input value was 38. Upon receiving this question, the student intensely looked at the problem before stating: Oh, I see it now. Okay, so I see if you multiply this by – each number [points at all the numbers in the left input column] by two and add 1, that’s the number on this side [points at all the numbers in the right output column]. So take 15 for example. 15 times 2 is 30, plus 1 is 31 and that is in the out. [15 and 31 correspond to each other in the table. 15 being in the input column and 31 being in the output column]. The student used this mapping between the numbers in the input column and the output column to determine 38 corresponds to 77. In this context, the student used a correspondence approach (Confrey & Smith, 1994) to determine the output when the input was 38. In conclusion, both tasks contained the same structure as linear functions. However, the student thought differently about how the functions “worked” when given a visual pattern of growth as opposed to when given an input-output table. This student showed the capacity to reason through covariation and correspondence while the context of the problem may ha","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":"78197675","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.42.2020-328
Paul N. Reimer, Aileen Rizo, Steve Pauls
Although the field has recognized the importance of early mathematics education for young children, many early childhood educators do not have access to high-quality, engaging professional development in math or science (McClure et al., 2017). In particular, educators in rural communities lack opportunities to grow professionally due to geographical isolation or under-resourced programs. In addition, many professional development opportunities offer prescribed programs that do not encourage active participation and are not connected to teachers’ existing practices or approaches (Kennedy, 2016). Building on frameworks for effective professional development (Fishman, Davis, & Chan, 2014) and embodied design (Abrahamson & Lindgren, 2014), we used video conferencing technology paired with tangible materials to engage a cohort of remote online early childhood educators in four sessions of professional development in early mathematics education. Each session was one hour in length and included opportunities for online teacher learners to engage with tangible materials (e.g., Froebel gifts, triangle construction materials) to explore mathematical concepts central to early childhood development. Our approach was designed to 1) engage teachers as learners with carefully designed materials to develop their own understanding, and 2) open pathways for mediated participation through the sharing of physical constructions via video conferencing. In this poster, we focus on the following research question: How do participants’ material constructions and interactions act as mediating resources in their participation in remote online professional learning? We video recorded two of the professional learning sessions, surveyed participants, and interviewed a sample of participants about their experiences in the professional learning. Our findings suggest that tangible materials allowed for common sense-making and active participation throughout the sessions. Furthermore, tangible materials served as resources for mathematical engagement and dialogue in spite of educators’ remote participation. This poster will highlight the professional development approach and suggest several implications for the use of tangible materials to enhance online professional development engagement.
{"title":"Remote engagement in early mathematics professional development: Using tangible artifacts to mediate participation","authors":"Paul N. Reimer, Aileen Rizo, Steve Pauls","doi":"10.51272/PMENA.42.2020-328","DOIUrl":"https://doi.org/10.51272/PMENA.42.2020-328","url":null,"abstract":"Although the field has recognized the importance of early mathematics education for young children, many early childhood educators do not have access to high-quality, engaging professional development in math or science (McClure et al., 2017). In particular, educators in rural communities lack opportunities to grow professionally due to geographical isolation or under-resourced programs. In addition, many professional development opportunities offer prescribed programs that do not encourage active participation and are not connected to teachers’ existing practices or approaches (Kennedy, 2016). Building on frameworks for effective professional development (Fishman, Davis, & Chan, 2014) and embodied design (Abrahamson & Lindgren, 2014), we used video conferencing technology paired with tangible materials to engage a cohort of remote online early childhood educators in four sessions of professional development in early mathematics education. Each session was one hour in length and included opportunities for online teacher learners to engage with tangible materials (e.g., Froebel gifts, triangle construction materials) to explore mathematical concepts central to early childhood development. Our approach was designed to 1) engage teachers as learners with carefully designed materials to develop their own understanding, and 2) open pathways for mediated participation through the sharing of physical constructions via video conferencing. In this poster, we focus on the following research question: How do participants’ material constructions and interactions act as mediating resources in their participation in remote online professional learning? We video recorded two of the professional learning sessions, surveyed participants, and interviewed a sample of participants about their experiences in the professional learning. Our findings suggest that tangible materials allowed for common sense-making and active participation throughout the sessions. Furthermore, tangible materials served as resources for mathematical engagement and dialogue in spite of educators’ remote participation. This poster will highlight the professional development approach and suggest several implications for the use of tangible materials to enhance online professional development engagement.","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":"77386240","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.42.2020-183
Jeffrey D. Pair, G. Calva
{"title":"Understanding the roles of proof through exploration of unsolved conjectures","authors":"Jeffrey D. Pair, G. Calva","doi":"10.51272/PMENA.42.2020-183","DOIUrl":"https://doi.org/10.51272/PMENA.42.2020-183","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":"80498375","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.42.2020-141
S. Kandasamy, Jennifer A. Czocher
In this theoretical paper we compare the Piagetian perspective on knowledge construction to mathematical model construction, with the aim to understand how mathematical modeling enables learning of mathematics and learning of science, as is often claimed. We do this by examining data through two lenses:(i) examining the role of cognitive conflict as it arises during validation of a model and (ii) viewing model validation as a reflection on activity-effect relationship. We explain why we chose to look deeply into model validation specifically, present examples for each lens, and consider implications.
{"title":"How mathematical modeling enables learning?","authors":"S. Kandasamy, Jennifer A. Czocher","doi":"10.51272/PMENA.42.2020-141","DOIUrl":"https://doi.org/10.51272/PMENA.42.2020-141","url":null,"abstract":"In this theoretical paper we compare the Piagetian perspective on knowledge construction to mathematical model construction, with the aim to understand how mathematical modeling enables learning of mathematics and learning of science, as is often claimed. We do this by examining data through two lenses:(i) examining the role of cognitive conflict as it arises during validation of a model and (ii) viewing model validation as a reflection on activity-effect relationship. We explain why we chose to look deeply into model validation specifically, present examples for each lens, and consider implications.","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":"81429261","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.42.2020-285
Brooke Max, Rachael M. Welder
{"title":"Textbook use of children’s thinking to support prospective elementary teachers’ geometric understanding","authors":"Brooke Max, Rachael M. Welder","doi":"10.51272/PMENA.42.2020-285","DOIUrl":"https://doi.org/10.51272/PMENA.42.2020-285","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":"81833421","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.42.2020-350
Hangil Kim
Proof plays significant roles in the context of school mathematics and is a tool for enhancing student’s understanding of mathematics. Lack of opportunities for proving in textbook has been documented. This study was conducted to consider an instructional way to make proving as everyday lesson by formulating more opportunities than did textbooks. The guiding assumption of this study is that conjectures which students come up with can be initiatives for learning how to prove. This preliminary study will show that problem posing is a strategic tool with potential to bridge everyday instruction and the practice of proving so as to teach how to prove more meaningfully and authentically.
{"title":"Embracing proving into everyday lesson by problem posing","authors":"Hangil Kim","doi":"10.51272/PMENA.42.2020-350","DOIUrl":"https://doi.org/10.51272/PMENA.42.2020-350","url":null,"abstract":"Proof plays significant roles in the context of school mathematics and is a tool for enhancing student’s understanding of mathematics. Lack of opportunities for proving in textbook has been documented. This study was conducted to consider an instructional way to make proving as everyday lesson by formulating more opportunities than did textbooks. The guiding assumption of this study is that conjectures which students come up with can be initiatives for learning how to prove. This preliminary study will show that problem posing is a strategic tool with potential to bridge everyday instruction and the practice of proving so as to teach how to prove more meaningfully and authentically.","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":"85134115","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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