Embedding Mathematics in Socio-Scientific Games: The Mathematical in Grappling with Wicked Problems

Abstract

This paper discusses the ways in which digitally enabled transformation in mathematics education could envisage a role for rationality in post-normal science and wicked problems. The scene is set firstly by reviewing the ways in which digital media have been designed and used in transformative mathematics education as a rationale for thinking about such media for wicked problem education. The problem is set in epistemological terms: can normal science approaches contribute to post-normal science? By considering the basic arguments regarding wicked problem education, I focus on the discussion of a specific constructionist digital tool called ‘ChoiCo: Choices with Consequences’, designed to embed mathematical ideas and facilitate mathematical reasoning, yet be about grappling with wicked problems. The final section discusses student discourse to set the scene for what such reasoning might look like in the context of grappling with wicked problems.

1. Wicked Problems as a Challenge for Transformation in Mathematics Education

In recent times, every one of us has felt exposed to wicked problems, those universal ill-defined, controversial, complex, value-laden socio-scientific issues such as the pandemic and climate change. Our society is replete with individual and shared stress, denial, and inertia, ultimately leading to an exponential augmentation of risk for wide-ranging consequences. In Europe, at least, there is a push for educational transformation aiming to provide students with experience in grappling with such issues in a knowledgeable contributory way. This paper discusses the potential role for using digital media to engage in rationality and mathematical thinking as a means of grappling with such issues. Pedagogical transformation is not new to mathematics education, albeit in different ways. So, could mathematical rationality in handling wicked problems be one of the transformation avenues worth addressing in mathematics education?

It has now been 50 years since Papert introduced the idea of fundamentally changing students’ experiences with mathematical reasoning through the use of digital media to express, explore, and generate mathematical meaning [1]. The need for transformation in mathematics education has since then been widely argued from many angles beyond the advent of digital technologies. It has been generally portrayed as a need to move away from overbearing ‘visiting the monuments’ paradigms, as Chevallard [2] would put it, where students are typically exposed to abstract mathematical truths in a rigid, control-oriented, time-bound setting aiming to strengthen their ability to respond to specially pre-designed tasks [3]. Instead, the push has been to find ways to provide students with agency [4], with experiences in mathematical reasoning for themselves, for meaning making in personally relevant individual and discursive settings. Digital media have been perceived as powerful tools to that end [5]. This powerful way in which learners use digital media to structure mathematical knowledge-in-use has been well researched and established as a goal and a challenge for transformation in mathematics education [6].

From an epistemological point of view, it is not the nature of mathematics that has been challenged but rather the ways in which the practice of mathematicians has been understood and communicated. Mathematical epistemology has not been fundamentally debated. Mathematics has been deservedly perceived as the ultimate scientific endeavor; it is the field where reasoning comes from, where rigor comes from, and where the ability to make connections, to deduce and to prove, to generalize, to be certain, or to gain accuracy and develop a sophisticated language about uncertainty come from. This is a science where we know when something is true, and we question whether something is true in very rigorous ways [7].

The transformation sought has thus to do with education, not the scientific paradigm. It is to provide learners with the opportunity to experience what it means to do mathematics, the same kind of experience mathematicians themselves go through. Mathematicians expose ideas and propositions to peer scrutiny and refutation attempts. They thus perceive mathematical ideas to be fallible, and only the ones which survive this scrutiny remain as mathematical certainty [7,8]. This means that most of the time they spend scrutinizing ideas by others or having their own ideas put in the frying pan so to speak. The scrutiny process is a fundamental part of doing mathematics, and transformational approaches in mathematics education argue that learners should be given much more space to engage in this kind of process.

Recently, however, and importantly highlighted by the era of the pandemic, we have realized that what has hit each of us in our society and everyday life is the engagement, pre-occupation and involvement at a personal level not with clean, potentially solvable, mathematical problems such as the ones designed for students to tackle in school. Instead, students join adults in sensing the increasing prevalence of very complex issues and problems that are around us: climate crisis, sustainability, sustainable cities, pandemics, and personal diet, combining health and well-being. These kinds of issues do not really have a solution in them, and there is no clever way in which humanity can find the way to deal with them, nor can we find how to cope with them and get rid of them in the end. Even though mathematicians produce endless models of such complex situations, none of them really explains the respective phenomenon in any comprehensive, resolving way. So, at the individual and social citizen level, what is required is that we become a little less stressed about these issues. How can we learn, as citizens, to grapple with them with an aim to survive within contexts where these issues apply? And how can mathematical reasoning and scrutiny maintain and enhance its perceived value in situations where it could play an important yet not primary role regarding the issues at hand?

This paper thus suggests that the following questions should be put to the mathematics education community:

• Is there a role for mathematics and mathematical thinking in coping with complex, contentious, socio-scientific issues replete with uncertainty?
• If yes, how can digital media be designed and used to introduce mathematical thinking and rationality in addressing and grappling with such issues?
• How can we think of pedagogical mathematical transformation with digital media to include grappling with wicked problems?

Such issues have played a central role in creating big currents pushing for change that affect educational systems in Europe and around the world. They have been connected to the ideas of cultivating 21st-century skills and action competence. Mathematics education researchers have connected those kinds of skills—such as creativity, computational thinking, collaboration and communication, and problem posing and solving—to mathematical reasoning. At the same time, however, the push for educational reform has originated from much wider educational and societal perspectives and is a part of the European strategy for education and equity. The EU strategic plan for 2021–2024 is oriented towards the twin green and digital transitions for a sustainable, fair, and more resilient economy and society [9]. The development of a high-performing education ecosystem through the digital transformation of educational paradigms is also one of the core goals of the EU Education Action Plan for 2021–2027 [10] and of UNESCO’s Education 2030 Framework for action [11].

So, is there a role for the mathematics education community to contribute to this wave of change, or is it better that we stay on the side in the hope that our siloed domain of mathematics and its beauty will remain and be respected as in the previous century? In other words, is there a role for mathematical thinking and rationality in socio-scientific issues? If yes, what kind of digital media can be used for expressing mathematical reasoning while grappling with wicked problems and issues replete with uncertainty? How can we design for added pedagogical value based on their use?

The following three sections provide a background to carefully address these questions. The first section sets the ground with respect to epistemology, how can traditional science approaches contribute to addressing socio-scientific issues. The second then sets the scene and basic arguments regarding wicked problem education. The third section discusses the ways in which digital media that have been designed and used in transformative mathematics education are analyzed as a rationale for thinking about such media for wicked problem education. What follows is the description of a specific constructionist digital tool called ‘ChoiCo: Choices with Consequences’, designed by our ‘Educational Technology Lab’ (http://eds.uoa.gr, accessed on 6 June 2024) to embed mathematical ideas and facilitate mathematical reasoning, yet be about grappling with wicked problems (http://etl.ppp.uoa.gr/choico, accessed on 6 June 2024). The final section discusses student discourse to set the scene for what such reasoning might look like in the context of grappling with wicked problems.

2. Epistemological Considerations: Normal vs. Post-Normal Science

The problem is with normal science. Scientists are perceived by the wider society with diminishing credibility and relevance, as people who will give you facts and truths that are not so relevant when you think of the issues at hand [12]. So, what happens? So how else can we think of science? Well, recently, there has been a movement termed ‘post-normal science’. It addresses complex issues of our time where disciplinary fragmentation and traditional scientific conduct appear to lack the necessary capacity to allow an integrated understanding of the issues. Transmitting simple truths does not help policy makers, and that does not help citizens and the individuals. We need to draw on epistemological principles that recognize uncertainty and develop ways of dealing with uncertainty, with value-laden agency.

Most current science thinking and research, centrally including mathematics, has been built on epistemological assumptions developed along the deliberate aim to moderate complexity and minimize uncertainties in the world so that problems and issues can be ultimately modeled. These ‘normal science’ paradigms [12], supported by appropriate conceptual modes of representing reality and specialized codes for studying it, narrow down the focus of their enquiry within the boundaries of specialized disciplinary fields such as mathematics to address attentively defined (but eventually simplified) ‘problems’, with the intent to generate valid and generalizable evidence-based knowledge to feed decision-making. However, when it comes to complex issues of our time, disciplinary fragmentation and traditional scientific conduct as encouraged by ‘normal science’ seem to be lacking the necessary capacity to allow an integrated understanding of the issues. Moreover, transmitting simple truths to policy making is rather inadequate when dealing with multi-faceted issues carrying a great degree of uncertainty [13]. There are many open questions as to how science can contribute to fostering social innovation and change in as many social groups as possible, rather than providing only expert-based knowledge to policy makers [14]. This is particularly the case with, for example, current crisis and sustainability challenges, recognized as complex, controversial, and value-laden issues by nature and, therefore, difficult to be dealt with in mono-disciplinary ways [15]. Complexity stems from their multi-faceted character and the requirement to apply various perspectives to grasp them more holistically. Different interpretations may lead to different implementations based on the context and the situation. These features render such wicked problems difficult for normal scientific practice to address and deal with [16]. To counteract these shortcomings, ‘post-normal science’ has emerged as an alternative paradigm of scientific enquiry and knowledge [17]. Drawing on epistemological principles that recognize uncertainty, value-laden agency, and context-specificity as intrinsic attributes of the contemporary, post-normal science promotes transdisciplinary approaches to framing and studying current complex issues and gaining an understanding of the world. Global crises, pandemics, and sustainability issues are among those most characteristic examples, the understanding of which necessitates the application of post-normal lenses and processes, such as the co-creation of diverse types of knowledge and the employment of participatory methods, designs, and tools, that facilitate the emergence of multiple representations and reflection to take place [18].

3. Wicked Problems and Wicked Problem Education

Consider the role of schooling to inspire lifelong citizen engagement with ‘wicked’ problems that can contribute to a democratic, socially engaging sustainable development practice, where experts and various groups of citizens with different perspectives engage in a dialogical inquiry on a complex, fuzzy, multi-faceted, contentious issue, such as sustainable living. This kind of issue has been called ‘a wicked problem’, i.e., a dysfunctionality within a complex system [19]. Wicked problems are difficult to contain and structure, are interconnected and interdependent, and are ill-defined and dynamic as their parameters are continually in flux [20,21]. Individuals often feel overwhelmed, and develop denial and resignation to such a problem, followed by inertia due to a sense of determinism, which permeates societies [22,23]. Yet wicked problems need action at many levels, and the present inertia risks the exponential growth of the problem and its consequences at high stakes [24]. For the individual, it is important to engage in becoming sensitive and knowledgeable on the problem and to also engage in actions such as taking care of individual footprint, challenging own actions, beliefs, and habits, and being interested not only in individual action but also in contributing to collective action at the level of the city or municipality [25]. A paradigm shift is needed, from solving well-defined siloed problems to a post-normal science approach [26]. So, consider a transformational stance to schooling in an attempt to integrate such a post-normal science approach in teaching and learning, addressing and perceiving students as young citizens [27]. Consider the challenge of harnessing wicked problem education to become syntonic and integrated with the innovative educational push towards cultivating key competencies for lifelong learning such as the following:

• Agency, ability to make own decisions, challenges not set by another [28];
• Action-in-context, when the context is not necessarily about mathematics in school;
• Beyond silo disciplinary approaches;
• Beyond timed, solvable, regulated challenges;
• Beyond integration: a competency in the service of another [29].

So, what is a wicked problem exactly? It is a problem impossible to solve, and that is because it is not well defined. It is contradictory. Different people have different views on it. It changes all the time. Some examples of wicked problems are poverty; urban renewal; school curriculum design; education, environmental, and natural resources policy; healthcare; climate change challenges; sustainable cities; diet; and individual and social challenges in times of world crisis.

These wicked problems cause problems to the individual. They often develop a denial about the problem, for example, assertions such as ‘come on, pandemic coronavirus! It’s easy, it’s just a flu!’. They create a resignation of the individual regarding the acknowledgment of the existence of the problem. We see such a point argued all over the media, ‘there’s no point in vaccinating since if you’re vaccinated you can still catch it’. Resignation is followed by inertia: ‘I’ll wait, I’ll wait for everybody else to get vaccinated and then see what happens’. But wicked problems need action at many levels because the inertia is a risk, and there is an exponential growth of the problem if people do not realize and do not start developing strategies. And the consequences are high stakes. So, for the individual, it is important to become sensitive and knowledgeable not only to find a solution by engaging in actions such as taking care of the individual footprint, but also by perceiving that they are a member of societies at different levels who are collectively addressing the problem [27]. This challenge has been tightly connected with the key competencies for life-long learning stated above. The question is, is there a role for rationality and mathematical thinking in the addressing of such a challenge?

To date, has transformation in mathematics education addressed the role of mathematics and mathematical thinking in post-normal science? What can a normal science such as mathematics provide to empower and support post-normal perspectives aiming to address controversial ill-defined, complex socio-scientific and value-laden issues? Fallibility in mathematics has been perceived in the context of the process to look for truth, for certainty, in a normal science setting. In post-normal science, however, the focus is not on the process of producing mathematical truth but could potentially be on the role such truth can play in, say, wicked problems. In this paper, I suggest that mathematical thinking and mathematical concepts do and should have a no less important role to play in post-normal science and that mathematics educators need to consider their involvement in the recent pedagogical wave demanding schooling to afford wicked problem education.

4. Digital Media to Transform Mathematics Education: An Example

Let us, in this section, look a little more deeply at the ways in which digital media have been perceived and designed to bring added pedagogical value in transforming mathematics education paradigms to cultivate mathematical reasoning [30]. Researchers seem to agree that there is particular value in digital media being used by students as tools with which to engage in mathematical reasoning, in putting mathematical concepts and ideas to use, in mathematical discourse and expressivity. When we have classrooms where students are given space to develop their own ideas and to work with these tools, we can see that the mathematical meanings that they develop and construct are unavoidably connected to the tools that they use. Researchers have witnessed this kind of reciprocal shaping of meanings and tools when mathematics is put to use to create and change mathematical models and representations [6,31]. The pedagogical value in meaning-making has been considered important enough to address the connection between those meanings and the abstract curricular mathematical concepts as a necessary educational task in the context of students having built a positive disposition towards and experience with mathematical reasoning.

As learners create models and representations with these tools, they progressively create ‘schemes of action’ as [32] would put it, i.e., individual and shared meanings of a tool’s functionality and kinds of use together with the kinds of mathematics cultivated during such use. Thus, the key aspects of focus in designing such tools and envisaging their usage are as follows:

• Mathematical expression, augmenting the representational repertoire and interdependencies;
• Engagement with mathematical thinking;
• Putting concepts to use;
• Reciprocal shaping, instrumentalization, constructionism, and creativity.

The task is to generate environments that are rich in opportunities for meaning-making, to perceive digital artifacts as media for expressing mathematical meaning, to access powerful mathematical ideas that are otherwise difficult or obscure with pencil and paper or with other representations and to engage teachers in taking part in the design of pedagogical added-value activities. The main concern in looking for pedagogical added value is thus connected to designing for innovation to adopt a transformative stance to education. The main thrust in this approach questions the way that mathematics is perceived and taught and the way that curricula are structured and looks for ways in which we can use technology within a transformation process. As researchers, we focus on technology for expression and meaning-making and we are developing theory on meaning-making processes and on teachers’ knowledge and teachers’ practices. Our main concerns have thus been to do the following:

• Generate environments rich in opportunities for mathematical meaning-making [1];
• Perceive digital artefacts as expressive media for mathematical meaning-making, a new literacy [5];
• Access powerful mathematical ideas otherwise obscured by traditional methods of expression [33,34];
• Engage teachers in designing added-value media and activities and dealing with professional, institutional, and societal traditions so as to generate such environments in the classroom [35];
• Develop media that is specially designed for questioning traditional practices and doing something different.

In this context, at the Educational Technology Lab, Dept. of Educ. Sciences, Sch. Of Philosophy at NKUA, we have been adopting a transformational approach by designing and using media for teachers and students to in turn design and tinker with models and representations. Over a period of more than 25 years, we have been engaged in design research to illuminate mathematical processes in respective educational practices and to contribute to the development of a ‘framework for action’ theory helping to both design and understand meaning-making [36]. In this venture, we found it most useful to combine and integrate diverse theoretical constructs, having been greatly influenced by our participation in the TELMA, ReMATH, and M C Squared European Research projects whose main aim was precisely to forge connectivity amongst constructs lying in fragmentation on a ‘theoretical landscape’, to use Artigue’s terms [37]. Our particular objective was not the practice of creating such connections per se but instead of considering how to best try to make sense of the environments we designed and studied. So, we found these particular four constructs, albeit widely diverse, to be pivotal in our approach:

• Conceptual fields [32];
• Restructurations [35];
• Half-baked artefacts [38];
• Reciprocal shaping of meaning and tool [39].

We found Vergnaud’s idea of conceptual fields centrally useful in the sense that, for mathematics education, it diverts priority from a mathematical concept to be ‘learned’, to all that makes it useable and communicable. To think, i.e., of a concept in educational design, it is necessary to place it in the center of a dense circle of related concepts and a set of representations that become the basis for resolving a set of problem situations. So, in education, this is the way we should be thinking. We should not be thinking of whether students learn how to factorize or learn how to solve a quadratic equation, but rather of situations resolvable by dense sets of concepts around a central one.

Restructuration is the exercise of questioning the structure of the curriculum and the kinds of mathematics to best approach mathematical problems. The current curriculum structures have been decided, established, and fixed in historical time before the advent of digital media and even before the advent of mathematics education research, for that matter. But mathematics is the discipline characterized by fluidity in the ways in which it makes sense to build structures; its nature is such that you can portray mathematical concepts in a very large number of different alternative structures. So now that we have technologies and we live in the technology world, it is time to rethink what kind of structure of mathematical concepts is now amenable for children to engage in mathematical thinking with these tools. Imagine, for instance, a section on ‘curvature’, periodicity, and the rate of change, inflation, compound interest, and approximation combined, or a section on mathematical complexity, or on gaming theory. Ask the question: which mathematical structures are good spaces for students to engage in meaning-making and mathematical reasoning, given digital media?

Half-baked artifacts: well, this is didactical design, or rather, engineering. It is when, from a pedagogical point of view, students are given problems, models, or representations that are incomplete or have faults in them and then invited to identify and correct them. Behind this, there is the epistemological idea of fallibility, the idea of questioning and the idea of not perceiving mathematics as a game of absolute truths but perceiving mathematics as a field where reasoning and questioning prevails.

So, the main concerns of the research community at large, and of our Educational Technology Lab, have seen mathematics as useable intellectual processes and traits in diverse situations. Educational transformation in mathematics education has perceived digital media as a pivotal tool and digital transformation in society as the global situation in which this educational transformation may start to materialize. In this wake, mathematics curricula and curricular structures have come into scrutiny, asking the following question: what kinds of structures can operate as fertile fields within which learners can develop mathematical reasoning? With respect to digital media, mathematical reasoning has been connected to constructionism, a kind of discursive low-stakes tinkering-style engineering co-evolving with computational thinking skills and competencies. However, still in all cases, the mathematics education research community has understandably perceived mathematics as the priority and the end target. The situations, the tools, and the restructuring have been a means to an end.

5. Digital Media to Grapple with Wicked Problems: The Case of ChoiCo

The first section in this paper contains a discussion of how digital media have been designed as tools to help with the pedagogical wave of transforming the mathematics education paradigm from what Chevallard [2] calls a ‘visiting the monuments’ paradigm to an experiential, questioning the world, creative, and discursive paradigm. The transformations, however, have so far maintained mathematics as the primary educational objective of the enterprise where the focus is on the modeling of mathematical objects and representations outright or at most the modeling of objects and behaviors directly and importantly embedding mathematical concepts [32,40,41,42,43]. Even in the case of media, which is primarily focused on computational thinking and creativity to create games such as scratch, mathematics educators have shaped microworlds and modeling exercises with a focus on the mathematical concepts inherent within [44,45]. The most well-known attempt to design a medium for students to engage with complex issues is NETLOGO [46]. Even there, however, the focus is on mathematics as a means to fully understand the phenomenon by modeling it, based on an albeit diverse kind of mathematics calling for a restructuring of our perception of mathematical curricula [35]. In this paper, the attempt is to consider digital media primarily in the role of tools to help grapple with wicked problems, yet, at the same time, embedding mathematical ideas and designed to cultivate rationality in such an enterprise.

It is in this context that we introduce a constructionist tool which we call ‘ChoiCo’, a digital medium specially designed for post-normal science education in the sense of [14]. ChoiCo is an acronym for ‘Choices with Consequences’ [47]. It is a system for authoring games embedding socio-scientific issues. The system leaves the choice and definition of such an issue up to the user. It is based on the gaming idea that there is a single gamer making choices amongst objects placed on a geo-coded map. Every choice has consequences across a pre-set range of fields, yet there is no clean choice, i.e., one which has only positive or negative consequences. The game ends when the player crosses some pre-set value in one of the fields, i.e., crosses a ‘red line’. So, the gamer needs to navigate through a field of choices, the point of the game being to stay on the game as long as they can, avoiding ‘red lines’. The more choices made, the better the player. Sustainability is key; the more the player can sustain making choices, the better. But most importantly, ChoiCo affords important transparency, leaving users, in the role of game creators or modifiers, to name as many fields as they wish, to set values for every choice, to program the starting values, the ‘red lines’, and a number of ‘warning messages’ and other rules via a block-based programming language. Field values can be numerical, fixed or random, and visible or hidden from the player (they can make the player need to infer the consequence of the field by observing some text, a video, or a picture).

ChoiCo is thus not just about designing or playing a game, it affords the user the ability to take on the role of a prosumer, someone who engages in in-game modding as well as design and play, interchangeably. Each game can be considered as a ‘living document’, to use the term coined by Trouche and his colleagues in their theory of addressing educational practice through the continual re-design of educational resources [48]. The main features of ChoiCo are based on the following design principles:

• Constructionist games, games affording access to the content and rules of the game and providing tools to define and to change them;
• Free climate, alternative reality, low stakes, so that users find a safe space to try out risk-free solutions to wicked problems and consider the consequences;
• A framework, or reference, for discussion and debate, the idea is that pairs or groups of students engage in the modding process;
• Rules and content of a game open to modding;
• Gaming rules: sustainability, i.e., stay on the game as long as you can;
• Interchangeable gameplay and game modding, i.e., the practice of adopting a binary role of player and designer of a game.

So far, ChoiCo has been used in schools spanning from year 3 to year 10 in the context of group project work either in design thinking or transdisciplinary projects or in domain-specific courses like mathematics. Typically, students are given an incomplete or faulty ChoiCo game [38] carefully designed to embed mathematical ideas in a realistic socio-scientific issue and asked to investigate and make changes by back and forth playing and modding the game. Alternatively, they take part in a design thinking project (https://extendt2.eu/, accessed on 6 June 2024) where their task is to collectively design and develop a ChoiCo game for others to play. ChoiCo has also been used in professional development and Masters’ courses where teachers engage in design thinking projects themselves or alternatively individually design games aimed to be given to their respective students to play and make changes to in turn. The site contains diverse sample games and ensuing textual and video resources. The system is designed to be transparent with respect to the context and the pedagogical agenda and meaning regarding its use, open to ethical and responsible educational designs and to approaches sensitive to cultural, socio-economic, and epistemological stances to the issues at hand. The content inserted by its users is what makes the affordances acquire meaning, for instance the value of a numerical consequence can be either arbitrary or scientifically accurate and can be mediated to the students as questionable or not.

What follows is a detailed description of three diverse show-case examples to make the above characteristics clearer to exemplify in the discussion of mathematics in socio-scientific issues. The first is about the affordances of a game regarding a citizen at the time of the COVID pandemic. The second analyses the design principles of a game included in the mathematics primary and secondary digital books, an infrastructure of the Ministry of Education in Greece freely available to everyone. It is about the shopping decisions of a citizen in a supermarket trying to balance currency, dietary needs, and pleasure. The third contains excerpts from group discussions over a game under question, to contribute to the discussion on ways in which mathematical concepts and mathematical thinking may appear in a debate over an environmental issue.

So, let us start with an example of a game that was designed by postdoc researchers at the Lab [49]. Consider a ChoiCo user in the role of the player, ‘Tina’. The game is called ‘COVID Survival’ and can be found amongst other example games on the ChoiCo site for anyone to play or make changes to. In ‘COVID Survival, Tina as the player is a citizen, and she lives in the COVID pandemic era. The choices available to her are about everyday activities such as ‘running’, ‘going to the mall’, ‘resting at the park’, etc. For instance, she can consider ‘walking with a friend’. If she does choose to do this, the game tells her what the consequences will be along a line of values. The values are ‘physical capacity’, ‘it’s fun’, ‘social’, ‘money’, and ‘risk of COVID infection’. So, the game tells Tina what is going to happen if she makes a choice before she decides to make it. If she chooses to take a walk with a friend, then the COVID risk would be a random number from minus five to minus fifteen, i.e., an equal probability in the respective range of values. So, there would be some risk of catching COVID, i.e., being thrown out of the game, but not a lot. In any case Tina would not be certain of what is going to happen. Her ‘physical condition’ would in this case improve by 20 points (points are arbitrary of course). Fun, social activities have positive consequences, and money has slightly negative consequences, since they might hypothetically go to a vending machine. So, upon clicking on the selection of the choice to walk with a friend, Tina observes the change in aggregate values in a respective panel on the screen. Then, she can try doing something else, considering how to manage the negative consequences of her previous choice. If she decides to go to the local store or the shopping mall, then she will greatly benefit in fun, social, and physical condition but she would be in grave danger of catching COVID and will get into financial trouble (Figure 1).

If, on the other hand, she decides to go running in the park, she would have no risk to catch COVID and gain in physical and fun, but then get a bit lonely and spend a little money. If, on the other hand, she decides to go get money from the bank she will put herself in grave danger, get depressed by the experience, talk to a couple of people maybe but gain minimal physical condition standing in the queue. So maybe she might decide on going home instead. Clicking on that choice brings up a different plane with another set of other choices such as work from home, shop online, etc. For instance, if she decides to play video games she gets a warning sign ‘you’re running out of money’ and if she continues making the same choice again, she loses, since her physical condition, social life, and money run out. So, the idea of the game is to stay on the game as long as possible, and there is no choice that has only positive or only negative consequences. Tina needs to find a way to navigate through everyday choices in order to stay in the game as long as possible. It is in this sense that this is a game about a citizen dealing with a wicked problem.

And now, we will consider a modding a game, taking on the role of a hacker. In the example of the COVID game, everything mentioned so far can be changed by Tina who can switch roles to become a game modder. A button click switches to the editing page where Tina herself or anyone else can make changes at all levels. She can change the picture; add new choices; and change the names of existing ones and thus their meaning–substance. Inserting a new choice makes it appear on the map and at the same time on a tabular representation of a relational database as a new record. Tina can give the choice a name and start putting values on the consequence fields. She can observe that there are diverse kinds of values she defines for each of the field columns. For instance, the ‘COVID risk’ field has been defined as containing random values in a range set by Tina. There are other formulas she can use to define field values, such as simple functions or just straight numerical values. Tina can also easily change or add fields (Figure 2). Finally, she can switch to a Blockly programming feature to write programs to set the initial values, set the ‘red lines’, put up warning texts or sounds when a certain value gets close to a red line, and anything else regarding the game rules (Figure 3).

All of these affordances have been designed to allow users to engage with a larger and more complex set of concepts, practices, and values than one may find in like-minded authoring systems in education. There are three distinct but also interconnected areas for these. One is, of course, the socio-systemic issue embedded in a game. The second is computational thinking, i.e., concepts and practices cultivated and employed in programming and the use of GIS, and relational databases. The third one may involve mathematical concepts and rationale. Before we elaborate on the latter, it is important to say a few words about computational thinking since this too can be connected to mathematical thinking in various ways [50]. ChoiCo affords the use of functionalities to do with geo-coded data with relational databases and with block-based programming, giving some emphasis on event handling and Boolean logic. This enables the main characteristics of computational thinking as originally defined by [51].

In a study of students using ChoiCo in such a way [49], we analyzed their progressive process of exploration (play the game), deconstruction (break down the game structure), analysis (analyze the game elements), synthesis, and construction. The process involved the integration of interacting with various affordances (graphics, story, rules, characters, etc.) and with the use and understanding of Blockly programming concepts such as conditionals, Boolean logic, and event handling, but also programming processes such as developing high thinking skills like iteration and refinement, debugging, error prediction, etc. [52,53].

6. The Mathematical in Grappling with Wicked Problems

So how can we think of the use of ChoiCo to develop dispositions to use rationality and mathematical thinking to grapple with wicked problems such as the consequences of the COVID pandemic? Let us first consider the COVID game—and ChoiCo more widely—as a mathematical microworld [54]. What mathematical concepts are or could be embedded in a game? And when students change the game, what mathematical reasoning could they engage in as they identify, question, and modify values and relations between them? As implicitly discussed in the previous section, the COVID game embeds proportional thinking, functions, probability, and mathematical issues related to programming. But, of course, these mathematical ideas adopt the status of affordances. When the game is put to use by learners, the identification of these ideas and the ways they may or may not be put to use is a process of instrumentation and instrumentalization [28].

Game modding in education has mainly been connected to computational thinking, but if the games have embedded mathematical ideas, then this computational thinking becomes connected to thinking mathematically. ChoiCo games are thus seen as productions; they are artifacts designed to be used by somebody else. They are fun and have many kinds of connections: connections to real issue debates, connections to gaming, and connections to entrepreneurship. They could therefore be considered as useful resources in this new era of 21st-century skills and equity and wanting to change this silo domain structure of the education system. And, they still retain some of the benefits that we have from teaching students domains and mathematics specifically.

ChoiCo games can be resources for learners to engage in and to grapple with socio-scientific issues in the context of post-normal science and they can be specifically designed to invite students to make changes. They need not be designed for changes to be difficult and for students to be terribly savvy. They can be designed for students to find it easy to engage in modding. From a pedagogical point of view, rather than resources designed to enhance student responsiveness to pure mathematical tasks, they can be considered as tools to cultivate mathematical disposition and competence, perceiving students as young citizens. Modding with games such as ChoiCo can then help students to recognize the value of mathematization, modelling, tinkering with models, and using rationality to grapple with wicked problems.

There are several freely available games already up there on the ChoiCo site, http://etl.ppp.uoa.gr/choico (accessed on 6 June 2024). A smaller number of these have been designed for more focus on mathematical ideas. So far these have been embedded in permutations of a specific gaming idea, which is shopping in a supermarket. Supermarket games have gained wide-scale visibility since they reside in the official recourse portal of the Ministry of Education in Greece (http://photodendro.edu.gr, accessed on 6 June 2024). This is an infrastructure called ‘the digital school’, which contains a large portal of digital artefacts for students to use in all subjects and at the same time has links to these artefacts residing inside the online version of the curriculum books from years 3 to 11. There are around 1600 such artefacts in mathematics, 1200 built with GeoGebra and 220 built with a 3D dynamic programmable modeler called MaLT2, also developed by us at the Educational Technology Lab (http://etl.ppp.uoa.gr/malt2, accessed on 6 June 2024). There are six versions of the Supermarket ChoiCo game spread from the end of primary to the beginning of secondary year books.

The links to ChoiCo games in the curriculum book are in a section about mathematical problems. Clicking onto the ChoiCo micro-experiment [55] gets you directly into the game, which is about doing things in a supermarket. Your values are the number of items, how much money you have, health, and pleasure (Figure 4).

So, if you buy chips (crisps) then you get the following: price is four euros, number of items is 2 times more than milk, and your health will deteriorate by 3 but your pleasure will increase by 3. So, if you keep eating crisps by choosing them consecutively, the game will warn you when your health is below 10 and throw you out when it is below zero on the grounds of poor health. If, instead, you buy lettuce or yogurt, your health gets better, but your pleasure is reduced and then you get thrown out due to depression. The idea here is that consequences do not only have direct values. Sometimes values need to be considered in connection to other things, so students need to think about units of measure and proportional relations and operations. The supermarket game is an example of shaping the design of a game to fit mathematization and the use of mathematical concepts more directly within a silo mathematics curriculum. But still, even now, there are important value-laden issues such as taste, health, pleasure eating, balanced spending, etc.

The idea behind ChoiCo is what has been termed as ‘black and white box design’ [56]. There are some digital objects and functionalities that are black boxes to the user, such as the database and the geo-coding functionality. ChoiCo has not been designed to get users to reprogram a relational database. It is unlikely for a piece of technology to contain programming and databases and a GIS at the same time. The design principle, however, is to prioritize a pedagogical perspective irrespective of whether it is easy to find the technological infrastructure. This is dealt with by finding available components, containing the development part mainly to gluing them together, and then building whatever else is needed on top. This means that the ‘white box’, or alternatively, the transparency element of the system, affords the content inserted, the consequences attached to the decisions, the start–end rules, and warning messages. In this way, the system allows for any type of consequence to be inserted, and therefore a mathematical consequence, e.g., a function of another consequence (bananas are double the price of oranges plus one), or a random consequence affording stochastic thinking, e.g., the probability of catching COVID. The embedded mathematics can be considered in two ways, the actual consequence dependencies and also the combinatoric thinking required for a player to make judgments and estimates before making the next decision.

7. When Mathematical Reasoning and Values Co-Exist

So, what does mathematical reasoning in the context of grappling with wicked problems sound like? In the example elaborated here, a group of three 13-year-olds are jointly modding a ChoiCo game designed by the researchers to include questionable perceptions about what it means for a citizen to live a life supporting sustainability in their city [57]. The students played the game to start with and then began to discuss the ideas embedded within. R is the researcher, S(x) is a student.

R: So who is the winner?

S8: We all are! We all finished the game.

S1: We won because we have the largest amount of money: 1200.

S8: Who said that money was most important for the game? All you did was to go from work to home and vice versa.

S5: I think we won ‘cause we have the highest energy levels and the highest social status.

S11: Wait, wait. The winner is the one who has the highest values in all these: money, energy, fun, social status, health, hygiene.

R: What kind of life do you have to live in order to achieve that?

S16: You have to do a lot of everything: have a lot of fun, have a lot of money, do not neglect your social life… This is all too much.

S3: You have to be a freak to live like this; you won’t have a moment of peace.

In the above dialogue generated by the researcher, the students discussed a gaming idea—what does it mean to win—connecting it primarily to the wicked problem at hand: is a citizen’s life worth living if they go for high achievement in all aspects all of the time? Within this cycle, the students considered numerical values and their aggregates, implicitly keeping in mind that each of the available choices had at least one undesirable value, a negative number in this respect. The underlying problem was how to increase the values of all of the fields even though every choice would unavoidably bring a decrease in at least one of them.

S(1) I am not sure about not having cars in the city.

S(2) I am telling you it has been done in Freiburg. Cars are related to pollution.

S(1) Yes, but imagine how much more time you need if you go to work by bicycle. You need to wake up at least one hour earlier.

S(3) Ok then, we will add time in the indicators. Taking the bicycle should have reduced pollution but raised time.

In discussing the pros and cons of ways to go to work, the students identified that pollution and time were conversely dependent and thus decided to insert another field, time. Here again, their focus was on the pollution problem primarily, and the inverse proportion idea was a tool with which to think about and argue for the citizen’s choice to take the car or the bicycle to work.

8. Discussion

Admittedly this paper has opened up many issues simultaneously: Can mathematics as a normal science be considered as a tool, skill, and competence in situations better understood through post-normal science, such as individuals and collectives up against wicked problems? How can the design of digital media for wicked problem education incorporate agendas from transformative mathematics education, such as agency, putting concepts to use in personally meaningful situations, considering mathematical arguments open to refutation? What kinds of skills, competencies, and dispositions regarding wicked problems can be cultivated with the help of mathematical reasoning? Can mathematical reasoning be cultivated in the context of making one’s own decisions, being creative, and adopting an individual and social active stance to wicked problems?

The COVID game examples and the short excerpts from students modding the ‘citizen in a sustainable city’ game were considered as contexts to bring some elaboration of potential educational and research endeavors. These two cases provided a glimpse of users grappling with socio-scientific issues while using some mathematical concepts as tools for the former. The mathematical ideas embedded in these two games ranged from simple operations to proportions and linear functional relationships to probability. This was only a beginning. The range of ideas that may be useful in wicked problem education has yet to be understood, but most of all, the interesting question is what kind of rationality may grow using mathematical reasoning and, in particular, how can this rationality be used in the quest to understand and grapple with wicked problems? What kind of weight should rationality have on post-normal science education and what form should it take? Far from claiming to resolve this huge issue, this paper aims to contribute to articulating the problem a little more succinctly. In the excerpts, we saw students reasoning at different levels, from articulating local arguments of aggregating or calculating values to re-considering the ‘big issues’ such as ‘what is the value of being a high scorer in everything’. There is much more to be learned about how reasoning can be used in such situations and how and when mathematical ideas and concepts might become useful. The COVID ChoiCo game could embed a larger range of mathematical concepts such as quadratic or exponential or periodic functional relations, and a pendulant course could be designed with respect to attention and focus from citizen habits to the mathematics underlying the consequences. The supermarket game embedded mathematics a little more clearly if crudely. The handling of currency, simple functional relations and item amounts were meant to be visibly intertwined with dietary issues. The design principle here was for the game to gain visibility as a resource in primary and early secondary mathematics education, which means that the intended perspective for teachers and students was more towards a mathematical exercise embedded in a realistic everyday issue. But this need not be so. Had the game been designed as, e.g., a transdisciplinary project not connected to mathematics curriculum, there might have been a larger number of value-laden fields such as social life (crisps is a plus when going to a movie, maybe broccoli has minus at a party, etc.). The design issue therefore is tightly connected not only with the normal vs. post-normal tension, but also, with the designers’ agenda concerning institutionalization. Is the approach to push transformation, to allow it, or to slow it down?

Critics of normal science complain that single domain approaches to real world problems are restrictive since they only provide facts and models providing tunnel vision perspectives of complexity. From the point of view of mathematics education, however, the process of considering and understanding facts and of building such models to engage in discourse concerning wicked problems is key to the cultivation of critical mathematical thinking in a transforming paradigm [2,14]. Engagement with mathematical thinking need not necessarily curtail the appreciation of wicked problem complexity, on the contrary, mathematics can be put to use to enhance competencies in grappling with and further understanding socio-scientific issues. From an epistemological point of view, mathematics and post-normal science are in many ways incompatible. In education, however, teaching the use and the role of rigor, generalization, proof, and argument need not be restricted to exposition with axiomatic systems, it can additionally be employed to address value-laden complex and conflicting issues to which citizens are exposed throughout their lives.

In any case, the paper aimed to call upon the mathematics education community to consider the role of mathematical reasoning for wicked problems and the challenge to develop an argument for cultivating such reasoning in this kind of transformative educational context.