Carol H. Wade, Sandra K. Cimbricz, G. Sonnert, Meagan Gruver, P. Sadler
Quantitative analysis of the Factors Influencing College Success in Mathematics (FICSMath) Survey data indicates that high school mathematics teachers’ abilities to teach for conceptual understanding is a significant and positive predictor of student performance in single- variable college calculus. To explore these findings further, we gathered and analyzed interview data gained from a representative sample of high school precalculus teachers from across the U.S., identified by their students as requiring high levels of conceptual understanding (n = 13). Seventeen themes were identified and then combined into five overarching phenomenological themes. These overarching themes suggest that teachers who teach for high conceptual understanding (a) support relational understanding during problem solving, (b) require students to learn how to study to build on prior knowledge and learn from mistakes, (c) use mathematical language and ask critical questions to support learning, (d) focus on content knowledge necessary to make connections, and (e) use technology to support learning concepts but limit calculator use. Comparison of these results to quantitative findings further illuminate that intentional development of disciplinary knowledge, cognition, and language are noteworthy points of intersection for teachers and researchers alike.
{"title":"The Secondary-Tertiary Transition in Mathematics: What High School Teachers Do to Prepare Students for Future Success in College-Level Calculus","authors":"Carol H. Wade, Sandra K. Cimbricz, G. Sonnert, Meagan Gruver, P. Sadler","doi":"10.7916/JMETC.V9I2.583","DOIUrl":"https://doi.org/10.7916/JMETC.V9I2.583","url":null,"abstract":"Quantitative analysis of the Factors Influencing College Success in Mathematics (FICSMath) Survey data indicates that high school mathematics teachers’ abilities to teach for conceptual understanding is a significant and positive predictor of student performance in single- variable college calculus. To explore these findings further, we gathered and analyzed interview data gained from a representative sample of high school precalculus teachers from across the U.S., identified by their students as requiring high levels of conceptual understanding (n = 13). Seventeen themes were identified and then combined into five overarching phenomenological themes. These overarching themes suggest that teachers who teach for high conceptual understanding (a) support relational understanding during problem solving, (b) require students to learn how to study to build on prior knowledge and learn from mistakes, (c) use mathematical language and ask critical questions to support learning, (d) focus on content knowledge necessary to make connections, and (e) use technology to support learning concepts but limit calculator use. Comparison of these results to quantitative findings further illuminate that intentional development of disciplinary knowledge, cognition, and language are noteworthy points of intersection for teachers and researchers alike.","PeriodicalId":30179,"journal":{"name":"Journal of Mathematics Education at Teachers College","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43567366","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}
Multiplicative reasoning is a key concept in elementary school mathematics. Item statistics reported by the National Assessment of Educational Progress (NAEP) assessment provide the best current indicator for how well elementary students across the U.S. understand this, and other concepts. However, beyond expert reviews and statistical analysis, there is relatively little evidence on the validity of NAEP items for assessing mathematical reasoning. The present study examined the convergent validity of several public-release fourth grade NAEP multiplication items. Findings indicate that most items assessed lack sufficient validity for examining children’s underlying conceptual knowledge. Rather, NAEP items may be less appropriate for assessing conceptual understanding and more appropriate for assessing procedural recall. Recommendations and implications are discussed.
{"title":"What Form of Mathematics Are Assessments Assessing? The Case of Multiplication and Division in Fourth Grade NAEP Items.","authors":"K. Kosko, Rashmi Singh","doi":"10.7916/JMETC.V9I1.597","DOIUrl":"https://doi.org/10.7916/JMETC.V9I1.597","url":null,"abstract":"Multiplicative reasoning is a key concept in elementary school mathematics. Item statistics reported by the National Assessment of Educational Progress (NAEP) assessment provide the best current indicator for how well elementary students across the U.S. understand this, and other concepts. However, beyond expert reviews and statistical analysis, there is relatively little evidence on the validity of NAEP items for assessing mathematical reasoning. The present study examined the convergent validity of several public-release fourth grade NAEP multiplication items. Findings indicate that most items assessed lack sufficient validity for examining children’s underlying conceptual knowledge. Rather, NAEP items may be less appropriate for assessing conceptual understanding and more appropriate for assessing procedural recall. Recommendations and implications are discussed.","PeriodicalId":30179,"journal":{"name":"Journal of Mathematics Education at Teachers College","volume":"9 1","pages":"1-8"},"PeriodicalIF":0.0,"publicationDate":"2018-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48974683","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}
Mathematicians are often inimically portrayed in popular culture, resulting in an abundance of non-mathematical identities in the classroom. Various tropes are propagated by the media that dominate our mental schemas of what makes a mathematician: the eccentric Einstein-like old man; the young, tortured genius; and the “genetically different” savant. However, these portrayals in popular culture can be used as a tool—rather than a hindrance—if teachers know how to present them properly in the classroom. Although the media often promotes mathematical myths, it can also be used to debunk them.
{"title":"The Portrayal of Mathematicians and Mathematics in Popular Culture.","authors":"K. Barba","doi":"10.7916/JMETC.V9I1.599","DOIUrl":"https://doi.org/10.7916/JMETC.V9I1.599","url":null,"abstract":"Mathematicians are often inimically portrayed in popular culture, resulting in an abundance of non-mathematical identities in the classroom. Various tropes are propagated by the media that dominate our mental schemas of what makes a mathematician: the eccentric Einstein-like old man; the young, tortured genius; and the “genetically different” savant. However, these portrayals in popular culture can be used as a tool—rather than a hindrance—if teachers know how to present them properly in the classroom. Although the media often promotes mathematical myths, it can also be used to debunk them.","PeriodicalId":30179,"journal":{"name":"Journal of Mathematics Education at Teachers College","volume":"9 1","pages":"9-14"},"PeriodicalIF":0.0,"publicationDate":"2018-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47169157","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}
Mathematics teachers frequently provide concrete manipulatives to students during instruction; however, the rationale for using certain manipulatives in conjunction with concepts may not be explored. This article focuses on area models that are currently used in classrooms to provide concrete examples of integer and binomial multiplication. The innovation of combining the representations for negative numbers with both algebra tiles and Algeblocks is provided, with a mathematical justification for its development. Teachers’ effective integration of tools such as these in mathematics instruction can help students develop conceptual understanding and procedural fluency.
{"title":"Selecting Tools to Model Integer and Binomial Multiplication","authors":"S. Pratt, Colleen M. Eddy","doi":"10.7916/JMETC.V8I2.595","DOIUrl":"https://doi.org/10.7916/JMETC.V8I2.595","url":null,"abstract":"Mathematics teachers frequently provide concrete manipulatives to students during instruction; however, the rationale for using certain manipulatives in conjunction with concepts may not be explored. This article focuses on area models that are currently used in classrooms to provide concrete examples of integer and binomial multiplication. The innovation of combining the representations for negative numbers with both algebra tiles and Algeblocks is provided, with a mathematical justification for its development. Teachers’ effective integration of tools such as these in mathematics instruction can help students develop conceptual understanding and procedural fluency.","PeriodicalId":30179,"journal":{"name":"Journal of Mathematics Education at Teachers College","volume":"8 1","pages":"31-40"},"PeriodicalIF":0.0,"publicationDate":"2017-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45687095","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}
This paper explores the viability and development of the first edition of the researcher’s mathematical card game, Groups , as a learning tool for elementary group theory, a topic in abstract algebra. Groups was play-tested by six undergraduate students in late 2016 who provided feedback on Groups from both utility-centric and design-centric perspectives. This paper addresses how well undergraduates with no prior group theory experience understand the fundamentals of group theory after playing several games of Groups , and how well undergraduates with prior knowledge of group theory related the Groups game play and mechanics to the fundamentals of group theory. Based on interview and questionnaire data, players found Groups to be an engaging and effective learning tool with both strengths and weaknesses in the tested iteration.
{"title":"Designing and Testing a Mathematics Card Game for Teaching and Learning Elementary Group Theory.","authors":"P. Galarza","doi":"10.7916/JMETC.V8I2.592","DOIUrl":"https://doi.org/10.7916/JMETC.V8I2.592","url":null,"abstract":"This paper explores the viability and development of the first edition of the researcher’s mathematical card game, Groups , as a learning tool for elementary group theory, a topic in abstract algebra. Groups was play-tested by six undergraduate students in late 2016 who provided feedback on Groups from both utility-centric and design-centric perspectives. This paper addresses how well undergraduates with no prior group theory experience understand the fundamentals of group theory after playing several games of Groups , and how well undergraduates with prior knowledge of group theory related the Groups game play and mechanics to the fundamentals of group theory. Based on interview and questionnaire data, players found Groups to be an engaging and effective learning tool with both strengths and weaknesses in the tested iteration.","PeriodicalId":30179,"journal":{"name":"Journal of Mathematics Education at Teachers College","volume":"8 1","pages":"9-18"},"PeriodicalIF":0.0,"publicationDate":"2017-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45664999","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}
In this article I explore some of the underlying mathematics of Pokemon battles and describe ways that teachers at the secondary level could explore concepts of mathematical game theory in this context. I discuss various ways of representing and analyzing a Pokemon battle using game theory and conclude with an example of applying concepts of expected value and solving systems of linear equations to find a mixed-strategy Nash equilibrium.
{"title":"Pokémon Battles as a Context for Mathematical Modeling","authors":"W. McGuffey","doi":"10.7916/JMETC.V8I1.798","DOIUrl":"https://doi.org/10.7916/JMETC.V8I1.798","url":null,"abstract":"In this article I explore some of the underlying mathematics of Pokemon battles and describe ways that teachers at the secondary level could explore concepts of mathematical game theory in this context. I discuss various ways of representing and analyzing a Pokemon battle using game theory and conclude with an example of applying concepts of expected value and solving systems of linear equations to find a mixed-strategy Nash equilibrium.","PeriodicalId":30179,"journal":{"name":"Journal of Mathematics Education at Teachers College","volume":"8 1","pages":"1-6"},"PeriodicalIF":0.0,"publicationDate":"2017-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45122730","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}
This paper analyzes the thought process involved in problem solving and its categorization as creative thinking as defined by psychologist R. Weisberg (2006). Additionally, the notion of insight, sometimes present in unconscious creative thinking and often leading to creative ideas, is discussed in the context of geometry problem solving. In particular, it is argued that solving insight problems used by psychologists to measure creativity and solving certain geometry problems share similar thought processes.
{"title":"Creativity and Insight in Problem Solving","authors":"Laura Golnabi","doi":"10.7916/JMETC.V7I2.795","DOIUrl":"https://doi.org/10.7916/JMETC.V7I2.795","url":null,"abstract":"This paper analyzes the thought process involved in problem solving and its categorization as creative thinking as defined by psychologist R. Weisberg (2006). Additionally, the notion of insight, sometimes present in unconscious creative thinking and often leading to creative ideas, is discussed in the context of geometry problem solving. In particular, it is argued that solving insight problems used by psychologists to measure creativity and solving certain geometry problems share similar thought processes.","PeriodicalId":30179,"journal":{"name":"Journal of Mathematics Education at Teachers College","volume":"7 1","pages":"27-29"},"PeriodicalIF":0.0,"publicationDate":"2016-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71368771","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}
Zuhairina Suhaimi, M. Shahrill, K. A. Tengah, N. Abbas
This study incorporated the use of writing-to-learn strategy, particularly journal writing, in Grade 10 mathematics lessons. Although part of a study conducted to investigate the effects of journal writing on academically lower-achieving learners with English as their second language, this paper will focus only on the students’ perceptions of writing journals in mathematics. The students’ perceptions are based on their experience from a 2-cycle action research study. A total of 35 students, ranging from age 15 to 16 years old, from two Grade 10 classes were involved in the study. Data from the student journal entries and questionnaires revealed that the majority of students found journal writing beneficial for the development of their mathematical learning.
{"title":"Incorporating the Use of Writing-to-Learn Strategy in Grade 10 Mathematics Lessons: The Students’ Perspectives","authors":"Zuhairina Suhaimi, M. Shahrill, K. A. Tengah, N. Abbas","doi":"10.7916/JMETC.V7I2.793","DOIUrl":"https://doi.org/10.7916/JMETC.V7I2.793","url":null,"abstract":"This study incorporated the use of writing-to-learn strategy, particularly journal writing, in Grade 10 mathematics lessons. Although part of a study conducted to investigate the effects of journal writing on academically lower-achieving learners with English as their second language, this paper will focus only on the students’ perceptions of writing journals in mathematics. The students’ perceptions are based on their experience from a 2-cycle action research study. A total of 35 students, ranging from age 15 to 16 years old, from two Grade 10 classes were involved in the study. Data from the student journal entries and questionnaires revealed that the majority of students found journal writing beneficial for the development of their mathematical learning.","PeriodicalId":30179,"journal":{"name":"Journal of Mathematics Education at Teachers College","volume":"7 1","pages":"11-20"},"PeriodicalIF":0.0,"publicationDate":"2016-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71368766","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}
For the past three years, the Program in Mathematics and Education at Teachers College, Columbia University, has, in coordination with its colloquium series, held a Columbus Day Symposium. Each year, the all day event has been focused on a particular theme in mathematics education, and has brought in various speakers and leaders in the field to share their work and insights. This has included a presentation of an award to a scholar who has made significant contributions to the field. This year, Dr. Geoffrey Howson received this award. Throughout his career, Dr. Howson has written, edited, and contributed to many important books and papers on mathematics, education and mathematics education. Among other things, he has also served as President of the Mathematical Association of Great Britain, and Secretary-General of ICMI. The theme for the symposium this past year was the teaching of the mathematically gifted, and Dr. Howson has also made contributions in this area. The following is an edited version of the remarks he gave at the Columbus Day Symposium.
{"title":"Some Thoughts on Educating More Able Students","authors":"G. Howson","doi":"10.7916/JMETC.V7I1.780","DOIUrl":"https://doi.org/10.7916/JMETC.V7I1.780","url":null,"abstract":"For the past three years, the Program in Mathematics and Education at Teachers College, Columbia University, has, in coordination with its colloquium series, held a Columbus Day Symposium. Each year, the all day event has been focused on a particular theme in mathematics education, and has brought in various speakers and leaders in the field to share their work and insights. This has included a presentation of an award to a scholar who has made significant contributions to the field. This year, Dr. Geoffrey Howson received this award. Throughout his career, Dr. Howson has written, edited, and contributed to many important books and papers on mathematics, education and mathematics education. Among other things, he has also served as President of the Mathematical Association of Great Britain, and Secretary-General of ICMI. The theme for the symposium this past year was the teaching of the mathematically gifted, and Dr. Howson has also made contributions in this area. The following is an edited version of the remarks he gave at the Columbus Day Symposium.","PeriodicalId":30179,"journal":{"name":"Journal of Mathematics Education at Teachers College","volume":"7 1","pages":"1-5"},"PeriodicalIF":0.0,"publicationDate":"2016-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71368735","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}
Carol H. Wade, G. Sonnert, P. Sadler, Z. Hazari, Cherrie Watson
This article compares the views of teachers and professors about the transition from secondary mathematics to tertiary calculus. Quantitative analysis revealed five categories where teachers and professors differed significantly in the relative frequency of addressing them. Using the rite of passage theory, the separation and incorporation phases were investigated by carrying out thematic analyses on these five categories. For the professors, the analysis revealed specific content within algebra and precalculus that they viewed as vital preparation for students’ tertiary calculus success. For the teachers, the analysis highlighted the classroom environment realities of teaching in the separation phase. The rite of passage and professional turf theories are used to discuss and interpret the findings.
{"title":"A Comparison of Mathematics Teachers’ and Professors’ Views on Secondary Preparation for Tertiary Calculus","authors":"Carol H. Wade, G. Sonnert, P. Sadler, Z. Hazari, Cherrie Watson","doi":"10.7916/JMETC.V7I1.782","DOIUrl":"https://doi.org/10.7916/JMETC.V7I1.782","url":null,"abstract":"This article compares the views of teachers and professors about the transition from secondary mathematics to tertiary calculus. Quantitative analysis revealed five categories where teachers and professors differed significantly in the relative frequency of addressing them. Using the rite of passage theory, the separation and incorporation phases were investigated by carrying out thematic analyses on these five categories. For the professors, the analysis revealed specific content within algebra and precalculus that they viewed as vital preparation for students’ tertiary calculus success. For the teachers, the analysis highlighted the classroom environment realities of teaching in the separation phase. The rite of passage and professional turf theories are used to discuss and interpret the findings.","PeriodicalId":30179,"journal":{"name":"Journal of Mathematics Education at Teachers College","volume":"7 1","pages":"1-10"},"PeriodicalIF":0.0,"publicationDate":"2016-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71368741","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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