Published Articles on Mathematics & Mathematics Education+recent drafts

See also other pages for Earlier Drafts, Selected Lectures, Curriculum Vitae.

My Mathematics Education PhD (1986) may be downloaded from here.

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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.
Go directly to papers published in 1969, 1975, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 2000, 01, 02, 03, 04, 05, 06, 07, 08, 09, 10, 11, 12, 13

Some papers listed below are also organised under themes or, where appropriate, with research students.
Any problems: e-mail david.tall@warwick.ac.uk.

See also:

Earlier draft papers | Selected lectures | Curriculum Vitae

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

2014 Ivy Kidron & David Tall (2014). The roles of embodiment and symbolism in the potential and actual infinity of the limit process. (Accepted for publication with minor corrections.)
2014b David Tall, Rosana Nogueira de Lima & Lulu Healy: Evolving a Three-world Framework for Solving Algebraic Equations in the Light of What a Student Has Met Before. (To appear in Journal of Mathematical Behavior.)
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. (To appear in Educational Studies in Mathematics.)
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.
2013f 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.
2013e

Nellie C. Verhoef, Fer Coenders, Jules M. Pieters, Daan van Smaalen, and David O. Tall (2013). Professional development through lesson study: teaching the derivative using GeoGebra. To appear in Professional Development in Education.

2013d

Mercedes McGowen & David Tall (2013). Flexible Thinking and Met-befores: Impact on learning mathematics, With Particular Reference to the Minus sign. Journal of Mathematical Behavior 32, 527–537.
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: 10.1007/978-94-007-7473-5_13.
2013b Nellie Verhoef & David Tall (2013). The Complexity of Lesson Study in a Dutch Situation. International Journal of Science and Mathematics Education. DOI: 10.1007/s10763-013-9436-6.

2013a

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, Ying-Hao 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.

2012s

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 29-May 3, 2012, Ben-Gurion University of the Negev, Beer Sheva, Israel.Overheads. (Original extended version.)
2012w David Tall (2012). Setting Lesson Study within a Long-Term Framework for Learning. Chapter for an APEC Lesson Study Book. Ed. Maitree Inprasitha, Patsy Wang Iverson, Yeap Ban Har.
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) 17-18.
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. 297-304. Turkey: University of Ankara.

2011a David Tall (2011). Crystalline concepts in long-term mathematical invention and discovery. For the Learning of Mathematics, 31 (1) March 2011, 3-8.
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, 21-28.
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 Met-before? 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: problem-solving 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 Mejia-Ramos (2009). The Long-Term 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, 207-258.

2008e

David Tall (2008). The Transition to Formal Thinking in Mathematics. Mathematics Education Research Journal, 2008, 20 (2), 5-24 [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, 45-50.
2008c A life-time’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 set-before and met-before. 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) 3-18.
2007x Masami Isoda & David Tall (2007). Long-term 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 long-term 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 APEC-Tsukuba International Conference, December 3–7, 2006, at the JICA Institute for International Cooperation (Ichigaya, Tokyo). (pre-publication 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 Mejia-Ramos (2006). The Long-Term 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. (pre-publication 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 80-7290-255-5
2006b David Tall (2006). A life-time’s journey from definition and deduction to ambiguity and insight. Retirement as Process and Concept: A Festschrift for Eddie Gray and David Tall, Prague. 275-288, ISBN 80-7290-255-5. [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. 468-475.
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. 23-35). 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, 5-7 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. 1-8. Available from http://www.bsrlm.org.uk/IPs/ip25-1/.
2005c Mejia-Ramos, J. P. and Tall, D. (2005), ‘Personal and public aspects of formal proof: a theory and a single-case study’, in D. Hewitt and A. Noyes (Eds), Proceedings of the sixth British Congress of Mathematics Education held at the University of Warwick, pp. 97-104. Available from http://www.bsrlm.org.uk/IPs/ip25-1/.
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. 128-135. Available from http://www.bsrlm.org.uk/IPs/ip25-1/.

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 Post-Calculus-Reform. 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.445-452, 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, 73-97.
2003b Giraldo, V., Carvalho, L. M. & Tall, D. O. (2003). Conflitos Teórico-Computacionais e a Imagem Conceitual de Derivada. In L.M. Carvalho and L.C. Guimarães, História e Tecnologia no Ensino da Matemática, vol. 1, pp. 153-164, 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. 1-28, Rio de Janeiro, Brasil.

2002m

Giraldo, V.; Carvalho, L. M. & Tall, D. O, (2002). Theoretical-Computational Conflicts and the Concept Image of Derivative. Proceedings of the BSRLM Conference. Nottingham, England, 22 (3), 37–42.
2002l Ehr-Tsung Chin, David Tall (2002), Proof as a Formal Procept in Advanced Mathematical Thinking, International Conference on Mathematics: Understanding Proving and Proving to Understand, 212-221. 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 Erh-Tsung 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 Long-Term 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 never-ending struggle, Educational Studies in Mathematics, 48 (2&3), 129–136.
2001n Thomas, M. O. J. & Tall, D. O., (2001). The long-term 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, 590-597.
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, 1470-1475, 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 Kit-Bags 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) Ehr-Tsung 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.

2001c

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.

2001b

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

2001a

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 Wei-Chi Yang, Sung-Chi Chu, Jen-Chung Chuan (Eds), Proceedings of the Fifth Asian Technology Conference in Mathematics, Chiang Mai, Thailand (pp. 3–20). ATCM Inc, Blackwood VA. ISBN 974-657-362-4.

2000g

Technology and Versatile Thinking In Mathematical Development. In Michael O. J. Thomas (Ed), Proceedings of TIME 2000, (pp. 33–50). Auckland, New Zealand

2000f

(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)].

2000e

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

2000d

(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 PME-NA, 1, 247–254.

2000c

(with Ehr-Tsung 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.

2000b

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

2000a

(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 18-20, 1999. (An unpublished collection of discussion points taken from other articles.)

1999k

(with Eddie Gray, Demetra Pitta, Marcia Pinto), Knowledge Construction and diverging thinking in elementary and advanced mathematics, Educational Studies in Mathematics. 38 (1-3), 111-133.

1999j

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.

1999i

(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).

1999h

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

1999g

(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.

1999f

(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.

1999e

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

1999d

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

1999c

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

1999b

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

1999a

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).

1998c

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.

1998b

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

1998a

(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.

1997f

Metaphorical objects in Advanced Mathematical Thinking, International Journal for Computers in Mathematics Learning, 1, 61–65.

1997e

(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.

1997d

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

1997c

(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.

1997a

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, 22-100.

1996i

Understanding the Processes of Advanced Mathematical Thinking, L’Enseignement des Mathématiques, 42, 395– 415. [original version]

1996h

(with Marcia Pinto), Student Teachers’ Conceptions of the Rational Numbers, Proceedings of PME 20, Valencia, 4, 139– 146.

1996g

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

1996f

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

1996e

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

1996d

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

1996c

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

1996b

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.

1996a

Advanced Mathematical Thinking and the Computer. Proceedings of the 20th University Mathematics Teaching Conference, Shell Centre, Nottingham, 8 pp. 1-8.


1995f

Cognitive Development, Representations & Proof, Justifying and Proving in School Mathematics, Institute of Education, London, 27- 38.

1995e

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

1995d

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

1995c

(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.

1995b

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.

1995a

Visual Organizers for Formal Mathematics. In R. Sutherland & J. Mason (Eds.), Exploiting Mental Imagery with Computers in Mathematics Education. Springer-Verlag: 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]

1994~

Cognitive difficulties in learning analysis. In Report on the Teaching of Analysis (ed. Barnard A.), for the TaLUM committee.

1994f

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. 3680-3681, 3686.

1994e

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.

1994d

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

1994c

(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.

1994b

(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.

1994a

(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, ICME-7, Québec, Canada, 75– 77. ISBN 2 920916 23 

1993k

Students’ Difficulties in Calculus, Plenary Address, Proceedings of Working Group 3 on Students’ Difficulties in Calculus, ICME-7, Québec, Canada, 13– 28. ISBN 2 920916 23 8.

1993j

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.

1993i

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

1993h

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

1993g

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

1993f

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

1993e

(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

1993d

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

1993c

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

1993b

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

1993a

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


1992m

Construction of Objects through Definition and Proof, PME Working Group on AMT, Durham, NH.

1992l

(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.

1992k

(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.

1992j

(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.

1992i

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

1992h

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.

1992g

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

1992f

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.

1992e

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.

1992d

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

1992c

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

1992b

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..

1991m

Reflections, in Tall D. O. (ed.), Advanced Mathematical Thinking, Kluwer: Holland, 251– 259.

1991l

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

1991k

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

1991j

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

1991i

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.

1991h

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

1991g

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

1991f

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

1991e

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

1991d

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

1991c

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

1991b

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

1991a

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.

1990e

(with John Mills & Michael Wardle) A quartic with a thousand roots, Mathematical Gazette, 74, 339– 346.

1990d

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

1990c

Misguided Discovery, Mathematics Teaching, 132 27– 29.

1990b

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

1990a

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.

1989e

Concept Images, Generic Organizers, Computers & Curriculum Change, For the Learning of Mathematics, 9 3, 37– 42.

1989d

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

1989c

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

1989b

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

1989a

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


1988i

The Nature of Advanced Mathematical Thinking, Papers of the Working Group of AMT.

1988h

(with John Mills) From the Visual to the Logical, Bulletin of the I.M.A. 24 11/12 Nov– Dec, 176– 183. [text only]

1988g

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

1988f

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

1988e

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

1988d

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

1988c

(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.

1988b

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.

1987e

Graphical Packages for Mathematics Teaching and Learning, Informatics and the Teaching of Mathematics, (ed. Johnson D.C. & Lovis F.), North Holland, 39– 47.

1987d

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

1987c

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.

1987a

Whither Calculus?, Mathematics Teaching, 117 50– 54.


1986q n/a

(with others) Algebra, Graphs & Programming, Proceedings of the Mathematical 11-16 Conference on Computers in the Mathematics Curriculum, Hertford.

1986p n/a

Chords [computer program], Secondary Mathematics with Micros: In-Service 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.

1986m

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.

1986k

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.

1986i

(with Michael Thomas) The value of the computer in learning algebra concepts, Proceedings of P.M.E. 10, London 313– 318.

1986h

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.

1986g

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.

1986f

(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 J-P), C.U.P., 107– 119.

1986e

(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.

1986c

Powerful functions on a modern micro, Micromath, 2, 1 19– 23.

1986b

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

1986a

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 A-level 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.

1985e

Arithmetic with large numbers [article and computer program], Micromath, 1, 2 48– 50.

1985d

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.

1985b

The gradient of a graph, Mathematics Teaching 111, 48– 52.

1985a

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.

1983a

(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, Problem-solving 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.

1982b

Elementary axioms and pictures for infinitesimal calculus, Bulletin of the IMA, 18, 43– 48.

1982a

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.

1981d

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.

1981c

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

1981b

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

1981a

(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.

1980e

The anatomy of a discovery in mathematical research, For the Learning of Mathematics, 1, 2 25– 30.

1980d

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

1980c

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

1980b

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

1980a

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.

1979d

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.

1979b

(with Ian Stewart) Calculations and canonical elements II, Mathematics in School 8, 5 5– 7.

1979a

(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.

1978b

Mathematical Thinking and the Brain, Osnabrücker Schriften zür Mathematik, 333– 343.

1978a

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


1977b n/a

Essay Review: Mathematics as an Educational Task, Instructional Science, 6 187– 198.

1977a

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.

1975a

A long-term 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, lambda-rings and the J-homomorphism, Topology 8, 253– 297.

last modified: Tuesday, June 28, 2011