See also other pages for Earlier Drafts, Selected Lectures, Curriculum Vitae.
My Mathematics Education PhD (1986) may be downloaded from here.
These files are in PDF (portable document format), readable on a PC, Mac, or Unix Machine with Adobe Acrobat Reader. Downloading a file depends on the settings of your browser. If you have an Acrobat plugin, the file opens in the browser window, otherwise it downloads to your disc, either to the desktop or to a folder specified in your preferences. To download to a folder of your choice, use right click (PC) or control click (Mac). 
The downloads below include all my papers available in electronic form. Published papers are listed for any year with suffixes a, b, c, ... Current drafts are listed as z, y, x, ... or have no letter. 

See also: 

Items marked x are drafts still under development and may later be 

2020a x  David Tall (2020). Making Sense of Mathematical Thinking over the Long Term: The Framework of Three Worlds of Mathematics and New Developments. Draft.To appear in Tall, D. & Witzke, I. (Eds.): MINTUS: Beiträge zur mathematischen, naturwissenschaftlichen und technischen Bildung. Wiesbaden: Springer. 
2019b x  David Tall (2019). From Biological Brain to Mathematical Mind: The Longterm Evolution of Mathematical Thinking. Draft for Interdisciplinary Approaches to Mathematical Thinking. ed. Marcel Danesi. 
2019a  David Tall (2019). Complementing supportive and problematic aspects of mathematics to resolve transgressions in longterm sense making. Fourth Interdisciplinary Scientific Conference on Mathematical Transgressions, Krakow, March 2019. 
2018a  David Tall (2018). Longterm sense making in arithmetic and algebra. Published online at http://maths4maryams.org/mathed/. Full address: http://maths4maryams.org/mathed/wpcontent/uploads/2018/07/DavidWikiLetter.pdf. Version in Farsi here. 
2017x  David Tall (2017). Making sense of elementary arithmetic and algebra for longterm success, Draft chapter for Japanese Elementary School Teachers. 
2017a  David Tall, Nic Tall, Simon Tall (2017). Problem posing in the long term conceptual development of a gifted child. In Martin Stein (Ed.): A Life’s Time for Mathematics Education and Problem Solving. On the Occasion of Andràs Ambrus’ 75th Birthday, pp. 445457. WTMVerlag, Münster. 
2016  See also selected lectures on Three worlds of mathematics and the brain and Three worlds and the calculus that discuss current developments of the theoretical framework as yet unpublished (April 2016). 
2016b  David Tall (2016). Long term effect of sensemaking and anxiety in algebra. In Stewart, S. (ed): And the rest is just algebra, (Springer, New York). 
2016a  Stephen Hegedus & David Tall (2016). Foundations for the future: the potential of multimodal technologies for learning mathematics. In English, L. D. & Kirshner D. (Eds): Handbook of International Research in Mathematics Education, 3rd Edition, pp. 543–562. Routledge. 
2015c  Nellie C. Verhoef, Fer Coenders, Jules M. Pieters, Daan van Smaalen, and David O. Tall (2015). Professional development through lesson study: teaching the derivative using GeoGebra. Professional Development in Education, 41 (1), 109126. DOI:10.1080/19415257.2014.886285 
2015b  David Tall (2015). Setting Lesson Study within a LongTerm Framework for Learning. In Maitree Inprasitha, Masami Isoda, Patsy Wang Iverson, Yeap Ban Har (Eds): Lesson Study: Challenges in Mathematics Education. 27–50. World Scientific, Singapore. 
2015a  Ivy Kidron & David Tall (2015). The roles of embodiment and symbolism in the potential and actual infinity of the limit process. Educational Studies in Mathematics 88: 183. doi:10.1007/s106490149567x. 
2014d  David Tall (2014) Will Chinese Teaching Methods help us improve UK performance in Mathematics? published in Prospect Magazine, May 2014, pp. 6465. 
2014c  Nellie Verhoef, David Tall, Fer Coenders, Daan Smaalen (2014). The complexities of a lesson study in a Dutch situation: Mathematics Teacher Learning. International Journal of Science and Mathematics Education 12: 859–881. doi: 10.1080/19415257.2014.88628. 
2014b  David Tall, Rosana Nogueira de Lima & Lulu Healy (2014). Evolving a Threeworld Framework for Solving Algebraic Equations in the Light of What a Student Has Met Before. Journal of Mathematical Behavior, 34, 113. 
2014a  David Tall & Mikhail Katz (2014). A Cognitive Analysis of Cauchy’s Conceptions of Function, Continuity, Limit, and Infinitesimal, with implications for teaching the calculus. Educational Studies in Mathematics, 86 (1), 97124. 
2013  How Humans Learn to Think Mathematically, Chapter I. [from the book, CUP (NY)]. The full book is now available for purchase from standard booksellers e.g. from here. 
2013d  Integrating History, Technology and Education in Mathematics. Plenary Presentation: História e Tecnologia no Ensino da Matemática, July 15, 2013, Universidade Federal de São Carlos, Brazil. 
2013c  David Tall (2013). Making Sense of Mathematical Reasoning and Proof. In Michael N. Fried & Tommy Dreyfus (Eds.), Mathematics & Mathematics Education: Searching for Common Ground. New York: Springer, Advances in Mathematics Education series. DOI: 10.1007/9789400774735_13. 
Mercedes McGowen & David Tall (2013). Flexible Thinking and Metbefores: Impact on learning mathematics, with particular reference to the minus sign. Journal of Mathematical Behavior 32, 527–537.  
David Tall (2013). The Evolution of Technology and the Mathematics of Change and Variation. In Jeremy Roschelle & Stephen Hegedus (eds), The Simcalc Vision and Contributions: Democratizing Access to Important Mathematics, (pp. 449–561). Springer.  
2012c  Kin Eng Chin & David Tall (2012). Making Sense of Mathematics through Perception, Operation & Reason: The case of Trigonometric Functions. Proceedings of PME36, Taipei, 2012 (one page summary as appearing in Proceedings). Full version. 
2012b  Mikhail Katz & David Tall (2012). The tension between intuitive infinitesimals and formal analysis. In Bharath Sriraman, (Ed.), Crossroads in the History of Mathematics and Mathematics Education, (The Montana Mathematics Enthusiast Monographs in Mathematics Education 12), pp. 71–90. 
2012a  David Tall, Oleksiy Yevdokimov, Boris Koichu, Walter Whiteley, Margo Kondratieva, YingHao Cheng (2011). The Cognitive Development of Proof. In In Hanna, G. and De Villiers, M. (Eds). ICMI 19: Proof and Proving in Mathematics Education, pp.13–49. 
David Tall (2012). Making Sense of Mathematical Reasoning and Proof. Plenary at Mathematics & Mathematics Education: Searching for Common Ground: A Symposium in Honor of Ted Eisenberg, April 29May 3, 2012, BenGurion University of the Negev, Beer Sheva, Israel.Overheads. (Original extended version.)  
2015b  David Tall (2015). Setting Lesson Study within a LongTerm Framework for Learning. In Maitree Inprasitha, Masami Isoda, Patsy Wang Iverson, Yeap Ban Har (Eds): Lesson Study: Challenges in Mathematics Education. 27–50. World Scientific, Singapore. 
2012z  David Tall (2012) A Sensible Approach to the Calculus. To appear in Handbook on Calculus and its Teaching, ed. François Pluvinage & Armando Cuevas. 
2011c  David Tall (2011). Looking for the bigger picture. For the Learning of Mathematics, 31 (2) 1718. 
2011b  Verhoef, N. C., & Tall, D. O. (2011). Lesson Study: The Effect on Teachers’ Professional Development. Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education, Vol. 4, pp. 297304. Turkey: University of Ankara. 
2011a  David Tall (2011). Crystalline concepts in longterm mathematical invention and discovery. For the Learning of Mathematics, 31 (1) March 2011, 38. 
2010d  David Tall (2010). Perceptions, Operations and Proof in Undergraduate Mathematics, CULMS Newsletter (Community for Undergraduale Learning in the Mathematical Sciences), University of Auckland, New Zealand, 2, November 2010, 2128. 
2010c  David Tall (2010). Mathematical and emotional foundations for lesson study in mathematics. Plenary presented at the APEC Lesson Study Conference, Chiang Mai, Thailand, November 2010. 
2010b  Mercedes McGowen & David Tall (2010). Metaphor or Metbefore? The effects of previous experience on the practice and theory of learning mathematics. Journal of Mathematical Behavior 29, 169–179. 
2010a  David Tall (2010). A Sensible Approach to the Calculus. (Plenary at The National and International Meeting on the Teaching of Calculus, 23–25th September 2010, Puebla, Mexico.) 
2010x  Nellie Verhoef & David Tall (2010). The effectiveness of lesson study on mathematical knowledge for teaching (draft). 
2010x  Mikhail Katz & David Tall (2010). Who invented Dirac’s Delta Function? (Draft.) 
2010x  David Tall & Rosana Noguiera de Lima (2010). An example of the fragility of a procedural approach to solving equations. (Draft.) 
2009d  David Tall (2009). The development of mathematical thinking: problemsolving and proof. In Celebration of the academic life and inspiration of John Mason. 
2009c  David Tall (2009). Dynamic mathematics and the blending of knowledge structures in the calculus. ZDM – The International Journal on Mathematics Education, 41 (4) 481–492. 
2009b  David Tall & Juan Pablo MejiaRamos (2009). The LongTerm Cognitive Development of Different Types of Reasoning and Proof, Hanna, G., Jahnke, H. N., & Pulte, H. (Eds.), Explanation and proof in mathematics: Philosophical and educational perspectives. New York: Springer. 
2009a 
David Tall (2009). Cognitive and social development of proof through embodiment, symbolism & formalism. Paper for the ICMI Conference on Proof, May 2009, Taipei. 
2008f 
David Tall, David Smith & Cynthia Piez (2008). Technology and Calculus. In M. Kathleen Heid and Glendon M Blume (Eds), Research on Technology and the Teaching and Learning of Mathematics, Volume I: Research Syntheses, 207258. 
2008e 
David Tall (2008). The Transition to Formal Thinking in Mathematics. Mathematics Education Research Journal, 2008, 20 (2), 524 [A summary of the framework of three worlds of mathematics as applied to the shift to formal thinking]. 
2008d  David Tall (2008). Using Japanese Lesson Study in Teaching Mathematics. The Scottish Mathematical Council Journal 38, 4550. 
2008c  A lifetime’s journey from definition and deduction to ambiguity and insight. Mediterranean Journal for Research in Mathematics Education, 7 (2), 183–196. 
2008x 
The Historical & Individual Development of Mathematical Thinking: Ideas that are setbefore and metbefore. Plenary at HTEM conference, Rio, May 5th 2008. Overheads. 
2008b 
James J Kaput (1942–2005): Imagineer and Futurologist of Mathematics Education. Educational Studies in Mathematics, 68 (2) 185–193. 
2008a  Rosana Nogueira de Lima and David Tall (2008). Procedural embodiment and magic in linear equations. Educational Studies in Mathematics. 67 (1) 318. 
2007x  Masami Isoda & David Tall (2007). Longterm development of Mathematical Thinking and Lesson Study. Prepared as a chapter for a forthcoming book on Lesson Study. 
2007f  Eddie Gray & David Tall (2007). Abstraction as a natural process of mental compression. Mathematics Education Research Journal 19 ( 2), 23–40. [Reflections and developments of our work together.] 
2007e 
Setting Lesson Study within a longterm framework of learning. Presented at APEC Conference on Lesson Study in Thailand, August 2007. 
2007d  Embodiment, Symbolism, Argumentation and Proof, Keynote presented at the Conference on Reading, Writing and Argumentation at National Changhua Normal University, Taiwan, May 2007. 
2007c  Teachers as Mentors to encourage both power and simplicity in active mathematical learning. Plenary at The Third Annual Conference for Middle East Teachers of Science, Mathematics and Computing, 17–19 March 2007, Abu Dhabi. [Overheads] 
2007b  David Tall (2007). Embodiment, Symbolism and Formalism in Undergraduate Mathematics Education, Plenary at 10th Conference of the Special Interest Group of the Mathematical Association of America on Research in Undergraduate Mathematics Education, Feb 22–27, 2007, San Diego, California, USA. [Overheads] 
2007a 
David Tall (2006). Developing a Theory of Mathematical Growth. To appear in International Reviews on Mathematical Education (ZDM). 
2006g  David Tall (2006).Encouraging Mathematical Thinking that has both power and simplicity. Plenary presented at the APECTsukuba International Conference, December 3–7, 2006, at the JICA Institute for International Cooperation (Ichigaya, Tokyo). (prepublication draft) [A presentation the overall framework of three worlds of mathematics to an audience interested in elementary school teaching, concentrating on the relationship between embodiment and symbolism.] 
2006f 
David Tall & Juan Pablo MejiaRamos (2006). The LongTerm Cognitive Development of Different Types of Reasoning and Proof, presented at the Conference on Explanation and Proof in Mathematics: Philosophical and Educational Perspectives, Essen, Germany. (prepublication draft). 
2006e  David Tall (2006). A Theory of Mathematical Growth through Embodiment, Symbolism and Proof. Annales de Didactique et de Sciences Cognitives, IREM de Strasbourg. 11 195–215. [The published version of 2005e.] 
2006d  Rosana Nogueira de Lima and David Tall (2006). The concept of equation: what have students met before? Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education, Prague, Czech Republic, vol. 4, 233–241. 
2006c  David Tall & Lillie Crowley (2006). Two Students: Why does one succeed and the other fail? Retirement as Process and Concept: A Festschrift for Eddie Gray and David Tall, Prague, 57–65, ISBN 8072902555 
2006b  David Tall (2006). A lifetime’s journey from definition and deduction to ambiguity and insight. Retirement as Process and Concept: A Festschrift for Eddie Gray and David Tall, Prague. 275288, ISBN 8072902555. [A celebration of those who have taught me almost everything I know.] 
2006a  Rosana Nogueira de Lima and David Tall (2006). What does equation mean? A brainstorm of the concept. Third International Conference on the Teaching of Mathematics, Istanbul. 
2005h  Pegg, J., & Tall, D. (2005). The fundamental cycle of concept construction underlying various theoretical frameworks. International Reviews on Mathematical Education (Zentralblatt für Didaktik der Mathematik), vol. 37, no.6, pp. 468475. 
2005g  Tall, D. (2005). The transition from embodied thought experiment and symbolic manipulation to formal proof. In M. Bulmer, H. MacGillivray & C. Varsavsky (Eds.), Proceedings of Kingfisher Delta’05, Fifth Southern Hemisphere Symposium on Undergraduate Mathematics and Statistics Teaching and Learning (pp. 2335). Fraser Island, Australia. 
2005f  Amir Asghari & David Tall (2005). Students’ experience of equivalence relations: a phenomenographic approach. Proceedings of PME29, Melbourne, Australia. 
2005e 
David Tall (2005) A Theory of Mathematical Growth through Embodiment, Symbolism and Proof. Plenary Lecture for the International Colloquium on Mathematical Learning from Early Childhood to Adulthood, Belgium, 57 July, 2005. 
2005d  Akkoç, H. and Tall, D. (2005), ‘A mismatch between curriculum design and student learning: the case of the function concept’, in D. Hewitt and A. Noyes (Eds), Proceedings of the sixth British Congress of Mathematics Education held at the University of Warwick, pp. 18. Available from http://www.bsrlm.org.uk/IPs/ip251/. 
2005c  MejiaRamos, J. P. and Tall, D. (2005), ‘Personal and public aspects of formal proof: a theory and a singlecase study’, in D. Hewitt and A. Noyes (Eds), Proceedings of the sixth British Congress of Mathematics Education held at the University of Warwick, pp. 97104. Available from http://www.bsrlm.org.uk/IPs/ip251/. 
2005b 
Poynter, A. and Tall, D. (2005). ‘What do mathematics and physics teachers think that students will find difficult? A challenge to accepted practices of teaching’, in D. Hewitt and A. Noyes (Eds), Proceedings of the sixth British Congress of Mathematics Education held at the University of Warwick, pp. 128135. Available from http://www.bsrlm.org.uk/IPs/ip251/. 
2005a 
Anna Poynter & David Tall (2005). Relating theories to practice in the teaching of mathematics. Fourth Congress of the European Society for Research in Mathematics Education. 
2004d 
David Tall (2004). Thinking Through Three Worlds of Mathematics. Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, Bergen, Norway, 4, 281–288. 
2004c  David Tall (2004). Reflections on research and teaching of equations and inequalities. Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, Bergen, Norway, 1, 158–161. 
2004b  David Tall, Juan Pablo Mejia Ramos (2004). Reflecting on PostCalculusReform. Plenary for Topic Group 12: Calculus, International Congress of Mathematics Education, Copenhagen, Denmark. 
2004a  David Tall (2004). Introducing Three Worlds of Mathematics. For the Learning of Mathematics, 23 (3). 29–33. 
2003e 
Giraldo V., Tall, D. O., Carvalho, L. M., (2003). Using Theoretical Computational Conflicts to Enrich the Concept Image of Derivative. Research in Mathematics Education, vol. 5, pp. 63–78. 
2003d  Giraldo, V., Carvalho, L. M. & Tall, D. O. (2003). Descriptions and Definitions in the Teaching of Elementary Calculus. In N.A. Pateman, B.J. Dougherty and J. Zilliox (eds.) Proceedings of the 27th Conference of the International Group for the Psychology of Mathematics Education, vol. 2, pp.445452, Honolulu, USA. 
2003c  Anna Watson, Panayotis Spirou, David Tall, (2003). The Relationship between Physical Embodiment and Mathematical Symbolism: The Concept of Vector. The Mediterranean Journal of Mathematics Education. 1 2, 7397. 
2003b  Giraldo, V., Carvalho, L. M. & Tall, D. O. (2003). Conflitos TeóricoComputacionais e a Imagem Conceitual de Derivada. In L.M. Carvalho and L.C. Guimarães, História e Tecnologia no Ensino da Matemática, vol. 1, pp. 153164, Rio de Janeiro, Brasil. 
2003a  David Tall, (2003). Using Technology to Support an Embodied Approach to Learning Concepts in Mathematics. In L.M. Carvalho and L.C. Guimarães História e Tecnologia no Ensino da Matemática, vol. 1, pp. 128, Rio de Janeiro, Brasil. 
2002m 
Giraldo, V.; Carvalho, L. M. & Tall, D. O, (2002). TheoreticalComputational Conflicts and the Concept Image of Derivative. Proceedings of the BSRLM Conference. Nottingham, England, 22 (3), 37–42. 
2002l  EhrTsung Chin, David Tall (2002), Proof as a Formal Procept in Advanced Mathematical Thinking, International Conference on Mathematics: Understanding Proving and Proving to Understand, 212221. National Taiwan Normal University, Taipei, Taiwan. 
2002k  David Tall, (2002). Differing Modes of Proof and Belief in Mathematics, International Conference on Mathematics: Understanding Proving and Proving to Understand, 91–107. National Taiwan Normal University, Taipei, Taiwan. 
2002j  Anna Watson (and David Tall), (2002). Embodied action, effect, and symbol in mathematical growth. In Anne D. Cockburn & Elena Nardi (Eds), Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education, (Norwich, UK), 4, 369–376. 
2002i  David Tall and ErhTsung Chin, (2002). University students embodiment of quantifier. In Anne D. Cockburn & Elena Nardi (Eds), Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education, (Norwich, UK), 4, 273–280. 
2002h  Shakar Rasslan & David Tall, (2002). Definitions and images for the definite integral concept. In Anne D. Cockburn & Elena Nardi (Eds), Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education, (Norwich, UK), 4, 89–96. 
2002g  John Pegg (& David Tall), (2002). Fundamental Cycles of Cognitive Growth. In Anne D. Cockburn & Elena Nardi (Eds), Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education, (Norwich, UK), 4, 41–48. 
2002f  Hatice Akkoc & David Tall, (2002). The simplicity, complexity and complication of the function concept. In Anne D. Cockburn & Elena Nardi (Eds), Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education, (Norwich, UK), 2, 25–32. 
2002e  Eddie Gray & David Tall, (2002). Abstraction as a natural process of mental compression. In Anne D. Cockburn & Elena Nardi (Eds), Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education, (Norwich, UK), 1, 115–120. 
2002d  Gary Davis & David Tall, (2000). What is a Scheme? In David Tall & Michael Thomas (eds), Intelligence, Learning and Understanding in Mathematics – A Tribute to Richard Skemp, pp. .... Flaxton, Australia: Post Pressed. 
2002c  David Tall, (2002). Continuities and Discontinuities in LongTerm Learning Schemas (reflecting on how relational understanding may be instrumental in creating learning problems). In David Tall & Michael Thomas (eds), Intelligence, Learning and Understanding in Mathematics – A Tribute to Richard Skemp, pp. 151–177. Flaxton, Australia: Post Pressed. 
2002b  David Tall & Michael Thomas, (2002). A Tribute to Richard Skemp. In David Tall & Michael Thomas (eds), Intelligence, Learning and Understanding in Mathematics – A Tribute to Richard Skemp, pp. i–iii. Flaxton, Australia: Post Pressed. 
2002a  Marcia Pinto and David Tall (2002). Building formal mathematics on visual imagery: a case study and a theory. For the Learning of Mathematics, 22(1) 2–10. 
2001p 
Conceptual and Formal Infinities, Educational Studies in Mathematics, 48 (2&3), 199–238. 
2001o  David Tall and Dina Tirosh, (2001). Infinity — The neverending struggle, Educational Studies in Mathematics, 48 (2&3), 129–136. 
2001n  Thomas, M. O. J. & Tall, D. O., (2001). The longterm cognitive development of symbolic algebra, International Congress of Mathematical Instruction (ICMI) Working Group Proceedings  The Future of the Teaching and Learning of Algebra, Melbourne, 2, 590597. 
2001m  Chae, S. D. and Tall, D. O. (2001). Students’ concept images for period doublings as embodied objects in chaos theory. Proceedings of the International Conference on Computers in Education, Vol. 3, 14701475, Seoul: Korea. 
2001l  David O. Tall, (2001). A Child Thinking about Infinity. Journal of Mathematical Behavior, 20, 7–19. 
2001k 
What Mathematics is Needed by Teachers of Young Children?, Proceedings of SEMT 01, Prague, Czech State. 
2001j 
Marcia Maria Fusaro Pinto and David Tall ,(2001). Following student’s development in a traditional university classroom, Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education, 4, 57–64. Utrecht, The Netherlands. 
2001i 
Eddie Gray & David Tall, (2001). Relationships between embodied objects and symbolic procepts: an explanatory theory of success and failure in mathematics, Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education, 3, 65–72. Utrecht, The Netherlands. 
2001h 
Lillie Crowley & David Tall, (2001). Attainment and Potential: Procedures, Cognitive KitBags and Cognitive Units, Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education, 1, 300. Utrecht, The Netherlands. Full 8 page version. 
2001g 
(Abe) EhrTsung Chin & David Tall, (2001). Developing Formal Mathematical Concepts over Time, Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education, 2, 241–248.Utrecht, The Netherlands. 
2001f 
Soo Duck Chae & David Tall, (2001). Students’ Concept Images for Period Doublings as Embodied Objects in Chaos Theory, Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education, 1, 294. Utrecht, The Netherlands. Full 8 page version. 
2001e 
David Tall, (2001). Reflections on Early Algebra, Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education, 1, 149–152. Utrecht, The Netherlands. 
2001d 
Barnard, A. D. & Tall, D. O. (2001) A Comparative Study of Cognitive Units in Mathematical Thinking, Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education, 2,89–96. Utrecht, The Netherlands. 
Chae, S. D., & Tall, D. O. (2001). Aspects of the construction of conceptual knowledge in the case of computer aided exploration of period doubling. In C. Morgan, T. Rowlands (Eds), Research in Mathematics Education, Volume 3. BSRLM Publications, Graduate School of Education, University of Bristol, England. 

David Tall (2001). Cognitive Development in Advanced Mathematics Using Technology, Mathematics Education Research Journal. 12 (3), 196–218. 

David Tall, Eddie Gray, Maselan Bin Ali, Lillie Crowley, Phil DeMarois, Mercedes McGowen, Demetra Pitta, Marcia Pinto, Michael Thomas, and Yudariah Yusof (2001). Symbols and the Bifurcation between Procedural and Conceptual Thinking, Canadian Journal of Science, Mathematics and Technology Education 1, 81–104. 

2000h 
Biological Brain, Mathematical Mind & Computational Computers (how the computer can support mathematical thinking and learning). In WeiChi Yang, SungChi Chu, JenChung Chuan (Eds), Proceedings of the Fifth Asian Technology Conference in Mathematics, Chiang Mai, Thailand (pp. 3–20). ATCM Inc, Blackwood VA. ISBN 9746573624. 
Technology and Versatile Thinking In Mathematical Development. In Michael O. J. Thomas (Ed), Proceedings of TIME 2000, (pp. 33–50). Auckland, New Zealand 

(with Silvia Di Giacomo). Cosa vediamo nei disegni geometrici? (il caso della funzione blancmange), Progetto Alice 1 (2), 321–336). [English version: What do we "see" in geometric pictures? (the case of the blancmange function)]. 

(with Mercedes McGowen and Phil DeMarois), Using the Function Machine as a Cognitive Root, Proceedings of PMENA, 1, 255–261. 

(with Mercedes McGowen and Phil DeMarois), The Function Machine as a Cognitive Root for building a rich concept image of the Function Concept, Proceedings of PMENA, 1, 247–254. 

(with EhrTsung Chin), Making, Having and Compressing Formal Mathematical Concepts. In T. Nakahara and M. Koyama (eds) Proceedings of the 24th Conference of the International Group for the Psychology of Mathematics Education (vol. 2, pp. 177–184). Hiroshima, Japan. 

(with Eddie Gray, Demetra Pitta), Objects, Actions And Images: A Perspective On Early Number Development. Journal of Mathematical Behavior,18, 4, 1– 13. 

(with Michael Thomas, Garry Davis, Eddie Gray, Adrian Simpson), What is the object of the encapsulation of a process?, Journal of Mathematical Behavior, 18 2, 1–19. 

1999u 
Technology and Cognitive Growth in Mathematics: A discussion paper for the Conference on Mathematics and New Technologies, Thessaloniki, Greece, June 1820, 1999. (An unpublished collection of discussion points taken from other articles.) 
(with Eddie Gray, Demetra Pitta, Marcia Pinto), Knowledge Construction and diverging thinking in elementary and advanced mathematics, Educational Studies in Mathematics. 38 (13), 111133. 

The Cognitive Development of Proof: Is Mathematical Proof For All or For Some? In Z. Usiskin (Ed.), Developments in School Mathematics Education Around the World, vol, 4, 117–136. Reston, Virginia: NCTM. 

(with Soo Duck Chae), Aspects of the construction of conceptual knowledge: the case of computer aided exploration of period doubling, Proceedings of BSRLM, 6 pages (to appear). 

(with Yudariah Binte Mohd Yusof), Changing Attitudes to University Mathematics through Problemsolving, Educational Studies in Mathematics (in press). 

(with Márcia Maria Fusaro Pinto), Student constructions of formal theory: giving and extracting meaning. In O. Zaslavsky (Ed.), Proceedings of the 23rd Conference of PME, Haifa, Israel, 4, 65– 73. 

(with Mercedes McGowen), Concept Maps & Schematic Diagrams as Devices for Documenting the Growth of Mathematical Knowledge. In O. Zaslavsky (Ed.), Proceedings of the 23rd Conference of PME, Haifa, Israel, 3, 281– 288. 

(with Phil DeMarois), Function: Organizing Principle or Cognitive Root? In O. Zaslavsky (Ed.), Proceedings of the 23^{rd} Conference of PME, Haifa, Israel, 2, 257– 264. 

(with Lillie Crowley), The Roles of Cognitive Units, Connections and Procedures in achieving Goals in College Algebra. In O. Zaslavsky (Ed.), Proceedings of the 23^{rd} Conference of PME, Haifa, Israel, 2, 225– 232. 

Reflections on APOS theory in Elementary and Advanced Mathematical Thinking. In O. Zaslavsky (Ed.), Proceedings of the 23^{rd} Conference of PME, Haifa, Israel, 1, 111– 118. 

Efraim Fischbein, 1920–1998, Founder President of PME  A Tribute. In O. Zaslavsky (Ed.), Proceedings of the 23^{rd} Conference of PME, Haifa, Israel, 1, 3–5. 

The Chasm between Thought Experiment and Mathematical Proof. In G. Kadunz, G. Ossimitz. W. Peschek, E. Schneider, B. Winkelmann (Eds.), Mathematische Bildung und neue Technologien, Teubner, Stuttgart, 319– 343. 

1998~ 
Original version of plenary presentation ‘Symbols and the Bifurcation between Procedural and Conceptual Thinking’ given at the International Conference on Teaching Mathematics at Pythagorion, Samos, Greece in July 1998. Subsequently published in a revised version as 2000c (above). 
Information Technology and Mathematics Education: Enthusiasms, Possibilities & Realities. In C. Alsina, J. M. Alvarez, M. Niss, A. Perez, L. Rico, A. Sfard (Eds), Proceedings of the 8th International Congress on Mathematical Education, Seville: SAEM Thales, 65–82. 

(with Adrian Simpson), Computers and the Link between Intuition and Formalism. In Proceedings of the Tenth Annual International Conference on Technology in Collegiate Mathematics. AddisonWesley Longman. pp. 417– 421. 

(with Liz Bills), Operable Definitions in Advanced Mathematics: The case of the Least Upper Bound, Proceedings of PME 22, Stellenbosch, South Africa, 2, 104– 111. 

1997g 
From School to University: the effects of learning styles in the transition from elementary to advanced mathematical thinking. In Thomas, M. O. J. (Ed.) Proceedings of The Seventh Annual Australasian BridgingNetwork Mathematics Conference, University of Auckland, 9– 26. 
Metaphorical objects in Advanced Mathematical Thinking, International Journal for Computers in Mathematics Learning, 1, 61–65. 

(with Eddie Gray & Demetra Pitta), The Nature of the Object as an Integral Component of Numerical Processes, Proceedings of PME 21, Finland, 1, 115– 130. 

(with Tony Barnard), Cognitive Units, Connections and Mathematical Proof, Proceedings of PME 21, Finland, 2, 41– 48. 

(with Garry Davis & Michael Thomas), What is the object of the encapsulation of a process?, In F. Biddulph & K. Carr (Eds.) People in Mathematics Education, MERGA 20, Aotearoa. 2,132– 139, MERGA Inc. 

1997b n/a 
Informatietechnologie en Wiskunde Onderwijs, Niewe Wiskrant, 16, 4, (June 1997), 4– 11. 
Functions and Calculus. In A. J. Bishop et al (Eds.), International Handbook of Mathematics Education, 289–325, Dordrecht: Kluwer. 

1997~ 
Making Research in Mathematics Education Relevant to Research Mathematicians, Paper presented to MER Session of AMS/MAA Conference, San Diego, January 1997. 
1997z  Davis, G.; di Giacomo, S.; Gray, E. M.; Hegedus, S.; McGowen, M.; Pinto, M. M. F.; Pitta, D.; Simpson, A. P.; Tall, D. O. (1997): The Object of the Encapsulation of a Distilled Spirit. Proceedings of Malt I, 22100. 
1996i 
Understanding the Processes of Advanced Mathematical Thinking, L’Enseignement des Mathématiques, 42, 395– 415. [original version] 
(with Marcia Pinto), Student Teachers’ Conceptions of the Rational Numbers, Proceedings of PME 20, Valencia, 4, 139– 146. 

(with Yudariah b. Muhammad Yusof), Conceptual and Procedural Approaches to Problem Solving, Proceedings of PME 20, Valencia, 4, 3– 10. 

(with Phil Demarois), Facets and Layers of the Function Concept, Proceedings of PME 20, Valencia, 2, 297– 304. 

(with Maselan bin Ali), Procedural and Conceptual Aspects of Standard Algorithms, Proceedings of PME 20, Valencia, 2, 19– 26. 

(with Robin Foster), Can all children climb the same curriculum ladder?, Mathematics in School, 25 3,8– 12. 

Can all children climb the same curriculum ladder?, The Mathematical Ability of School Leavers, Gresham College, London, 23– 32. 

A Versatile Theory of Visualisation and Symbolisation in Mathematics, Plenary Presentation, Proceedings of the 46th Conference of CIEAEM, Toulouse, France (July, 1994), 1, 15– 27. 

Advanced Mathematical Thinking and the Computer. Proceedings of the 20th University Mathematics Teaching Conference, Shell Centre, Nottingham, 8 pp. 18. 

1995f 
Cognitive Development, Representations & Proof, Justifying and Proving in School Mathematics, Institute of Education, London, 27 38. 
Mathematical Misconceptions and Music of the Spheres (Inaugural Lecture). In E. M. Gray (Ed.), Thinking about Mathematics and Music of the Spheres, Mathematics Education Research Centre, Warwick, 42– 52 

The Psychology of Symbols and Symbol Manipulators, Proceedings of the Seventh Annual International Conference on Technology in College Mathematics Teaching, AddisonWesley, 453– 457. [longer original version as presented] 

(with Yudariah Binte Mahommad Yusof), Professors’ perceptions of students’ mathematical thinking: Do they get what they prefer or what they expect? In L. Meira & D. Carraher, (Eds.), Proceedings of PME 19, Recife, Brazil, II, 170– 177. 

Mathematical Growth in Elementary and Advanced Mathematical Thinking, plenary address. In L. Meira & D. Carraher, (Eds.), Proceedings of PME 19, Recife, Brazil, I, 61– 75. 

Visual Organizers for Formal Mathematics. In R. Sutherland & J. Mason (Eds.), Exploiting Mental Imagery with Computers in Mathematics Education. SpringerVerlag: Berlin. 52– 70. 

1994g n/a 
Understanding the Processes of Advanced Mathematical Thinking, Abstracts of Invited Talks, International Congress of Mathematicians, Zurich, August 1994, 182– 183. [Full lecture] 
Cognitive difficulties in learning analysis. In Report on the Teaching of Analysis (ed. Barnard A.), for the TaLUM committee. 

Calculus and Analysis. In Dina Tirosh (Ed.), Mathematical Topics of Instruction, in T. Husen & T. N. Postlethwaite, (Eds.) The International Encyclopaedia of Education, Second Edition, Pergamon Press. pp. 36803681, 3686.  
The Psychology of Advanced Mathematical Thinking: Biological Brain and Mathematical Mind, Abstracts of the Working group on Advanced Mathematical Thinking (A.M.T.), PME 18, Lisbon, 33– 39. 

(with Lillie Crowley & Michael Thomas), Algebra, Symbols, and Translation of Meaning, Proceedings of PME18, Lisbon, II, 240– 247. ISBN 972 8161 00 X. 

(with John Monaghan & Shyashiow Sun), Construction of the Limit Concept with a Computer Algebra System, Proceedings of PME 18, Lisbon, III, 279– 286. ISBN 972 8161 00 X. 

(with Yudariah Bt Mohammed Yusof), Changing Attitudes to Mathematics through Problem Solving, Proceedings of PME 18, Lisbon, IV, 401– 408. ISBN 972 8161 00 X. 

(with Eddie Gray). Duality, Ambiguity and Flexibility: A Proceptual View of Simple Arithmetic, The Journal for Research in Mathematics Education, 26 (2), 115– 141. 

1993 n/a 
Foreword, Computer Algebra Systems in the Classroom, (eds. J. Monaghan, T.Etchells), Centre for Sudies in Science & Mathematics Education, University of Leeds, 4– 5. 
1993l n/a 
Technology in the teaching of calculus, Proceedings of Working Group 3 on Students’ Difficulties in Calculus, ICME7, Québec, Canada, 75– 77. ISBN 2 920916 23 
Students’ Difficulties in Calculus, Plenary Address, Proceedings of Working Group 3 on Students’ Difficulties in Calculus, ICME7, Québec, Canada, 13– 28. ISBN 2 920916 23 8. 

Computer environments for the learning of mathematics, Didactics of Mathematics as a Scientific Discipline – The State of the Art, ed R. Biehler, R. Scholtz, R. W. Sträßer, B. Winkelmann. Dordrecht: Kluwer, 189– 199. 

Interrelationships between mind and computer: processes, images, symbols, Advanced Technologies in the Teaching of Mathematics and Science (ed. David L. Ferguson), New York: SpringerVerlag, 385– 413. 

Real Mathematics, Rational Computers and Complex People, Proceedings of the Fifth Annual International Conference on Technology in College Mathematics Teaching, 243– 258. 

Success & Failure in Arithmetic and Algebra, New Directions in Algebra Education, Queensland University of Technology , Brisbane, 232– 245. 

The Transition from Arithmetic to Algebra: Number Patterns or Proceptual Programming?, New Directions in Algebra Education, Queensland University of Technology, Brisbane, 213– 231. 

(with Lan Li), Constructing Different Concept Images of Sequences and Limits by Programming, Proceedings of PME 17, Japan, 2, 41– 48.End of papers 1992/93 

(with Mohamad Rashidi Razali), Diagnosing Students’ Difficulties in Learning Mathematics, Int J. Math Ed, Sci & Techn., 24 2, 209– 202. 

School Algebra and the Computer, Micromath, 9 1, 38– 41. 

(with Eddie Gray), Success and Failure in Mathematics: The Flexible Meaning of Symbols as Process and Concept, Mathematics Teaching, 142, 6– 10. 

Mathematicians Thinking about Students Thinking about Mathematics, (summary), Newsletter of the London Mathematical Society, 202, 12– 13 [full version available as preprint]. 

1992m 
Construction of Objects through Definition and Proof, PME Working Group on AMT, Durham, NH. 
(with Eddie Gray): Success and Failure in Mathematics: Procept and Procedure – Secondary Mathematics, Workshop on Mathematics Education and Computers, Taipei National University, April 1992, 216– 221. 

(with Eddie Gray): Success and Failure in Mathematics: Procept and Procedure – A Primary Perspective, Workshop on Mathematics Education and Computers, Taipei National University, April 1992, 209– 215. 

(with Beverly West) Graphic Insight in Mathematics, The influence of computers and informatics on mathematics and its teaching, (ed. Cornu, B., & Ralston, A.) UNESCO, Paris, 117– 123. 

Conceptual Foundations of the Calculus, Proceedings of the Fourth International Conference on College Mathematics Teaching, 73– 88. 

Mathematical Processes and Symbols in the Mind, in Z. A. Karian (ed.) Symbolic Computation in Undergraduate Mathematics Education, MAA Notes 24, Mathematical Association of America, 57– 68. 

(with John Mills) Modelling Irrational Numbers in Analysis using Elementary Programming, The Mathematical Gazette, 76, 243– 250. 

Current difficulties in the teaching of mathematical analysis at university: an essay review of Victor Bryant Yet another introduction to analysis, Zentralblatt für Didaktik der Mathematik, 92/2, 37– 42. 

The Transition to Advanced Mathematical Thinking: Functions, Limits, Infinity, and Proof, in Grouws D.A. (ed.) Handbook of Research on MathematicsTeaching and Learning, Macmillan, New York, 495– 511. 

Success and failure in arithmetic and algebra, Mathematics Teaching 1991, Edinburgh University, September 1991, 2– 7. 

(with Md Nor Bakar) Students’ Mental Prototypes for Functions and Graphs, Int. J. Math Ed Sci & Techn., 23 1, 39– 50. 

Visualizing differentials in two and three dimensions, Teaching Mathematics and its Applications, 11 1, 1– 7. 

1992a n/a 
Enseignement de l’analyse à l’âge de l’informatique, L’ordinateur pour enseigner les mathématiques, Nouvelle Encyclopédie Diderot, 159– 182. 
1991n n/a 
DIY Mathematics Tools, MicroMath, 7 1, 41–42.. 
Reflections, in Tall D. O. (ed.), Advanced Mathematical Thinking, Kluwer: Holland, 251– 259. 

(with Ed Dubinsky), ‘Advanced Mathematical Thinking and the Computer’, in Tall D. O. (ed.), Advanced Mathematical Thinking, Kluwer: Holland, 231– 248. 

The Psychology of Advanced Mathematical Thinking, in Tall D. O. (ed.) Advanced Mathematical Thinking, Kluwer: Holland, 3– 21. 

Visualizing Differentials in Integration to Picture the Fundamental Theorem of Calculus, Mathematics Teaching, 137, 29–32. 

Recent developments in the use of the computer to visualize and symbolize calculus concepts, The Laboratory Approach to Teaching Calculus , M.A.A. Notes Vol. 20, 15– 25. 

(with Eddie Gray), Duality, Ambiguity & Flexibility in Successful Mathematical Thinking, PME 15, Assisi, 2 72– 79. 

(with Norman Blackett) Gender and the Versatile Learning of Trigonometry Using Computer Software, PME15, Assisi, 1 144– 151. 

(with Md Nor Bakar) Students’ Mental Prototypes for Functions and Graphs, PME 15, Assisi, 1 104– 111. 

To prove or not to prove, Mathematics Review 1 3, 29– 32. 

Setting the Calculus Straight, Mathematics Review, 2 1, 2– 6. 

(with Michael Thomas) Encouraging Versatile Thinking in Algebra using the Computer, Educational Studies in Mathematics, 22 2, 125– 147. 

(with Guershon Harel) The General, the Abstract, and the Generic in Advanced Mathematics, For the Learning of Mathematics, 11 1, 38– 42. 

Intuition and rigour: the role of visualization in the calculus, Visualization in Mathematics (ed. Zimmermann & Cunningham), M.A.A., Notes No. 19, 105– 119. 

1990f 
The Reality of the Computer in the Classroom, in Fraser R and Dubinsky E (Eds), Computers and the Teaching of Mathematics, Shell Centre, Nottingham, 32– 38. 
(with John Mills & Michael Wardle) A quartic with a thousand roots, Mathematical Gazette, 74, 339– 346. 

A Versatile Approach to Calculus and Numerical Methods, Teaching Mathematics and its Applications, 9 3 124– 131. 

Misguided Discovery, Mathematics Teaching, 132 27– 29. 

Inconsistencies in the Learning of Calculus and Analysis, 12 3&4, 49– 63. Focus. 

Using Computer Environments to Conceptualize Mathematical Ideas, Proceedings of Conference on New Technological Tools in Education, Nee Ann Polytechnic, Singapore, 55– 75. 

1989g n/a 
(with Frank Knowles) Using the Algebraic Calculator in the Sixth Form [computer program and text] in Secondary Mathematics with Micros – A Resource Pack, Mathematical Association. 
1989f n/a 
(with Michael Thomas) Dynamic algebra [computer program and lesson plans], in Secondary Mathematics with Micros – A Resource Pack, Mathematical Association. 
Concept Images, Generic Organizers, Computers & Curriculum Change, For the Learning of Mathematics, 9 3, 37– 42. 

(with Michael Thomas) Verbal Evidence for Versatile Understanding of Variables in a Computer Environment, Proceedings of P.M.E., Paris, volume 3, 213– 220. 

(with Michael Thomas) Versatile Learning and the Computer, Focus, 11, 2 117– 125. 

New Cognitive Obstacles in a Technological Paradigm, Research Issues in the Learning and Teaching of Algebra, N.C.T.M., 87– 92. 

The nature of mathematical proof, Mathematics Teaching, 127, 28– 32. 

1988i 
The Nature of Advanced Mathematical Thinking, Papers of the Working Group of AMT. 
(with John Mills) From the Visual to the Logical, Bulletin of the I.M.A. 24 11/12 Nov– Dec, 176– 183. [text only] 

(with Bernard Winkelmann) Hidden algorithms in the drawing of discontinuous functions, Bulletin of the I.M.A., 24 111– 115. [text only] 

Promoting versatile learning of higher order concepts in algebra using the computer, Proceedings of B.S.R.L.M., Warwick May 1988. 

Concept Image and Concept Definition, Senior Secondary Mathematics Education, (ed. Jan de Lange, Michiel Doorman), OW&OC Utrecht, 37– 41. 

Mathematics 15 – 19 in a Changing Technological Age, Senior Secondary Mathematics Education, (ed. Jan de Lange, Michiel Doorman), OW&OC Utrecht, 2–12. 

(with Michael Thomas) Longer Term Effects of the Use of the Computer in the Teaching of Algebra, Proceedings of P.M.E.12. Hungary, 601– 608. 

Inconsistencies in the Learning of Calculus and Analysis, The Role of Inconsistent Ideas in Learning Mathematics, AERA, New Orleans April 7 1989, published by Department of Math Ed, Georgia, 37– 46. 

1988a n/a 
Seeing is Believing, Voordrachten en Werkgroepen van het Vijfde Congres Van de Vlaamse Vereniging Wiskunde Leraars Neerpelt , Juli 1987, 221– 240. 
1987h 
Readings in Mathematical Education: Understanding the calculus, A.T.M. (collected articles: 1985a, 1985b, 1985d, 1986a, 1986k,1987a) [see separate articles] 
1987g n/a 
For competence, Mathematics Teaching, 117, 54– 55. 
1987f n/a 
(with F.R. (Joe) Watson) Computer languages for the mathematics classroom, Mathematical Gazette, 71, 275– 285. 
Graphical Packages for Mathematics Teaching and Learning, Informatics and the Teaching of Mathematics, (ed. Johnson D.C. & Lovis F.), North Holland, 39– 47. 

Algebra in a Computer Environment, Proceedings of the Eleventh International Conference of P.M.E., Montreal, I, 262– 271. 

Constructing the concept image of a tangent, Proceedings of the Eleventh International Conference of P.M.E., Montreal, III, 69– 75. 

1987b n/a 
The reality of the computer in the secondary mathematics classroom, Mathematics in School, 16, 1 44– 45. 
Whither Calculus?, Mathematics Teaching, 117 50– 54. 

1986q n/a 
(with others) Algebra, Graphs & Programming, Proceedings of the Mathematical 1116 Conference on Computers in the Mathematics Curriculum, Hertford. 
1986p n/a 
Chords [computer program], Secondary Mathematics with Micros: InService Pack, M.E.P. 
1986o n/a 
My teacher’s car’s an old banger, Micromath, 1, 36. 
1986n n/a 
A paradigm for developing the use of computer technology in mathematics education, I.D.M. Bielefeld, Occasional Papers 83. 
The complementary roles of short programs and prepared software for mathematics learning, Bulletin of the I.M.A., 23, 128– 133. 

1986l n/a 
Talking about fractions, [article and computer program] Micromath, 2, 2 8– 10. 
Lies, damn lies and differential equations, Mathematics Teaching, 114 54– 57. 

1986j n/a 
The Calculus Curriculum in the Microcomputer Age, Mathematical Gazette, 70, 123– 128. 
(with Michael Thomas) The value of the computer in learning algebra concepts, Proceedings of P.M.E. 10, London 313– 318. 

Using the computer as an environment for building and testing mathematical concepts: A tribute to Richard Skemp, in Papers in Honour of Richard Skemp, 21– 36, Warwick. 

Using the computer to represent calculus concepts, Plenary lecture, Le IVème École d’Été de Didactique des Mathématiques, Orléans, Recueil des Textes et Comptes Rendus, 238– 264. 

(with Beverley West) Graphic insight into calculus and differential equations, in The Influence of Computers and Informatics on Mathematics and its Teaching (ed. Howson G. & Kahane JP), C.U.P., 107– 119. 

(with Norman Blackett) Investigating graphs and the calculus in the sixth form, Exploring Mathematics with Microcomputers (ed. Nigel Bufton) C.E.T., 156– 175. 

1986d n/a 
(with Michael Thomas) Playing algebra with the computer, Exploring Mathematics with Microcomputers (ed. Nigel Bufton) C.E.T., 59– 74. 
Powerful functions on a modern micro, Micromath, 2, 1 19– 23. 

Drawing implicit functions, Mathematics in School, 15, 2 33– 37. 

A graphical approach to integration and the fundamental theorem, Mathematics Teaching, 113 48– 51. 

1985h n/a 
contributions to Proceedings of the Mathematical Association Primary Conference, Leicester. 
1985g n/a 
(with others) The calculus curriculum, in Proceedings of the Mathematical Association Conference on Microcomputers in the Alevel Curriculum. 
1985f n/a 
Visualising calculus concepts using a computer, The Influence of Computers and Informatics on Mathematics and its Teaching: Document de Travail, I.C.M.I., Strasbourg, 203– 212. 
Arithmetic with large numbers [article and computer program], Micromath, 1, 2 48– 50. 

Tangents and the Leibniz notation, Mathematics Teaching, 112 48– 52. 

1985c n/a 
Using computer graphics as generic organisers for the concept image of differentiation, Proceedings of PME 9, Holland, 1, 105– 110. 
The gradient of a graph, Mathematics Teaching 111, 48– 52. 

Understanding the calculus, Mathematics Teaching 110 49– 53. 

1984b n/a 
The Mathematics Curriculum and the Micro, Mathematics in School, 13, 4 7– 9. 
1984a n/a 
Continuous mathematics and discrete mathematics are complementary, not alternatives, College Mathematics Journal, 15 389– 391. 
1983b n/a 
Introducing algebra on the computer: today and tomorrow, Mathematics in School, 12, 4 37– 40. 
(with Geoff Sheath) Visualizing higher level mathematical concepts using computer graphics, Proceedings of the Seventh International Conference for the Psychology of Mathematics Education, Israel, 357– 362. 

1982f n/a 
(with David Pimm) The algebra of complex numbers (tape/slide presentation), Audio Learning. 
1982e n/a 
Qualitative protocol analysis, Problemsolving protocols: a task oriented method of analysis, by Hillel J. & Wheeler D. 178– 181. 
1982d n/a 
(with Walter Ledermann) Sequences and series, Handbook of Applicable Mathematics Volume IV, (chapter 1) (ed. W. Ledermann & S. Vajda) 1– 47. 
1982c n/a 
(with Shlomo Vinner) Existence statements and constructions in mathematics, with some consequences for mathematics teaching, American Mathematical Monthly, 89, 10 752– 756. 
Elementary axioms and pictures for infinitesimal calculus, Bulletin of the IMA, 18, 43– 48. 

The blancmange function, continuous everywhere but differentiable nowhere, Mathematical Gazette, 66 11– 22. 

1981e 
Infinitesimals constructed algebraically and interpreted geometrically, Mathematical Education for Teaching, 4, 1 34– 53. 
The mutual relationship between higher mathematics and a complete cognitive theory of mathematical education, Actes du Cinquième Colloque du Groupe Internationale P.M.E., Grenoble, 316– 321. 

Intuitions of infinity, Mathematics in School, 10, 3 30– 33. 

Comments on the difficulty and validity of various approaches to the calculus, For the Learning of Mathematics, 2, 2 16– 21. 

(with Shlomo Vinner) Concept image and concept definition in mathematics, with special reference to limits and continuity, Educational Studies in Mathematics, 12 (2), 151– 169. 

1980f n/a 
Arithmetic, Handbook of Applicable Mathematics Volume I (Chapter 3). (ed. by W. Ledermann & S. Vajda) 77– 99. 
The anatomy of a discovery in mathematical research, For the Learning of Mathematics, 1, 2 25– 30. 

Mathematical intuition, with special reference to limiting processes, Proceedings of the Fourth International Conference for the Psychology of Mathematics Education, Berkeley, 170– 176. 

Intuitive infinitesimals in the calculus, Abstracts of short communications, Fourth International Congress on Mathematical Education, Berkeley, page C5. 

The notion of infinite measuring number and its relevance in the intuition of infinity, Educational Studies in Mathematics, 11 271– 284. 

Looking at graphs through infinitesimal microscopes, windows and telescopes, Mathematical Gazette, 64 22– 49. 

1979e 
Qualitative thought processes in clinical interviews, Proceedings of the Third International Conference for the Psychology of Mathematics Education, Warwick, 206– 207. 
Cognitive aspects of proof, with special reference to the irrationality of the square root of 2, Proceedings of the Third International Conference for the Psychology of Mathematics Education, Warwick, 203– 205. 

1979c n/a 
The calculus of Leibniz, an alternative modern approach, Mathematical Intelligencer, 2 54– 55. 
(with Ian Stewart) Calculations and canonical elements II, Mathematics in School 8, 5 5– 7. 

(with Ian Stewart) Calculations and canonical elements I, Mathematics in School, 8, 4 2– 5. 

1978c 
(with R. L. E. Schwarzenberger) Conflicts in the learning of real numbers and limits, Mathematics Teaching, 82, 44– 49. 
Mathematical Thinking and the Brain, Osnabrücker Schriften zür Mathematik, 333– 343. 

The dynamics of understanding mathematics, Mathematics Teaching, 81, 50– 52.(Longer version as submitted.) 

1977b 
Essay Review: Mathematics as an Educational Task, Instructional Science, 6 187– 198. 
Cognitive conflict in the learning of mathematics, paper presented at the first meeting of the International Group for the Psychology of Learning Mathematics, Utrecht, Holland. 

1976a 
Conflicts and catastrophes in the learning of mathematics, Mathematical Education for Teaching 2,4 2– 18. 
1975b 
The Mathematics Teaching Degree at Warwick University, Times Educational Supplement 18/4/75. 
A longterm learning schema for calculus/analysis, Mathematical Education for Teaching, 2, 5 3– 16. 

1970 n /a 
(with G.C. Wraith) Representable functors and operations on rings, Proc. London Math. Soc. 3, 20 619– 643. 
1969 
(with M.F. Atiyah) Group Representations, lambdarings and the Jhomomorphism, Topology 8, 253– 297. 
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