1 Introduction

This research contributes to conversations around the role university mathematics can play in supporting teacher practice (Wasserman, et al., 2019) and fostering awareness of mathematical connections (Bass, 2022), structures (Taylor, 2018), and ways of being (Mason & Davies, 2013). In particular, we are inspired by the power of geometric models (Whiteley, 2019) and their potential to address the “double discontinuity” (Klein, 1945) between university and secondary school mathematics, which continues to be problematic (Liang, et al., 2022). To address the discontinuity within teacher education, there is a need to foster a general awareness of mathematics as a connected landscape (Bass, 2022), in addition to fostering specific connections within and outside of mathematics, termed intra- and extra-mathematical connections, respectively (De Gamboa, et al., 2022). In our research, we conceptualize this awareness as part of teachers’ knowledge at the mathematical horizon (KMH), which is characterized by an understanding of mathematical structure, practices, and values that allow for an interconnected view of the mathematical world and how to be within it (Zazkis & Mamolo, 2011; Mamolo & Taylor, 2018).

In our review of the literature, we found extensive research which highlights the power of, and competencies for, teaching mathematical modelling for fostering extra-mathematical connections (Kaiser & Schukajlow, 2022; Kaiser et al., 2022), yet we found less attention paid toward the intra-mathematical connections made possible via modelling approaches. Nevertheless, intra-mathematical connections are recognized as similarly motivational for learners (Schukajlow, et al., 2022). In this paper, we seek to extend understanding of how intra-mathematical connections between concepts and practices can be fostered through the creation and use of a tangible geometric model.

In line with Sullivan et al.’s (2013) notion of a purposeful representational task, we consider how geometric models can provide opportunities to represent and broaden connections between core mathematical concepts, such as permutations and reflections. We also consider how geometric models can provide opportunities to connect school mathematics practices to universal ways of being enacted by mathematicians, such as seeking alternative representations, visualization, and play (Taylor, 2018; Whitelely, 2019).

2 Review of the literature

2.1 Mathematical models for mathematics education

The history of model making and use in mathematics dates back to the early 18th century. As Schubring (2010) recounts, the Modellkammer, a collection of mathematical models introduced by J.A. Segner that emphasizes applications of mathematics to modern technologies, was an early feature of the progressive teaching approach at Göttingen University. By the 20th century, however, the Modellkammer was dissolved by then-director H.A. Schwarz. Instead, Schwarz began to establish a collection of geometric models, which Klein lobbied to expand, noting the “necessity of “Raumanschauung” – spatial intuition – for successful mathematics teaching” (ibid.). For Klein, “The tendency to crowd intuition completely off the field and to attain to really pure logical investigations seems to me not completely feasible” (1945, p.14). As Cumino et al. (2021) note, the latter half of the 19th century saw progress in the study of geometry, which led to the creation of models for algebraic surfaces and curves. Yet, the shift to more abstract and analytic approaches of the early 20th century saw a decline in model use and production. Nevertheless, they note that scholars pushed back against verbal and abstract modes of teaching to advocate for “a multimodal learning of mathematics, developed (especially for geometry) through the use of concrete materials and movement” (Cumino, et al., 2021, p.153). Klein also emphasized the importance of teachers’ accruing experiences building and using models (Mattheis, 2019).

Model building has at least two senses in the mathematics education literature: building physical models in the sense of making objects, and working with mathematical models of real-world scenarios. We note that Klein was interested in models in both of these senses; the physical models were held in the Modellkammer and the importance of modelling competencies was stressed in Klein’s Meraner reform (Mattheis, 2019). Research regarding modelling in this second sense is flourishing (Cevikbas, et al., 2022), and attention has focused on conceptualizing and fostering modelling competencies (Garcia, et al., 2006; Kaiser & Schukajlow, 2022; Maaß, 2006; Niss and Højgaard, 2019). The role of teachers, and the importance of supporting and enhancing their modelling competencies for teaching, is a key theme. This theme resonates with Schubring’s (2010) outlook on the Modellkammer and Klein’s reform initiatives: “one has to return to teacher formation as key prerequisite for any sustainable improvement of mathematics instruction” (p.8). Mathematical modelling plays a role in connecting school mathematics to the broader world, where the possible connections are influenced by the sense in which modelling is interpreted (Garcia, et al., 2006).

In our research, we seek to extend conversations around mathematical modelling in teacher education by exploring learning opportunities that can be fostered through building a physical geometric model, which we see as related to the competency of solving mathematical problems within a mathematical model (Cevikbas, et al., 2022). In our case, the problems to be solved relate to the algebraic structure captured within the spatial visual representation of the geometric model.

2.2 Spatial visual models and teacher education

Spatial visualization is central to mathematical reasoning (Arcavi, 2003; Cumino, et al., 2021) and “has always been a vital capacity for human action and thought, but has not always been identified or supported in schooling” (Whiteley, et al., 2015, p.3). Recognized challenges in geometry instruction have been linked to procedural and rote approaches that emphasize deductive aspects of the subject while neglecting underlying spatial sense (Del Grande, 1990). Subsequently, research on teaching and learning of geometry has focused on how to foster spatial-visual approaches through uses of technologies, as well as on the professional development experiences needed to support teachers (Jones & Tzekaki, 2016).

Nagar et al. (2022) note that teachers’ past experiences with mathematics influence their interactions with, and expectations for, geometric models. Sinclair et al. (2011) found that preservice teachers who had previously experienced mathematics via teaching methods that tackled topics separately and disjointedly encountered difficulties reasoning with, and inferring from, geometric models. They recommended that teachers encounter more robust and extended experiences exploring geometric models of typically algebraic tasks so as to “develop exploration strategies, build connections, and learn to notice mathematical details [of the models]” (p.156). They call for learning opportunities that can help “teachers revise their approach to mathematics” (ibid. p.157).

Recent research investigating the double discontinuity between university and secondary school mathematics highlights the importance of coherent experiences and values amongst learning environments, in particular because such experiences inform how preservice teachers envision their future roles (Liang, et al., 2022). The use of tactile models, or manipulatives, has been recognized as a useful pedagogical approach to help scaffold difficult lessons or concepts (Anghileri, 2006; Whiteley, 2019) though their use for exploring mathematics at the post-secondary level is not as well accepted. The predominance of lecture-based approaches in university mathematics courses leads students to believe that the use of tangible models is no longer appropriate as mathematics becomes more advanced and abstract. Incorporating instructional methods that support computational fluency with geometric intuition at the undergraduate level via the use of purposeful representational tools (Sullivan, et al., 2013) can help legitimize the relevance of such approaches to mathematical activity in general. As Watson and Ohtani (2015) observe, “Tasks shape the learners’ experience of the subject and their understanding of the nature of mathematical activity” (p.3). Mamolo et al. (2015) observe that geometric and spatial visual models provide learners of varying mathematical sophistication with a concrete tool through which to both represent and advance their conceptual understanding of major disciplinary ideas that cut across curricula. They emphasize that conceptual thinking involves connections across mathematical strands and a flexibility and fluency in navigating amongst multiple representations of the same idea. Geometric models in which mathematical structure appears tangible to the learner can provide a mechanism for fostering such connected and conceptual understandings of mathematics.

In this research, we investigate an instructional approach informed by the Erlangen Program (Klein, 1893; 1945), which uses tactile models as purposeful representational tools of symmetric groups in abstract algebra. Abstract algebra is commonly required in teacher preparation programs as it engages “students in the mathematical activity of defining a structure as a means of learning about it” (Zbiek & Heid, 2018, p.190). In particular, we are interested in how a tactile model can be used to cultivate geometric and spatial-visual intuition of algebraic structure, to help legitimize the epistemological value of model-making in mathematics, and to foster a sense of cohesion of ideas within the greater mathematical world.

2.3 Structural understanding and abstract algebra

Researchers have pointed to the relevance of structures in abstract algebra to school teaching, noting that an ultimate goal of school algebra is to foster understanding of mathematical structures (Usiskin, 1988). Groups were among the first algebraic structures to be axiomatized in the nineteenth century (Wussing, 2007). Historically, the discovery of group theory proceeded from particular isolated groups (e.g. of permutations) towards a general theory of abstract groups (Kleiner, 1986). In modern abstract algebra courses, students are introduced to group, field, and ring theories, often as mandatory part of mathematics degrees and teacher education. Lee and Heid (2018) advocate for instructional approaches that focus on developing a structural perspective of abstract algebra, which they articulate as “recognizing and having a tendency to look for mathematical objects, as well as knowing and using the relationships among mathematical objects” (p.295). Nevertheless, learners struggle with abstract algebra (Dubinsky, et al., 1994) and the challenges of reasoning with and acting upon mathematical objects at such a high level of abstraction (Hazzan, 1999). The struggles often relate to the process-object duality recognized in the APOS Theory (Dubinsky & McDonald, 2001), and articulated by Sfard (1991) as operational versus structural understandings, respectively.

In APOS terms, the process-object duality refers to the dual nature of mathematical ideas and representations as both procedures to be carried out (processes) and as encapsulated entities with their own properties (objects). For instance, when considering reflective symmetries, an individual with a process conception might think of a reflection as a procedure of flipping in order to determine a new image. In contrast, an individual with an object or structural conception might think of a reflection as a group element with order two on which other actions or processes may be applied. Symmetries admit other connections to object-process duality as well. Breive (2022) discusses a tension between how individuals perceive symmetry and how we describe it. Individuals can visually appreciate symmetry as a whole, but “the linearity of speech does not allow us to refer simultaneously to two different points, rather a symmetrical interaction between two entities is transformed into one entity that interacts with the other entity” (Breive, 2022, p.318).

We suggest that tangible geometric models which can portray symmetries without relying on the linearity of speech may help facilitate shifts in attention needed to begin to conceptualize symmetries structurally as objects in addition to processes. With respect to Sfard’s (1991) work, a structural understanding relates to a trans-object level of mathematical analysis, in which “relations across the objects are established in the learner’s mind and a collection of the object and others cohere and form a developed schema” (Lee & Heid, 2018, p.293). Such an understanding connects with the disciplinary knowledge for teaching that fosters a sense of the interconnectedness of mathematical ideas, practices, relationships, and domains – their knowledge at the mathematical horizon.

3 Theoretical framework

Klein’s Erlangen Program promoted a conceptualization of mathematics that emphasized its connections. Klein was interested in showing “the mutual connection between problems in the various fields” (1945, p.1–2) and supporting teachers in developing “the ability to draw (in ample measure) from the great body of knowledge there put before you a living stimulus for your teaching” (ibid., p.2). This connects to Klein’s idea of fostering an advanced perspective of elementary mathematics applicable to teaching situations. As Schubring (2010, 2019) notes, the English language translation skews some of the original intent of Klein’s work, misrepresenting notions of “elementary” and “advanced” as “simple” and “academic”, respectively. In contrast, Schubring (2019) asserts that Klein’s perspective of an advanced standpoint is one that provides a higher view of mathematical interconnectedness, including connections across disparate mathematical strands and amongst school and university mathematics; his notion of elementary mathematics is one that speaks to the fundamental ideas of the discipline.

Following Klein, Zazkis and Mamolo (2011) argued that the mathematical subject matter knowledge acquired in university studies can help provide a higher view of fundamental ideas in (school) mathematics. They conceptualize knowledge at the mathematical horizon (KMH) as an advanced understanding of mathematical structure, practices, and values that situates the mathematics of the moment within a greater mathematical world, and which can be leveraged in teaching situations in order to support and promote students’ mathematical activities. They extend the work of Ball and Bass (2009), who introduced KMH as knowledge that “engages those aspects of mathematics that… illuminate and confer a comprehensible sense of the larger significance of what may be only partially revealed in the mathematics of the moment” (p.5).

As per Ball and Bass (2009), KMH includes four elements: (i) a sense of the mathematical environment surrounding the mathematics of the moment, (ii) major disciplinary ideas and structures, (iii) key mathematical practices, and (iv) core mathematical values and sensibilities. In the perspective developed by Zazkis and Mamolo (2011; also Mamolo and Taylor, 2018), KMH is closely connected to a teacher’s focus of attention and “her ability to flexibly shift attention such that relevant properties, generalities, or connections, which embed particular mathematical content in a greater structure [can be] accessed in teaching situations” (Mamolo & Taylor, 2018, p.434). Shifts in attention are key mechanisms through which individuals develop mathematical awareness for teaching that includes both subject matter mastery and an ability to articulate how to foster such mastery in others (Mason, 1998).

To understand some of the shifts of attention necessary for mathematical understanding, Zazkis and Mamolo (2011) connected the four components of KMH to Husserl’s philosophical notions of inner and outer horizon, extending it to encompass the abstractness of a mathematical object. An object’s inner horizon is conceptualized as specific features of the object itself and the attributes evoked by its particular representation. As such, an object’s inner horizon depends on our focus of attention, and may shift as attention also shifts. The outer horizon represents the “greater mathematical world”, including the general structures, values, and sensibilities, which are relevant to, and exemplified by, the particular object. Mamolo and Taylor (2018) articulated connections across conceptualizations of KMH, as depicted in Table 1.

Table 1 Interpreting components of KMH (Mamolo & Taylor, 2018, p.435)
Full size table

With these links, Mamolo and Taylor (2018) analyze how group theoretic ideas are a valuable component of mathematical knowledge for teaching. They articulate pathways from specific “locations” of the mathematical landscape of the moment, through the greater mathematical world, and back again, analysing examples that connect group structures to inverse functions (Zazkis & Marmur, 2018), isomorphisms to functions (Wasserman & Galarza, 2018), and Pythagorean triples to ring structures (Cuoco, 2018). Zazkis and Marmur (2018) suggest that group theory “can serve as a mathematical guide for teachers in situations of contingency” (p.379), while Mamolo and Taylor (2018) elaborate on the central importance of paying explicit attention to the mathematical structures of abstract algebra.

In abstract algebra, “we compare and contrast structures, exemplify and extend them, investigate implications, push boundaries, and play with relationships, all with a sort of directness and cohesion” (Mamolo & Taylor, 2018, p. 438). These ways of being mathematical are connected to schoolteachers’ practices and decision-making. Planning decisions such as which definitions or properties to address, what emphases to place (Wasserman & Galarza, 2018), task design decisions that illuminate rather than cloud underlying solving methods (Cuoco, 2018), and in-the-moment responses to unanticipated student questions or confusions (Zazkis & Marumur, 2018) are shaped and influenced by the underlying structure, and disciplinary practices and values embodied in abstract algebra.

While much of Klein’s considerations for developing connected understandings focused on specific mathematical ideas such as logarithms, he also took more general approaches when considering the higher standpoints afforded by geometry (Allmendinger, 2019). In this paper, we also take a general approach and consider how a geometric model can foster connections between abstract algebra and geometry, as well as between permutations and group elements, and between mathematical ways of being that are experienced in early schooling and at university levels. In line with Zazkis and Mamolo (2011), we conceptualize KMH “as an advanced perspective on elementary knowledge, that is, as advanced mathematical knowledge in terms of concepts (inner horizon), connections between concepts (outer horizon), and major disciplinary ideas and structures (outer horizon)” (p.12). We suggest that an advanced standpoint, or higher view, of fundamental ideas can foster KMH, both in terms of teachers’ knowledge of disciplinary ideas, practices and sensibilities, as well as how ideas are connected and situated within a greater mathematical world. In this paper, we consider how tangible, geometric explorations of group theory may provide opportunities to foster KMH, with particular attention to importance of mathematical structure and interconnectedness. As such, we explore the research question:

In what ways can tactile experiences with geometric models in undergraduate mathematics provide for opportunities to foster knowledge at the mathematical horizon?

4 Methodology

4.1 Design of the dihedral calculator

Klein introduced an approach to geometry, which was independently developed by Poincaré and Lie (Hawkins, 1984), that focused on classifying geometries by their symmetry groups. This changed the emphasis of geometry from the study of particular spaces and their associated metrics, to the study of spaces and their symmetry groups. Norton (2019) notes that such an approach “focuses our attention on the dynamic operations that define mathematical objects rather than the static figures that we use to represent them” (p.27). Piaget (1970) noted this as a “radical change of the traditional representational geometry into one integrated system of transformations” (p.22). This shift situates geometry within the framework of algebra and brings it in line with contemporary undergraduate education. For Klein, the “strong development of space perception, above all, will always be a prime consideration” (ibid., p.4). Indeed, strengthening learners’ “spatial intuition” was a key objective of Klein’s (Mattheis, 2019, p.93), as well as of his contemporary Peter Treutlein (Weiss, 2019), who leveraged paper folding to foster spatial intuition and teach modern approaches to geometry. Folded-paper models are notable as the transformation involved in producing a 3D shape from a 2D sheet of paper has precise mathematical meaning (Cumino, et al., 2021). The Dihedral Calculator is an example of paper craft with a deliberate purposeful design, which we developed to facilitate exploration of fundamental ideas in abstract algebra.

The dihedral groups \({D}_{n}\) are one of the first examples of groups studied in undergraduate abstract algebra, and represent the group of symmetries of regular n-gons in the plane. The group structure of \({D}_{n}\) can be depicted geometrically as polygons with dotted lines marking the lines of reflection (see Fig. 1 by Keith Conrad).

Fig. 1
figure 1

Geometric representations of D3, D4, D5, and D6

Full size image

The purpose of the Dihedral Calculator is to make the geometric transformations of the dihedral group palpable by allowing learners to experiment with rotations and reflections. As a model, the dihedral calculator consists of two regular n-gons: the base and the disk, made out of paper. The front and back of the disk are labelled, as in Fig. 2, and each vertex of the disk is assigned a unique label. Notice that the vertex labels on both sides of the disk agree at the vertices. The vertices of the base are labelled like the vertices of the disk, and there is a pair of reflections s and t. To assemble the calculator, place the disk on top of the base. The disk should be able to move freely, rotate, and be turned over. The model is designed so that positions of the disk, which correspond to symmetries of the n-gon, can be viewed and described as permutations of the vertex set and as products of the reflections s and t. Writing the reflections s and t as permutations in cycle notation, we have \(s=\left(25\right)\left(34\right)\) and \(t=\left(12\right)\left(35\right)\).

Fig. 2
figure 2

The disk (left and center) and the base (right) with their labels

Full size image

Consider the position of the disk illustrated in Fig. 3 (right). This position can be written as a permutation of the vertices of the disk as follows: \(p=\left(15\right)\left(24\right)\). Notice that we map the vertices on the base to the values on the disk; the importance of this convention is discussed in Subsection 5.1. One can verify that \(sts=p\) by direct calculation with permutations. The physical model allows individuals to instantiate concepts from abstract algebra, such as permutations and cycles, in a concrete physical form. This physical representation builds a familiarity with core concepts of abstract algebra while encapsulating dihedral structure and structure-preserving mappings that can act on the regular n-gon. It also helps to develop a sense that groups naturally occur as the symmetries of real world objects.

Fig. 3
figure 3

The dihedral calculator assembled (left) and after the transformation sts (right)

Full size image

The Dihedral Calculator provides an instance of a group admitting distinct presentations – via permutations and via reflections, fostering awareness of the mathematical value of multiple perspectives on a single problem. As elements of the symmetry group, the reflections s and t have order two, however, this is not the only relation they satisfy. Experimenting with the physical model shows that st produces a counter-clockwise rotation by 2\(\pi\)/5 about the center of the disk. If the transformation st is iterated five times, then the disk rotates through 2\(\pi\) radians and returns to its initial position. This highlights an additional relation among the generators s and t. Working at the level of permutations, we see that:\(st=\left[\left(25\right)\left(34\right)\right]\left[\left(12\right)\left(35\right)\right]=\left(54321\right)\)which is a cycle of order five. Thus, we can present the group by its generators and relations as:

$${D_5} = \left\langle {s,t|{s^2} = {t^2} = {{\left( {st} \right)}^5} = e} \right\rangle$$

4.2 Data collection and analytic approach

This study took place in a fourth year geometry course at a large Canadian university. The course is a degree requirement for all mathematics majors and its prerequisites include introduction to proof, linear algebra, and group theory. The textbook used was Sosinskiĭ (2012), which defines a “geometry in the sense of Klein” to be a pair (X : G) where X is a set and G is a group of automorphisms of that set. To make the high level of abstraction inherent in this perspective more concrete, a component of the course involved model building to foster tactile experiences of the groups involved. We focus this paper on one model from the course. A written description of how to assemble the model and a short silent video showing two hands performing the main steps of the construction were provided to students. Students were asked to construct a Dihedral Calculator from paper and use their construction to answer questions about group elements and structure.

This course ran September to December 2020; during the height of the COVID-19 Pandemic and all instruction and research activity was shifted to remote online environments. Given the nature of the online constraints during this time, data collection was limited to participant artefacts, which were submitted through the course’s learning management system. Our data consists of participants’ responses (N = 39) to the written component of the tasks along with photographs of their models; their written work was in response to both mathematical activities as well as feedback on their engagement with the model. For the calculational part of written component, questions addressed the behaviour of combining reflection and rotations. For instance, participants were asked to use their models to compare and contrast the group elements srk and rks, where \(r=st\)is a rotation, and their respective behaviours. In addition, participants were asked to explore equivalency classes of D5, as well as to pose their own original question based on using the model to solve a related mathematical problem. The tasks were designed so that students would provide multiple representations of their responses – briefly with fifty-words-or-less written descriptions, computationally with algebraic notation, and visually with diagrammatic representations and photographs of their physical model. Photographs of the physical models were taken at various stages of the exploration, including when the model was in the identity position and when it was positioned to depict non-trivial group elements. Given limitations imposed on our data collection procedures that prevented us from observing students interact with the model in real time, we relied on comparisons of photographs taken at multiple stages of the exploration for our analyses. For instance, to infer how the model could have been used in order to depict a nontrivial group element, we compared photographs of the final position of the model with photographs of the identity position. We then recreated possible steps that would yield the position of the nontrivial group element from the identity position. The axiomatic structure of group theory, and the uniqueness of a group’s identity element and inverse elements, allowed us to conduct this analysis with a high degree of confidence and reliability.

We thematically examined the data, first coding independently and then meeting several times to compare. We analysed the photographic data of how participants constructed, labelled, and depicted their models, focusing on spatial visual features of the models, such as the position of the disk and orientation of the labels. We crossed analysed these features with the respective drawn diagrams, written descriptions, and algebraic calculations. We first coded with respect to components (i) and (ii) of Table 1, examining structural similarities and discrepancies amongst the photographed models, the diagrams, and the algebra, seeking to interpret what possible connections within the mathematical horizon could be afforded by the model. We attended to participants’ notational choices, both in how they depicted their models and images, as well as in their algebraic calculations. Subtleties in how the models were built (with what materials), positioned (disk relative to base), and labelled (disk and base), provided insight into participants’ reasoning and different aspects of the mathematical horizon which could be fostered through an Erlangen-inspired approach to abstract algebra. We then re-examined the data and coded with respect to components (iii) and (iv) of Table 1, analyzing the data for evidence of connections beyond their particular model to general ideas, such as the relevance of modelling practices and spatial visual reasoning in mathematics.

5 Findings

We examine ways in which tactile experiences with a physical geometric model of the Dihedral Calculator can provide for opportunities to develop KMH. We analyze our data with respect to the four components of KMH: (i) the mathematical environment surrounding the current ‘location’ of learning; (ii) major disciplinary ideas and structures; (iii) key mathematical practices; and (iv) core mathematical values and sensibilities. As we are concerned with teachers’ horizon, we include in our analyses consideration of inner and outer horizons with an eye toward how our purposeful representational tool can facilitate shifts in how students attended to the mathematics at hand (inner horizon) and what aspects of the greater mathematical world were brought into view as a result (outer horizon).

When considering KMH through the lens of inner and outer horizons, we see the interconnected nature of the four components of Ball and Bass’s (2009) conceptualization. For instance, a structural understanding of mathematics contributes to an individual’s knowledge of the mathematical environment, and awareness of mathematical sensibilities guides ways of working with specific practices. As such, we organize our findings in two sections. Section 5.1Highlighting key concepts with purposeful representational tools analyses instances of structural understanding that can emerge from students’ engagement with the Dihedral Calculator and what aspects of the mathematical landscape were brought into view by the purposeful design of the model. Section 5.2Exemplifying the duality of mathematical ideas and their representations analyses the mathematical sensibilities that can be fostered through exploration with the model and the key mathematical practices that can be brought into view.

5.1 Highlighting key concepts with purposeful representational tools

Our findings revealed that the experience exploring group theoretic ideas via a tactile model provided new opportunities for students to make connections amongst geometry and algebra, as well as amongst specific ideas within algebra, corresponding to “major disciplinary ideas and structures” (Table 1, component ii). In general, students (n = 27) reported in their feedback that they “learned a lot” from making the model, and they appreciated the surprising connections amongst abstract algebra and geometry. For instance, students remarked on how the course drew their attention to “how different branches of math can relate” and how “things tend to fit together nicely in almost a magical way” suggesting a broadening of their outer horizons and expectations for the interconnectedness of the mathematical landscape.

An important aspect of the mathematical landscape (Table 1, component i) captured by the Dihedral Calculator is the connection amongst permutations, reflections, and rotations. In particular, the model provides students a way to develop a feel for how the group operations of D5 behave, emphasizing structural aspects that exist in the outer horizon of every group. To answer the questions, students needed to play with the disk, spinning it or flipping it, to intuit (in the sense of Descartes, see Garber (1998) the position of the disk and structure of the accompanying group element. Depicted in Fig. 4 is a typical paper construction of the Dihedral Calculator, depicting the identity position. In general, the position of the disk corresponds to a permutation of the vertices of the disk, and in this way, the model captures the group structure and provides a tactile mechanism to determine the behaviour of group elements, which correspond to major disciplinary ideas (Table 1, component ii).

Fig. 4
figure 4

A student’s construction of the dihedral calculator, marking an axis of reflection, s

Full size image

The model required students to read the permutations as maps from the base to the disk so that each generator could be replaced by its representation in cycle notation in order to obtain a permutation that accurately described the position of the disk. The calculator was used to explore equivalence classes, another major disciplinary idea, by rotating or flipping the disk and noting its final position, which offered an embodied experience connecting permutations, reflections, and rotations. For instance, when asked to calculate \({r}^{2}sr\), students were required to physically rotate the disk, flip the disk along the axis of reflection, and then rotate it twice to find the final position. One such example is provided in Fig. 5, where the student used their physical model to determine the final position of \({r}^{2}sr\), which they wrote as (15)(24). In group-theoretic terms, the convention of reading from the base to the disk is consistent with having both the group generated by r and s and the permutation group act on the right as group actions, which was a strategic design choice for this purposeful representational tool. The design decision to read from base to disk also allowed for students to self-check their work – students who read the permutation from the disk to the base (as opposed to from the base to the disk) obtained cycles that were inverses of the intended permutations, instantiating a common challenge learners face when composing group elements. The majority of students (approximately two thirds) either labelled their disks correctly right away or noticed a discrepancy when self-checking and adjusted their labelling, which was evidenced by erasing marks on the models or diagrams and crossed out written work in the algebraic computations. For the calculator to produce consistent results as permutations, the two sides of the disk must also have their vertices labeled consistently. That is, each vertex must have the same label on both sides of the disk. A handful of students labeled the disk so that the numbers appeared in clockwise order on both sides of the disk. This produced a model where the vertices were given different labels on either side of the disk. We suggest that students who made this labeling error struggled to connect permutation and generator/relations approaches to generating a group. Thus, while the Dihedral Calculator can help connect these ideas within the horizon for some students, for other students a more explicit articulation of these connections and their relationship seemed needed.

Fig. 5
figure 5

Connecting permutations, reflections, and rotations via the position of\({r}^{2}sr\)

Full size image

The physical representation of the Dihedral Calculator allowed students to view operations in a spatial visual way and as such apply actions on these operations, a key consequence that marks and results from encapsulating processes into mathematical objects (Dubinsky & McDonald, 2001). Shifting between process and object conceptions of group operations connects to both inner and outer horizons of the dihedral group. On one hand, the ontological shift from process to object can be described as a shift in attention between intended properties of the operation (inner horizon), such as reflection as something to carry out (process) or as something upon which actions may be carried out (object). On the other hand it also instantiates a phenomenon that exists across the greater mathematical world (outer horizon): the dual nature of mathematical ideas and their representations. In what follows, we highlight instances where students internalized the physical actions carried out with the models and in doing so engaged with key mathematical practices, and we link an understanding of mathematical structure with an awareness of core mathematical sensibilities.

5.2 Exemplifying the duality of mathematical ideas and their representations

Our findings showed evidence of students (approx. one third) internalizing the actions of rotating and flipping the disk into an imagined process, which could then be executed to understand the group element in a structural way as a combination of operations. We see this as engaging in “key mathematical practices” within KMH (see Table 1, component iii). We view the internalization of model elements and actions as indication of student learning, as it allowed students to express their KMH without the need of an external physical guide. For instance, the example depicted in Fig. 6 is interesting, as this student chose to depict their model virtually, without a physical model with rotating disk. We suggest that this student could separate their physical model from their conceptual representation of it, and make this separation in a way that acknowledged the relevant group of symmetries. In this virtual model, the student demonstrated the flipped orientation of the disk with backwards (reflected) text for the bottom and its numbered vertices. In the paper model (Fig. 5) depicting \({r}^{2}sr\), the student positioned the calculator such that the disk was oriented with the vertex 1 pointing up, while the base was rotated. In contrast, the virtual model (Fig. 6) keeps the base constant and imagines the computations acting on the disk to depict its new orientation for the element \({r}^{2}sr\). Internalizing the action of rotating the disk involved coordinating multiple steps and keeping track of how each step influenced the position of the disk. We note that the labels of the disk are reflected yet not rotated, suggesting that the student enacted the group operations internally and then labelled the diagram, rather than creating a virtual model in which the image was rotated using computer software.

Fig. 6
figure 6

A virtual model depicting the position of\({r}^{2}sr\)

Full size image

Another example of internalizing algebraic structure was observed when students were asked to draw the ten dihedral symmetries of a regular pentagon, and a student decided to take a short cut (see Fig. 7). An accurate drawing of the physical model would show the labels of the vertices and faces rotating. In drawing the ten positions of the disk, this particular student began by rotating the “top” label, as depicted in their second drawing. They then chose to label the subsequent faces of the disk but not orient them consistently with the physical model. This student could abstract from the physical model to convey the necessary information to depict the required position of the disk, while ignoring non-mathematical details such as the labels “top” and “bottom”, which prior research with pre-service teachers suggests is challenging to do (e.g., Sinclair, et al., 2011). In a comment below the figures, the student acknowledged that they were making an aesthetic choice that was inconsistent with the physical model, but still conveyed the essential structural information of the model.

Fig. 7
figure 7

Student visualizes rotating the disk while leaving the labels invariant. Note the student’s comment: Number and “Top” and “Bottom” are not accurately written for the sake of tediousness

Full size image

The experience reasoning with a physical model in an undergraduate course also broadened students’ perspectives about what mathematical practices and sensibilities are inherent in advanced mathematics, including leveraging visual reasoning to inform proving approaches and to represent group structures. Many students (n = 14) went on to pose interesting problems related to building models. As one student put it:

I enjoyed making the models in this course because I get to take a break from just doing math problems, and I think it’s very helpful to visualize what we are learning… because it gets confusing just imagining it in my head, for example counting rotational symmetries.

In this comment, we observe the common notion that mathematics is about “doing problems… in my head”. The model making experience offered “a break” from math in the student’s point of view, yet in Klein’s (1893) perspective, the model is the math. This student’s comments highlighted the epistemological value of the model, as “it’s very helpful to visualize what we are learning”, thus giving credence to the use of models for exploring and explaining higher mathematics and broadening understanding of key mathematical practices and sensibilities for solving problems (see Table 1, components iii and iv). Further, the model provided opportunities for participants to generalize their reasoning. For instance, we noted participants (n = 19) speculating on the applicability of such a calculator for other groups, both of higher and lower order. One participant thought to “Design a Calculator for a 6 faces cube” and “find the corresponding rotational axis for all the symmetries”. Consideration of how to extend the calculator to higher dimension suggests an expectation of an analogous group structure, despite the added complexities of modeling symmetries of 3D shapes, as well as of the appropriateness of using a model for such advanced mathematics. While tactile models, or manipulatives, tend to be recognized for their scaffolding properties and support for early mathematical experiences, the use of models at more advanced levels is less commonly seen or appreciated. Part of the horizon includes understanding of key mathematical practices, such as model making and visual reasoning, and when they can be applied (see Table 1, component iii). In addition, the physical representation can support students’ generalizations about the structure of group elements such as \(s{r}^{k}\) and \({r}^{k}s\) by imagining how the calculator would apply. The majority of students articulated the general structure of both elements in terms of the model, and described how they would manipulate the disk to determine the final positions of each element. We saw evidence that this revealed for them equivalences amongst group elements, as exemplified in Fig. 8. The connections made across physical, virtual / visual, and symbolic representations of the group structure are interpreted as a broadening of participants’ KMH and enacting “core mathematical values and sensibilities” (Table 1, component iv).

Fig. 8
figure 8

Student intuits general structure of group elements and equivalences

Full size image

6 Discussion

In this paper, we examined ways in which tactile geometric models in undergraduate mathematics can help bring into view aspects of the mathematical horizon that are relevant for school teaching. Tactile geometric models can draw attention to mathematical structure, foster conceptual understanding, reveal and reinforce intra-mathematical connections, and make complex abstract ideas tangibly accessible (Mamolo, et al., 2015). In the context of teacher education, they broaden and strengthen reasoning skills and pedagogical approaches, giving credence to geometric sensibilities and ways of being in mathematics (Sinclair, et al., 2011). The Dihedral Calculator is a model that leverages connections between algebra and geometry that were advocated for in Klein’s Erlangen Program and was designed to provide a tactile means through which to construct, represent, and act upon the dihedral group. Groups are recognized as major disciplinary structures within the mathematical horizon, and knowledge of group theory has contributed positively to teachers’ abilities to respond to and support pupils’ mathematical activity (Zazkis & Mamolo, 2011; Zazkis & Marumur, 2018). Our findings suggest that tactile models, when designed as purposeful representational tools, can provide for opportunities to develop KMH by bringing into view different features, connections, and practices associated with the object being modelled. In terms of inner and outer horizons, we note that the inner horizon encompasses the features of an object that are evoked, or brought into view, by the specific representation of the object, and these features help situate that object within a greater mathematical world (the outer horizon). We suggest that the Dihedral Calculator can serve as a tool to broaden both inner and outer horizons for undergraduate students and future teachers.

6.1 Broadening the view of the horizon

Through its design, the Dihedral Calculator brought to the foreground key components of the mathematical horizon, including the duality of mathematical representations and the connectedness of mathematical ideas. The construction of the Dihedral Calculator required students to coordinate permutations, reflections, and rotations simultaneously, and shift attention amongst these ideas. To use the model, the students needed to physically rotate and reflect the disk, and then interpret the result of these actions in terms of permutations in cycle notation. See Fig. 5 which shows the position of a student’s model after performing three rotations and a flip. We saw evidence that at least some of these actions were internalized, such as in Fig. 6, where the position and structure of the object \({r}^{2}sr\) was depicted without first having (physically) executed processes of rotating and reflecting. We suggest that the connection between the position of the disk and the structure of the accompanying group element can support a shift in attention between process and object conceptions of group operations, which is a well-established challenge in abstract algebra (Dubinsky & McDonald, 2001). This shift can broaden the inner horizon through awareness of reflection as a process to carry out and as an object upon which other processes may act, and it can broaden the outer horizon through awareness of this dual nature of mathematical ideas, symbols, and representations.

Our findings suggested a further connection made visible by the Dihedral Calculator: the notion that model-building and visualization are mathematical activities applicable to algebra. This was novel for many participants, who viewed it as a helpful break from their typical computation-focused problems. Students were no longer “just imagining” the group structure, the model helped them visualize what they were computing and brought into view “how different branches of math can relate”. In our experience, both building the Dihedral Calculator and manipulating it required a coordination of algebraic and geometric ideas. On the one hand, building the model required interpreting the geometric structure algebraically. On the other hand, playing with it required interpreting algebraic structure geometrically.

Our research suggests that a tangible geometric model can offer more than pedagogical scaffolding and can help strengthen the expectation that the mathematical landscape is a connected one. Further, in prior research about KMH, attention to practices and sensibilities has focused primarily on elements such as conjecturing, generalizing, precision, and questioning conventions (Ball & Bass, 2009). We add to this literature by attending to model-building and visualization, respectively, as practices and sensibilities relevant for providing a higher view of the connections within the mathematical landscape. We suggest that models designed as purposeful representational tools can help reveal that ideas connect, as well as what and how ideas connect within the mathematical landscape.

6.2 Fostering KMH from outer to inner

Fostering a connected view of mathematical ideas is seen as an essential and valuable role that undergraduate mathematics education can play in the formation of future teachers (Bass, 2022). Through its design and use, the Dihedral Calculator can bring into view relationships that situate groups in a landscape that includes links amongst permutations, reflections, rotations, group actions, group elements, algebra, and geometry. Strengthening such links can contribute to knowledge at the mathematical horizon in two ways. First, it can provide for a more robust outer horizon by forging new connections across mental schemas of previously disparate ideas. A schema of processes and objects (Sfard, 1991; Dubinsky & McDonald, 2001) developed in a learner’s mind for permutations can establish a new pathway to a schema for symmetry and a new pathway to a schema for group elements and operations. Such pathways can contribute to a more connected world in which the mathematics of the moment is situated. This greater world is understood as the outer horizon and informs what teachers view as relevant or helpful for their students’ learning trajectories (Zazkis & Mamolo, 2011). Second, being able to access a richer collection of connections can influence what ideas or strategies (mathematical or pedagogical) may come into view when considering a specific question or problem, thus contributing to a richer periphery, or inner horizon, of related ideas, properties, practices, and so on. A robust inner horizon allows teachers to anchor their broad knowledge of mathematical ways of being in the moment and shift attention amongst these different relevant ways (Mamolo & Taylor, 2018); we suggest it can support Klein’s (1945) goal to foster teachers’ abilities to draw amply from a great body of mathematical knowledge and experience.

Our data was drawn from students’ written, pictorial, and physical artefacts and our analyses attended to ways in which the Dihedral Calculator could provide for opportunities to develop KMH. We focused our analyses on components of KMH that were inherent in the model itself and that were suggested by students’ decisions constructing, labeling, and positioning their models in connection with their algebraic and written descriptions. While our study was limited by restrictions that prevented us from physically observing participants create and interact with the models, we nevertheless suggest there is insight that can be gained into ways in which a tactile model can foster knowledge at the mathematical horizon.

We propose that the epistemological value of the Dihedral Calculator is that it captures and can convey aspects of the greater mathematical world (outer horizon) that can influence what ideas, connections, or practices are evoked and attended to (inner horizon). For instance, the sensibility of visualization exists in the outer horizon and is associated with mathematical practices such as visual reasoning, visual discernment, or visual proof. We hypothesize that the spatial visual representation of the dihedral group can direct attention to the applicability of spatial visual practices in algebraic contexts and thus broaden what connections become available (or in view) in the inner horizon. Although spatial visualization has been described as central to mathematical reasoning, the dearth of spatial visual approaches in secondary and university mathematics leaves learners ill equipped to reason geometrically in algebraic contexts (Sinclair, et al., 2011). We suggest that engaging with tangible geometric models at an advanced level can offer important experiences for future teachers whose stereotype of mathematics is that of a computational or procedural subject. We also suggest that the Dihedral Calculator can support generalizations from acting on D5 physically to intuiting the group structure for arbitrary elements such as \(s{r}^{k}\) (see Fig. 8), as well as in imagining extensions from identifying symmetries of 2D figures to ones of 3D shapes, such as designing an analogous calculator for the symmetries of a cube. Thus, affording opportunities to develop a higher view of fundamental group theoretic ideas, while also providing opportunities to advance understanding of the interconnected nature of mathematics. There is a need for more research which can capture and unpack students’ physical interactions with the model, as well as instances where they may have internalized structural aspects of the model. Research that includes the use of video recording and follow-up interviews, for example, could provide further insight into students’ reasoning with tactile purposeful representational models in undergraduate mathematics and its relationship to developing KMH.