Equity and Access in Mathematics Education

Mathematics Curriculum Evaluation

Preface and Acknowledgements.- Introduction.- Mathematics and Mathematics Education: Searching for common ground: Michael N. Fried.- Chapter 1. A Dialogue on a Dialogue.- Mathematics and Mathematics Education: Beginning a Dialogue in an Atmosphere of Increasing Estrangement: Michael N. Fried.- Some of my pet-peeves with mathematics education: Ted Eisenberg.- Mathematics at the Center of Distinct Fields: A Response to michael and Ted: Norma Presmeg.- Chapter 2. Mutual Expectations.- Mutual Expectations between Mathematicians and Mathematics Educators : Tommy Dreyfus.-With contributions by:Uri Onn, Joanna Mamona-Downs, Stephen Lerman.- Chapter 3. History of Mathematics, Mathematics Education, and Mathematics.- History in mathematics education. A hermeneutic approach: Hans Niels Jahnke.- Reflections on History of Mathematics: History of Mathematics and Mathematics Education: Luis Radford.- With contributions by:Alain Bernard, Michael N. Fried , Fulvia Furinghetti, Nathalie Sinclair.- Chapter 4. Problem-Solving: A Problem for Both Mathematics and Mathematics Education.- Reflections on Problem-Solving: Problem solving in mathematics and in mathematics education: Boris Koichu.- With contributions by: Gerald A. Goldin, A. Israel Weinzweig, Shlomo Vinner, Roza Leikin.- Chapter 5. Mathematical What Is It and How is It Determined?.-Mathematical Literacy: An Inadequate Metaphor: E. Paul Goldenberg.- Reflections on Mathematical literacy : What's new, why should we care, and what can we do about it? : Anna Sfard.- With contributions by:Abraham Arcavi, Ron Livne, Iddo Gal, Anna Sfard, Hannah Perl.- Chapter 6. Visualization in Mathematics and Mathematics Education: Visualization in Mathematics and Mathematics Education: A Historical Overview : M. A. (Ken) Clements.- Visualization in mathematics and mathematics education: Elena Nardi (University of East Anglia).- With contributions by: Rina Hershkowitz, Raz Kupferman , Norma Presmeg, Michal Yerushalmy.- Chapter 7. Justification and Proof.- Making Sense of Mathematical Reasoning and Proof: David Tall.- Reflections on Justification and Proof: Justification and Proof in Mathematics and Mathematics Education: Keith Weber.- With contributions by: Gila Hanna, Guershon Harel, Ivy Kidron, Annie Selden and John Selden.- Chapter 8. Policy: What Should We Do, and Who Decides?.- Mathematics and mathematics education policy: Mogens Niss.- Reflections on Policy: Mathematics and Mathematics Education Policy-Searching for Common Ground : Nitsa Movshovitz-Hadar.- With contributions by:Jonas Emanuelsson, Davida Fischman, Azriel Levy, Zalman Usiskin.- Chapter 9. Collaboration.- Mathematics and Education: Collaboration in Practice: Hyman Bass.- Deborah Loewenberg Ball.- Reflections on Collaboration between Mathematics and Mathematics Education: Patrick W. Thompson.- With contributions by: Michele Artigue, Gunter Torner, Ehud de Shalit.- Postscript.- We Must Cultivate Our Common Ground: Jeremy Kilpatrick.- Appendix 1. Ted Eisenberg, Teacher, Colleague, and Friend.- Ted as advisor and colleague: Tommy Dreyfus.- Thank you, Ted!: Francis Lowenthal.- Annotated bibliography of Ted Eisenberg's Major Publications: Tommy Dreyfus.- Appendix 2. Reprints of the Dialogues between Presmeg, Eisenberg, and Fried from ZDM 41(1-2).-Index

Introduction I suggest a simple thought experiment. Science fiction books occasionally mention an imaginary device: a replicator. It consists of two boxes; you put an object in a box, close the lid, and instantly get its undistinguishable fully functional copy in the second box. In particular, a replicator can replicate smaller replicators. Now imagine the economy based on replicators. It needs two groups of producers: a very small group of engineers who build and maintain the biggest replicator and a very diverse, but still small, group of artisans, designers, and scientists who produce a single original prototype of each object. This hypothetical economy also needs service sector, mostly waste disposal. Next, try, if you can, imagine a sustainable, stable, equal, and democratic model of education that supports this lopsided economy. But this apocalyptic future is already upon us – in the information sector of economy, where computers act as replicators of information. Mathematics, due to its special role in the information technology, is the most affected part of human culture. The new patterns of division of labour split mathematics for makers from mathematics for users and trigger a crisis of mathematics education. The latter increasingly focuses on mathematics for users and undermines itself because sustainable reproduction of mathematics requires teachers educated as makers. The ultimate replicating machines I borrowed the title of this section from a chapter in my book [1]. I argue there that the essence of mathematics is its precise replicability which imitates the stability of laws of the physical universe, that Mathematics is the ultimate in the technology transfer. [2] A mathematical theorem needs to be proved only once – and then used for centuries. An algorithm needs to be developed only once – and then it can serve, as the Google Ranking Algorithm does, as a kingpin of a global information system. In previous historic epochs, every use of a mathematical result required participation of humans, who had to understand what they were doing and therefore had to be mathematically educated; the criterion of understanding was the ability to reproduce the proof. Nowadays, mathematics is used mostly by computers, not by people, and used in an instantly replicable way. This creates a completely different socio-economic environment for mathematics. Division of labour As I argue in my paper [3], the history of human civilisation is the history of division of labour. By the start of the 21st century, the ever deepening division of labour has reached a unique point when 99% of people have not even the vaguest idea about the workings of 99% of technology in their immediate surrounding. This transformation is deeper than the Great Industrial Revolution of 18th and 19th centuries, and its social consequences have a chance to be more dramatic. Mathematics and mathematics education are the proverbial canaries in the mine, they are more sensitive to this technological change. It costs to make ("replicate") a smartphone, it costs to write an app for smartphone, but the per unit cost of mathematics encoded and hardwired within the phone converges to zero. There are more mobile phones in the world now than toothbrushes. But the mathematics built into mobile communication systems is beyond the understanding of most universities' graduates. This creates a paradox: mathematics is used in everyday life millions of times more intensively than 50 or even 10 years ago – but remains invisible. Meanwhile, mathematical results and concepts involved in practical applications are much deeper and more abstract and difficult than ever before. The cutting edge of mathematics research moves further away from the stagnating mathematics education. From the point of view of an aspiring PhD student, mathematics looks like New York in the Capek Brothers' book A Long Cat Tale [4] (and notice that Karel Capek was the man who coined the word "robot"): And New York – well, houses there are so tall that they can't even finish building them. Before the bricklayers and tilers climb up them on their ladders, it is noon, so they eat their lunches and start climbing down again to be in their beds by bedtime. And so it goes on day after day. Investment cycles and research-and-development cycles in many modern industries are just two years long. On the other hand, proper mathematics education still takes at least 15 years from the age of 5 to the age of 20 – or even 20 years if postgraduate studies are needed. As I argue in [3], mathematics education is being undermined by this tension between the ever deepening specialisation of labour and ever increasing length of specialised training required for jobs at the increasingly sharp cutting edge of technology. If banks and insurance companies were interested in having numerate customers, we would witness the golden age of school mathematics – fully funded, enjoying cross-party political support, promoted and popularised by the best advertising companies in all forms of mass and social media. But they are not; banks and insurance companies need numerate workforce – and even more so they need innumerate customers. 25 years ago in the West, the benchmark of arithmetic competence at the consumer level was the ability to balance a chequebook. Nowadays, bank customers can instantly get full information about the state of their accounts from an app on a mobile phone – together with timely and tailored to individual circumstances advice on the range of available financial products. As Anna Sfard [5] put it, It is enough to take a critical look at our own lives to realize that we do not, in fact, need much mathematics in our everyday lives. In short, the present model of "mathematics education for all" is unsustainable and, not surprisingly, first cracks have started to appear. On the other hand, the reproduction cycle of mathematics primary school – high school – university – teacher training – a teacher's return to school is 20 years long, and it is not clear at all whether the current model of education could be smoothly and peacefully replaced by the new one, aimed at in-depth mathematics education of a much smaller stratum of people. Assessments of this situation from the opposite ends of the political spectrum are instructive: Failure in achieving a meaningful mathematics education is not a malfunction which could be solved through better research and a proper crew, but is endemic in capitalist schooling. (Alexandre Pais [6]) While there is an upside limit to the average intellectual capabilities of population, there is no upper limit to the complexity of technology. … With ... an apparently inbred upper limit to human IQ, are we destined to have an ever smaller share of our workforce staff our ever more sophisticated high-tech equipment and software? (Alan Greenspan [7]) Mathematics education When previously meaningful social activities (and social institutions supporting them) loose their economic purpose, they either collapse or transform themselves into a complex of rituals, "cargo cult," in the words of Richard Feynman. In the "cargo cult" environment, everything goes. This is why we see the explosive growths in the number of various approaches and methods tried at school – because there are no objective bottom-line criteria to distinguish between them. Here, I want to touch on a popular myth: that the same computer technology that kills demand for mathematics will save mathematics education. First of all, we have to distinguish between education and training. As a famous saying goes, "For those of you with daughters, would you rather have them take sex education or sex training?" This witticism makes it clear what is expected from education as opposed to training: the former should give a student ability to make informed and responsible decisions. This is the old class divide that tears many education systems apart: education is for people who are expected to make decisions and give orders; training is for ones who take orders. However it is increasingly accepted that modern mathematics education is not even training of workforce for future employment (this model of education is so 20th century), it is filtering of workforce by means of mathematical tests – even if no mathematics is needed at the actual workplace. Computers could be very efficient tools for training students to pass tests – I do not dispute that. However, although the skill of passing a mathematics test remains personally important, it becomes increasingly redundant at the scale of the economy as a whole. An exam at the end of the course should test students' ability to perform certain tasks – but in case of school and college mathematics, these tasks now are much better performed by computers – see a detailed discussion of that in [3]. Then what is the aim of training? The ability to imitate robots? Are students' skills assessed are of any economic (or "real life") value if computers can pass the tests in an instant and with better scores than humans? Makers and Users So far I was looking at the emerging new social environment of mathematics. Now a few words on consequences for mathematics itself. The new patterns of division of labour split mathematics for makers from mathematics for users. How t describe the two? The replicability of mathematics mirrors the stability of laws of the physical universe, which is captured by the apocryphal formula: Mathematics is the language of contracts with Nature which Nature accepts as binding. It is dangerous to replace, in this formulation, "Nature" by "Computer" – but it appears that this increasingly frequently happens in practice. Therefore, in my understanding, Mathematics for Makers is mathematics that cannot be entrusted to computers, mathematics for those whose duty is writing contracts with Nature, in the process inventing new mathematics and new ways to apply mathematics. In terms of the "universal replicator" simile from the Introduction, these are people who produce the originals for subsequent replication. The mainstream mathematics education increasingly focuses on mathematics for users. But sustainable reproduction of mathematics requires teachers educated as makers – on that point, I refer the reader to my paper [8]. Conclusions The expansionist model of mathematics education is dying because the technological changes in the wider economy lead to the shift of demand for mathematically competent workers: smaller numbers are needed, but much better educated. Compression cracks are more destructive and less predictable than expansion gaps – for the obvious reason: where should the excessive mass go? Potential social consequences bring to mind the apocryphal curse May you live in interesting times; It looks as if interesting times are already upon us. But I do not takes sides in the increasingly politicised debate. In my view, most policies in mathematics education can be divided in two categories: rearranging chairs on the deck of Titanic (the preferred option of the political Right); helping disadvantaged passengers to get better chairs on the deck of Titanic (the preferred option of the political Left). My role is different, I am with my fellow teachers in the famous band that continues to play regardless. Not the first violin, of course; I am in the back row, with a tuba: "Boop, boop, boop, boop." I am a mathematician; I will play to the end. Disclaimer The author writes in his personal capacity; his views do not necessarily represent the position of his employer or any other person, corporation, organisation or institution. References and Notes Borovik, A. V. Mathematics under the Microscope: Notes on Cognitive Aspects of Mathematical Practice, American Mathematical Society, Providence, USA, 2010; pp. 217 – 245. Stewart, I. Does God Play Dice? The Mathematics of Chaos. Penguin, London, UK, 1990. Borovik, A. V. Calling a spade a spade: Mathematics in the new pattern of division of labour, to appear. A pdf file: http://goo.gl/TT6ncO Capek, K.; Capek, J. A Long Cat Tale, Albatros, Prague, The Czech Republic, 1996; p. 44. Sfard, A. Why Mathematics? What Mathematics? In The Best Writings on Mathematics, Pitici M., Ed.; Princeton University Press, Princeton, USA, 2013; pp. 130-142. Pais, A. An ideology critique of the use-value of mathematics, Stud. Math., 2013, vol. 84, pp. 15 – 34. Greenspan, A. The Map and the Territory: Risk, Human Nature and the Future of Forecasting, Allen Lane, USA, 2013. Borovik, A. V. Didactic transformation in mathematics teaching, http://www.academia.edu/189739/Didactic_transformation_in_mathematics_teaching

Preface.- Section I - Theoretical Perspectives. 1. Mathematical Literacy and Globalisation.- 2. Epistemological Issues in the Internationalization and Globalization of Mathematics Education .- 3. All around the World: Science Education, Constructivism, and Globalization.- 4. Geophilosophy, Rhizomes and Mosquitoes: Becoming Nomadic in Global Science Education Research. 5. Science Education and Contemporary Times: Finding Our Way through the Challenges.- 6. Social (In)Justice and International Collaborations in Mathematics Education.- 7. Globalisation, Ethics and Mathematics Education.- 8. The Politics and Practices of Equity, (E)quality and Globalisation in Science Education: Experiences from Both Sides of the Indian Ocean.- Section II - Issues in Globalisation and Internationalisation 9. Context or Culture: Can TIMSS and PISA Teach Us about What Determines Educational Achievement in Science?.- 10. Quixote's Science: Public Heresy/Private Apostasy.- 11. The Potentialities of (ethno) Mathematics Education: An Interview with Ubiratan D' Ambrosio.- 12. Ethnomathematics in the Global Episteme: Quo Vadis?.- 13. POP: A Study of the Ethnomathematics of Globalisation Using the Sacred.- 14. Internationalisation as an Orientation for Learning and Teaching in Mathematics.- 15. Contributions from Cross-National Comparative Studies to the Internationalization of Mathematics Education: Studies of Chinese and U.S. Classrooms.- 16. International Professional Development as a Form of Globalisation.- 17. Doing Surveys in Different Cultures: Difficulties and Differences - A Case from China and Australia.- 18. The Benefits and Challenges for Social Justice in International Exchanges in Mathematics and Science Education.- 19. Globalisation, Technology, and the Adult Learner of Mathematics.- Section III - Perspectives from Different Countries. 20. Balancing Globalisation and Local Identity in the Reform of Education in Romania.- 21. Voices from theSouth: Dialogical Relationships and Collaboration in Mathematics Education.- 22. Globalization and its Effects in Mathematics and Science Education in Mexico: Implications and Challenges for Diverse Populations.- 23. In between the Global and the Local: The Politics of Mathematics Education Reform in a Globalized Society.- 24. Singapore and Brunei Darussalam: Internationalisation and Globalisation through Practices and a Bilateral Mathematics Study.- 25. Lesson Study (JYUGYO KENKYU), from Japan to South Africa: A Science and Mathematics Intervention Program for Secondary School Teachers.- 26. The Post-Mao Junior Secondary School Chemistry Curriculum in the People's Republic of China: A Case Study in the Internationalization of Science Education.- 27. Globalisation/Localisation in Mathematics Education: Perception, Realism and Outcomes of an Australian Presence in Asia.- Biographical Notes.

Following up on efforts to improve the quality and quantity of international publications of lecturers and students of UPI (Indonesia University of Education) Postgraduate Schools, Master Program in Chemistry, Physics, Biology, Science and Mathematics Education and Doctor Program in Science and Mathematics Education collaboratively conducted International Conference on Mathematics and Science Education 2019 on Saturday 29 June 2019 at the Grand Mercure Setiabudi Bandung.The theme of the conference was “Mathematics and Science Education Research for Sustainable Development”, with coverage of Mathematics Education, Physics Education and STEM (Science, Technology, Engineering and Mathematics).The main objective of this conference is to improve the academic atmosphere within the UPI environment, particularly at the UPI Postgraduate School and strengthen the lecturer and student publications through the International Conference on Mathematics and Science Education (ICMScE )2019. Specific objectives to be achieved regarding this conference are (1). Increase the number of scientific publications of lecturers and Postgraduate students in conference proceedings, and (2). Increase the number of citation index lecturers and students of the UPI Graduate School in the Master Program in Chemistry, Physics, Biology, Science and Mathematics Education and Doctor Program in Science and Mathematics Education.List of Committees and Conference Photographs are available in this PDF.

From a traditional perspective, induction and deduction have been discussed as key ways to generate new knowledge. On the other hand, Charles Sanders Peirce introduced the notion of abduction, distinct from deduction and induction, to avoid the pitfalls of empiricism and rationalism. Abduction is the process of forming an explanatory hypothesis on an observed surprising result (C.P. 5.171). Peirce emphasized that abductive reasoning is the only way of creating new ideas, and both epistemologists and educational researchers have attempted to tackle this notorious problem, the so-called learning paradox, from a Peircean perspective on knowledge generation (Prawat, 1999). With mathematics educators’ recent interests in semiotic approaches, there has been growing attention to the importance of investigating abductive reasoning in mathematics education research. Studies have attempted to clarify the forms and uses of abductive reasoning in students’ mathematical inquiries in order to identify how students generate new mathematical ideas. Researchers also consider that investigating students’ abductive reasoning may help to interpret and understand what occurs in mathematics classrooms. Thus, exploring abductive reasoning in mathematics education may provide a more helpful cornerstone in understanding how mathematics teaching and learning progresses. This special issue of EURASIA Journal of Mathematics, Science and Technology Education aims to share current and future issues on abductive reasoning in mathematics education. In inviting the contributions for this special issue, we intend to offer the reader, original elements of reflection from a wide range of issues on abductive reasoning in mathematics education. David Reid gives an overview of the discussion of abductive reasoning in mathematics education researches based on a meta-analysis of the state-of-the-art literature. He presents the origins of the concept of abductive reasoning and identifies the most significant approaches in mathematics education literature that refers to abductive reasoning. He then proposes a framework in which the different approaches taken in the research literature can be placed and compared. Michael Hoffmann presents very fundamental issues regarding abductive reasoning. Hoffmann tackles two crucial questions related to knowledge creation from a Peircean perspective: Can diagrammatic reasoning indeed be conceived as a foundation of abductive creativity? What is the relationship between abduction and diagrammatic reasoning? To answer these questions, he clarifies the Peircean concept of diagram and diagrammatic reasoning and analyzes the significance of a consistent system of representation for diagrammatic reasoning. He then examines how diagrammatic reasoning and creation of abduction are related. Three further contributions address how abduction is related to various contexts of mathematics learning. Bettina Pedemonte presents the role of abduction in the proving process of students solving a geometrical problem. She focuses on two types of rules in problem solving: strategic rules and definitory rules. She then compares two types of abductions that are related to these two rules, and analyzes the relationship between these two types of abductions and the deductive proof. Ferdinand Rivera examines elementary students’ pattern generalization. He focuses on identifying multiple abductive actions in their pattern generalization, and shows how multiple abductions can be coordinated and how this coordination is related to pattern generalization. He also analyzes the relationship between elementary children’s structural incipient generalizations and the natural emergence of their understanding of functions, especially the central role of abduction in such an understanding. JinHyeong Park and Kyeong-Hwa Lee investigate the abductive nature of mathematical modeling and the characteristics of mathematical inquiries triggered by mathematical modeling. They identify four characteristics of mathematical inquiries triggered by mathematical modeling based on an analysis of didactical and historical cases: abductive, recursive, analogical, and context-dependent. Michael Meyer presents various task-design options that can be used to support students in discovering mathematical properties that refer to abductive reasoning. These task-design options are identified using intense scrutiny of the processes of discovering and verifying mathematical properties from a Peircean perspective on learning and knowledge creation. He also presents concrete examples of task design and empirical findings on the implementation of tasks. From theoretical issues to more practical issues, these contributions in this special issue present state-of-the-art issues in abductive reasoning in the mathematics education research community. We hope that these offer an informative insight into the lively research on abductive reasoning in mathematics education.

The Web-based mathematics education framework (WME) aims to create a Web for mathematics education. WME empowers mathematics teachers, learning content developers, as well as dynamic mathematics computation and education service providers, to deliver an unprecedented mathematics learning environment to students and educators. Main WME components include the Mathematics Education Markup Language (MeML), the MeML processor (Woodpecker, browser plug-in), and on-Web Mathematics Education Services. MeML provides effective and expressive markup elements to represent and structure mathematics education pages that may also contain XHTML and MathML elements. Woodpecker enables regular Web browsers to process MeML pages and to interact with a wide variety of mathematics computation and education services deployed on the Web.

Previous articleNext article No AccessAttitudes of Prospective Teachers toward ArithmeticWilbur H. DuttonWilbur H. Dutton Search for more articles by this author PDFPDF PLUS Add to favoritesDownload CitationTrack CitationsPermissionsReprints Share onFacebookTwitterLinkedInRedditEmail SectionsMoreDetailsFiguresReferencesCited by The Elementary School Journal Volume 52, Number 2Oct., 1951 Article DOIhttps://doi.org/10.1086/459310 Views: 8Total views on this site Citations: 32Citations are reported from Crossref Copyright 1951 The University of ChicagoPDF download Crossref reports the following articles citing this article:Rosetta Zan, Pietro Di Martino Students’ Attitude in Mathematics Education, (Feb 2020): 813–817.https://doi.org/10.1007/978-3-030-15789-0_146Kyoung Whan Choe, Jalisha B. Jenifer, Christopher S. Rozek, Marc G. Berman, Sian L. 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This article describes a laboratory supplementary program that was integrated into a traditional mining engineering mathematics unit. The practical classes consisted of computer investigations designed to help develop mathematical concepts. The program described here was mainly web based and did not directly rely on a computer algebra system for its implementation. An evaluation of the program is included. References S. Cunningham. The visualization environment for mathematics education. In Visualization in Teaching and Learning Mathematics , ed. W. Zimmermann and S. Cunningham, 67--76. USA, Mathematical Association of America, 1991. A. Franco, P. Franco, A. Garcia, F. Garcia, F. J.Gonzalez, S. Hoya, G. Rodriguez, and A. de la Villa. Learning calculus of several variables with new technologies. The International Journal of Computer Algebra in Mathematics Education , 7 (4), 295--309, 2000. B. E. Garner and L. E. Garner. Retention of concepts and skills in traditional and reformed applied calculus. Mathematics Education Research Journal , 13 (3), 165--184, 2001 S. Habre. Visualization enhanced by technology in the learning of multivariate calculus. The International Journal of Computer Algebra in Mathematics Education , 8 (2), 115--130, 2001. B. H. Hallet. Visualization and calculus reform. In Visualization in Teaching and learning Mathematics , ed. W. Zimmermann and S. Cunningham, 121--126, 1991. USA, Mathematical Association of America F. Marton and R. Saljo. Approaches to learning. In eds. F. Marton, D. Hounsell and N. Entwistle, The Experience of Learning , 36--55, 1984. Scottish Academic Press, Edinburgh. R. Moreno and R. Mayer. Verbal redundancy in multimedia learning; When reading helps listening. Journal of Educational Psychology , 94 (1), 153--163, 2002. L. D. Murphy. Computer algebra systems in calculus reform, MSTE, University of Illinois at Urbana-Champaign, 1999. http://mste.illinois.edu/users/Murphy/Papers/CalcReformPaper.html M. Pemberton. Integrating web-based maple with a first year calculus and linear algebra course. Proceedings of the 2nd International Conference on the Teaching of Mathematics , Hersonissos, Greece, July 2002. http://www.math.uoc.gr/ ictm2/Proceedings/pap316.pdf R. Pierce and K. Stacey. Observations on students' responses to learning in a cas environment. Mathematics Education Research Journal , 13 (1), 28--46, 2001. M. D. Roblyer. Integrating Educational Technology Into Teaching (4th Ed.), 2006. Pearson, New Jersey, USA. J. Stewart. Calculus (5th Ed.), 2003. Brooks/Cole, Belmont, USA. E. J. Tonkes, B. I. Loch and A. W. Stace. An innovative learning model for computation in first year mathematics. International Journal of Mathematical Education in Science and Technology , 36 (7), 751--759, 2005. L. M. Villarreal. A step in the positive direction: Integrating a computer laboratory component into developmental algebra courses. Mathematics and Computer Education , 37 (1), 72--78, 2003. S. Vinner. The pseudo-conceptual and the pseudo-analytical thought processes in mathematics learning. Educational Studies in Mathematics , 34 (2), 97--129, 1997. P. Vlachos and A.K. Kehagias. A computer algebra system and a new approach for teaching business calculus. The International Journal of Computer Algebra in Mathematics Education . 7 (2), 87--104, 2000.

Part 1: Background: Doctoral production in mathematics education in the United States: 1960-2005 by R. Reys, R. Glasgow, D. Teuscher, and N. Nevels Doctoral programs in mathematics education in the United States: 2007 status report by R. Reys, R. Glasgow, D. Teuscher, and N. Nevels Report of a 2007 survey of U. S. doctoral students in mathematics education by D. Teuscher, N. Nevels, and C. Ulrich Part 2: Developing stewards of the discipline: core elements: Creating a broader vision of doctoral education: Lessons from the Carnegie Initiative on the Doctorate by C. M. Golde What core knowledge do doctoral students in mathematics education need to know? by J. Ferrini-Mundy Breakout sessions: The mathematical education of doctorates in mathematics education by D. Chazan and W. J. Lewis Curriculum as core knowledge by R. M. Zbiek and C. R. Hirsch Making policy issues visible in the doctoral preparation of mathematics educators by E. Silver and E. Walker Preparing teachers in mathematics education doctoral programs: Tensions and strategies by P. S. Wilson and M. Franke Doctoral programs in mathematics education: Diversity and equity by E. V. Taylor and R. Kitchen Using technology in teaching and learning mathematics: What should doctoral students in mathematics education know? by M. K. Heid and H. S. Lee Part 3: Developing stewards of the discipline: delivery systems: Program delivery issues, opportunities, and challenges by D. S. Mewborn Breakout sessions: Doctoral preparation of researchers by J. A. Middleton and B. Dougherty Key components of mathematics education doctoral programs in the United States: Current practices and suggestions for improvement by W. S. Bush and E. Galindo On-line delivery graduate courses in mathematics education by M. Burke and V. M. Long Mathematics education doctoral programs: Approaches to part-time students by G. Kersaint and G. A. Goldin Induction of doctoral graduates in mathematics education into the profession by B. J. Reys, G. M. Lloyd, K. Marrongelle, and M. S. Winsor Part 4: Doctoral programs in mathematics education: Some international perspectives: Doctoral programs in mathematics education: An international perspective by J. Kilpatrick Doctoral studies in mathematics education: Unique features of Brazilian programs by B. S. D'ambrosio Nordic doctoral programs in didactics of mathematics by B. Grevholm Japanese doctoral programs in mathematics education: Academic or professional by M. Koyama Post-graduate study program in mathematics education at the University of Granada (Spain) by L. Rico, A. Fernandez-Cano, E. Castro, and M. Torralbo Part 5: Accreditation: Accreditation of doctoral programs: A lack of consensus by G. Lappan, J. Newton, and D. Teuscher Part 6 Reflections from within: Preparing the next generation of mathematics educators: An assistant professor's experience by A. Tyminski Mathematics content for elementary mathematics education graduate students: Overcoming the prerequisites hurdle by D. Kirshner and T. Ricks Intellectual communities: Promoting collaboration within and across doctoral programs in mathematics education by D. Teuscher, A. M. Marshall, J. Newton, and C. Ulrich Part 7: Closing commentary: Reflecting on the conference and looking toward the future by J. Hiebert, D. Lambdin, and S. Williams Appendices.

Children's Mathematics Learning: The Struggle to Link Form and Understanding

Background: Mathematics education in the United States: Origins of the field and the development of early graduate programs by E. F. Donoghue Doctoral programs in mathematics education in the U.S.: A status report by R. E. Reys, B. Glasgow, G. A. Ragan, and K. W. Simms Reflections on the match between jobs and doctoral programs in mathematics education by F. Fennell, D. Briars, T. Crites, S. Gay, and H. Tunis International perspectives on doctoral studies in mathematics education by A. J. Bishop Core components: Doctoral programs in mathematics education: Features, options, and challenges by J. T. Fey The research preparation of doctoral students in mathematics education by F. K. Lester, Jr. and T. P. Carpenter The mathematical education of mathematics educators in doctoral programs in mathematics education by J. A. Dossey and G. Lappan Preparation in mathematics education: Is there a basic core for everyone? by N. C. Presmeg and S. Wagner The teaching preparation of mathematics educators in doctoral programs in mathematics education by D. V. Lambdin and J. W. Wilson Discussions on different forms of doctoral dissertations by L. V. Stiff Beyond course experiences: The role of non-course experiences in mathematics education doctoral programs by G. Blume Related issues: Organizing a new doctoral program in mathematics education by C. Thornton, R. H. Hunting, J. M. Shaughnessy, J. T. Sowder, and K. C. Wolff Reorganizing and revamping doctoral programs--Challenges and results by D. B. Aichele, J. Boaler, C. A. Maher, D. Rock, and M. Spikell Recruiting and funding doctoral students by K. C. Wolff The use of distance-learning technology in mathematics education doctoral programs by C. E. Lamb Emerging possibilities for collaborating doctoral programs by R. Lesh, J. A. Crider, and E. Gummer Reactions and reflections: Appropriate preparation of doctoral students: Dilemmas from a small program perspective by J. M. Bay Perspectives from a newcomer on doctoral programs in mathematics education by A. Flores Why I became a doctoral student in mathematics education in the United States by T. Lingefjard Policy--A missing but important element in preparing doctoral students by V. M. Long My doctoral program in mathematics education-A graduate student's perspective by G. A. Ragan Ideas for action: Improving U. S. doctoral programs in mathematics education by J. Hiebert, J. Kilpatrick, and M. M. Lindquist

Contents: D. Williams, A Framework for Thinking About Research in Mathematics and Science Education. R. Zevenbergen, Ethnography in the Mathematics and Science Classrooms. J.S. Schaller, K. Tobin, Establishing Credibility and Authenticity in Ethnographic Studies. J. Truran, K. Truran, Using Clinical Interviews in Qualitative Research. R. Bleicher, Classroom Interactions: Using Interactional Sociolinguistics to Make Sense of Recorded Classroom Talk. P. Taylor, V. Dawson, Critical Reflections on a Problematic Student-Supervisor Relationship. G. Leder, H. Forgasz, J. Landvogt, Higher Degree Supervision: Why It Worked. L. White, Teacher, Researcher, Collaborator, Student: Multiple Roles and Multiple Dilemmas. F.E. Crawley, Guiding Collaborative Action Research in Science Education Contexts. J.A. Malone, On Supervising and Being Supervised at a Distance. W-M. Roth, M.K. McGinn, Legitimate Peripheral Participation in the Training of Researchers in Science and Mathematics Education. A. Begg, B. Bell, V. Compton, E.A. McKinley, Supervision in a Graduate Centre. T. Cooper, A.R. Baturo, L. Harris, Scholarly Writing in Mathematics and Science Education Higher-Degree Courses. J. Hourcade, H. Anderson, Writing for Publication. D. Squires, The Impact of New Developments in Information Technology on Postgraduation Research and Supervision. P. Rillero, B. Gallegos, Databases: A Gateway to Research in Mathematics and Science Education Research.

(1982). Mathematics Education and Educational Research in the USA and USSR: Two Comparisons Compared. Journal of Curriculum Studies: Vol. 14, No. 2, pp. 109-126.

Mathematics Education in Different Cultural Traditions: A Comparative Study of East Asia and the West.- Mathematics Education in Different Cultural Traditions: A Comparative Study of East Asia and the West.- Research on Affect in Mathematics Education: A Reconceptualisation.- Mathematics Education in East Asia and the West: Does Culture Matter?.- Context of Mathematical Education.- A Traditional Aspect of Mathematics Education in Japan.- From Wasan to Yozan.- Perceptions of Mathematics and Mathematics Education in the Course of History - A Review of Western Perspectives.- Historical Topics as Indicators for the Existence of Fundamentals in Educational Mathematics.- From Entering the Way to Exiting the Way: In Search of a Bridge to Span Basic Skills and Process Abilities.- Practice Makes Perfect: A Key Belief in China.- The Origins of Pupils' Awareness of Teachers' Mathematics Pedagogical Values: Confucianism and Buddhism - Driven.- Curriculum.- Some Comparative Studies between French and Vietnamese Curricula.- An Overview of the Mathematics Curricula in the West and East.- Classification and Framing of Mathematical Knowledge in Hong Kong, Mainland China, Singapore, and the United States.- Comparative Study of Arithmetic Problems in Singaporean and American Mathematics Textbooks.- A Comparative Study of the Mathematics Textbooks of China, England, Japan, Korea, and the United States.- A Comparison of Mathematics Performance Between East and West: What PISA and TIMSS Can Tell Us.- Case Studies on Mathematics Assessment Practices in Australian and Chinese Primary Schools.- Philippine Perspective on the ICMI Comparative Study.- Teaching and Learning.- The TIMSS 1995 and 1999 Video Studies.- Proposal for a Framework to Analyse Mathematics Education in Eastern and Western Traditions.- Cultural Diversity and the Learner's Perspective: Attending to Voice and Context.- Mathematics Education in China: From a Cultural Perspective.- Mathematics Education and Information and Communication Technologies.- Cultural Awareness Arising from Internet Communication between Japanese and Australian Classrooms.- The International Distance Learning Activities of HSARUC.- Distance Learning between Japanese and German Classrooms.- Values and Beliefs.- Comparing Primary and Secondary Mathematics Teachers' Beliefs about Mathematics, Mathematics Learning and Mathematics Teaching in Hong Kong and Australia.- The Impact of Cultural Differences on Middle School Mathematics Teachers' Beliefs in the U.S. and China.- U.S. and Chinese Teachers' Cultural Values of Representations in Mathematics Education.- A Comparison of Mathematical Values Conveyed in Mathematics Textbooks in China and Australia.- Values and Classroom Interaction: Students' Struggle for Sense Making.- Trip for the Body, Expedition for the Soul: An Exploratory Survey of Two East Asian Teachers of Mathematics in Australia.- Conceptualising Pedagogical Values and Identities in Teacher Development.- Outlook and Conclusions.- Elements of a Semiotic Analysis of the Secondary Level Classroom in Japan.- Other Conventions in Mathematics and Mathematics Education.- What Comes After This Comparative Study - More Competitions or More Collaborations?.