Challenges and suggestions regarding the aesthetic experience of mathematics when teaching and learning fully online

The digital transformation of mathematics education has long been discussed, debated and, generally, encouraged (Hoyles, 2018). Discussions have largely centered on how students may learn with the use of digital tools and how to exploit this potential (e.g., Drijvers et al., 2018). Much less is known, however, about the more immersive experience of learning mediated almost entirely through the use of digital technologies ¹ . As a significant pedagogical innovation, this paper addresses the rapidly growing area of fully online mathematics education where all teacher–student interactions associated with the formal education process are mediated by the online medium and tend to be asynchronous ² (Trenholm & Peschke, 2020).

Since the creation of the internet almost 40 years ago, the teaching and learning of mathematics fully online has grown steadily (Blair et al., 2018). More recently this growth has markedly accelerated in response to environmental constraints presented by the COVID-19 pandemic, thus pushing this modality to the fore (CBMS, 2022; Lederman, 2021; Seaman et al., 2021). This development has led many to speculate if we are on the precipice of profound transformation to the field of education in general, and mathematics education, in particular (e.g., García-Morales et al., 2021).

Driving this transformation, a variety of forces are exerting pressure on the delivery of education. Perhaps first among these, educationalists and educational institutions continue their unending search for efficiencies, typically centered on financial costs and the production of “graduates” (e.g., Baltodano, 2012), often characterized in education as neoliberalism (Connell, 2013). This quest has led many to consider the replacement of human- with computer-mediated teaching—an obvious attraction given human resources are perhaps the costliest element in the educational process. In this regard, some may question whether fully online education is a harbinger for what is yet to come—that is, a kind of computer-driven, automated teaching system. Indeed, steady advances in learning analytics and artificial intelligence suggest there is at least some truth to this, with some hypothesizing super intelligent computers may replace human teachers in only a few decades (e.g., Haw, 2019).

Relatedly, socio-cultural changes continue to influence the delivery of education. We are, for example, witnessing the pursuit of truth, historically a fundamental goal of education, increasingly replaced by the pursuit of material wealth (e.g., Kilbourne et al., 2009). Reflecting this trend, more and more educational institutions are being transformed into places mainly for vocational training (Beekman & Gal, 2019) with the push for so-called STEM education as one current illustration (e.g., Axelrod, 2017; Woodard, 2019). Yet, to be clear, while the pursuit of financial stability is no doubt desirable, as an end in and of itself it reflects an increasingly materialistic and nihilistic culture, linked to both a rejection of spirituality (Pratt, 2022) as well as rising mental health issues (e.g., Dreher, 2019; Kasser, 2002)—the latter also being a feature of heavy digital media use (Twenge & Campbell, 2019).

In this context, we consider the transformation of mathematics education into fully online mathematics education. Early research has provided a roadmap for understanding this transformation with some arguing fully online education is simply an innovation to be exploited (Engelbrecht & Harding, 2005a, 2005b). Others, in contrast, argue fully online education will lead to a distorted or limited view of what it means to learn (Maiese, 2021). Alongside advances in AI many are raising serious questions how all these changes will affect us (e.g., Selwyn, 2019). Indeed, questions persist about how the use of fully online mathematics education may be shaping the way teachers and students interact with each other and mathematical content. As one potential issue, and as an important educational goal, this paper focuses on aesthetic experiences in mathematics education which will be shown to be associated with the development of deep mathematical thinking.

We undertake this discussion in three parts. First, I lay out three assumptions, providing evidence and arguments highlighting the importance of these experiences. Second, I discuss three challenges related to the fully online delivery of mathematics education. Third and finally, I provide three suggestions to aid in the delivery of high quality fully online mathematics education aimed at enriching both the minds and spirits of our students.

1. Three assumptions

The nineteenth century French mathematician, Henri Poincaré, is thought to have first claimed the essence of the mathematical experience is aesthetic (Mack, 2006, p. 1). As a first assumption, I extend this claim to suggest learning mathematics at a deep level involves some form of aesthetic experience, such as experiencing beauty or wonder. In this thinking, we are not merely biological machines and education is not about being “programmed.” On the contrary, we are unlike all other living creatures (cf. Ananthaswamy, 2014). We seek meaning. As human beings, we wonder about our existence and mathematics, as the foundation of logic and reason, is critical to that search.

The importance of these experiences and the idea of a search for greater meaning is attested to in the lives of professional mathematicians. To them, mathematical ideas are described as, for example, “transcendent, powerful, astonishing, [and] majestic” (Sinclair & Watson, 2001, as cited in Mack, 2006, p. 12). Proofs are described as “elegant” (Hanna & Mason, 2014). Many founding mathematicians were also philosophers in their own right with deep spiritual interests. They include Riemann, Gauss, Newton, and Pascal. The list is long. Their lives testify to some kind of “spiritual nature” associated with mathematical experiences (see also, Kessler, 2022). Even professed atheists, those who may not recognize any spiritual dimension, have acknowledged some deep sense of wonder. Bertrand Russell, for example, referred to the “supreme beauty” possessed by mathematics (Kessler, 2019, p. 55). In short, aesthetic mathematical experiences have a long association with learning mathematics deeply. To quote Henri Poincaré, “Mathematicians do not study pure mathematics because it is useful; they study it because they delight in it and they delight in it because it is beautiful.” It is, perhaps, understandable many stress the value of incorporating aesthetics as part of the learning process (Girod et al., 2003; Sinclair, 2003; Wickman et al., 2022).

Following this, a second assumption emphasizes the need for human interactions to potentiate these experiences. In simple terms, human teachers are critical to the effective and holistic development of humans and their mathematical thinking. They are critical because the nature of human interactivity in the teaching and learning process as well as the self-reflection it engenders, is what enables our students to learn and experience mathematics deeply.

This second assumption has some basis in current mathematics education theory regarding the development of mathematical thinking. Skemp (1979), for example, considered the growth in mathematical understanding as a product of cycles of “alternation” in mathematical discussions. Similarly, Sfard (2008) coined the term commognition, theorizing that both learning mathematics and developing mathematical thinking involve a process of communication and cognition within oneself and with others.

Moreover, the importance of human interactivity may be understood in the way that aesthetic experiences in mathematics are revealed as much in how the mathematics is taught than in the mathematics itself. In this view, a human agent teaching mathematics is seen as an artist performing their artistry where the human onlooker sees beyond the immediate visual sign, even past what it signifies, to ponder ideas related to, for example, order and beauty (Richardson, 2015). In this way, the human discovery of objective mathematical truths may be a highly subjective human experience. In contrast, fundamentally, a computer is only able to recognize and communicate elements of mathematical thinking much the same as it recognizes elements of a piece of artwork as, for example, a collection of pixels. In this regard, artificially intelligent systems will never completely emulate good teaching which “is far too complex to be reduced solely to data analysis and algorithms” (Selwyn, 2016, p. 106) ³ . Arguments to the contrary tend to ignore the reality of subjective experiences which point to “wonder” and, for some, a spiritual world beyond our own. Indeed, it is doubtful computers alone can effectively provide the holistic pastoral care needed to address mathematics anxiety common to learning mathematics. Only human teachers who have struggled, wondered, and have themselves experienced moments of discovery can effectively empathize and holistically communicate the wonder of mathematics. As some have argued, our own experience of wonder acts as a “sublime structuring agent” for all of our mathematical discourse (Mack, 2006, p. 4). Such arguments regarding the limitations of computer technology appear consistent with findings that computer systems work best at helping students achieve a procedural level of understanding (Paterson, 2002). Moreover, a complete reliance on computer systems is thought to lead to a form of homogenization which fails to recognize the diversity of learners and learning experiences (Darragh, 2021). In sum, while it remains unclear how student–computer content interactivity enables any form of aesthetic experience in mathematics, teacher–student interactivity continues to hold the greatest promise for leading students to a deeper sense of wonder and learning in mathematics.

A third and final assumption presupposes a transhumanist vision of education as ultimately failing humanity (e.g., Leonhard, 2016). This is where, for example, technological devices may be surgically implanted to cognitively enhance a student's mathematical thinking. Though such efforts may raise concerns about safety, choice, and control (Eisenberg, 2019), fundamentally they raise questions about the human essence. In relation to fully online teaching and learning, efforts to advance a transhumanist vision of education give rise to questions about both the potential and limitations of technological mediation in education (Bostrom, 2001). An in-depth discussion of this issue is beyond the scope of this paper, though here I make explicit the view our essence extends beyond being mere biological computers that process information (Dahlin, 2012). Rather, our human intellectual development also involves what may be considered intuitive experiences of wonder and beauty that enhance the learning process.

2. Three challenges

Given these assumptions, we turn to discuss the state of fully online mathematics teaching and learning where, due to its stage of development, most research has been focused at the tertiary level (e.g., Allen & Seaman, 2007). Nevertheless, despite obvious differences between primary, secondary, and tertiary level mathematics teaching, this research has important implications for all mathematics teaching.

As one important lens into the state of fully online mathematics teaching and learning, large-scale research has made comparisons on the basis of measures such as student satisfaction, academic performance, and retention. In a recent large-scale review of this research, fully online mathematics teaching was found not to be as successful as traditional face-to-face mathematics teaching (Trenholm et al., 2019). Furthermore, consistent with this work, as a second important lens, recent systematic comparisons of the use of recorded video versus live classroom teaching found increased use of video recordings associated with poorer academic performance (Lindsay & Evans, 2021; Trenholm et al., 2012). Though potentially correlative, follow-up work suggested a causal link: students learn mathematics better when attending live face-to-face teaching sessions versus relying on video recordings of those sessions ⁴ (Trenholm, 2022; Trenholm et al., 2019).

There are at least three reasons why the fully online modality is challenging for teaching and learning mathematics. First, the nature of interactivity available in the fully online teaching context is considered suboptimal for learning mathematics. As previously discussed, the development of mathematical thinking has been theorized as a cyclical process of reflection and communication. Prior research has shown that mathematics teachers struggle with teaching fully online, where all interactivity tends to be asynchronous, as they cannot realize the kind of short cycle question-answer-feedback interactivity considered necessary for the effective development of students’ mathematical thinking (Trenholm, 2013). Instead, it may be days between the time when a student poses a question and their teacher responds. This means a student's train of thought may be lost in that time, threatening the crystalline growth of their mathematical thinking (Tall, 2013).

Moreover, in consideration of communicating about mathematics, the typical flow of communication in the fully online teaching context is not considered to be as free-flowing or facile as what may be experienced in a live classroom setting. For example, semiotic resources common to face-to-face teaching may be severely limited or non-existent when teaching fully online (Arzarello et al., 2009). In particular, there is a growing recognition of the need for multiple modes of communication in mathematics including verbal cues, hand gestures, and facial expressions, all channels emerging research suggests may act as cognitive supports enabling the learning process (Abrahamson et al., 2020). While similar communication challenges have been noted in other disciplines (Hrastinski & Stenbom, 2013), some claim a greater need for these resources exists for mathematics given its abstract nature and the higher cognitive demands required to learn it deeply (Edwards, 2009).

As a result, little human-to-human mathematical discourse appears to be taking place in fully online as compared to traditional face-to-face mathematics education (Trenholm & Peschke, 2020). Indeed, prior research has found distance teaching of school mathematics to be primarily a “one-on-one” experience with little use of collaboration (Lowrie & Jorgensen, 2012). It may then come as little surprise, as will be discussed next, that lower levels of human interactivity have been found to coincide with increasing levels of human–computer activity (Trenholm et al., 2015). In sum, the fully online teaching context presents significant challenges for both students and teachers to effectively engage in mathematical discourse.

Second, relatedly, the nature of the typical fully online teaching context makes it difficult to judge and monitor students’ developing understanding of mathematics. Regarding formative assessment practices, when fully online mathematics teachers are unable to read, for example, students’ facial gestures or monitor student attentiveness (e.g., via eye contact), it may be difficult or almost impossible to accurately judge and monitor the state of students’ developing understanding of any new mathematics (Trenholm, 2013). This compared to, for example, teaching in a lecture theatre where a lecturer may quickly observe student faces and gain a sense of how they understood (or not) a new concept being introduced. These informal lecture or classroom practices may be replaced fully online with frequent computer-based assessments that typically focus on the product and not necessarily the process of mathematical thinking (Trenholm et al., 2015). In short, an experienced teacher in a traditional classroom setting may, for example, intuit both the nature and timing of their feedback based on subtle cues such as the tone of voice or facial gestures of their students. In stark contrast, in the typical fully online teaching context, computer-based assessment systems may only provide a limited repertoire of feedback based on single alphanumeric expressions inputted into the computer for each question/problem. That is, the feedback is likely to be more focused on the product and not necessarily the process of students’ mathematical thinking.

Furthermore, regarding summative assessment practices, the use of invigilated assessment instruments (e.g., tests or exams), which dominate approaches to assessment in mathematics (Iannone & Simpson, 2011) are often discouraged, if not restricted. This is because fully online teaching is typically marketed as “anytime, anywhere” (or “100% online”) learning (e.g., https://www.open.edu.au/your-studies/getting-started/covid-exams). So, when human supervision is not available, either all summative assessments must be designed as “open book” or new remote computer invigilation systems must be used, despite ongoing concern about both assessment approaches (Balash et al., 2021; Iannone, 2020). Indeed, some research has found fully online mathematics teachers are increasing their use of formative assessment instruments, particularly via computer-based assessment systems, to compensate for decreased use of summative assessment instruments. This feedback appears to be more geared to verifying academic integrity (i.e., authenticating student work by requiring consistent ongoing course assessment interactions) rather than simply focused on advancing student learning (Trenholm et al., 2015). In short, it is unclear how such mediational efforts are accurately judging and effectively advancing student learning as compared to traditional approaches to assessment in mathematics (Englander et al., 2011).

As a third and final reason, the online environment may challenge student learning of a subject already considered challenging by many, where deep learning is seen to require high levels of cognitive processing (Henningsen & Stein, 1997; McCabe et al., 2010). The online medium, for example, has been characterized as an ecosystem of distraction technologies (Carr, 2010). In this regard, it is not difficult to assume that students often multitask while studying online, either monitoring or using multiple applications such as social media and email, practices linked to depressed learning (Junco & Cotton, 2012). Moreover, similar to the cognitive experience of watching television, there is some evidence the screen-based medium may be cuing students to relax their mental engagement (Trenholm, 2022). In sum, as compared to live face-to-face instruction, the high-level cognitive processing necessary for deep learning of mathematics may be more difficult to realize when interactivity is mediated via the online medium.

There is no suggestion this list is exhaustive; however, I suggest these three issues are critical to understanding current challenges and therefore solutions.

To be clear, there are several clear benefits to providing online instruction. Perhaps the single greatest benefit is improved equity of access (Stephens, 2020). Students who, for example, could not afford to undertake further study that requires meeting at a single physical location can now meet together online. Similarly, students are afforded much greater flexibility when the teaching is entirely asynchronous (e.g., https://online.illinois.edu/articles/online-learning/item/2017/06/05/5-benefits-of-studying-online-(vs.-face-to-face-classroom)). Though some contest these benefits are universal (Sublett, 2020), for many others fully online education may be more efficient and less expensive for both institutions and students.

As some have argued, online education is a disruptive innovation which promises to transform the nature of education for generations to come (Meyer, 2011). Yet, while there is little doubt the online medium is disrupting the delivery of mathematics education, it is not necessarily as some envision.

3. Three strategies for the future

Fully online mathematics teaching and learning is here to stay yet we still know little about how to teach mathematics effectively fully online. Some argue technological solutions are emergent. For example, recent research has already brought about advancements in computer-based assessment systems (e.g., intelligent tutors, adaptive prescriptive software…) and remote invigilation. While these developments may help, they generally represent new teaching tools and the question remains, how do we effectively teach mathematics fully online in a way that enriches both the minds and spirits of our students?

To help address this challenge I suggest three fundamental strategies.

First and foremost, we must allow ourselves to be human. Most of us will not learn overnight to effectively transition from traditional synchronous (face-to-face, same place, same time) teaching to teaching that is more often asynchronous and always technologically mediated. This transition may be particularly challenging given it involves a fundamental shift from traditional mathematics teaching which emphasizes transmitting knowledge and understanding to teaching that emphasizes the orchestration of learning experiences. Vitally, this is a creative process with the teacher, as conductor, directing the orchestration of instruments/tools and activities to create music which is the silence between the notes—that is, the students’ own reflections on process and developing mathematical thinking. In this way, teachers create activities or opportunities for students to reflect deeply on the mathematics, a process which should attend to both students’ affective and cognitive needs.

As a creative process learning to teach online must be viewed as an incremental trial and error approach where we recognize the potentialities and limitations of the medium while being attentive to student feedback as well as our own abilities (Trenholm & Peschke, 2020). While prior instructional technology expertise may help, the learning curve is often very steep. Unless we are willing to grow, learn, and sometimes fail, our new pedagogical approach is likely to reflect a kind of digital version of a correspondence course: resources are simply posted online for use in completing assignments which are then submitted online. It cannot be emphasized enough that good teaching is an art form which instructional technologies can help facilitate, but also potentially hinder. We must resist using “shiny new objects,” which new technologies sometimes resemble, unless they serve our teaching objectives. The ultimate educational goal is to help students learn mathematics deeply.

A second strategy is to maximize human interactivity. Prior theorizations about online teaching have posited there are at least three forms of interactivity: student–teacher, student–content, and student–student (Anderson, 2004; Moore, 2013). Consistent with current theory, research suggests problems with human interactivity in fully online mathematics are leading to a greater dependence on student–content interactions, particularly a dependence on the use of computer-based assessment systems which, as previously mentioned, only appear helpful in achieving lower-level procedural learning, though the technology is evolving (Sangwin et al., 2010; Shute, 2008; Simmons & Cope, 1993). Moreover, the limitations of these systems and importance of human interactivity were evident in a recent meta-analysis on the effects of technology on students’ mathematics achievement in K-12 classrooms: “The largest significant moderator effects were found when technology was used to design and support collaborative and communicative environments (ḡ  =  0.49; Ran et al., 2022, p. 258). In other words, computer technology is most effective when it is used to facilitate human interactivity and mathematical discourse.

The importance of mathematical discourse has been well established (e.g., Skinner, 2003). In order to facilitate this discourse and advance students’ developing mathematical cognition, some pedagogical approaches should be targeted. In particular, careful design of student–student (i.e., peer to peer) interactivity holds potential for guiding students to a deeper understanding of mathematics (Crouch & Mazur, 2001; Stein et al., 2008), as students may be exposed to and reflect upon multiple solution paths (Große, 2014). Moreover, as the kind of short cycle interactivity common to live classroom teaching is unlikely to be realized fully online (e.g., the wait time between a student's question and the teacher's response; cf. Rowe, 1986), longer cycle interactivity may be leveraged by careful teacher-orchestrated peer interactions which enable deeper reflection on and thus learning of the associated mathematical concepts (Trenholm et al., 2016). These technologies may be used, for example, to randomly and anonymously place students in groups to share and even judge each other's problem-solving work, uploaded either as static text or, given its increasing ubiquity, as a dynamic video production (Jones & Alcock, 2014). Bringing all of this together, the development of a community of mathematical learners should be a priority (e.g., Rovai, 2002) alongside the maximization of peer-to-peer learning opportunities as a critical instructional design goal. In so doing we may thus achieve Rodd's (2003) “‘spiritual’ purpose in gathering the mass of students for teaching… [which] can contribute to students’ success in their undergraduate mathematics education because this witness-participation can help develop students’ identities as mathematicians and can inspire mathematical imagination” (p.15).

Third, we must recognize our students are human. They too need to learn new ways which means part of our job will be preparing them for and guiding them through what may be an entirely new experience of learning mathematics. Notably, this requires students to be much more engaged in their learning, not passively acting as repositories of knowledge. As this may suggest, students will require skills in self-regulation, self-discipline, and communication, all critical to their success. As a first course of action at the onset of teaching it may thus be important to administer a self-graded online course readiness quiz (e.g., https://online.uark.edu/students/readiness-quiz.php). This quiz may direct students to reconsider taking a fully online mathematics course for the time being or recommend a short online course directed at inculcating the study skills needed for success. Ideally, any study skills course should be focused on learning mathematics fully online. Whatever action is taken, it is imperative students are made fully aware of what is expected of them. Some online teachers may choose to have students sign contracts further acknowledging teaching and learning expectations. Following this, appropriate signposts, which may be text- and/or video-based instructions, should be placed throughout the course to further guide students in their studies.

Finally, drawing together all three strategies as implications for both instructors and students, an important part of guiding your students is the ongoing formative assessment of students’ mathematics work. This may involve activities such as regularly monitoring students’ computer-based assessment results or regularly reading, and occasionally participating in, online forums set up for the course. Students must be active participants in the learning process which instructors must creatively orchestrate. This process will produce a stream of information which should inform your immediate work of developing your students’ thinking as well as future course redesigns. Of course, it is unrealistic to look at all student work in detail. As an additional means of gaining insight into your students’ developing understanding, you may choose to intently read the work of randomly selected students at scheduled intervals throughout the course. Because of the “human” work involved, pedagogical decisions surrounding the formative assessment of your students may present a nexus of pressure. This is where strong considerations regarding the ease of automated assessment systems may undermine direct human involvement in the teaching process. Whatever happens in the end, our pedagogical approach should resemble more a work of art which, when done well through our own trial and error efforts, may reveal the depth and wonder of mathematical understanding.

4. Conclusion

The forces arrayed against this work include a growing materialism and nihilism as well as ongoing pressures to teach more efficiently, often by leveraging advancing computer technologies. How we deal with these pressures will inevitably affect the quality of students’ experience of mathematics, their learning and, ultimately, their intellectual development. I suggest we are in a battle for the heart and soul of our discipline, a battle of strategic importance given mathematics and mathematical thinking are fundamental to education, vital to becoming an informed citizen and foundational to what it means to be human. Your fully online mathematics teaching could fall victim to this battle, but only if you let it. Yes, teaching fully online involves a fundamental pedagogical shift from what tends to be a teacher-led approach to one that is more student-led but teacher-guided. However, when designed carefully, your online teaching may not only engage students in higher level thinking but also experiences of the beauty and wonder of mathematics. To be clear, good teacher-orchestration in the fully online modality remains as much a human art than science.

The early 20th century warning, “technology is a useful servant, but dangerous master,” appears even more relevant today (Lange, 1921). In the present context, it may be stated our efforts to leverage online education possibilities must avoid potential pitfalls that diminish our common human essence. In particular, we must actively preserve the potential to experience wonder and beauty through intellectual pursuits, of which mathematical discovery is critical. In advancing online mathematics education, the strategies proposed in this paper are aimed at striking the correct balance of taking advantage of this technology without allowing it to diminish who we are.