On the Negative Numbers from the Historical and Educational Perspective

Piotr Błaszczyk, Mirosława Sajka

Abstract


We identify two ways of introducing negative numbers. In the first one, a totally ordered set (L, $\prec $) is presupposed, an element 0 in L is arbitrarily taken, and a number a is negative when a 0. In the second one, a negative number is defined by the formula a + (−a) = 0. From a mathematical perspective, the first method involves the idea of a totally ordered group (G,+, 0,<), while the second one considers the idea of the algebraic group (G,+, 0) alone. Through the analysis of source texts, we show that the first model originates in John Wallis’ 1685 Treatise of Algebra, while the second one comes to form the theory of polynomials, as developed by Descartes in his 1637 La Géométrie. In mathematical education, the first model is applied in the overwhelming majority. Still, we identify a theory that applies to the second model. We show how to develop it further and simplify the representation of the operation a + (−a) = 0 by turning the second model into a tablet game.

Keywords


Negative numbers, History of mathematics, Mathematics education

References


Artin, E., Schreier, O.: 1926, Algebraische Konstruktion reeller Körper, Abhandlungen aus dem Mathematischen Seminar der Hamburgischen Universität 5, 85-99.

Arystoteles: 1831, Physica, w: I. Bekker (red.), Aristotelis Opera, Georg Reimer, Berlin.

Błaszczyk, P., Mrówka, K.: 2013, Euklides Elementy. Księgi V-VI teoria proporcji i podobieństwa, Copernicus Center Press, Kraków.

Błaszczyk, P., Mrówka, K.: 2015, Kartezjusz, Geometria. Tłumaczenie i komentarz, Wydawnictwo Universitas, Kraków.

Battista, M. T.: 1983, A complete model for operations on integers, Arithmetic Teacher 30(9), 26–31.

Bruner, J.: 1978, Poza dostarczone informacje, Mroziak, B. (przeł.), PWN, Warszawa.

Chodnicki, J., Dąbrowski, M., Pfeiffer, A.: 2014, Matematyka 2001. Klasa 6. Podręcznik

do szkoły podstawowej, WSiP, Warszawa.

Dedekind, R.: 2017, Ciągłość i liczby niewymierne, Annales Universitatis Paedagogicae

Cracoviensis Studia ad Didacticam Mathematicae Pertinentia 9, §2.

Dehaene, S.: 1997, The Number Sense, Oxford University Press, Oxford.

Descartes: 1637, La Géométrie, l’Imprimerie de Jan Maire, Leyde.

Frand, J., Granville, E. B.: 1978, Theory and Application of Mathematics for Teachers, 2 edn, Wadsworth Publishing Company, Belmont, Calif.

Freudenthal, H.: 1983, Didactical Phenomenology of Mathematical Structures, Kluwer Academic Publishers, Dordrecht.

Glaeser, G.: 1981, Epistemologie des nombres relatifs, Recherches en Didactiques des Mathematiques 2(3), 303–346.

Grady, M. B.: 1978, A Manipulative Aid for Add-ing and Subtracting Integers, Arithmetic Teacher 26(3).

Hejný, M.: 2008, Scheme oriented educational strategy in mathematics, w: B. Maj, M. Pytlak, E. Swoboda (red.), Supporting Independent Thinking Through Mathematical Education, 1st edn, Wydawnictwo Uniwersytetu Rzeszowskiego, Rzeszów, 40–48.

Howden, H.: 1985, Algebra tiles for the overhead projector, Cuisenaire Company of America, New York.

Klakla, M.: 1982, Procesy psychiczne związane z tworzeniem pojeć i struktur, w: I. Gucewicz-Sawicka (red.), Podstawowe zagadnienia dydaktyki matematyki, PWN, Warszawa, 32–45.

Lakoff, G., Núñez, R.: 2000, Where Mathematics Comes From. How the Embodied Mind Brings Mathematics into Being, Basic Books, New York.

Linchevski, L., Williams, J.: 1999, Using intuition from everyday life in ’filling’ the gap in children’s extension of their number concept to include the negative numbers, Educational Studies in Mathematics 39, 131–147.

Matłosz, B.: 2012, Teaching and learning the elements of arithmetic and algebra using the multiple intelligence theory, Annals of the Polish Mathematical Society, Series 5, Didactica Mathematicae 34(20), 69–84.

MEN, ORE: 2017, Podstawa programowa kształcenia ogólnego z komentarzem. Szkoła podstawowa. Matematyka, Dobra Szkoła.

Moroz, H.: 2018, Wykład 16. numeracja i system pozycyjny, Cz. III, w: B. Nowecki (red.), Materiały pomocnicze do nauczania matematyki w szkole podstawowej. Wykłady telewizyjne, WSiP, Warszawa.

Papy, G.: 1969, Minicomputer, Educational Studies in Mathematics 2(2/3), 333–345.

Peled, I., Mukhopadhyay, S., Resnick, L. B.: 1989, Formal and informal sources of mental models for negative numbers, w: J. G. Vergnaud, G. ans Rogalski, M. Artigue (red.), Proceedings of the Thirteenth International Conference for the Psychology of Mathematics Education, Vol. III, PME, Paris, France, 106–110.

Piaget, J.: 1966, Studia z psychologii dziecka, PWN, Warszawa.

Piaget, J.: 1970, Genetic Epistemology, W.W. Norton, New York.

Sfard, A.: 1991, On the dual nature of mathematical conceptions: reflections on processes and objects as different sides of the same coin, Educational Studies in Mathematics 22, 1–36.

Tanton, J.: 2009, ”Exploding dots”. Advanced version for personal fun. http://testing.mathteacherscircle.org/assets/session-materials/

JTantonExplodingDots_EducatorsVersion.pdf.

Thompson, P. W., Dreyfus, T.: 1988, Integers as transformations, Journal for Research in Mathematics Education 19, 115–133.

Turnau, S.: 1990, Wykłady o nauczaniu matematyki, PWN, Warszawa.

Wallis, J.: 1685, Treatise of Algebra, John Playford, London, przeł. P. Błaszczyk, K. Mrówka (2015).

Zródła internetowe:

https://minecraft-pl.gamepedia.com/Koordynaty

https://tinyurl.com/yb8h4vws


Full Text: PDF (Polski)

e-ISSN: 2450-341X, ISSN: 2080-9751

AUPC SDMP is on the List of the Ministry’s scored journals (part B) with 5 points for 2016