2,185 research outputs found
ERME column
No full textIn this contribution we introduce three classical theoretical stances within the field of mathematics education regarding representations. Our aim is to highlight what we consider to be an interesting shift in how representations are conceived and studied in the field of mathematics education, and how this could impact both the practice of teaching and learning mathematics, and on further theorizing mathematical representation. We also indicate potential directions in which to develop ways to talk about newer forms of dynamic interactive representation
Introduction to the Special Issue “Supporting Transitions Within, Across and Beyond Digital Experiences for the Teaching and Learning of Mathematics”
No full textWhether we consider digital resources, such as graphing software, computer algebra systems, or dynamic geometry environments as “vehicles for mathematical ways of thinking”, or as a means for providing an increasing variety of visual, dynamic and linked computer-based representations that can support the learning of mathematics, “transitioning” between such resources becomes crucial particularly when considering the flexibility, variety, and multimodality of digital environments in combination with all non-digital forms of learning.
For this special issue, we invited papers that focused specifically on such transitions, highlighting in particular three types of transition related to digital resources, within, beyond and across. We received a high number of proposals and accepted thirteen papers, collected in the two volumes dedicated to this Special Issue
Grouping passengers: A microgenetic case study of a struggling student's representational strategies for quotitive division
No full textInternational audienceThis paper focuses on the arithmetical understandings and behaviours of one fourteen-year old student with a history of very low attainment in mathematics, as she worked on a sequence of scenario-based quotitive division (grouping) tasks with individually-tailored verbal and visuospatial support. The student's independent and co-created visuospatial representations of arithmetical structures, along with verbal comments, were analysed qualitatively using a multimodal microgenetic approach. This paper uses selected illustrative examples to discuss certain arithmeticalrepresentational changes (e.g. employment of pictorial, iconic and symbolic elements), some of the particular difficulties that may be experienced by students with impaired memory for arithmetical relationships and procedures, and potential compensatory strategies. It is intended to stand as a companion piece to the case study presented at the previous CERME TWG24 (Finesilver, 2019)
An introduction to TWG24: Representations in mathematics teaching and learning
No full textCERME13 was the fourth full conference to feature TWG24, Representations in mathematics teaching and learning. This year we accepted 18 papers and 1 poster. In terms of participants, Italy and Germany were strongly represented, with the group also including researchers from Australia, Denmark, Israel, Mexico, Netherlands, Switzerland, Turkey, United Arab Emirates, and United
Kingdom. This year over half the group were early career researchers. All papers were shared with the participants in advance of the conference, then presented by the author(s), followed by group discussions. This year we introduced a semi-formal ‘response’ procedure, where each participant was assigned two papers for which they were asked to prepare a question or comment to start the group discussion. This ensured the voices of every member of the group were heard in discussion, including those who may be less confident in such settings, and/or who require more time to prepare a response. For this year’s final ‘workshop’ session in breakout groups, we asked each group to identify, discuss and present an emerging theme of the week that they had found particularly engaging. These themes chosen by the TWG participants are used to structure this report
Emerging and developing multiplicative structure in students' visuospatial representations: Four key configuration types
No full textInternational audienceVisuospatial representations of quantities and their relations are widely used to support the understanding of basic arithmetic, including multiplicative relationships. These include drawn imagery and concrete manipulatives. This paper defines four particular configurations of nonstandard representation according to the spatial organization of their visual elements. These are: unit containers, unit arrays, array-container blends, and number containers, all of which have been observed to support developing multiplicative thinking, allowing low-attaining students to work with the equal-groups structures of natural number multiplication-and division-based tasks. Student-created examples are discussed, and pedagogical and diagnostic implications considered. In their early encounters with quantitative relationships, children become aware of concepts such as conservation of number, counting, etc., through interactions with collections of objects. For example, addition as the joining of collections and subtraction as removing a subset of objects from a collection-in which the ordering of individual objects is unimportant-can be considered conceptual 'grounding metaphors' (Lakoff & Núñez, 2000). Various models of children's arithmetical problem-solving development indicate a broadly similar progression from early concrete/enactive-based reasoning, to imagic/iconic, to abstract/symbolic reasoning (e.g. Bruner, 1974; Piaget, 1952). Within this broad outline, the actual external representations of learners' thinking during problem-solving include many possible sub-varieties (e.g., sets of actual objects, pictures of objects, tally marks in different configurations, dot arrays, etc.), and many possible categorizations of these for analytical purposes. The construction of appropriate analytical frameworks is necessary for the discerning of inter-individual differences and intra-individual trajectories (Meira, 1995; Voutsina, 2012). This is particularly the case when studying atypically-developing learners (Fletcher et al., 1998). This aim of this paper is to share one aspect from the qualitative analytical framework for student-and co-created visuospatial data used in Finesilver (2014), delineating four particular types of visuospatial representation and demonstrating their use with selected examples. The project took an essentially grounded analytical approach, and so whilst this paper does not report results as such, a sample of research data is included with brief description of the process. Theoretical background To understand multiplication and division represents a significant qualitative change in learners' thinking compared to understanding addition and subtraction (Nunes & Bryant, 1996). These authors, amongst others, have recommended a replications model of multiplication, which is highly relevant both to counting-based strategies and to unitary drawn or modelled representations of multiplicative relationships. A central concept for considering this particular aspect of representation is spatial structuring: Thematic Working Group 24 Proceedings of CERME10 392
Oral History Interview, Carla Trujillo (1504)
No full textIn this interview, Carla Trujillo discusses her roots, which include being born in New Mexico and growing up in Northern California. Carla received her M.S. and Ph.D. degrees in Educational Psychology from UW-Madison and became an established author. To learn more about this oral history, download & review the index first (or transcript if available). It will help determine which audio file(s) to download & listen to.Carla Trujillo was born to a working class family in New Mexico and grew up in Northern California. Her extended family and roots are New Mexican (Chicana). She received her B.S. degree in Human Development from UC Davis, and her M.S. and Ph.D. degrees in Educational Psychology from the University of Wisconsin, Madison. Her dissertation focused on assessing differential treatment of underrepresented students in college classrooms. She is the editor of Living Chicana Theory and Chicana Lesbians: The Girls Our Mothers Warned Us About (Third Woman Press), winner of a Lambda Book Award and the Out/Write Vanguard Award. Her first novel, What Night Brings (Curbstone Press 2003), won the Miguel Marmol prize focusing on human rights. What Night Brings also won the Paterson Fiction Prize, the Latino Literary Foundation Latino Book Award, Bronze Medal from Foreword Magazine, Honorable Mention for the Gustavus Meyers Books Award, and was a LAMBDA Book Award finalist. Carla has also written various articles on identity and higher education. Her latest novel, Faith and Fat Chances, was a finalist for the 2012 PEN Bellwether Prize for socially engaged fiction and is forthcoming from Curbstone/Northwestern University Press. Carla works as the Assistant Dean for Graduate Diversity Program at U.C. Berkeley and has focused some of her recent activities on improving the work and classroom climate using Interactive Theater. She has lectured in Ethnic Studies at U.C. Berkeley and Mills College, and in Women’s Studies at S.F. State University. She has also taught fiction for the Sandra Cisneros Macondo Writers Program and the Lambda Literary Foundation’s Emerging Writers Program
Writers Talk Featuring Carla Buckley, Sarah Gridley, Paula McLain
No full textFeaturing Paula McLain, author of the memoir Like Family: Growing Up in Other People's Houses; poet Sarah Gridley; and Carla Buckley, author of the novel The Things that Keep us Here.The media can be accessed here: http://streaming.osu.edu/knowledgebank/cstw11/New_Voices-Carla_Buckley_Sarah_Gridley_Paula_McLain.mp3Ohio State University. Center for the Study and Teaching of Writin
An introduction to TWG24: Representations in mathematics teaching and learning
No full textRepresentations in mathematics teaching and learning: an introduction to TWG24 of the Thirteenth Congress of the European Society for Research in Mathematics Educatio
Spatial structuring, enumeration and errors of S.E.N. students working with 3D arrays
No full textInternational audienceThe move from understanding and working with arithmetical structures in one dimension (i.e. additive) to two or more dimensions (i.e. multiplicative) requires a significant change in children’s thinking. This paper investigates the varied and developing strategies and understandings of young people struggling with that change, through a series of 3D array enumeration tasks. Participants relied heavily on counting-based strategies, and a new analytical framework is proposed with which to diagnose initial (mis-)conceptions and observe microprogressions on the path towards multiplicative understanding
Low-attaining students’ representational strategies: tasks, time, efficiency, and economy
No full textThere are many potential ways to represent arithmetical tasks, but students’ choices may be limited by beliefs that only certain standardised representations are 'legitimate' in school mathematics. Furthermore, concern for the quantity and speed of ‘work done’ can override opportunities for meaningful engagement with the content. This paper draws on a sample of the informal representational strategies observed during a microanalytic study of 11-15 year old students with low prior attainment in mathematics. In absence of pressure to provide quick answers, or to obtain them in a prescribed manner, students worked flexibly, participating in arithmetical reasoning, attempting and succeeding in tasks they were previously unable to engage with. The relationships between representational strategies, economy and efficiency are discussed in relation to multiplicative thinking. These have pedagogical implications for the representational expectations placed on students with difficulties in mathematics, particularly in learning support and intervention contexts
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