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, Mathematics Teaching 111, 48-52.
1985c Using computer graphics as generic organisers for the concept image of differentiation, Proceedings of PME 9, Holland, 1, 105-110.
1985d Tangents and the Leibniz notation, Mathematics Teaching, 112 48-52.
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.
1986b Drawing implicit functions, Mathematics in School, 15, 2 33-37.
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.
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.
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.
1986j The Calculus Curriculum in the Microcomputer Age, Mathematical Gazette, 70, 123-128.
1986k Lies, damn lies and differential equations, Mathematics Teaching, 114 54-57.
1987a Whither Calculus?, Mathematics Teaching, 117 50-54.
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.
1991d Setting the Calculus Straight, Mathematics Review, 2 1, 2-6.
1991j 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.
1996e (with Maselan bin Ali), Procedural and Conceptual Aspects of Standard Algorithms, Proceedings of PME 20, Valencia, 2, 19-26.
1997a Functions and Calculus. In A. J. Bishop et al (Eds.), International Handbook of Mathematics Education, 289-325, Dordrecht: Kluwer.
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. 417421.
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.
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 Martins 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)]
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.
2001f (with Soo Duck Chae) Students’ Concept Images for Period Doublings as Embodied Objects in Chaos Theory, In Marja van den Heuvel-Panhuizen (Ed.) Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education 1, 294. Utrecht, The Netherlands, as a short presentation.
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.) Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education 4, 57-64. Utrecht, The Netherlands.
2002a (with Marcia Maria Fusaro Pinto), Building formal mathematics on visual imagery: a case study and a theory. For the Learning of Mathematics, 22(1).
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.
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.
2009c David Tall (2009). Dynamic mathematics and the blending of knowledge structures in the calculus. ZDM The International Journal on Mathematics Education, 41 (4) 481492.
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, 2325th September 2010, Puebla, Mexico.)
to be published as:
2011e David Tall (2011). A Sensible Approach to the Calculus. To appear in Handbook on Calculus and its Teaching, ed. François Pluvinage & Armando Cuevas.