Abstract
In this paper we present a study of one of the best-known types of descriptive word fraction problems. These problems have disappeared from today’s textbooks but are hugely important for developing arithmetic thinking. The aim of this paper is to examine the historical solution methods for these problems and discuss the analytical readings suggested by the authors. On the basis of this analysis we have conducted a preliminary study of the performance of 35 Spanish students who are highly trained in mathematics. Our results show that these students have a preference for algebraic reasoning, are reluctant to use arithmetic methods, and have reading comprehension difficulties that are reflected in their translations, from literal language to symbolic language, of the relationship between the parts expressed in the problem statement.
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Notes
- 1.
A collection originally compiled by Metrodorus, probably around the 6th century and later on greatly enriched by C. Cephalas in the 10th century and M. Planudes in the 13th century.
References
Aurel, M. (1552). Libro primero, de arithmetica algebraica. Valencia: En casa de Ioan de Mey Flandro.
Bachet C. G. (1612). Problêmes plaisans et délectables qui se font par les nombres. Alton: Chez Pierre Rigaud.
Bruño, G. (1940). Aritmética Razonada. Curso Superior. Madrid: Editorial Bruño.
Cerdán, F. (2008). Estudios sobre la Familia de Problemas Aritmético-Algebraicos. València: Servei de Publicacions de la Universitat de València.
Chemla, K., & Guo, S. (Eds.). (2005). Les Neuf Chapitres, le classique mathématique de la Chine ancienne et ses commentaires. Paris: Dunod.
Colebrooke, H. T. (Trans.). (1817). Algebra, with arithmetic and mensuration: From the Sanscrit of Brahmegupta and Bhascara. London: John Murray (Original: 1150 AD).
De Ortega, J. (1552). Tractado subtilissimo de Arismetica y de Geometria. Compuesto por el reverendo padre fray Juan de Hortega de la orden de los predicadores. Ahora de nuevo enmendado con mucha diligencia por Gonzalo Busto de muchos errores que havia en algunas impressiones pasadas. Sevilla: Juan Canalla.
Descartes, R. (1701). Opuscula posthuma physica et mathematica. Amsterdam: Typographia P. & J. Blaev.
Euler, L. (1822). Elements of algebra. Translated from the French with the notes of M. Bernouilli, &c and the additions of M. De La Grange. Third edition, carefully revised and corrected (J. Hewlett, Trans.). London: Longman, Hurst, Rees, Orme, and Co (Original work published in 1770).
Goldin, G. A., & McClintock, C. E. (1979). Task variables in mathematical problem solving. Philadelphia: Franklin Institute Press.
Gómez, B. (2003). La investigación histórica en didáctica de las matemáticas. In E. Castro, P. Flores, T. Ortega, L. Rico, & A. Vallecillos A. (Eds.), Proceedings VII Simposio de la Sociedad Española de Investigación en Educación Matemática (SEIEM): Investigación en Educación Matemática (pp. 79–85). Granada: Universidad de Granada.
Gómez, B., Sanz, M., & Huerta, I. (2016). Problemas Descriptivos de fracciones. Boletim de Educaçao Matematica (Bolema), 30(55), 586–604.
Hutton, C. (1844). Recreation in science and natural philosophy: Dr. Hutton’s translation of Montucla’s edition of Ozanam. The present new edition of this celebrated work is revised by Edward Riddle, master of the mathematical school, Royal Hospital, Greenwich, who has corrected it to the present era, and made numerous additions. London: Thomas Tegg (Original work of Ozanam published in 1692).
Jacobs, F. (Trans.). (1863). Anthologie Grecque. Traduite sur le texte publié d’après le manuscrit palatin par Fr. Jacobs avec des notices biographiques et littéraires sur les poètes de l’anthologie. Tome second. Paris: Hachette.
Migne, J. P. (Ed.). (1850). Venerabilis Bedae Opera Omnia: Patrologia Cursus Completus, Series Latina (Vol. 90, columns 665–676). Paris: Petit-Montrouge (Original work from 641 AD).
Migne, J. P. (Ed.). (1863). Beati Flacci Albini seu Alcuini Opera Omnia: Patrologia Cursus Completus, Series Latina (Vol. 101, columns 1143–1160). Paris: Petit-Montrouge (Original work from 775 AD).
Ministerio de Educación y Cultura (MEC). (2014). Real Decreto 126/2014, de 28 de febrero, por el que se establece el currículo de la Educación Primaria. Boletin Oficial del Estado (BOE), 52, 19349–19420.
Pérez de Moya, J. (1562). Arithmetica practica y speculativa. Salamanca: Mathias Gast.
Phadke, N. H., Patwardhan, K. S., Naimpally, S. A., & Singh, S. L. (Eds.). (2001). Līlāvatī of Bhāskarācārya: A treatise of mathematics of Vedic tradition: With rationale in terms of modern mathematics largely based on N.H. Phadke’s Marāthī translation of Līlāvatī (Bhāskarācārya, book 1). Delhi: Motilal Banarsidass.
Polya, G. (1966). Mathematical discovery (2 Vols.). New York: Wiley.
Puig, A. (1715). Arithmetica especulativa y practica y Arte de Algebra. Barcelona: Joseph Giralt. Impresor, y Librero.
Puig, L. (2003). Historia de las ideas algebraicas: componentes y preguntas de investigación desde el punto de vista de la matemática educativa. In E. Castro, P. Flores, T. Ortega, L. Rico, & A. Vallecillos A. (Eds.), Proceedings VII Simposio de la SEIEM: Investigación en Educación Matemática (pp. 97–108). Granada: Universidad de Granada.
Sánchez, E., & Cobos, J. M. (Trans.). (1996). Juan Martínez Silíceo. Ars Arithmética. Madrid: Universidad de Extremadura, Servicio de Publicaciones (Original work published in 1513).
Sarasvati, S. S. P., & Jyotishmati U. (Eds.). (1979). The Bakhshali manuscript. An ancient treatise of Indian arithmetic. Allahabad: Arvind Printer (Original work from 300).
Shen, K., Crossley, J. N., & Lun, A. W.-C. (1999). The nine chapters on the mathematical art: Companion and commentary. Oxford and Beijing: Oxford University Press and Beijing Science Press.
Sigler L. E. (2002). Fibonacci’s Liber Abaci: A translation into modern English of Leonardo Pisano’s book of calculation. New York: Springer (Original work from 1202).
Singmaster, D. (1998). Chronology of recreational mathematics. http://utenti.quipo.it/base5/introduz/singchro.htm. Accessed August 9, 2017.
Swetz, F. J. (2014). Expediciones matemáticas. La aventura de los problemas matemáticos a través de la historia (J. Migual Parra, Trad.). Madrid: La esfera de los libros.
Tahan, M. (1993). The man who counted: A collection of mathematical adventures (L. Clark & A. Reid, Trans.). New York: W. W. Norton.
Vinot, J. (1860). Récréations mathématiques. Paris: Larousse et Boyer.
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Sanz, M.T., Gómez, B. (2018). Missing Curious Fraction Problems. In: Clark, K., Kjeldsen, T., Schorcht, S., Tzanakis, C. (eds) Mathematics, Education and History . ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-319-73924-3_10
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