**CALCULUS (AND COMPUTERS)**

My initial forays into the teaching of analysis and calculus were based on consideration of teaching the mathematical ideas in a meaningful way (eg 1975a). Having found inherent difficulties in the limit concept, (1978c with R. L. E. Schwarzenberger, 1980b 1980d ), I sought a new method of introducing ideas in the calculus that used the limit concept *implicitly*, rather than making it the explicit foundation of the theory for beginners. Having studied infinitesimals in non-standard analysis and proposing an axiomatic form of infinitesimal calculus (1982b), I was able to use the notion of high magnification of a differentiable curve to lead to the idea that ‘**local straightness**’ of the graph (which generalises to the notion of locally euclidean in manifold theory) is a good cognitive root on which to build the calculus. To achieve this purpose, I designed the Graphic Calculus software to enable the student to interact with examples of the ideas. These include magnifying a portion of the graph to see it look straight under high magnification, and tracing the gradient numerically along the graph. This allows the concept of differential equations to be seen as determining locally straight graphs which have known gradient at each point (given by the equation). Software (the solution sketcher) allows the student to point the mouse at a place in the plane and the software to draw a short line segment of the given gradient. By enactively sticking such segments end to end, an approximate solution can be constructed physically and visually. The locally straight approach has been filled out in my papers to cover a wide range of ideas including visual representations of differentials, partial derivatives, tangent planes, parametric curves, composite functions, implicit functions, area and integral, numerical methods of solving equations etc.

More recently, as I have developed the theoretical framework of Three Worlds of Mathematics, the clear distinction between the natural locally straight approach to the calculus and the formal epsilon-delta approach to analysis has become clearly defined. In particular, it clarifies the distinction between natural embodiment in calculus and logical linguistic proof in analysis and contrast with the APOS theory approach using symbolic programming to encapsulate the limit process as the derivative. Instead of encapsulating a potentially infinte limiting *process* as a limit *object*, it operates on an* object *to conceptualise a new *object* found by looking along the original graph to see the stabilised graph of its changing slope. Inherently focusing on visual objects is more basic than imagining the product of a never ending process. (See my recent papers in 2009, 2010.)

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

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

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

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

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

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.

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

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

1985b The gradient of a graph,

1985c Using computer graphics as generic organisers for the concept image of differentiation,

1985d Tangents and the Leibniz notation,

1985f Visualising calculus concepts using a computer,

1986a A graphical approach to integration and the fundamental theorem,

1986b Drawing implicit functions,

1986e (with Norman Blackett) Investigating graphs and the calculus in the sixth form,

1986f (with Beverley West) Graphic insight into calculus and differential equations, in

Using the computer to represent calculus concepts, Plenary lecture, Le IVème École d'Été de Didactique des Mathématiques, Orléans,

1986j The Calculus Curriculum in the Microcomputer Age,

1986k Lies, damn lies and differential equations,

1987a Whither Calculus?,

1987c Constructing the concept image of a tangent,

1988b Inconsistencies in the Learning of Calculus and Analysis,

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

1990d A Versatile Approach to Calculus and Numerical Methods,

1991a Intuition and rigour: the role of visualization in the calculus,

1991i Recent developments in the use of the computer to visualize and symbolize calculus concepts,

1991d Setting the Calculus Straight,

1991j Visualizing Differentials in Integration to Picture the Fundamental Theorem of Calculus,

1992a Enseignement de l'analyse à l'âge de l'informatique, L'ordinateur pour enseigner les mathématiques,

1992b Visualizing differentials in two and three dimensions,

1996e (with Maselan bin Ali), Procedural and Conceptual Aspects of Standard Algorithms,

1997a Functions and Calculus. In A. J. Bishop et al (Eds.),

1998b (with Adrian Simpson), Computers and the Link between Intuition and Formalism. In

1999g (with Márcia Maria Fusaro Pinto), Student constructions of formal theory: giving and extracting meaning. In O. Zaslavsky (Ed.),

1999i (with Soo Duck, Chae), Aspects of the construction of conceptual knowledge: the case of computer aided exploration of period doubling,

2000g (with Silvia Di Giacomo), Cosa vediamo nei disegni geometrici? (il caso della funzione blancmange)

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),

2001f (with Soo Duck Chae) Students’ Concept Images for Period Doublings as Embodied Objects in Chaos Theory, In Marja van den Heuvel-Panhuizen (Ed.)

2001~ (with Anna Watson) Schemas and processes for sketching the gradient of a graph, unpublished.

2001j (with Marcia Maria Fusaro Pinto) Following student’s development in a traditional university classroom. In Marja van den Heuvel-Panhuizen (Ed.)

2002a (with Marcia Maria Fusaro Pinto), Building formal mathematics on visual imagery: a case study and a theory.

2003a Using Technology to Support an Embodied Approach to Learning Concepts in Mathematics. (Talk given in Rio de Janiero). This is the first presentation of ideas that refer to my idea of ‘the three worlds of mathematics’, showing how embodied foundations relate to symbolic manipulations and formal theory.

**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.**

2010x David Tall & Mikhail Katz (2010). A Cognitive Analysis of Cauchy’s Conceptions of Continuity, Limit, and Infinitesimal, with Implications for Teaching The Calculus. (Submitted for publication.)

2010y 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.)**
**to be published as:

2011e David Tall (2011). A Sensible Approach to the Calculus. To appear in

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