Pub Date : 2020-12-23DOI: 10.51272/PMENA.42.2020-49
Leslie Dietiker, Rashmi Singh, Meghan Riling, Hector I. Nieves
How can we design mathematical lessons that spark student interest? To answer this, we analyzed teacher-designed and enacted lessons that students described as interesting for how the content unfolded. When compared to those the same students describe as uninteresting, multiple distinguishing characteristics are evident, such as the presence of misdirection, mathematical questions that remain unanswered for extended time, and a greater number of questions that are unanswered at each point of the lesson. Low-interest lessons did not contain many special narrative features and mostly had questions that were answered immediately. Our findings offer guidance for the design of lessons that can shift student mathematical dispositions.
{"title":"What makes a mathematics lesson interesting to students?","authors":"Leslie Dietiker, Rashmi Singh, Meghan Riling, Hector I. Nieves","doi":"10.51272/PMENA.42.2020-49","DOIUrl":"https://doi.org/10.51272/PMENA.42.2020-49","url":null,"abstract":"How can we design mathematical lessons that spark student interest? To answer this, we analyzed teacher-designed and enacted lessons that students described as interesting for how the content unfolded. When compared to those the same students describe as uninteresting, multiple distinguishing characteristics are evident, such as the presence of misdirection, mathematical questions that remain unanswered for extended time, and a greater number of questions that are unanswered at each point of the lesson. Low-interest lessons did not contain many special narrative features and mostly had questions that were answered immediately. Our findings offer guidance for the design of lessons that can shift student mathematical dispositions.","PeriodicalId":68089,"journal":{"name":"数学教学通讯","volume":" 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72381122","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.42.2020-178
María José Aviña González, Angelina Alvarado Monroy
{"title":"Report of a classroom experience for the development of distribution models / Experiencia en el aula para el desarrollo de modelos para el reparto","authors":"María José Aviña González, Angelina Alvarado Monroy","doi":"10.51272/pmena.42.2020-178","DOIUrl":"https://doi.org/10.51272/pmena.42.2020-178","url":null,"abstract":"","PeriodicalId":68089,"journal":{"name":"数学教学通讯","volume":"43 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72412513","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.42.2020-192
Irma E. Stevens
Researchers have identified both the affordances of engaging students in symbolization activities and students’ difficulties in meaningfully representing contexts through algebraic expressions/formulas. In a semester-long teaching experiment, two pre-service teachers demonstrated their conflicting meanings for formulas with their images of a context when engaging in a task about a dynamic geometric object. The two students could construct both normative formulas by reasoning with a context and descriptions of covariational relationships between quantities within the context, but both still struggled to relate their formulas and quantitative relationships to one another. This result highlights the importance of attending to what students’ formulas mean to them, which for the students in this study, could be either a way of “solving” or “relating” quantities.
{"title":"“Solving versus relating”: Pre-service teachers’ conflicting images of formulas and dynamic contexts","authors":"Irma E. Stevens","doi":"10.51272/PMENA.42.2020-192","DOIUrl":"https://doi.org/10.51272/PMENA.42.2020-192","url":null,"abstract":"Researchers have identified both the affordances of engaging students in symbolization activities and students’ difficulties in meaningfully representing contexts through algebraic expressions/formulas. In a semester-long teaching experiment, two pre-service teachers demonstrated their conflicting meanings for formulas with their images of a context when engaging in a task about a dynamic geometric object. The two students could construct both normative formulas by reasoning with a context and descriptions of covariational relationships between quantities within the context, but both still struggled to relate their formulas and quantitative relationships to one another. This result highlights the importance of attending to what students’ formulas mean to them, which for the students in this study, could be either a way of “solving” or “relating” quantities.","PeriodicalId":68089,"journal":{"name":"数学教学通讯","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84376149","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.42.2020-116
Brendan T Dames, Corey Webel
{"title":"Evolution of elementary math leaders’ collaborative plans for school-level change","authors":"Brendan T Dames, Corey Webel","doi":"10.51272/PMENA.42.2020-116","DOIUrl":"https://doi.org/10.51272/PMENA.42.2020-116","url":null,"abstract":"","PeriodicalId":68089,"journal":{"name":"数学教学通讯","volume":"73 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81927740","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.42.2020-193
Omar Arenas Bonifacio, E. Sánchez
The first fundamental theorem of calculus relates differential and integral calculus, one of its important aspects according to Bressoud (2011) is that it shows the existence of two ways of calculating an integral: with the limit of a Riemann sum and by an antiderivative. Larsen, Marrongelle, Bressoud and Graham (2017) indicate that calculus is a barrier to the academic progress of many students and that there is a need for research that seeks to develop proposals for instruction to improve the understanding of its concepts. Therefore, with the idea of carrying out this type of research in the future, the present study seeks to identify the common interpretation of the first theorem of calculus and whether it is useful in solving contextual problems. Answering these questions will provide some elements to develop a proposal for instruction. This study involved 18 students between the ages of 18 and 21 from engineering careers at a university located in Mexico City, who had completed a calculus course. The instrument was a set of three problems, in two, we propose contextual situations that can be solved by applying the first fundamental theorem of the calculus or by performing integration and derivation operations (one situation is about the ratio of change of the volume of water contained in a tank, with respect to time, where water falls to a variable ratio; the other is about the ratio of change of the volume of water contained in a cylindrical tank with respect to the height of water). In the last problem, the same type of situation is posed in abstract form: If F(x) = f t dt ! ! , obtain F′(x), justify your answer.
微积分的第一个基本定理与微分和积分有关，根据Bressoud(2011)的说法，它的一个重要方面是它表明存在两种计算积分的方法:用黎曼和的极限和用不定积分。Larsen, Marrongelle, Bressoud和Graham(2017)指出，微积分是许多学生学业进步的障碍，需要进行研究，寻求制定教学建议，以提高对其概念的理解。因此，在未来开展这类研究的想法下，本研究试图确定微积分第一定理的共同解释，以及它是否有助于解决上下文问题。回答这些问题将提供一些元素，以制定指导建议。这项研究涉及18名年龄在18到21岁之间的学生，他们来自墨西哥城的一所大学，从事工程工作，他们完成了微积分课程。该仪器是一组三个问题，其中两个，我们提出了上下文情况，可以通过应用微积分的第一基本定理或通过执行积分和求导操作来解决(一个情况是关于水箱中水的体积变化的比例，相对于时间，其中水下降到一个可变的比例;另一个是关于圆柱形水箱中水的体积与水的高度的变化率。在最后一个问题中，同样的情况以抽象的形式提出:如果F(x) = F t dt !！ ，得到F ' (x)，证明你的答案。
{"title":"First fundamental theorem of calculus: How do engineering students interpret and apply it? / Primer teorema fundamental del cálculo: ¿cómo lo interpretan y aplican estudiantes de ingeniería?","authors":"Omar Arenas Bonifacio, E. Sánchez","doi":"10.51272/PMENA.42.2020-193","DOIUrl":"https://doi.org/10.51272/PMENA.42.2020-193","url":null,"abstract":"The first fundamental theorem of calculus relates differential and integral calculus, one of its important aspects according to Bressoud (2011) is that it shows the existence of two ways of calculating an integral: with the limit of a Riemann sum and by an antiderivative. Larsen, Marrongelle, Bressoud and Graham (2017) indicate that calculus is a barrier to the academic progress of many students and that there is a need for research that seeks to develop proposals for instruction to improve the understanding of its concepts. Therefore, with the idea of carrying out this type of research in the future, the present study seeks to identify the common interpretation of the first theorem of calculus and whether it is useful in solving contextual problems. Answering these questions will provide some elements to develop a proposal for instruction. This study involved 18 students between the ages of 18 and 21 from engineering careers at a university located in Mexico City, who had completed a calculus course. The instrument was a set of three problems, in two, we propose contextual situations that can be solved by applying the first fundamental theorem of the calculus or by performing integration and derivation operations (one situation is about the ratio of change of the volume of water contained in a tank, with respect to time, where water falls to a variable ratio; the other is about the ratio of change of the volume of water contained in a cylindrical tank with respect to the height of water). In the last problem, the same type of situation is posed in abstract form: If F(x) = f t dt ! ! , obtain F′(x), justify your answer.","PeriodicalId":68089,"journal":{"name":"数学教学通讯","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79679262","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.42.2020-93
Lili Zhou, S. Richardson
{"title":"A narrative inquiry of GEMS women’s experience with STEM","authors":"Lili Zhou, S. Richardson","doi":"10.51272/PMENA.42.2020-93","DOIUrl":"https://doi.org/10.51272/PMENA.42.2020-93","url":null,"abstract":"","PeriodicalId":68089,"journal":{"name":"数学教学通讯","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84665418","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.42.2020-187
A. Mirin
By analyzing the responses of 100 introductory calculus students to two questions, this study addresses how students understand the fundamental theorem of calculus as it relates to function identity. One question involves students’ understandings of the fundamental theorem of calculus, and the other involves their concept definitions of function sameness. This analysis aims to better understand students’ concept images of function sameness, both in the context of the fundamental theorem of calculus and in general.
{"title":"Function identity and the fundamental theorem of calculus","authors":"A. Mirin","doi":"10.51272/PMENA.42.2020-187","DOIUrl":"https://doi.org/10.51272/PMENA.42.2020-187","url":null,"abstract":"By analyzing the responses of 100 introductory calculus students to two questions, this study addresses how students understand the fundamental theorem of calculus as it relates to function identity. One question involves students’ understandings of the fundamental theorem of calculus, and the other involves their concept definitions of function sameness. This analysis aims to better understand students’ concept images of function sameness, both in the context of the fundamental theorem of calculus and in general.","PeriodicalId":68089,"journal":{"name":"数学教学通讯","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85296026","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.42.2020-396
Carolyn Mitten
Formative assessment has been identified as one way that teachers can gather critical information about a student’s level of understanding in order to make informed instructional adaptations that meet the needs of all students (NCTM, 2000; Shepard et al., 2005). Over several decades, research has shown the potential of formative assessment to effectively improve student achievement (Black & Wiliam, 1998; Kingston & Nash, 2011). Despite its potential, issues in preparing teachers to implement formative assessment practices has kept its potential from being realized (Schoenfeld, 2015) and many teachers have limited understanding of its use (Shepard et al., 2005). In order to better understand how to prepare teachers to use formative assessment, a trajectory describing how teachers develop formative assessment knowledge and practice is needed. The Formative Assessment Levels of Appropriation (FALA) framework evolved from a larger qualitative research study on the evolution of formative assessment knowledge and practice of novice teachers from teacher preparation through their third year of teaching. Grounded in activity theory, the FALA framework describes the levels of appropriation (Grossman et al., 1998) for the five aspects of formative assessment defined by Black & Wiliam (2009). Table 1 provides an example of the framework for one aspect of formative assessment—clarifying intentions and criteria for success.
形成性评估被认为是教师收集有关学生理解水平的关键信息的一种方式，以便做出符合所有学生需求的明智的教学调整(NCTM, 2000;Shepard et al.， 2005)。几十年来，研究表明形成性评价有潜力有效地提高学生的成绩(Black & william, 1998;Kingston & Nash, 2011)。尽管它有潜力，但在准备教师实施形成性评估实践方面的问题使其潜力无法实现(Schoenfeld, 2015)，许多教师对其使用的理解有限(Shepard et al.， 2005)。为了更好地理解如何让教师准备好使用形成性评价，需要一个描述教师如何发展形成性评价知识和实践的轨迹。形成性评估拨款水平(FALA)框架是从一项更大规模的定性研究演变而来的，该研究是关于新手教师从教师准备到第三年教学的形成性评估知识和实践的演变。FALA框架以活动理论为基础，描述了Black & william(2009)定义的形成性评估的五个方面的挪用水平(Grossman et al.， 1998)。表1提供了形成性评估的一个方面的框架的例子——澄清成功的意图和标准。
{"title":"FALA framework: A learning progression for novice teachers’ use of formative assessment","authors":"Carolyn Mitten","doi":"10.51272/PMENA.42.2020-396","DOIUrl":"https://doi.org/10.51272/PMENA.42.2020-396","url":null,"abstract":"Formative assessment has been identified as one way that teachers can gather critical information about a student’s level of understanding in order to make informed instructional adaptations that meet the needs of all students (NCTM, 2000; Shepard et al., 2005). Over several decades, research has shown the potential of formative assessment to effectively improve student achievement (Black & Wiliam, 1998; Kingston & Nash, 2011). Despite its potential, issues in preparing teachers to implement formative assessment practices has kept its potential from being realized (Schoenfeld, 2015) and many teachers have limited understanding of its use (Shepard et al., 2005). In order to better understand how to prepare teachers to use formative assessment, a trajectory describing how teachers develop formative assessment knowledge and practice is needed. The Formative Assessment Levels of Appropriation (FALA) framework evolved from a larger qualitative research study on the evolution of formative assessment knowledge and practice of novice teachers from teacher preparation through their third year of teaching. Grounded in activity theory, the FALA framework describes the levels of appropriation (Grossman et al., 1998) for the five aspects of formative assessment defined by Black & Wiliam (2009). Table 1 provides an example of the framework for one aspect of formative assessment—clarifying intentions and criteria for success.","PeriodicalId":68089,"journal":{"name":"数学教学通讯","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85359494","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.42.2020-373
Michael S. Meagher, Jennifer N. Lovett, Allison W. McCulloch
Middle school students (n=144) worked with an applet specially designed to introduce the concept of function without using algebraic representations. The purpose of the study was to examine whether the applet would help students to understand function as a relationship between a set of inputs and a set of outputs and to begin to develop a definition of function based on that relationship. Results indicate that, by focusing on consistency of the outputs the students, at a rate of approximately 80%, are able to distinguish functions from non-functions. Also, students showed some promise in recognising constant functions as functions, a known area of common misconceptions.
{"title":"Middle school students’ development of an understanding of the concept of function","authors":"Michael S. Meagher, Jennifer N. Lovett, Allison W. McCulloch","doi":"10.51272/PMENA.42.2020-373","DOIUrl":"https://doi.org/10.51272/PMENA.42.2020-373","url":null,"abstract":"Middle school students (n=144) worked with an applet specially designed to introduce the concept of function without using algebraic representations. The purpose of the study was to examine whether the applet would help students to understand function as a relationship between a set of inputs and a set of outputs and to begin to develop a definition of function based on that relationship. Results indicate that, by focusing on consistency of the outputs the students, at a rate of approximately 80%, are able to distinguish functions from non-functions. Also, students showed some promise in recognising constant functions as functions, a known area of common misconceptions.","PeriodicalId":68089,"journal":{"name":"数学教学通讯","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81687800","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.42.2020-292
Joe Champion, Angela R. Crawford, Michele B. Carney
{"title":"Articulating effective middle grades instructional practices in a teacher-researcher alliance","authors":"Joe Champion, Angela R. Crawford, Michele B. Carney","doi":"10.51272/PMENA.42.2020-292","DOIUrl":"https://doi.org/10.51272/PMENA.42.2020-292","url":null,"abstract":"","PeriodicalId":68089,"journal":{"name":"数学教学通讯","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84054992","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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