For a full list of papers available for download from the web, consult David Tall’s University Page

**Publications on Calculus (and computers) in chronological order**

1975a A long-term learning schema for calculus/analysis,* Mathematical Education for Teaching*, 2, 5 3-16.

My first attempt at analysing the learning of calculus, using indistinguishability relations.

1980a Looking at graphs through infinitesimal microscopes, windows and telescopes, *Mathematical Gazette*, 64 22-49.

My first paper looking at magnification of curves.

1981b Comments on the difficulty and validity of various approaches to the calculus, *For the Learning of Mathematics*, 2, 2 16-21.

An analysis of several different approaches to the calculus.

1982a The blancmange function, continuous everywhere but differentiable nowhere, *Mathematical Gazette*, 66 11-22.

My first publication about a function which is not differentiable anywhere.

1982b Elementary axioms and pictures for infinitesimal calculus, *Bulletin of the IMA*, 18, 43-48.

My first publication formulating a simple axiomatic approach to infinitesimal calculus, based on a course I taught several times at Warwick University.

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.

My first publication with research into a Graphic Approach to the Calculus.

1984a Continuous mathematics and discrete mathematics are complementary, not alternatives, *College Mathematics Journal*, 15 389-391.

A paper contributed to a discussion on the declared opposition of continuous and discrete mathematics.

1985a Understanding the calculus, *Mathematics Teaching* 110 49-53.

The first of six papers on the calculus in the Journal Mathematics Teaching. An overview.

1985b The gradient of a graph, *Mathematics Teaching* 111, 48-52.

The second of six papers on the calculus in the Journal Mathematics Teaching. Introducing the gradient visually.

1985c Using computer graphics as generic organisers for the concept image of differentiation, Proceedings of PME 9, Holland, 1, 105-110.

A second publication concerning research into a Graphic Approach to the Calculus.

1985d Tangents and the Leibniz notation, Mathematics Teaching, 112 48-52.

The third of six papers on the calculus in the Journal Mathematics Teaching. Reviewing Leibniz‘ definition of differentiation.

1985f 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.

1986a A graphical approach to integration and the fundamental theorem, Mathematics Teaching, 113 48-51.

The fourth of six papers on the calculus in the Journal Mathematics Teaching. Area, integration, and the fundamental theorem.

1986b Drawing implicit functions, Mathematics in School, 15, 2 33-37.

A whimsical program for drawing implicit functions inspired by a walk in the English Peak District.

1986k Lies, damn lies and differential equations, Mathematics Teaching, 114 54-57.

The fifth of six papers on the calculus in the Journal Mathematics Teaching. A visual approach revealing the glaring errors in the usual symbolic approach.

1987a W(h)ither Calculus?, Mathematics Teaching, 117 50-54.

The last of six papers on the calculus in the Journal Mathematics Teaching. A reflection. Will calculus wither? Whither will it go?

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.

Using Graphic Calculus in a classroom.

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.

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.

A plenary lecture giving an overview to my PhD thesis.

1986j The Calculus Curriculum in the Microcomputer Age, Mathematical Gazette, 70, 123-128.

1987c Constructing the concept image of a tangent, Proceedings of the Eleventh International Conference of P.M.E., Montreal, III, 69-75.

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.

1990b Inconsistencies in the Learning of Calculus and Analysis, 12 3&4, 49-63. *Focus.*

1990d A Versatile Approach to Calculus and Numerical Methods, *Teaching Mathematics and its Applications*, 9 3 124-131.

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.

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.

1991j Setting the Calculus Straight, *Mathematics Review*, 2 1, 2-6.

1991n Visualizing Differentials in Integration to Picture the Fundamental Theorem of Calculus, *Mathematics Teaching*, 137, 29-32.

1992a Enseignement de l'analyse à l'âge de l'informatique, L'ordinateur pour enseigner les mathématiques, Nouvelle Encyclopédie Diderot, 159-182.

1992b Visualizing differentials in two and three dimensions,* Teaching Mathematics and its Applications*, 11 1, 1-7.

Viewing differentials as the components of the tangent vector, giving new insight into partial differentials.

1996e (with Maselan bin Ali), Procedural and Conceptual Aspects of Standard Algorithms, *Proceedings of PME 20*, Valencia, 2, 19-26.

How different students cope with the procedures of differentiation.

1997a Functions and Calculus. In A. J. Bishop et al (Eds.), International Handbook of Mathematics Education, 289-325, Dordrecht: Kluwer.

A literature review.

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.

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.

How students cope differently with the beginnings of formal analysis.

1999i (with Soo Duck, Chae), Aspects of the construction of conceptual knowledge: the case of computer aided exploration of period doubling, Proceedings of BSRLM Conference, 5th June, St Martin’s Lancaster, 13–18.

2000g (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)]

2000i 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.

An analysis of how we think about mathematics, with an extended discussion on calculus.

2001~ (with Soo Duck, Chae), Aspects of the construction of conceptual knowledge: the case of computer aided exploration of period doubling, (to appear).

2001~ (with Soo Duck Chae) Students’ Concept Images for Period Doublings as Embodied Objects in Chaos Theory, to appear in *PME 25* as a short presentation. (The full version is on the web).

2001~ (with Anna Watson) Schemas and processes for sketching the gradient of a graph, *in preparation*

2001~ (with Marcia Maria Fusaro Pinto) Following students’ development in a traditional university classroom, to appear in *PME 25
*

2001~ (with Marcia Pinto), Building formal mathematics on visual imagery: a theory and a case study.* in preparation*.