Reals: motivation, construction and representation as infinite decimal sequences.
Some ideas on how one thinks, learns and teaches the concepts of function, limit, continuity, derivative and integral. Technical insight and history of the formation of such concepts.
Applications to problems in convexity.
Material collected from various sources (and mostly available in e-form) on the MOODLE page of the course
Learning Objectives
Think about the didactical aspects of the main concepts of calculus
Deep knowledge and comprehension of the reals, their construction and their representation
Know how we learn and think mathematics
Strenghten through exercises in class the capacity of applying knowledge and comprehension
Strenghten the communicative abilities through preparing and giving a lecture
Prerequisites
Knowledge of elementary algebraic structures (groups, rings and fields), of the fundamental concepts in the differential geometry of curves and surfaces in 3d space, of the definition and properties of Lebesgue's integral, and first concepts in topology and measure theory.
Good mastery of definitions and properties in differential and integral calculus for functions of one or more variables and in the study of series of numbers and functions.
Teaching Methods
Frontal Lectures
Exercises done by students in front of the class
Use of the computer lab to know and practice softwares and applets useful for explaining the concepts
Further information
Material on the subject of the course is available at http://web.math.unifi.it/users/bianchi/programma_corso_didattica_calcolo_in_linea.html
Type of Assessment
Give a lecture: students, in pairs, give a lecture on a topics from a calculus course in front of the rest of the claas. Final critical discussion
Oral exam:
Some questions are asked and an exercise has to be solved. The exam is structured so to verify the knowledge and the degree of comprehension of the basis concepts of differential and integral calculus and of the theories presented during the lectures. The capacity of the student to have autonomous critical ideas regarding the difficulties of teaching mathematics is evaluated. The exercises are chosen so to evaluate the ability of the student to apply the main ideas and results of calculus in elementary problems
The lecture has no influence on the final vote. The questions and the exercise have the same influence on the vote.
Course program
Concept image and concept definition
Readings: S. Vinner, Concept definition, concept image and the notion of function, Int. J. Math. Educ. Sci. Technol. 14 (1983), 293–305;
D. Tall e S. Vinner, Concept image and concept definition in mathematics with particular reference to Limits and continuity, Educational Studies in Mathematics 12 (1981), 151–169;
Education Committee of the EMS, Solid Findings: Concept images in students’ mathematical reasoning, Newsletter of the European Mathematical Society, September 2014, 50-52;
Education Committee of the EMS, Solid Findings: It is necessary that teachers are mathematically proficient, but is it sufficient?, Newsletter of the European Mathematical Society, March 2012, 46-50.
Numbers.
Peano axioms for natural numbers. Construction of the real numbers through the naturals and through axioms. Decimal representation of the real numbers. Irrationality of e and π. Liouville's number. Real numbers and continuity: The intermediate value theorem. Logarithms. Cardinality.
Readings: T. Gowers, A dialogue concerning the need for the real number system;
Chapter 6 and section 2 in chapter 10 from the book C. H. Edwards, The historical development of the calculus, Springer (disponibile in biblioteca);
N. Marras, Come costruirsi con un foglio di carta (e come si usa) un regolo calcolatore;
Chapter 2, part III from the book F. Klein, Elementary Mathematics from an advanced standpoint: Arithmetic, Algebra, Analysis, Dover (1945)
Concepts of function, limit and derivative.
Readings: I. Kleiner, Evolution of the function concept, a brief survey, The College Mathematics Journal, 1989, 20, 282–300;
T.L. Hankins, Jean D’alembert- Science, CRC Press (1990), 47-48;
Education Committee of the EMS, Solid Findings: Student's over reliance on linearity, Newsletter of the European Mathematical Society, March 2015, 51-53;
M. Berni, Note per un corso di Analisi zero, L’insegnamento della matematica e delle scienze integrate 25 (2002);
J. Marsden, A. Weinstein, Calculus Unlimited, online;
J.V. Grabiner, The changing concept of change: the derivative from Fermat to Weierstrass, Mathematics magazine 56 (1983);
D. Tall, The Blancmange Function Continuous Everywhere but Differentiable Nowhere, The Mathematical Gazette , 66, n. 435 (1982), 11-22.
Integration.
Equivalence between Riemann and Darboux integration. Examples: nonintegrability of the Dirichlet's function, integrability of the function on [0,1] defined as f(x) = 1∕n when x = m∕n (with m and n coprime) and 0 elsewhere, in spite of its discontinuity on Q. Methods of numerical integration (midpoints, trapezoidal, Simpson) and error estimate. The fundamental theorem of calculus. Integrability condition for functions.
Readings: Sezione 3.5 del libro W.F. Trench, Introduction to Real analysis, Open Textbook Initiative;
M. Bramanti, Una proposta didattica: come e perché insegnare gli integrali, Emmeciquadro, 36 (2009), 47–53.
Inequalities and convexity.
Comparison of different definitions of convexity for functions and sets. Properties of convex functions of one variable. Steiner and Schwarz symmetrizations for convex sets. The isoperimetric theorem in the plane.
Readings: Chapter 2 from the book Talenti, Colesanti, Salani, Un'introduzione al Calcolo delle Variazioni", UMI (2016);
J.M. Steele, The Cauchy-Schwarz master class;
H. Chen, Excursions in classical analysis;
P. Duren, Invitation to classical analysis.