David M. Bressoud
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Naturwissenschaften, Medizin, Informatik, Technik / Mathematik
How our understanding of calculus has evolved over more than three centuries, how this has shaped the way it is taught in the classroom, and why calculus pedagogy needs to change
Calculus Reordered takes readers on a remarkable journey through hundreds of years to tell the story of how calculus evolved into the subject we know today. David Bressoud explains why calculus is credited to seventeenth-century figures Isaac Newton and Gottfried Leibniz, and how its current structure is based on developments that arose in the nineteenth century. Bressoud argues that a pedagogy informed by the historical development of calculus represents a sounder way for students to learn this fascinating area of mathematics.
Delving into calculus’s birth in the Hellenistic Eastern Mediterranean—particularly in Syracuse, Sicily and Alexandria, Egypt—as well as India and the Islamic Middle East, Bressoud considers how calculus developed in response to essential questions emerging from engineering and astronomy. He looks at how Newton and Leibniz built their work on a flurry of activity that occurred throughout Europe, and how Italian philosophers such as Galileo Galilei played a particularly important role. In describing calculus’s evolution, Bressoud reveals problems with the standard ordering of its curriculum: limits, differentiation, integration, and series. He contends that the historical order—integration as accumulation, then differentiation as ratios of change, series as sequences of partial sums, and finally limits as they arise from the algebra of inequalities—makes more sense in the classroom environment.
Exploring the motivations behind calculus’s discovery, Calculus Reordered highlights how this essential tool of mathematics came to be.
Variable (mathematics), Georg Cantor, Carl Friedrich Gauss, Derivative, Counterexample, Notation, Liber Abaci, Binomial series, Subset, Zermelo–Fraenkel set theory, Astronomy, Parallelogram, Proportionality (mathematics), Logarithm, Quantity, Mathematics, Lipman Bers, On the Sphere and Cylinder, Trigonometric functions, Riemann integral, Error, Power series, Summation, Tangent, Adequality, Circumference, Cartesian coordinate system, Leonhard Euler, Basel problem, Jacob Bernoulli, Continuous function (set theory), Hyperbolic function, Marin Mersenne, Cardinality, Maxima and minima, Inequality (mathematics), Continuum hypothesis, For All Practical Purposes, Differential equation, Geometric series, Quadratic equation, The Whetstone of Witte, Fourier series, Arithmetic, Cauchy sequence, Area of a circle, Function (mathematics), Differential calculus, Newton's method, Algebraic equation, Mean value theorem, Theorem, Rational number, Exponential function, Mathematician, Velocity, Partial differential equation, Nicole Oresme, Elliptic function, Calculation, Hilbert space, Ratio, Algebra, Euler's formula, Cours d'Analyse, Method of exhaustion, Equation, Karl Weierstrass, Taylor's theorem, Special case, Scientific notation, Srinivasa Ramanujan, Continuous function, Horologium Oscillatorium, Infinitesimal, Lebesgue integration, Coefficient, Addition, Arc length, Interval (mathematics), Pierre de Fermat, Fundamental theorem, Isaac Barrow, Rectangle, Approximation, Instant, Bernhard Riemann, Natural number, Series (mathematics), Analytic geometry, Geometry, Real number, Polynomial, Complex plane, Natural logarithm, Taylor series, Ratio test, Antiderivative, Fourier, David Hilbert