Abstract
The latest reform of the French high school education system leads to changes in the content of the curricula. In mathematics, a new theme entitled algorithmic and programming aims at initiating pupils (7th–9th grades) to “write, develop and run a simple program.” To achieve this, the curriculum offers several class activities centered on “games in a maze, …, Nim game and Tic-Tac-Toe.” As the mathematical solution of Nim relies on the binary system, easily characterized by bistable circuits, the first electromechanical Nim playing machines were built in the 1940s, followed later by smaller and purely mechanical machines. This article presents these inventions—which claimed pedagogical purposes—and considers their use in class as a recreational application to tackle the algorithmic and programming theme.
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Notes
- 1.
The content of this new theme can be found on the Ministry of National Education website: http://cache.media.eduscol.education.fr/file/Algorithmique_et_programmation/67/9/RA16_C4_MATH_algorithmique_et_programmation_N.D_551679.pdf.
- 2.
A preliminary version of the present chapter can be found in Radford et al. (2016).
- 3.
As far as I know, Colipan and the group she worked with during her Ph.D. are the only ones, in France, who experimented combinatorial games in class and published didactical results on it. The federative structure “Maths à Modeler” (whose aim is to propose workshops to the general public to discover fundamental computer sciences and mathematics) and “Plaisir Maths” (structure of mathematics popularization which gather animators, teachers and researchers to create and organize playful and didactical mathematical projects) also use combinatorial games in their actions of scientific dissemination.
- 4.
In the same way that there are objects in physics and chemistry, there exist mathematical objects whose handling gives a meaning to theoretical mathematical concepts, and there exist studies on the construction and analysis of such mathematical objects, for instance Caroline Poisard’s thesis on calculation instruments (Poisard 2005, p. 9).
- 5.
In the misère play convention the player who finds himself unable to play wins.
- 6.
- 7.
In “la course à 20” (“the race to 20”) each player adds 1 or 2 to the previous result and the first who reaches 20 wins.
- 8.
In practice, the analysis can be very complex, because of the high number of possible positions in most games.
- 9.
This approach is not new: the oldest analyses of games—which we would qualify nowadays as combinatorial—have been found in recreational mathematics books in the 16th century. Their main purpose was to “tickle curiosity” (Barbin 2007, p. 22), but also to use a playful dimension to acquire mathematical knowledge.
- 10.
For instance, when they play Nim game , pupils quickly start to analyze whether (1, 1), (1, 2) or (1, 1, 1) are winning or losing positions. These observations have been made on a group of 12 pupils (14–15 years old) in the context of a “mathematical summer camp” organized by Plaisir Maths in June 2016.
- 11.
E.g., once pupils have understood that (1, 1) and (2, 2) are losing positions, they can figure out that (n, n) is also a losing position, for any n. The same occurs with (1, 1, n), which is a winning position for any n.
- 12.
The booklet, released in 1951, The Ferranti Nimrod Digital Computer, is available at the following website: http://goodeveca.net/nimrod/NIMROD_Guide.html.
- 13.
Other machines designed to play Nim were created between 1941 and 1958, but in a more mathematical sphere. In 1941, an assistant professor of mathematics at the University of California in Los Angeles (Gardner 1959, p. 156), Raymond Moos Redheffer, improved considerably the Nim-playing machine (Redheffer 1948). To our knowledge, Redheffer’s machines were not exhibited to a broad public, consequently they were less known. In 1952, engineers from W.L. Corporation, Hubert Koppel, Eugene Grant and Howard Bailer, developed a lighter machine than Nimatron or Nimrod, as it weighed less than 25 kg and cost $2000 to build. We would like to note that the machines mentioned in this note had no clearly expressed pedagogical or educational aspirations and were probably not much widespread. Nevertheless, Pollack’s DEBICON (1958) can be found on Popular Electronics magazine cover, which soon became the “World’s Largest-Selling Electronics Magazine” (see Fig. 4, left).
- 14.
His journal Computers and Automation (1951–1973) was the first journal for computer professionals.
- 15.
Simple Simon was exhibited in New York, Seattle, Philadelphia, Boston, Washington, Detroit, Minneapolis, Pittsburgh, and other smaller cities. The fact that Berkeley could take Simon from place to place meant that students and other non-experts could have firsthand contact with automatic computing equipment “for real”.
- 16.
Daniel R. Davies was executive director between 1954 and 1959 of the UCEA (University Council for Educational Administration), an organization aimed to improve the professional preparation of educational administrators.
- 17.
For example, Popular Electronics or Galaxy Science Fiction magazines. A Ngram research in the English Google books corpus shows a net increase of the use of the term “Geniac” between 1955 and 1960: https://books.google.com/ngrams/graph?content=Geniac&year_start=1940&year_end=2000&corpus=15&smoothing=3&share=&direct_url=t1%3B%2CGeniac%3B%2Cc0.
- 18.
For example, during the early 1960s, Berkeley created other electric brain machines with Brainiacs and Tyniacs kits.
- 19.
In honor of Janis Joplin’s rock group Big Brother and the Holding Company (Levy 1984, p. 136).
- 20.
Dr. Nim has been tested with 14–15 years old teenagers during the math summer camp organized by Plaisir Maths in June 2016, and we noticed that, after they studied and understood the strategy of the Nim game , they were curious to know what were the mechanisms of the Dr. Nim machine.
References
Ahl, D. (1978). BASIC computer games. New York: Workman Publishing.
Barbin, E. (2007). Les Récréations: des mathématiques à la marge. Pour la Science, février-avril, 2007, 22–25.
Berkeley, E. C., & Jensen, R. A. (1950). World’s smallest electric brain. Radio Electronic, 1950, 29–30.
Bouton, C. (1901). Nim, a game with a complete mathematical theory. Annals of Mathematics, 3(1/4), 35–39 (second series). doi:10.2307/1967631.
Brougère, G. (1995). Jeu et education. Paris: l’Harmattan.
Brousseau, G. (1998). Théorie des situations didactiques. Grenoble: La Pensée sauvage.
Cohen, H. A. (1980). Modeling Boolean algebra with Tinkertoy or Meccano: How to construct Nim playing machine. In P. Williamson (Ed.), Learn to love Mathematics. Melbourne: Mathematics Association of Victoria.
Colipan, X. (2014). Étude didactique des situations de recherche pour la classe concernant des jeux combinatoires de type Nim. Doctoral thesis, Institut Fourier, Grenoble, France. Retrieved from https://tel.archives-ouvertes.fr/tel-01121726/.
Colipan, X. This volume.
Condon, E. U. et al. (1940). U.S. Patent Number 2,215,544. Machine to play game of Nim. Washington, D.C.: U.S. Office.
Condon, E. U. (1942). The Nimatron. The American Mathematical Monthly, 49(5), 330–332.
Cundy, M. H. (1958). A demonstration binary adder. The Mathematical Gazette, 42(342), 272–274.
Davies, D. R. (1956). The impending breakthrough. The Phi Delta Kappan, 37(7), 275–281.
Nim. (1966). Dr. Nim manual of instructions.
Du Bosque, C. Jr. (1962). U.S. Patent Number 3,048,403. Instructional amusement device. Washington, D.C.: U.S. Office.
Durand-Guerrier, V. (2007). La résolution de problèmes, d’un point de vue didactique et épistémologique. In L. Trouche, V. Durand-Guerrier, C. Margolinas, & A. Mercier (Eds.), Actes des journées mathématiques de l’INRP.
Ellis, A. B. (2011). Generalizing-promoting actions: How classroom collaborations can support students’ mathematical generalizations. Journal for Research in Mathematics Education, 42(4), 308–345.
Gardner, M. (1959). Hexaflegons and other mathematical diversions. The first scientific American book of puzzles and games. Chicago: The University of Chicago Press.
Geniacs. (1955a). Geniacs: Simple electric brain machines and how to make them. Copyright 1955 by Oliver Garfield. Retrieved from: http://www.computercollector.com/cgi-bin/exec/compcol/menu-st.cgi?directory=/archive/geniac/manual.
Geniacs. (1955b). Supplementary wiring diagrams for the Geniac No. 1 electrical brain construction kit. Copyright 1955 by Oliver Garfield. Retrieved from: http://www.computercollector.com/cgi-bin/exec/compcol/menu-st.cgi?directory=/archive/geniac/supplement.
Geniac. (1958). Geniac, an interesting kit builds circuits that solve problems and play games. Popular Electronics, 9(4), 27–29.
Giroud, N. (2011). Étude de la démarche expérimentale pour les situations de recherche pour la classe. Doctoral thesis, Institut Fourier, Grenoble, France. Retrieved from https://tel.archives-ouvertes.fr/tel-00649159.
Godfrey, J. T. (1968). U.S. Patent Number 3,390,471. Binary digital computer. Washington, D.C.: U.S. Office.
Godot, K. (2005). Situations recherche et jeux mathématiques pour la formation et la vulgarisation. Exemple de la roue aux couleurs. Doctoral thesis, Institut Fourier, Grenoble, France. Retrieved from https://tel.archives-ouvertes.fr/tel-00102171.
Levy, S. (1984). Hackers heroes of the computer revolution. New York, NY: Anchor Press/Doublebay.
Longo, B. (2015). Edmund Berkeley and the social responsibility of computers professionals. Morgan & Claypool Publishers.
Morris, D. (1971). U.S. Patent Number 3,606,331. Computer-controlled apparatus for playing game of Nim. Washington, D.C.: U.S. Office.
Nimrod. (1951). The Ferranti Nimrod digital computer. Booklet available at the following address: http://goodeveca.net/nimrod/NIMROD_Guide.html.
Pelay, N. (2011). Jeu et apprentissages didactiques: élaboration du concept de contrat didactique et ludique en contexte d’animation scientifique. Doctoral thesis, Université Claude Bernard, Lyon, France. Retrieved from https://tel.archives-ouvertes.fr/tel-00665076.
Poisard, C. (2005). Ateliers de fabrication et d’étude d’objets mathématiques, le cas des instruments à calculer. Doctoral thesis, Université de Provence Aix-Marseille I, France. Retrieved from https://tel.archives-ouvertes.fr/tel-00011850/.
Radford, L., Furinghetti, F., & Hausberger, T. (Eds.) (2016). In Proceedings of the 2016 ICME satellite meeting of the international study group on the relations between the history and pedagogy of mathematics. Montpellier, France: IREM de Montpellier.
Redheffer, R. (1948). A Machine for Playing the Game Nim. The American Mathematical Monthly, 55(6), 343–349.
Rougetet, L. (2014). Des récréations arithmétiques au corps des nombres surréels et à la victoire d’un programme aux Échecs. Une histoire de la théorie des jeux combinatoires au XX e siècle. Doctoral thesis, Université des Sciences et Technologies de Lille, France. Retrieved from https://ori-nuxeo.univ-lille1.fr/nuxeo/site/esupversions/85c50b29–5227-4616-98a4-2821730f5900.
Rougetet, L. (2016). A prehistory of Nim. In M. Pitici (Ed.), Best writing on mathematics 2015 (pp. 207–214). Princeton, Oxford: Princeton University Press.
Weisbecker, J. A. (1968). US Patent Number 3,338,483. Computer-type game and teaching aid. Washington, D.C.: U.S. Office.
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Rougetet, L. (2018). Machines Designed to Play Nim Games (1940–1970): A Possible (Re)Use in the Modern French Mathematics Curriculum?. In: Hart, E., Sandefur, J. (eds) Teaching and Learning Discrete Mathematics Worldwide: Curriculum and Research. ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-319-70308-4_15
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