The young learner recapitulates the learning process of mankind, though in a modified way. He repeats history not as it actually happened but as it would have happened if people in the past would have known something like what we do know now. It is a revised and improved version of the historical learning process that young learners recapitulate.
‘Ought to recapitulate’ – we should say. In fact we have not understood the past well enough to give them this chance to recapitulate it. (1983b, p. 1696)
1. Blaadjes en vruchten aan de boom
In an ancient Indian story, which was finally written down in the fourth century AD,
bee thought to worke according to reason, who to find out the place where the marke
stood, shall seeke out the middle place amongst all the arrowes: so amongst many different obseruations, the middlemost is likest to come nearest the truth.
(Eisenhart, 1974, p. 52)
8. Vermoedens over foutenverdelingen
In 1756, Simpson made this shift to looking at the law of errors in astronomy when he used simple probability functions to argue that the mean of several observations was a better estimate of a true value than a single observation (Steinbring, 1980). The first distribution of errors he proposed was a discrete uniform distribution, that is with equal probabilities for all error values -v, -v+1,...-1, 0, 1,..., v (Figure below a). Next, he assumed a discrete isosceles triangle distribution with probabilities proportional to 1, 2,..., v-1, v, v+1, v,..., 2, 1 (Figure b), from which he obtained a continuous isosceles triangle distribution one year later (Figure c).
a. discrete uniform (1756)
b discrete isosceles triangle (1756)
c. continuous isosceles triangle (1757)
Soon after Simpson had launched his idea of probability distributions, other scientists proposed alternative laws of error (Figure below). Among them were Lagrange, Lambert, Daniel Bernoulli, Laplace, and Gauss.
a Lambert (1765): flattened semicircle
b Lagrange (1776): continuous uniform
c Lagrange (1776): continuous parabolic
d Lagrange (1781): cosine function
e Laplace (1781): log function
f Laplace (1774): double exponential
g Gauss (1809), Laplace (1810):
Galton lyrisch over de normale verdeling
It is difficult to understand why statisticians commonly limit their inquiries to Averages, and do not revel in more comprehensive views. Their souls seem as dull to the charm of variety as that of the native of one of our flat English counties, whose retrospect from Switzerland was that, if its mountains could be thrown into its lakes, two nuisances would be got rid of at once. An Average is but a solitary fact, whereas if a single other fact be added to it, an entire Normal Scheme, which nearly corresponds to the observed ones, starts potentially into existence. (Natural Inheritance, Galton, 1889, p. 62)
I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the “Law of Frequency of Error.” The law would have been personified by the Greeks and deified, if they had known of it. It reigns with serenity and in complete self-effacement amidst the wildest confusion. The huger the mob, and the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of Unreason. Whenever a large sample of chaotic elements are taken in hand and marshalled in the order of their magnitude, an unsuspected and most beautiful form of regularity proves to have been latent all along. (ibid., p. 66)
Freudenthal over generaal Knapp
In de 19de eeuw, liet Generaal Knapp zijn recruten op een rij staan in de veronderstelling dat hij een normale verdeling zou zien.
Toen hij [generaal Knapp) tot zijn schrik bemerkte, dat de kruinen niet naar links èn rechts afliepen, dus niet zoiets als een normale kromme aftekenden, liet hij ze heel eigenaardige exercities uitvoeren, waarvan de finesses mij niet duidelijk zijn geworden.
Bakker, A. (2001). De mediaan in het middelpunt. Nieuwe Wiskrant, 21(1), 28-31.
Bakker, A. (2003). The early history of statistics and implications for education. Journal of Statistics Education, 11(1). www.amstat.org/publications/jse
Bakker, A. (2004). Design research in statistics education: On symbolizing and computer tools. Utrecht, the Netherlands: CD Beta Press.
Bakker, A., & Gravemeijer, K. P. E. (2006). An historical phenomenology of mean and median. Educational Studies in Mathematics, 62(2), 149-168.
Cournot, A. A. (1843/1984). Exposition de la théorie des chances et des probabilités [Treatise of the theory of chances and probabilities]. Paris: Librairie philosophique J. Vrin.
David, H. A. (1995). First (?) occurences of common terms in mathematical statistics. The American Statistician, 49, 121-133.
David, H. A. (1998). First (?) occurences of common terms in probability and statistics -A second list, with corrections. The American Statistician, 52, 36-40.
Eisenhart, C. (1974). The development of the concept of the best mean of a set of measurements from antiquity to the present day. 1971 ASA Presidential Address. Unpublished manuscript.
ESS (1981). Encyclopedia of statistical sciences (Eds. Kotz, S. & Johnson, N.L.). New York: Wiley & Sons.
Freudenthal, H. (1983b). The implicit philosophy of mathematics: History and education. In Proceedings of the International Congress of Mathematicians (pp. 1695-1709). Warsaw.
Galton, F. (1889). Natural inheritance. London: Macmillan.
Hacking, I. (1990). The taming of chance. Cambridge: Cambridge University Press.
Rabinovitch, N. L. (1973). Probability and statistical inference in ancient and medieval Jewish literature. Toronto: University of Toronto Press.
Stigler, S. M. (1977). Eight centuries of sampling inspection: The trial of the Pyx. Journal of the American Statistical Association, 72, 439-500.
Stigler, S. M. (1986). The history of statistics. The measurement of uncertainty before 1900. Cambridge, MA: Harvard University Press.
Stigler, S. M. (1999). Statistics on the table. The history of statistical concepts and methods. Cambridge, MA: Harvard University Press.
Boek van Bakker is verkrijgbaar bij firstname.lastname@example.org (E 17,50) maar als pdf ook op internet: http://igitur-archive.library.uu.nl/dissertations/2004-0513-153943/UUindex.html