Factorial number system as an example of substantial learning environment for pre- and in-service teachers of mathematics

Marek Janasz, Barbara Pieronkiewicz


This article draws on the work of Wittmann and his followers who conceived and developed the notion of substantial learning environment (SLE). The paper contains a proposal of a teaching unit based on the definition of Factorial Number System (FNS). First, we illustrate the process of conversion from FNS to the Decimal Number System (DNS) and back. Secondly, we provide theorems on the divisibility rules for several numbers in FNS. The main aim of this paper is to present FNS as an example of a~mathematically rich environment wherein pre-service teachers of mathematics may be actively engaged in the process of discovering subjectively new mathematics.


factorial number system, divisibility rule, substantial learning environment


Bruner, J. S.: 1966, Toward a theory of instruction, Belkapp Press, Cambridge, Mass.

Bruner, J. S.: 1977, The process of education, Harvard University Press, Cambridge, Mass.

Cajori, F.: 1911, Horner’s method of approximation anticipated by Ruffini, Bulletin of the American Mathematical Society 17(8), 409–414.

Fomin, S. V.: 1974, Number Systems, University of Chicago Press.

Goldin, G. A.: 2002, Representation in mathematical learning and problem solving, in: L. D. English (ed.), Handbook of international research in mathematics education, Lawrence Erlbaum Associates, Mahwah, New Jersey, London, 197–218.

Górowski, J., Łomnicki, A.: 2006, O cechach podzielnosci liczb i odkrywaniu twierdzen, Annales Universitatis Paedagogicae Cracoviensis Studia ad Didacticam Mathematicae Pertinentia 1, 67–73.

Granberg, C.: 2016, Discovering and addressing errors during mathematics problemsolving – A productive struggle?, The Journal of Mathematical Behavior 42, 33–48.

Horner, W. G.: 1819, XXI. A new method of solving numerical equations of all orders, by continuous approximation, Philosophical Transactions of the Royal Society of London 109, 308–335.

Jonsson, B., Kulaksiz, Y. C., Lithner, J.: 2016, Creative and algorithmic mathematical reasoning: effects of transfer-appropriate processing and effortful struggle, International Journal of Mathematical Education in Science and Technology 47(8), 1206–1225.

Jonsson, B., Norqvist, M., Liljekvist, Y., Lithner, J.: 2014, Learning mathematics through algorithmic and creative reasoning, The Journal of Mathematical Behavior 36, 20–32.

Kieran, C.: 2018, Teaching and Learning Algebraic Thinking with 5-to 12-Year-Olds: The Global Evolution of an Emerging Field of Research and Practice, ICME-13 Monographs, Springer.

Knuth, D. E.: 1981, The Art of Computer Programming, volume 2: Seminumerical Algorithms, Addison-Wesley, 2nd edition.

Krauthausen, G., Scherer, P.: 2013, Manifoldness of tasks within a substantial learning environment: Designing arithmetical activities for all, in: J. Novotná, H. Moraová (ed.), Proceedings of the International Symposium Elementary Maths Teaching: Tasks and tools in elementary mathematics, Univerzita Karlova v Praze, Pedagogická fakulta, Praha, 171–179.

Laisant, C. A.: 1888, Sur la numération factorielle, application aux permutations, Bulletin de la Société Mathématique de France 16, 176–183.

Lesh, R., Behr, M., Post, M.: 1987a, Representations and Translations among Representations in Mathematics Learning and Problem Solving, in: C. Janvier (ed.), Problems of Representations in the Teaching and Learning of Mathematics, Lawrence Erlbaum Associates, Hillsdale, NJ, 31–40.

Lesh, R., Behr, M., Post, M.: 1987b, Rational number relations and proportions, in: C. Janvier (ed.), Problems of Representations in the Teaching and Learning of Mathematics, Lawrence Erlbaum Associates, Hillsdale, NJ, 41–58.

Lithner, J.: 2008, A research framework for creative and imitative reasoning, Educational Studies in Mathematics 67(3), 255–276.

Lithner, J.: 2017, Principles for designing mathematical tasks that enhance imitative and creative reasoning, ZDM 49(6), 937–949.

Norqvist, M.: 2016, On mathematical reasoning: Being told or finding out, (Doctoral dissertation). Retrieved from: http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-100999.

Nührenbörger, M., Rösken-Winter, B., Fung, C., Schwarzkopf, R., Wittmann, E. C., Akinwunmi, K., Lensing, F., Schacht, F.: 2016, Design Science and Its Importance in the German Mathematics Educational Discussion, ICME-13 Topical Surveys, Springer, Cham.

Pehkonen, E.: 1992, Using problem fields as a method of change, The mathematics educator 3(1), 3–6.

Puchalska, E., Semadeni, Z.: 1988, Systemy pozycyjne niedziesiatkowe i potegi, in: Z. Semadeni (ed.), Nauczanie poczatkowe matematyki, Wydawnictwa Szkolne i Pedagogiczne, Tom 4, 77–160.

Qwillbard, T.: 2014, Less information, more thinking?: How attentional behavior predicts learning in mathematics, (dissertation). Retrieved from http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-100999.

Scardamalia, M.: 2002, Collective cognitive responsibility for the advancement of knowledge, Liberal education in a knowledge society 97, 67–98.

Solvang, R.: 1994, Thoughts about the free phase in connection to the use of generator problem, Selected Topics From Mathematics Education 3, 56–67.

Tripathi, P. N.: 2008, Developing mathematical understanding through multiple representations, Mathematics Teaching in the Middle School 13(8), 438–445.

Wardrop, R. F.: 1972, Divisibility rules for numbers expressed in different bases, The Arithmetic Teacher 19(3), 218–220.

Winter, H.: 1975, Allgemeine Lernziele für den Mathematikunterricht, Zentralblatt für Didaktik der Mathematik 7(10), 106–116.

Wirebring, L. K., Lithner, J., Jonsson, B., Liljekvist, Y., Norqvist, M., Nyberg, L.: 2015, Learning mathematics without a suggested solution method: Durable effects on performance and brain activity, Trends in Neuroscience and EducationTrends in Neuroscience and Education 4(1–2), 6–14.

Wittmann, E. C.: 1984, Teaching units as the integrating core of mathematics education, Educational studies in Mathematics 15(1), 25–36.

Wittmann, E. C.: 1995, Mathematics education as a ’design science’, Educational studies in Mathematics 29(4), 355–374.

Wittmann, E. C.: 1998, Standard number representations in the teaching of arithmetic, Journal für Mathematik-Didaktik 19(2–3), 149–178.

Wittmann, E. C.: 2001, Developing mathematics education in a systemic process, Educational Studies in Mathematics 48(1), 1–20.

Wittmann, E. C.: 2005, Mathematics as the science of patterns-a guideline for developing mathematics education from early childhood to adulthood, Plenary Lecture at International Colloquium’ Mathematical learning from Early Childhood to Adulthood, (Belgium, Mons).

Wittmann, E. C.: 2009, Operative proof in elementary mathematics, Proceedings of the ICMI Study 19, 251–256.

Zazkis, R., Gadowsky, K.: 2001, Attending to transparent features of opaque representations of natural numbers, w: A. Cuoco (red.), The roles of representation in school mathematics, VA: NCTM, Reston, 146–165.

Zazkis, R., Sirotic, N.: 2010, Representing and defining irrational numbers: Exposing the missing link, Research in Collegiate Mathematics Education 7, 1–27.

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