28 results
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2. Developing Teacher Understanding of Early Algebraic Concepts Using Lesson Study
- Author
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Mathematics Education Research Group of Australasia and Hunter, Jodie
- Abstract
This paper reports on the use of lesson study as a professional development tool. In particular the paper focuses on the way in which the teachers increased their understanding of how tasks, classroom activity and teacher actions scaffolded student learning of early algebraic reasoning of equivalence and the commutative principle. Teacher voice is used to illustrate how lesson study cycles caused the teachers to reflect and review their own understandings of early algebraic concepts and how their students considered the concepts.
- Published
- 2012
3. Developing a 'Conjecturing Atmosphere' in the Classroom through Task Design and Enactment
- Author
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Mathematics Education Research Group of Australasia and Hunter, Jodie
- Abstract
In recent years there has been an increased emphasis on algebraic reasoning in primary school classrooms. This includes introducing students to the mathematical practices of making conjectures, justifying and generalising. Drawing on findings from a classroom-based study, this paper explores one teacher's journey in shifting her task design and enactment to develop a "conjecturing atmosphere" in the classroom. The findings affirm the important role of the teacher in introducing mathematical practices. Careful task design and enactment, teacher questioning, and noticing and responding to student reasoning were important elements in facilitating conjecturing, justifying and generalising.
- Published
- 2014
4. Commognitive Analysis of Undergraduate Mathematics Students' Responses in Proving Subgroup's Non-Emptiness
- Author
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Mathematics Education Research Group of Australasia and Ioannou, Marios
- Abstract
Proving that a given set is indeed a subgroup, one needs to show that it is non-empty, and closed under operation and inverses. This study focuses on the first condition, analysing students' responses to this task. Results suggest that there are three distinct problematic responses: the total absence of proving this condition, the problematic understanding of subgroup's definition, and the inaccurate application of the relevant metarules. For the purposes of this study there has been used the Commognitive Theoretical Framework.
- Published
- 2016
5. The Space between the Unknown and a Variable
- Author
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Hewitt, Dave
- Abstract
The meaning given to letters is significant for students' ability to be successful with algebraic tasks. Recent studies have noted that even when students have a sense of generalised number, they often have a natural number bias in the values they think a letter can take. This study analyses interviews from 13 students across two schools to explore the meaning they had for letters. The responses supported the idea that some students have a natural number bias and also that the notion of a letter representing a fraction is problematic. In addition, three other factors emerged which affected the meaning given to a letter: what was mentally stressed; the desire to avoid "messy" calculations; and viewing an equation as an example of a wider class of equations. [For the complete proceedings, see ED597799.]
- Published
- 2014
6. Proceedings of the Conference of the International Group for the Psychology of Mathematics Education (30th, Prague, Czech Republic, July 16-21, 2006). Volume 2
- Author
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International Group for the Psychology of Mathematics Education., Novotna, Jarmila, Moraova, Hana, Kratka, Magdalena, and Stehlikova, Nad'a
- Abstract
This document contains the second volume of the proceedings of the 30th Annual Conference of the International Group for the Psychology of Mathematics Education. Conference presentations are centered around the theme "Mathematics at the Centre." This volume features 60 research reports by presenters with last names beginning between Abr and Dri: (1) The Odds of Understanding the Law of Large Numbers: A Design for Grounding Intuitive Probability in Combinatorial Analysis (Dor Abrahamson and Rose M. Cendak); (2) Imaginary-Symbolic Relations, Pedagogic Resources and the Constitution of Mathematics for Teaching in In-Service Mathematics Teacher Education (Jill Adler and Zain Davis); (3) Relationship between Pre-Service Mathematics Teachers' Teaching and Learning Beliefs and Their Practices (Hatice Akkoc and Feral Ogan-Bekiroglu); (4) Teachers' Awareness of Dimensions of Variation: A Mathematics Intervention Project (Thabit Al-Murani); (5) The Student Teacher and the Others: Multimembership on the Process of Introducing Technology in the Classroom (Nelia Amado and Susana Carreira); (6) Improving Student Teachers' Understanding of Fractions (Solange Amorim Amato); (7) Autodidactic Learning of Probabilistic Concepts through Games (Miriam Amit and Irma Jan); (8) Graduate Students' Processes in Generating Examples of Mathematical Objects (Samuele Antonini); (9) Reasoning in an Absurd World: Difficulties with Proof by Contradiction (Samuele Antonini and Maria Alessandra Mariotti); (10) Will Penelope Choose Another Bridegroom? Looking for an Answer through Signs (Ferdinando Arzarello, Luciana Bazzini, Francesca Ferrara, Ornella Robutti, Cristina Sabena, and Bruna Villa); (11) Motivation and Perceptions of Classroom Culture in Mathematics of Students across Grades 5 to 7 (Chryso Athanasiou and George N. Philippou); (12) Deductive Reasoning: Different Conceptions and Approaches (Michal Ayalon and Ruhama Even); (13) The Tendency to Use Intuitive Rules among Students with Different Piagetian Cognitive Levels (Reuven Babai); (14) Coming to Appreciate the Pedagogical Uses of CAS (Lynda Ball and Kaye Stacey); (15) Students' Conceptions of "m" and "c": How to Tune a Linear Function (Caroline Bardini and Kaye Stacey); (16) A Contradiction between Pedagogical Content Knowledge and Teaching Indications (Ibrahim Bayazit and Eddie Gray); (17) Identifying and Supporting Mathematical Conjectures through the Use of Dynamic Software (David Benitez Mojica and Manuel Santos Trigo); (18) Students Constructing Representations for Outcomes of Experiments (Palma Benko and Carolyn A. Maher); (19) Logarithms: Snapshots from Two Tasks (Tanya Berezovski and Rina Zazkis); (20) Trying to Reach the Limit--The Role of Algebra in Mathematical Reasoning (Christer Bergsten); (21) Semiotic Sequence Analysis--Constructing Epistemic Types Empirically (Angelika Bikner-Ahsbahs); (22) Service Teaching: Mathematical Education of Students of Client Departments (Erhan Bingolbali, John Monaghan, and Tom Roper); (23) Students' Thinking about the Tangent Line (Irene Biza, Constantinos Christou, and Theodossios Zachariades); (24) Habermas' Theory of Rationality as a Comprehensive Frame for Conjecturing and Proving in School (Paulo Boero); (25) Extending Students' Understanding of Decimal Numbers via Realistic Mathematical Modeling and Problem Posing (Cinzia Bonotto); (26) Different Media, Different Types of Collective Work in Online Continuing Teacher Education: Would You Pass the Pen, Please? (Marcelo C. Borba and Rubia B. A. Zulatto); (27) Reformulating "Mathematical Modelling" in the Framework of the Anthropological Theory of Didactics (Marianna Bosch, Fco. Javier Garcia, Josep Gascon, and Luisa Ruiz Higueras); (28) Students' Impressions of the Value of Games for the Learning of Mathematics (Leicha A. Bragg); (29) The Transition from Arithmetic to Algebra: To Reason, Explain, Argue, Generalize and Justify (Trygve Breiteig and Barbro Grevholm); (30) Resisting Reform Pedagogy: Teacher and Learner Contributions (Karin Brodie); (31) Manifestations of Affordances of a Technology-Rich Teaching and Learning Environment (TRTLE) (Jill P. Brown); (32) Types of Representations of the Number Line in Textbooks (Alicia Bruno and Noemi Cabrera); (33) Educational Neuroscience: New Horizons for Research in Mathematics Education (Stephen R. Campbell); (34) Variability in a Probability Context: Developing Pre-Service Teachers' Understanding (Daniel L. Canada); (35) Implementing a Reform-Oriented Mathematics Syllabus: A Survey of Secondary Teachers (Michael Cavanagh); (36) Student's Modelling with a Lattice of Conceptions in the Domain of Linear Equations and Inequations (Hamid Chaachoua, Marilena Bittar, and Jean-Francois Nicaud); (37) Using Reading and Coloring to Enhance Incomplete Prover's Performance in Geometry Proof (Ying-Hao Cheng and Fou-Lai Lin); (38) Aspects of Teachers' Pedagogical Content Knowledge for Decimals (Helen Chick, Monica Baker, Thuy Pham, and Hui Cheng); (39) Collaborative Action Research on Implementing Inquiry-Based Instruction in an Eighth Grade Mathematics Class: An Alternative Mode for Mathematics Teacher Professional Development (Erh-Tsung Chin, Yung-Chi Lin, Yann-Tyng Ko, Chi-Tung Chien, and Hsiao-Lin Tuan); (40) Routine and Novel Mathematical Solutions: Central-Cognitive or Peripheral-Affective Participation in Mathematics Learning (Mei-Shiu Chiu); (41) The Role of Self-Generated Problem Posing in Mathematics Exploration (Victor V. Cifarelli and Jinfa Cai); (42) A Longitudinal Study of Children's Mental Computation Strategies (Barbara Clarke, Doug M. Clarke, and Marj Horne); (43) Assessing Fraction Understanding Using Task-Based Interviews (Doug M. Clarke, Michal Sukenik, Anne Roche, and Annie Mitchell); (44) Evaluation of a Teaching Concept for the Development of Problem Solving Competences in Connection with Self-Regulation (Christina Collet and Regina Bruder); (45) Developing Probability Thinking in Primary School: A Case Study on the Constructive Role of Natural Language in Classroom Discussions (Valeria Consogno, Teresa Gazzolo, and Paulo Boero); (46) Collaboration with Teachers to Improve Mathematics Learning: Pedagogy at Three Levels (Tom J. Cooper, Annette R. Baturo, and Edlyn J. Grant); (47) "Aim High--Beat Yourself": Effective Mathematics Teaching in a Remote Indigenous Community (Tom J. Cooper, Annette R. Baturo, Elizabeth Warren, and Edlyn J. Grant); (48) Development of Children's Understanding of Length, Area, and Volume Measurement Principles (Margaret Curry, Michael Mitchelmore, and Lynne Outhred; (49) Mathematics-for-Teaching: The Cases of Multiplication and Division (Brent Davis, Elaine Simmt, and Dennis Sumara); (50) Generative Concept Images (Gary E. Davis and Catherine A. Pearn); (51) Developmental Assessment of Data Handling Performance Age 7-14 (Pauline Davis, Maria Pampaka, Julian Williams, and Lawrence Wo); (52) The Effect of Different Teaching Tools in Overcoming the Impact of the Intuitive Rules (Eleni Deliyianni, Eleni Michael, and Demetra Pitta-Pantazi); (53) Investigating Social and Individual Aspects in Teacher's Approaches to Problem Solving (Fien Depaepe, Erik De Corte, and Lieven Verschaffel); (54) Maths Avoidance and the Choice of University (Pietro Di Martino and Francesca Morselli); (55) Primary Students' Reasoning about Diagrams: The Building Blocks of Matrix Knowledge (Carmel M. Diezmann); (56) Integrating Errors into Developmental Assessment: "Time" for Ages 8-13 (Brian Doig, Julian Williams, Lawrence Wo, and Maria Pampaka); (57) Vygotsky's Everyday Concepts/Scientific Concepts Dialectics in School Context: A Case Study (Nadia Douek); (58) Creating Mathematical Models with Structures (Katherine Doyle); (59) Mechanisms for Consolidating Knowledge Constructs (Tommy Dreyfus, Nurit Hadas, Rina Hershkowitz, and Baruch Schwarz); and (60) Reconciling Factorizations Made with CAS and with Paper-and-Pencil: The Power of Confronting Two Media (Paul Drijvers, Carolyn Kieran, Andre Boileau, Fernando Hitt, Denis Tanguay, Luis Saldanha, and Jose Guzman). (Individual papers contain references.)
- Published
- 2006
7. Proceedings of the 2005 Annual Meeting of the Canadian Mathematics Education Study Group = Actes de la Rencontre Annuelle 2005 du Groupe Canadien d'Etude en Didactique des Mathematiques (29th, Ottawa, Ontario, Canada, May 27-31, 2005)
- Author
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Canadian Mathematics Education Study Group (CMESG) and Liljedahl, Peter
- Abstract
This submission contains the Proceedings of the 2005 Annual Meeting of the Canadian Mathematics Education Study Group (CMESG), held at the University of Ottawa in Ottawa, Ontario. The CMESG is a group of mathematicians and mathematics educators who meet annually to discuss mathematics education issues at all levels of learning. The aims of the Study Group are: to advance education by organizing and coordinating national conferences and seminars to study and improve the theories of the study of mathematics or any other aspects of mathematics education in Canada at all levels; and to undertake research in mathematics education and to disseminate the results of this research. These proceedings include plenary lectures, working group reports, topic session descriptions, new PhD reports, and summaries of ad hoc sessions. Papers include: (1) Learning Mathematics as Developing Identity in the Classroom (Stephen Lerman); (2) Formative Influences (Jean E. Taylor); (3) Mathematics Education, Society, and Peace (Arthur Powell and A. J. Dawson); (4) Learning Mathematics in the Early Years (pre-K-3) (Ann Anderson, Laurent Theis, and Ruth Dawson); (5) Discrete Mathematics (Leo Jonker and David Lidstone); (6) Socio-Cultural Dimensions of Mathematics Learning (Kathy Nolan and Elaine Simmt); (7) Partition of Integers and Their Reciprocals as Hidden K-12 Mathematics Curriculum (Sergei Abramovich and Peter Brouwer); (8) Mathematiques et musique (Chantal Buteau) [Written in French]; (9) Students' Understanding of the Completeness Property of the Set of Real Numbers (Analia Berge); (10) Study of Two Teaching Approaches Focusing on Spatial Sense with Three Different Profiles of High School Students (Patricia Marchand); (11) Communicating Mathematics Online: The Case of Online Help (Dragana Martinovic); (12) Attending in Mathematics: A Dynamic View about Students' Thinking (Immaculate Kizito Namukasa); (13) Silence and Voice in the Mathematics Classroom (David Wagner); (14) What Can We Learn from Learner-Generated Examples: A Case of Linear Algebra? (Marianna Bogomolny); (15) Reconstructing Foundational Mathematical Knowledge: Experiences of Math-Anxious Elementary Teachers in a Graduate Course (Rina Cohen); (16) Roadkill, Skeletons, and Other Mathematical Metaphors (Julie Long and Gladys Sterenberg); (17) Mathematics in Waldorf Education (John Grant McLoughlin); (18) Using Mathematics as a Source When Creating Metaphors or Images for Teaching and Learning (Joyce Mgombelo and Dave Hewitt); (19) Refining the Canadian Survey Questions for the "Census at School" Survey to Provide Richer Mathematical Learning (Joel Yan, Mary Townsend, and Florence Glanfield); and (20) Undergraduate Students' Errors That May Be Related to Confusing a Set with Its Elements (Kalifa Traore, Caroline Lajoie, and Roberta Mura). Appended are: (1) Working Groups at Each Annual Meeting; (2) Plenary Lectures at Each Annual Meeting; and (3) Proceedings of Annual Meetings. Individual papers contain references, figures, and tables. Individual papers contain references, figures, and tables. [Abstract modified to meet ERIC guidelines. For the 2004 proceedings, see ED529563.]
- Published
- 2006
8. Chinese Whispers - Algebra Style: Grammatical, Notational, Mathematical and Activity Tensions
- Author
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Hewitt, Dave
- Abstract
This paper analyses students' written work from an activity based on two well known games in the UK: Chinese Whispers and Consequences. Within this activity students were asked to translate formal algebraic equations into word statements and vice versa. Using the framework of affordances and constraints to offer an account for what the students' wrote, the author identified some tensions between four different aspects of the activity: the grammar of the word statements; notational conventions; mathematical sense-making; and the rules of the activity itself. Through an increased awareness of these tensions the author surmises that such tensions are not special to this activity and may be taking place during daily mathematical activity in classrooms.
- Published
- 2005
9. Just Do It: Flipped Lecture, Determinants and Debate
- Author
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Kensington-Miller, Barbara, Novak, Julia, and Evans, Tanya
- Abstract
This paper describes a case study of two pure mathematicians who flipped their lecture to teach matrix determinants in two large mathematics service courses (one at Stage I and the other at Stage II). The purpose of the study was to transform the passive lecture into an active learning opportunity and to introduce valuable mathematical skills, such as debate, argument and disagreement. The students were told in advance to use the online material to prepare, which had a short handout on matrix determinants posted, as the lesson would be interactive and would rely on them having studied this. At the beginning of the lesson, the two mathematicians worked together to model the skill of professional disagreement, one arguing for the cofactor expansion method and the other for the row reduction method. After voting for their preferred method, the students worked in small groups on examples to defend their choice. Each group elected a spokesperson and a political style debate followed as the students argued the pros and cons of each technique. Although one lecture does not establish whether the flipped lecture model is preferable for student instruction, the paper presents a case study for pursuing this approach and for further research on incorporating this style of teaching in Science, Technology, Engineering and Mathematics subjects.
- Published
- 2016
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10. Selection of Content in High School Mathematics Textbooks: An International Comparison
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Wang, J. and Lu, X.
- Abstract
As a component of the ongoing development of the mathematics curriculum in China, we compare the country's high school mathematics textbooks with those of several other countries. We base our analysis on the assumption that textbooks, as primary printed teaching resources, are key tools for interpreting educational policy. In this paper, we compare what content is selected and how that content is presented in series of high school textbooks from China, France, Germany, Japan, Russia, the United Kingdom, and the United States according to four core domains: algebra, geometry, statistics and probability, and calculus. We then discuss the implications of the analysis for the reform of the high school mathematics curriculum in China, particularly as it applies to the development of textbooks within that process of reform. The comparative results provide us with the opportunity to recognise the distinguishing features of the content and presentation of the Chinese mathematics curriculum and lead to some suggestions for future curriculum development.
- Published
- 2018
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11. Spreadsheets, Pedagogic Strategies and the Evolution of Meaning for Variable
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Wilson, Kirsty, Ainley, Janet, and Bills, Liz
- Abstract
We report on one aspect of a longitudinal study which seeks insight into the ways in which spreadsheet experience and teachers' pedagogic strategies shape pupils' construction of meaning for algebra. Using data from stimulated recall interviews we analyse the evolution of meaning for variable through the mediation of the variable cell and the mediation of naming a column. We discuss metaphors of change and dragging, together with the process of naming. [For complete proceedings, see ED496851.]
- Published
- 2005
12. Perspectives on Adults Learning Mathematics: Research and Practice. Mathematics Education Library.
- Author
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Coben, Diana, O'Donoghue, John, FitzSimons, Gail E., Coben, Diana, O'Donoghue, John, and FitzSimons, Gail E.
- Abstract
This book contains 22 papers that are designed to situate research and practice in adults learning mathematics within the wider field of lifelong learning and lifelong education. The following papers are included: "Introduction" (Diana Coben, Gail E. FitzSimons, John O'Donoghue); "Review of Research on Adults Learning Mathematics" (Diana Coben); "Section I: Perspectives on Research on Adults Learning Mathematics" (Gail E. FitzSimmons, Gail L. Godden); "Mathematics or Common Sense? Researching 'Invisible' Mathematics through Adults' Mathematics Life Histories" (Diana Coben); "Researching Adults' Knowledge through Piagetian Clinical Exploration--The Case of Domestic Work" (Juan Carlos Llorente); "Understanding Their Thinking: The Tension Between the Cognitive and the Affective" (Janet Duffin, Adrian Simpson); "Section II: Adults, Mathematics, Culture, and Society" (John O'Donoghue); "Mathematics: Certainty in an Uncertain World?" (Roseanne Benn); "Ethnomathematics and Political Struggles" (Gelsa Knijnik); "Statistical Literacy: Conceptual and Instructional Issues" (Iddo Gal); "The Roles of Feelings and Logic and Their Interaction in the Solution of Everyday Problems" (Dhamma Colwell); "Section III: Adults, Mathematics and Work" (Gail E. FitzSimons); "Women, Mathematics and Work" (Mary Harris); "Technology, Competences, and Mathematics" (Tine Wedege); "Mathematics and the Vocational Education and Training System" (Gail E. FitzSimons); "Section IV: Perspectives in Teaching Adults Mathematics" (John O'Donoghue); "Algebra for Adult Students: The Student Voices" (Katherine Safford); "Exploration and Modelling in a University Mathematics Course: Perceptions of Adult Students" (Barbara J. Miller-Reilly); "Assessing Numeracy" (John O'Donoghue); "Adult Mathematics and Everyday Life: Building Bridges and Facilitating Learning 'Transfer'" (Jeff Evans); "Teaching 'Not Less Than Maths, but More': An Overview of Recent Developments in Adult Numeracy Teacher Development in England--With a Sidelong Glance at Australia" (Diana Coben, Noyona Chanda); and "Postscript: Some Thoughts on Paulo Freire's Legacy for Adults Learning Mathematics" (Diana Coben). Most papers include substantial bibliographies. (MN)
- Published
- 2000
13. Constructing Meanings and Utilities within Algebraic Tasks
- Author
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Ainley, Janet, Bills, Liz, and Wilson, Kirsty
- Abstract
The Purposeful Algebraic Activity project aims to explore the potential of spreadsheets in the introduction to algebra and algebraic thinking. We discuss two sub-themes within the project: tracing the development of pupils' construction of meaning for variable from arithmetic-based activity, through use of spreadsheets, and into formal algebra, and tracing the ways in which children construct utilities for algebraic activity. Our analysis of pupils' activity suggests that tasks which offer opportunities to construct different utilities may also be associated with the construction of different meanings for variable. [For complete proceedings, see ED489632.]
- Published
- 2004
14. Comparing Competence in Transformational and Generational Algebraic Activities
- Author
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Wilson, Kirsty, Ainley, Janet, and Bills, Liz
- Abstract
The Purposeful Algebraic Activity Project is a longitudinal study of the development of pupils' algebraic activity in the early years of their secondary schooling. Here, we report on our empirical findings from the initial semi-structured interviews. We analyse the responses of three pairs of 12-year-old pupils to a range of algebra questions. In our analysis, we identify broad similarities in the "answers" pupils gave to transformational questions and quite significant differences in the pupils' responses to generational questions. We consider the implications for assessment, and discuss the potential of spreadsheets for developing pupils' appreciation of the need for an algebra-like notation. (Contains 1 footnote, 1 figure, and 2 tables.) [The Purposeful Algebraic Activity Project is funded by the Economic and Social Research Council. For complete proceedings, see ED500860.]
- Published
- 2003
15. Asymmetry in Student Achievement on Multiple-Choice and Constructed-Response Items in Reversible Mathematics Processes
- Author
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Sangwin, Christopher J. and Jones, Ian
- Abstract
In this paper we report the results of an experiment designed to test the hypothesis that when faced with a question involving the inverse direction of a reversible mathematical process, students solve a multiple-choice version by verifying the answers presented to them by the direct method, not by undertaking the actual inverse calculation. Participants responded to an online test containing equivalent multiple-choice and constructed-response items in two reversible algebraic techniques: factor/expand and solve/verify. The findings supported this hypothesis: Overall scores were higher in the multiple-choice condition compared to the constructed-response condition, but this advantage was significantly greater for items concerning the inverse direction of reversible processes compared to those involving direct processes.
- Published
- 2017
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16. To What Extent Are Students Expected to Participate in Specialised Mathematical Discourse? Change over Time in School Mathematics in England
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Morgan, Candia and Tang, Sarah
- Abstract
From a discursive perspective, differences in the language in which mathematics questions are posed change the nature of the mathematics with which students are expected to engage. The project The Evolution of the Discourse of School Mathematics (EDSM) analysed the discourse of mathematics examination papers set in the UK between 1980 and 2011. In this article we address the issue of how students over this period have been expected to engage with the specialised discourse of school mathematics. We explain our analytic methods and present some outcomes of the analysis. We identify changes in engagement with algebraic manipulation, proving, relating mathematics to non-mathematical contexts and making connections between specialised mathematical objects. These changes are discussed in the light of public and policy domain debates about "standards" of examinations.
- Published
- 2016
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17. The Definition of the Scalar Product: An Analysis and Critique of a Classroom Episode
- Author
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Foster, Colin and de Villiers, Michael
- Abstract
In this paper, we present, analyse and critique an episode from a secondary school lesson involving an introduction to the definition of the scalar product. Although the teacher attempted to be explicit about the difference between a definition and a theorem, emphasizing that a definition was just an arbitrary assumption, a student rejected the teacher's definition in favour of his own alternative. With reference to this particular case, we seek to explore some ways in which teachers can introduce mathematical definitions to students so as to support, rather than attempt to circumvent, their mathematical sense making. In this regard, we believe that it is important to develop learning opportunities for students which help them to gain some appreciation of important structural and historical reasons that underpin the definitional choices made.
- Published
- 2016
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18. Independent Study Workbooks for Proofs in Group Theory
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Alcock, Lara, Brown, Gavin, and Dunning, Clare
- Abstract
This paper describes a small-scale research project based on workbooks designed to support independent study of proofs in a first course on abstract algebra. We discuss the lecturers' aims in designing the workbooks, and set these against a background of research on students' learning of group theory and on epistemological beliefs and study habits in higher education. We organise our analysis of student responses around three emerging themes: 1) structured support provided by the workbooks, 2) productive forced study of lecture notes, and 3) engaging with proofs. Discussion of our data in terms of these themes suggests several considerations for the design of tasks for independent study of advanced mathematics.
- Published
- 2015
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19. Examining the Use of Computer Algebra Systems in University-Level Mathematics Teaching
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Lavicza, Zsolt
- Abstract
The use of Computer Algebra Systems (CAS) is becoming increasingly important and widespread in mathematics research and teaching. In this paper, I will report on a questionnaire study enquiring about mathematicians' use of CAS in mathematics teaching in three countries; the United States, the United Kingdom, and Hungary. Based on the responses from 1100 mathematicians, I will give an overview of the current extent of CAS use in universities and offer some examples of mathematicians' classroom use of CAS. I will particularly focus on responses from participants who reported employing CAS in their teaching practice. In addition, I attempt to explain the reasons behind these practices and highlight the importance of further research on university-level mathematics education. Accordingly, I will pinpoint possible research directions in this field that could further assist in the successful integration of CAS both by mathematicians and mathematics departments. (Contains 6 notes.)
- Published
- 2009
20. Using Animation to Support the Teaching of Computer Game Development Techniques
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Taylor, Mark John, Pountney, David C., and Baskett, M.
- Abstract
In this paper, we examine the potential use of animation for supporting the teaching of some of the mathematical concepts that underlie computer games development activities, such as vector and matrix algebra. An experiment was conducted with a group of UK undergraduate computing students to compare the perceived usefulness of animated and static learning materials for teaching such concepts. Undergraduate computer game development courses are currently a growing area of UK higher education. Computer game development can often involve the use of mathematical modelling of two-dimensional and three-dimensional computer game objects and their interactions. Overall, it appeared that animated learning materials appeared to be perceived as being more useful to undergraduate computer games students than traditional learning materials for learning such concepts.
- Published
- 2008
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21. Patterns of Interactions as Affected by Graphing Software: Developing a Theoretical Framework
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Gibbs, Marie Joubert
- Abstract
This paper extends and develops theories of mathematical learning to provide a framework for the analysis of classroom video data of students working at a computer in a task aimed at increasing understanding of multiple representations of quadratic functions. Student interactions are coded using novel software tools in the process of analysis, and for the presentation of results. In combination with the theoretical framing these techniques provide a rigorous approach to the analysis of the data. The analysis traces the development of mathematical learning over the course of one lesson and highlights the importance of feedback in the process. Of particular importance is the role of the computer in providing this feedback. (Contains 7 figures.)
- Published
- 2006
22. Levels of Understanding in Mathematics and Their Application to an Algebra Test.
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Jesson, D. F. St. John
- Abstract
The Concepts in Secondary Mathematics and Science Project isolated four "levels" of understanding in algebra. These were derived from a national study. The research reported here set out to assess how realistic it was to apply the notion of "level" to individual schools. (Author/SSH)
- Published
- 1983
23. With or without U(K): A pre-Brexit network analysis of the EU ETS.
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Borghesi, Simone and Flori, Andrea
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CARBON pricing ,FOREIGN exchange market ,EMISSIONS trading ,LINEAR algebra ,PHYSICAL sciences ,NETWORK hubs ,TRANSACTION systems (Computer systems) - Abstract
The European Emission Trading System (EU ETS) is commonly regarded as the key pillar of the European climate policy and as the main unifying tool to create a unique carbon price all over Europe. The UK has always played a crucial role in the EU ETS, being one of the most active national registry and a crucial hub for the exchange of allowances in the market. Brexit, therefore, could deeply modify the number and directions of such exchanges as well as the centrality of the other countries in this system. To investigate these issues, the present paper exploits network analysis tools to compare the structure of the EU ETS market in its first two phases with and without the UK, investigating a few different scenarios that might emerge from a possible reallocation of the transactions that have involved UK partners. We find that without the UK the EU ETS network would become in general much more homogeneous, though results may change focusing on the type of accounts involved in the transactions. [ABSTRACT FROM AUTHOR]
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- 2019
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24. APRIORI-SD: ADAPTING ASSOCIATION RULE LEARNING TO SUBGROUP DISCOVERY.
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Kavšek, Branko and Lavrač, Nada
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A priori ,ALGORITHMS ,GROUP theory ,LOGIC ,THEORY of knowledge ,REASONING ,ALGEBRA ,TRAFFIC accidents - Abstract
This paper presents a subgroup discovery algorithm APRIORI-SD, developed by adapting association rule learning to subgroup discovery. The paper contributes to subgroup discovery, to a better understanding of the weighted covering algorithm, and the properties of the weighted relative accuracy heuristic by analyzing their performance in the ROC space. An experimental comparison with rule learners CN2, RIPPER, and APRIORI-C on UCI data sets demonstrates that APRIORI-SD produces substantially smaller rulesets, where individual rules have higher coverage and significance. APRIORI-SD is also compared to subgroup discovery algorithms CN2-SD and SubgroupMiner. The comparisons performed on U.K. traffic accident data show that APRIORI-SD is a competitive subgroup discovery algorithm. [ABSTRACT FROM AUTHOR]
- Published
- 2006
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25. EFFECTIVE RATES OF PROTECTION FOR UNITED KINGDOM PRODUCTION: A COMMENT.
- Author
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Kitchin, P. D.
- Subjects
EQUATIONS ,ESTIMATES ,ESTIMATION theory ,ALGEBRA - Abstract
This article comments on an article about effective rates of protection for production in Great Britain, which contains an error concerning the formulation of the estimating equations and which makes the numerical estimates incorrect.
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- 1975
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26. The relative age effect in European elite soccer: A practical guide to Poisson regression modelling.
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Doyle, John R. and Bottomley, Paul A.
- Subjects
POISSON regression ,REGRESSION analysis ,PHYSICAL sciences ,SOCCER ,SOCIAL sciences ,AGE discrimination in employment - Abstract
Many disciplines of scholarship are interested in the Relative Age Effect (RAE), whereby age-banding confers advantages on older members of the cohort over younger ones. Most research does not test this relationship in a manner consistent with theory (which requires a decline in frequency across the cohort year), instead resorting to non-parametric, non-directional approaches. In this article, the authors address this disconnect, provide an overview of the benefits associated with Poisson regression modelling, and two managerially useful measures for quantifying RAE bias, namely the Indices of Discrimination and Wastage. In a tutorial-like exposition, applications and extensions of this approach are illustrated using data on professional soccer players competing in the top two tiers of the “Big Five” European football leagues in the search to identify paragon clubs, leagues, and countries from which others may learn to mitigate this form of age-discrimination in the talent identification process. As with OLS regression, Poisson regression may include more than one independent variable. In this way we test competing explanations of RAE; control for unwanted sources of covariation; model interaction effects (that different clubs and countries may not all be subject to RAE to the same degree); and test for non-monotonic versions of RAE suggested in the literature. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
27. Cognitive Test Scores in UK Biobank: Data Reduction in 480,416 Participants and Longitudinal Stability in 20,346 Participants.
- Author
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Lyall, Donald M., Cullen, Breda, Allerhand, Mike, Smith, Daniel J., Mackay, Daniel, Evans, Jonathan, Anderson, Jana, Fawns-Ritchie, Chloe, McIntosh, Andrew M., Deary, Ian J., and Pell, Jill P.
- Subjects
COGNITIVE testing ,BIOBANKS ,DATA reduction ,PHENOTYPES ,HEALTH outcome assessment - Abstract
UK Biobank includes 502,649 middle- and older-aged adults from the general population who have undergone detailed phenotypic assessment. The majority of participants completed tests of cognitive functioning, and on average four years later a sub-group of N = 20,346 participants repeated most of the assessment. These measures will be used in a range of future studies of health outcomes in this cohort. The format and content of the cognitive tasks were partly novel. The aim of the present study was to validate and characterize the cognitive data: to describe the inter-correlational structure of the cognitive variables at baseline assessment, and the degree of stability in scores across longitudinal assessment. Baseline cognitive data were used to examine the inter-correlational/factor-structure, using principal components analysis (PCA). We also assessed the degree of stability in cognitive scores in the subsample of participants with repeat data. The different tests of cognitive ability showed significant raw inter-correlations in the expected directions. PCA suggested a one-factor solution (eigenvalue = 1.60), which accounted for around 40% of the variance. Scores showed varying levels of stability across time-points (intraclass correlation range = 0.16 to 0.65). UK Biobank cognitive data has the potential to be a significant resource for researchers looking to investigate predictors and modifiers of cognitive abilities and associated health outcomes in the general population. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
28. `The emergency which has arrived': The problematic history of nineteenth-century British algebra...
- Author
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Fisch, Menachem
- Subjects
HISTORY of mathematics ,ALGEBRA ,PHILOSOPHY - Abstract
Discusses the difficulties encountered by British mathematicians during the nineteenth century. Reassertion of native traditions in mathematics; Resolution by downgrading the decade of keen mathematical reflection; `Problem situation' according to William Rowan Hamilton; Algebra as a connected and logical system of propositions; Euclidean representations.
- Published
- 1994
- Full Text
- View/download PDF
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