**CALCULUS (AND COMPUTERS)**

My initial forays into the teaching of calculus and analysis were based on consideration of teaching mathematical ideas in a meaningful way (e.g. 1975a). Having found inherent difficulties in the limit concept (1978c with Rolph Schwarzenberger, and in 1980b and 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 (which is far to subtle to use as a foundation for beginners), I realised that a differentiable function is one which 'looks straight' when highly magnified and continues to look straight as it is magnified further. I used this idea to develop Graphic Calculus software to enable the learner to interact dynamically with these ideas. This included 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 in terms of knowing the slope of a function at each point (given by the differential equation) and determining the original function. 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. (See papers below.)

Over the years I have developed a fully fledged theoretical framework of three distinct ways that mathematical thinking develops: first through physical and mental perception and thought experiment that I termed 'conceptual embodiment', then through the use of 'operational symbolism' in arithmetic, algebra and symbolic calculus and later through 'axiomatic formalism' involving set-theoretic definition and formal proof. This theoretical framework of 'Three Worlds of Mathematics', makes a clear distinction between the natural locally straight approach to the calculus built through embodiment and symbolic operations and the formal epsilon-delta approach to analysis.

While the limit approach involves a potentially infinite limiting *process* being reconceived as a limit *object, *a locally straight approach starts with an *object* (the graph) and dynamically traces along it to conceptualize the changing slope of a locally straight graph as a new *object*: the graph of the slope function, which is symbolically computed as the derivative.

Over the years I have used this approach to give simple formal interpretations of the Leibniz notation as a quotient dy over dx where dx and dy are the components of the tangent vector. This extends to simple interpretations in multi-dimensions and in partial derivatives (which simply relate to the components of the tangent vector).

The theory provides meaningful visual interpretations of *continuity*, *differentiation*, *integration*, *the fundamental theorem of calculus* and new interpretations of historical developments of the calculus. These are summarized in chapter 11 of my book *How Humans Learn to Think Mathematically* and extended to simple new ways of visualizing the number line to include not only real numbers but also infinitesimal quantities that can be pictured by linear magnification.

Therefore I offer not only a constructive meaningful visualization of calculus concepts suitable for the beginner, but also a formal theoretical extension of mathematical analysis that gives visual and symbolic meaning to the fundamental ideas of calculus.

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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.

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