The meaningful study of calculus has evolved since I first studied the subject at school in 1956. My research into teaching and learning began in 1975 when computers allowed us to draw graphs dynamically to different scales and I wrote software to magnify a differentiable function to reveal it as being 'locally straight' under high magnification. The derivative is now the slope of the graph itself.

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, symbolic operations to derive the formula for the derivative 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. Since the publication of that book, the theory has broadened to include the manner in which our eyes and brain follow moving objects to give a meaning to 'arbitrarily small quantities', expanding the 'number line' to give a larger continuum containing infinitesimals, interpreting algebraic expressions flexibly as process or concept to give new meanings for symbolic differentiation and integration and extending formal mathematics to more sophisticated forms of embodiment and symbolism using 'structure theorems'.

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.


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. 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.
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)]
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) 481–492.
2011e David Tall (2011). A Sensible Approach to the Calculus. Handbook on Calculus and its Teaching, ed. François Pluvinage & Armando Cuevas.
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.
2013 See David Tall How Humans Learn to Think Mathematically, CUP, chapters 11 and 13.
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. Educational Studies in Mathematics, 86 (1), 97-124.
2014c Nellie Verhoef, David Tall, Fer Coenders, Daan Smaalen (2014). The complexities of a lesson study in a Dutch situation: Mathematics Teacher Learning. International Journal of Science and Mathematics Education 12: 859–881. doi: 10.1080/19415257.2014.88628.
2014 See David Tall & Ian Stewart Foundations of Mathematics, 2nd edition. ISBN: 9780198531654.
2015a Ivy Kidron & David Tall (2015a). The roles of embodiment and symbolism in the potential and actual infinity of the limit process. Educational Studies in Mathematics 88: 183. doi:10.1007/s10649-014-9567-x
2015c Nellie C. Verhoef, Fer Coenders, Jules M. Pieters, Daan van Smaalen, and David O. Tall (2015). Professional development through lesson study: teaching the derivative using GeoGebra. Professional Development in Education, 41 (1), 109-126. DOI:10.1080/19415257.2014.88628
2019b Tall, D. O. (2019b). From Biological Brain to Mathematical Mind: The Long-term Evolution of Mathematical Thinking. In Danesi, M. (Ed.): Interdisciplinary Perspectives on Math Cognition, pp.1–28. Springer.
2019d David Tall (2019d). The Evolution of Calculus: A Personal Experience 1956-2019. Conference on Calculus in Upper Secondary and Beginning University Mathematics, Norway August 2019. Paper based on presentation. Video of presentation on YouTube entitled Making Human Sense of Calculus.
2021 David Tall (2021) Long-term principles for meaningful teaching and learning of mathematics. To appear in Sepideh Stewart (ed.): Mathematicians’ Reflections on Teaching: A Symbiosis with Mathematics Education Theories. 

last modified: Monday, September 13, 2010