Pub Date : 2020-12-23DOI: 10.51272/PMENA.42.2020-261
Luis E. Montero-Moguel, Verónica Vargas-Alejo, I. Rodríguez-González
{"title":"Conceptual systems when implementing model-eliciting activities / Sistemas conceptuales al implementar actividades provocadoras de modelos","authors":"Luis E. Montero-Moguel, Verónica Vargas-Alejo, I. Rodríguez-González","doi":"10.51272/PMENA.42.2020-261","DOIUrl":"https://doi.org/10.51272/PMENA.42.2020-261","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":"78801265","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.42.2020-277
Melike Kara, Kimberly Corum
Creating, connecting, and translating multiple representation are “important cognitive processes that lead students to develop robust mathematical understandings” (Huntley, Marcus, Kahan, & Miller, 2007, p. 117). These cognitive processes are also considered to be crucial elements of preservice teachers’ (PSTs’) pedagogical content knowledge (Dreher, Kuntze, & Lerman, 2016). In order to investigate PSTs’ cognitive processes regarding multiple representations, we collected data from 73 PSTs, who enrolled in a mathematics content course for elementary education majors in Spring 2019. We analyzed PSTs’ solutions to an assessment task following seven weeks of instruction related to the use of strip diagrams, double number lines, and algebraic equations to solve problems involving ratio and proportional relationships and word problems (Beckmann, 2014). The PSTs were asked to determine the total number of cookies Bonnie baked when given information about the cookie types (e.g., 1/3 of the cookies were chocolate chip, 1/6 were peanut butter, 1/6 were oatmeal raisin, and 24 were cinnamon) in two ways: using a strip diagram and writing and solving an algebraic equation. We used an error analysis technique (Radatz, 1979) to sort and interpret the responses based on fluency with strip diagram and algebraic solutions. The PSTs who exhibited complete reasoning were able to use both representations and the PSTs who exhibited incomplete reasoning were unable to use at least one of the representations. The preliminary analysis of solutions revealed the following themes in the PSTs’ strategies (Table 1).
{"title":"Pre-service elementary education teachers’ solutions for word problems: Using strip diagram vs. algebraic approach","authors":"Melike Kara, Kimberly Corum","doi":"10.51272/PMENA.42.2020-277","DOIUrl":"https://doi.org/10.51272/PMENA.42.2020-277","url":null,"abstract":"Creating, connecting, and translating multiple representation are “important cognitive processes that lead students to develop robust mathematical understandings” (Huntley, Marcus, Kahan, & Miller, 2007, p. 117). These cognitive processes are also considered to be crucial elements of preservice teachers’ (PSTs’) pedagogical content knowledge (Dreher, Kuntze, & Lerman, 2016). In order to investigate PSTs’ cognitive processes regarding multiple representations, we collected data from 73 PSTs, who enrolled in a mathematics content course for elementary education majors in Spring 2019. We analyzed PSTs’ solutions to an assessment task following seven weeks of instruction related to the use of strip diagrams, double number lines, and algebraic equations to solve problems involving ratio and proportional relationships and word problems (Beckmann, 2014). The PSTs were asked to determine the total number of cookies Bonnie baked when given information about the cookie types (e.g., 1/3 of the cookies were chocolate chip, 1/6 were peanut butter, 1/6 were oatmeal raisin, and 24 were cinnamon) in two ways: using a strip diagram and writing and solving an algebraic equation. We used an error analysis technique (Radatz, 1979) to sort and interpret the responses based on fluency with strip diagram and algebraic solutions. The PSTs who exhibited complete reasoning were able to use both representations and the PSTs who exhibited incomplete reasoning were unable to use at least one of the representations. The preliminary analysis of solutions revealed the following themes in the PSTs’ strategies (Table 1).","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":"78870879","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.42.2020-1
C. Hoyles, Carolyn Kieran, T. Rojano, A. I. Sacristán, M. Trigueros
ion process” (p. 1). Relevant to the questions addressed to this panel, she adds that computational thinking can be defined as “the thought processes involved in formulating a problem and expressing (with a linguistic representation) its solution in such a way that a computer – human or machine – can effectively carry it out” (p. 1). Interestingly, Andy diSessa (2018) – one of the two authors of Turtle Geometry back in 1981 – has taken issue with this point and has argued that noncomputer scientists rarely map out exactly how a problem can be solved before actually doing the solving. But is he right? In opposition to diSessa, and more in line with Wing, Al Cuoco (2018) in a paper on mathematical practice offers three examples. The first of these (see Fig. 1) relates to Wing’s emphasis on the process of abstraction and her point about formulating a problem and expressing its solution in a way that a computing being or machine can carry it out. This example involves what Cuoco refers to as “the dreaded algebra word problem,” where he insists that we think of the answer to the algebra problem as an equation rather than a number – in a method that involves abstracting from numerals. The problem is as follows: “Mary drives from Boston to Chicago, travels at an average rate of 60 MPH on the way down and 50 MPH on the way back. The total driving time takes 36 hours, how far is Boston from Chicago?” Figure 1. Arriving at an equation from abstracting the regularity in numerical guesses (Cuoco, 2018, p. 3) The method that Cuoco suggests builds upon students’ ability to solve similar problems in middle school (note: they have already learned the relationship between speed, time, and distance) and is as follows: Take a guess – but the aim is not intended to get closer to the answer with each succeeding guess; rather it is to arrive at an equation, not a number. The idea is to carry out enough guesses so as to see the regularity of the calculations that allow for checking the guesses – in Cuoco’s words: Develop “a generic ‘guess checker’ that is the desired equation”. The processes of mathematical Reflections on digital technologies in mathematics education across cultures 76 practice that are employed here, and which are ones that Cuoco declares he uses all the time in his own mathematical work, are: 1. Abstract regularity from repeated calculations, and 2. Use precise language (and algebraic symbolism) to give a generic and general description – the equation – for how you check your guesses. (Cuoco, 2018, p. 4) The conclusion to be drawn from this example is that these two processes of mathematical practice fit well with the programming and thinking-like-a-programmer characteristics of computational thinking (Wing, 2006, 2014), and that students who are currently engaged in using digital technologies (e.g., laptops, robots) to code with visual (e.g., Scratch) or text-based languages are participating in mathematical practices. Nevertheless, other research (
{"title":"Reflections on digital technologies in mathematics education across cultures","authors":"C. Hoyles, Carolyn Kieran, T. Rojano, A. I. Sacristán, M. Trigueros","doi":"10.51272/pmena.42.2020-1","DOIUrl":"https://doi.org/10.51272/pmena.42.2020-1","url":null,"abstract":"ion process” (p. 1). Relevant to the questions addressed to this panel, she adds that computational thinking can be defined as “the thought processes involved in formulating a problem and expressing (with a linguistic representation) its solution in such a way that a computer – human or machine – can effectively carry it out” (p. 1). Interestingly, Andy diSessa (2018) – one of the two authors of Turtle Geometry back in 1981 – has taken issue with this point and has argued that noncomputer scientists rarely map out exactly how a problem can be solved before actually doing the solving. But is he right? In opposition to diSessa, and more in line with Wing, Al Cuoco (2018) in a paper on mathematical practice offers three examples. The first of these (see Fig. 1) relates to Wing’s emphasis on the process of abstraction and her point about formulating a problem and expressing its solution in a way that a computing being or machine can carry it out. This example involves what Cuoco refers to as “the dreaded algebra word problem,” where he insists that we think of the answer to the algebra problem as an equation rather than a number – in a method that involves abstracting from numerals. The problem is as follows: “Mary drives from Boston to Chicago, travels at an average rate of 60 MPH on the way down and 50 MPH on the way back. The total driving time takes 36 hours, how far is Boston from Chicago?” Figure 1. Arriving at an equation from abstracting the regularity in numerical guesses (Cuoco, 2018, p. 3) The method that Cuoco suggests builds upon students’ ability to solve similar problems in middle school (note: they have already learned the relationship between speed, time, and distance) and is as follows: Take a guess – but the aim is not intended to get closer to the answer with each succeeding guess; rather it is to arrive at an equation, not a number. The idea is to carry out enough guesses so as to see the regularity of the calculations that allow for checking the guesses – in Cuoco’s words: Develop “a generic ‘guess checker’ that is the desired equation”. The processes of mathematical Reflections on digital technologies in mathematics education across cultures 76 practice that are employed here, and which are ones that Cuoco declares he uses all the time in his own mathematical work, are: 1. Abstract regularity from repeated calculations, and 2. Use precise language (and algebraic symbolism) to give a generic and general description – the equation – for how you check your guesses. (Cuoco, 2018, p. 4) The conclusion to be drawn from this example is that these two processes of mathematical practice fit well with the programming and thinking-like-a-programmer characteristics of computational thinking (Wing, 2006, 2014), and that students who are currently engaged in using digital technologies (e.g., laptops, robots) to code with visual (e.g., Scratch) or text-based languages are participating in mathematical practices. Nevertheless, other research (","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":"76229379","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":"Technological knowledge of mathematics pre-service teachers at the beginning of their methodology courses / Conocimiento tecnológico de los futuros maestros de matemáticas al iniciar sus cursos de metodología","authors":"Yency Edith Choque Dextre, Juliette Moreno-Concepción, Omar Hernández-Rodríguez, Wanda Villafañe-Cepeda, G. González","doi":"10.51272/PMENA.42.2020-135","DOIUrl":"https://doi.org/10.51272/PMENA.42.2020-135","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":"88028207","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.42.2020-304
Suzanne Harper, Dana C. Cox
This paper reports on a longitudinal study of mathematics teachers’ development of a vision of teaching with technology where we document professional events and activities that point to continued evolution and devolution of those beliefs. We extend earlier work and ask participants to reflect on the experiences they have had as early career teachers, and how they have influenced their beliefs since graduation. We find that there are significant opportunities for professional learning after graduation, and recommend continued development of graduate-level coursework that is technology-dependent. We also find that the use of Desmos is particularly influential in changing beliefs about the role of technology.
{"title":"Influences on early-career mathematics’ teachers vision of teaching with technology: a longitudinal study","authors":"Suzanne Harper, Dana C. Cox","doi":"10.51272/PMENA.42.2020-304","DOIUrl":"https://doi.org/10.51272/PMENA.42.2020-304","url":null,"abstract":"This paper reports on a longitudinal study of mathematics teachers’ development of a vision of teaching with technology where we document professional events and activities that point to continued evolution and devolution of those beliefs. We extend earlier work and ask participants to reflect on the experiences they have had as early career teachers, and how they have influenced their beliefs since graduation. We find that there are significant opportunities for professional learning after graduation, and recommend continued development of graduate-level coursework that is technology-dependent. We also find that the use of Desmos is particularly influential in changing beliefs about the role of technology.","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":"88261555","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.42.2020-150
S. Otten, Mitchelle M Wambua, R. Govender
Reasoning-and-proving is viewed by many scholars to be a crucial part of students’ mathematical experiences in secondary school. There is scholarly debate, however, about the necessity of formal proving. In this study, we investigated the notion of “proof for all” from the perspective of secondary mathematics teachers and we analyzed, using the framework of practical rationality, the justifications they gave for whether or not all students should learn proof. Based on interviews with twenty-one secondary teachers from a socioeconomically-diverse set of schools, we found that teachers do not share the same opinion on who should learn proving but they expressed obligations toward individual student learning as justifications both for teaching proving to all students and for not teaching proving to some students.
{"title":"Secondary teachers’ differing views on who should learn proving and why","authors":"S. Otten, Mitchelle M Wambua, R. Govender","doi":"10.51272/PMENA.42.2020-150","DOIUrl":"https://doi.org/10.51272/PMENA.42.2020-150","url":null,"abstract":"Reasoning-and-proving is viewed by many scholars to be a crucial part of students’ mathematical experiences in secondary school. There is scholarly debate, however, about the necessity of formal proving. In this study, we investigated the notion of “proof for all” from the perspective of secondary mathematics teachers and we analyzed, using the framework of practical rationality, the justifications they gave for whether or not all students should learn proof. Based on interviews with twenty-one secondary teachers from a socioeconomically-diverse set of schools, we found that teachers do not share the same opinion on who should learn proving but they expressed obligations toward individual student learning as justifications both for teaching proving to all students and for not teaching proving to some students.","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":"86157160","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.42.2020-266
Cody L. Patterson, Hiroko K. Warshauer, Max L. Warshauer
This study examines a professional development (PD) program, set in a summer mathematics program for middle grades students with a research-based curriculum, where preservice and inservice teachers collaborated in interpreting and responding to student thinking. We investigated the resources that participants contributed to this collaboration, and the opportunities the nontraditional PD setting afforded for the sharing of these resources. Our embedded case study consisted of two classes, where participants taught and then engaged in video-recorded debriefing sessions each day. Their discussions focused on what they noticed in class and how they responded in the moment or anticipated responding in future lessons. We find that participants’ observations from class catalyzed the sharing of resources, both from the program and from outside experiences, that contributed to the analysis of student thinking.
{"title":"Resources that preservice and inservice teachers offer in collaborative analysis of student thinking","authors":"Cody L. Patterson, Hiroko K. Warshauer, Max L. Warshauer","doi":"10.51272/PMENA.42.2020-266","DOIUrl":"https://doi.org/10.51272/PMENA.42.2020-266","url":null,"abstract":"This study examines a professional development (PD) program, set in a summer mathematics program for middle grades students with a research-based curriculum, where preservice and inservice teachers collaborated in interpreting and responding to student thinking. We investigated the resources that participants contributed to this collaboration, and the opportunities the nontraditional PD setting afforded for the sharing of these resources. Our embedded case study consisted of two classes, where participants taught and then engaged in video-recorded debriefing sessions each day. Their discussions focused on what they noticed in class and how they responded in the moment or anticipated responding in future lessons. We find that participants’ observations from class catalyzed the sharing of resources, both from the program and from outside experiences, that contributed to the analysis of student thinking.","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":"82803028","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.42.2020-76
C. Quinn, Susan B. Empson, V. Jacobs
Gender differences in fourth and fifth grade students’ strategy use for a fraction story problem were investigated using multinomial logistic regression on a sample of 193 written student strategies. Gender was not a significant predictor of type of strategy used, in contrast to earlier studies finding that boys tended to use more abstract strategies whereas girls tended to use more concrete strategies or the standard algorithm.
{"title":"Gender differences in number strategy use for students solving fraction story problems","authors":"C. Quinn, Susan B. Empson, V. Jacobs","doi":"10.51272/PMENA.42.2020-76","DOIUrl":"https://doi.org/10.51272/PMENA.42.2020-76","url":null,"abstract":"Gender differences in fourth and fifth grade students’ strategy use for a fraction story problem were investigated using multinomial logistic regression on a sample of 193 written student strategies. Gender was not a significant predictor of type of strategy used, in contrast to earlier studies finding that boys tended to use more abstract strategies whereas girls tended to use more concrete strategies or the standard algorithm.","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":"82956183","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.42.2020-357
Amanda T. Sugimoto, Eva Thanheiser, K. Melhuish, M. A. Sorto, Autumn Pham
Untangling the relationships between teaching, learning, and content is complex. This study focuses on one aspect of these relationships, i.e., the at times challenging role that language can play in mathematical tasks, discussions, and student access. The authors analyze two video banks to identify and operationalize combinations of teacher and student actions that support student access to mathematical tasks and language.
{"title":"Operationalizing access for students: making meaning of tasks, contexts and language","authors":"Amanda T. Sugimoto, Eva Thanheiser, K. Melhuish, M. A. Sorto, Autumn Pham","doi":"10.51272/PMENA.42.2020-357","DOIUrl":"https://doi.org/10.51272/PMENA.42.2020-357","url":null,"abstract":"Untangling the relationships between teaching, learning, and content is complex. This study focuses on one aspect of these relationships, i.e., the at times challenging role that language can play in mathematical tasks, discussions, and student access. The authors analyze two video banks to identify and operationalize combinations of teacher and student actions that support student access to mathematical tasks and language.","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":"89206270","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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