Abstract

In this article, I propose a reconsideration of mathematical content for teaching high school geometry for preservice teacher education based on my own experience teaching geometry courses. The Conference Board report on the mathematical preparation of teachers recommended that future teachers complete three courses focused on school mathematics from an advanced viewpoint (CBMS, 2012). As regards to geometry, they argued that preservice teachers’ preparation should enable them to stand above the content of high school geometry. Kilpatrick (2019) brought to the attention what Felix Klein had called the double discontinuity between university-to-school mathematics and the triple approach Klein had proposed to address it by a unified approach to show how problems in branches of mathematics are connected (e.g., geometry, algebra) and how they are related to the problems of school mathematics. As defined by Klein (2004) and Usiskin, Peressini, Marchisotto, and Stanley (2003), an advanced standpoint on geometry can be developed by focusing on alternative definitions of familiar geometric objects, their extensions, and connections. Here I present my experience conducting classroom experiments on developing an advanced perspective for future teachers exemplifying ways to revamp College Geometry courses preparing future teachers to teach high school geometry.

Here, an advanced standpoint is built by unifying three strands, each of which consists of activities involving the making of connections towards developing a more unified perspective about the geometry content teachers are teaching. The three unifying strands are 1. connections within geometry, 2. connections between geometry and other subjects, and 3. connections between the alternative perspectives on proof and justification practices in geometry. The first strand of activities explores the connections within geometries. Adopting an inquiry-based approach to learning geometry courses, familiar geometrical objects, such as square, rhombus, or parabola, are reconsidered in different geometries with their defining properties and relationships. Students are asked to consider how we can define a geometric object so that once we translate the definition into another geometry, such as hyperbolic, it will still hold. The first set of bridging task sequences builds an inquiry into extensibility of definitions of familiar geometric objects such as quadrilaterals and parabolas across Euclidean and non-Euclidean geometries. Students develop the family relationships among quadrilaterals in different geometries based on their definitions. 

The extensibility of geometric objects, such as rhombus or parabola, across Euclidean and non-Euclidean geometries is investigated to gain a higher standpoint. Teachers explore whether the definitions of geometric objects are extensible by exploring what their definitions would generate in alternative geometries. They discover that rhombus is a shared parent object for equilateral quadrilaterals subsuming squares in Euclidean and quasi-squares in spherical and hyperbolic geometries. In all three geometries, they observe that rhombi have diagonals that are perpendicular and bisect each other. Students use this property as a defining characteristic of squares/quasi-squares in Euclidean and Non-Euclidean geometries. An extensible definition of a square is therefore developed across alternative geometries reconceiving the square as a rhombus with congruent diagonals (See Fig 1). 

Figure 1. Quasi-Squares from rhombus as equilateral quadrilaterals with congruent diagonals extensible to Hyperbolic and Spherical Geometries

Students gain a higher perspective by revising a familiar geometric object in alternative geometries and acquire new meanings for that object by extending the geometric object into other geometries, redefining it through its viable manifestations. The various definitions of parabola are discussed as a case of defining a common geometric object by employing their constructive protocols to consistently build and contrast the features of the produced objects across geometries. Extensibility of the transformation approach from Euclidean into other geometries is discussed by recontextualizing the practice of isometries, symmetries, and dilations in Spherical, Hyperbolic, and other geometries.  

The second sequence of bridging tasks involves connections across geometry, statistics, and irrational numbers. I introduce future teachers to dual modeling task sequences in which they model a common problem, first using a geometric approach and then using another approach, such as statistics. The classroom experiments presented here build a scholarship of teaching and learning mathematics over three years, deliberately modifying and revising the targeted instructional tasks across three classes regularly taught by the author. I do this in various courses including College Geometry (for secondary mathematics teachers), Fundamentals of Mathematics II (Proportional Reasoning, Statistics and Probability), and Fundamentals of Mathematics III (which is a Geometry course for Elementary and Middle Grades Mathematics Teachers). I design, assess, and revise local instructional theories to help to coordinate students’ modeling work and actions across the multiplicity of these mathematical frames. This helps future teachers develop coherent meanings across fields, such as geometry and statistics building on proportional reasoning. In an inquiry-oriented instructional setting, students are intended to discover the mathematics behind modeling stars mainly by statistical and geometric perspectives. In geometrical modeling, students construct pentagons first by using GeoGebra’s given construction and then create the corresponding pentagrams as stars by connecting the diagonals. Students discover the similarity of triangles DAB, ABH, and BHI with a golden proportion as the similarity ratio as seen in Figure 2a below. Then students build their own constructions of pentagons and stars by creating segments with golden proportion as seen in Figure 2b. In statistical modeling, students produce 24 hand-drawn pentagrams as best as they can. Students then measure and find the proportion of the average arm lengths and average distances between the vertices for each star. Students discover that the distribution of these proportions yields the median of 1.617, which is a close approximation to golden ratio. This approach is designed to offer students opportunities to build connections and develop coherent meanings with the geometrical and statistical patterns that emerge in modeling starlike objects.

Figure 2. (a) Discovering golden proportions by connecting the Golden Star, pentagon, and golden triangles using the golden ratio; (b) constructing pentagram from golden proportions; (c) Statistical modeling of a star with a discovery of 1.617 from the distribution of the ratios of average arms over average side across 24 imperfect stars drawn by students 

The third sequence of bridging tasks is about contrasting geometric arguments based on the standpoints of different axiomatic systems such as Euclid’s, Hilbert’s, and SMSG’s (School Mathematics Study Group 1961) and transformational approach. Students investigate and compare the justifications of their arguments as they conjecture, build, justify and validate arguments by creating versions of their theorems and proofs from different standpoints including a transformational approach. Future teachers adjust the level of rigor and justification of the arguments they offer or expect from middle and high school students. 

This paper illustrates the use of three unifying strands for teaching geometry by building an advanced standpoint in geometry for preservice teachers, providing a more unified approach to geometry practice integrating with other disciplines.  

References

Kilpatrick, J. (2019). A double discontinuity and a triple approach: Felix Klein’s perspective on mathematics teacher education. In H.-G. Weigand et al. (eds.), The Legacy of Felix Klein, ICME-13 Monographs, (pp.215-226). Springer. https://doi.org/10.1007/978-3-319-99386-7_15

Klein, F. (2004). Elementary mathematics from an advanced standpoint: Geometry. Mineola, NY: Dover Publications

Matsuzaki, A., & Saeki, A. (2013). Evidence of a Dual Modelling Cycle: Through a Teaching Practice Example for Pre-service Teachers. In G. Stillman, G. Kaiser, W. Blum, & J. Brown (Eds.), Teaching mathematical modelling: Connecting to research and practice (pp. 195–205). New York: Springer.

Usiskin, Z, Peressini, A., Marchisotto, E., & Stanley, D. (2003). Mathematics for high school teachers: An advanced perspective. New York: Prentice Hall. 

School Mathematics Study Group (SMSG). (1961). Geometry. Yale Univ Press.