Cuisenaire Art: Modeling Figurate Number Sequences and Gnomonic Structures in a History of Mathematics Classroom

Introduction

All the successive numbers which serve to form the triangular,
quadrangular or polygonal numbers, are called gnomons.

– Theon of Smyrna (XXIII, Lawlor & Lawlor, 1979, p. 25)

This article highlights the author’s teaching experience in a university history of mathematics classroom focusing on the exploration of figurate numbers. Three mathematics history textbooks were adopted throughout the semester (Heath, 1921; NCTM, 1989; Katz, 2009); the present report focuses on students’ exploration of the figurate numbers using the historical resources as outlined in Thomas Heath's A History of Greek Mathematics: From Thales to Euclid (1921), together with the popular mathematical manipulative, Cuisenaire rods. The History of Mathematics course in which these activities were used consisted of students majoring in Mathematics and Mathematics Education (prospective secondary mathematics teachers) who had completed the prerequisite courses of Calculus & Analytic Geometry and College Geometry. The activities were conducted during two 75-minute class periods.

Figure 1. Cuisenaire rods

When modeling counting numbers from 1 to 10 with Cuisenaire rods, each colored rod represents a different number: the white rod represents 1, the red rod represents 2, the light green rod represents 3, the purple represents 4, and so on. This way, rods from white to orange are allotted whole number values 1 to 10, respectively (see Figures 1 and 2).

Figure 2. Cuisenaire rods