An examination of pre-service mathematics teachers ’ ethical reasoning in big data with considerations of access to data

Implementations of Big Data analysis are reshaping society. The novel ways mathematics operate in society warrants new efforts for mathematics education, both in teaching the new technology and in providing an ethical and critical awareness of its implications. This interview study investigates pre-service teachers ’ ethical reasoning in data science contexts, focusing on aspects of access to the data that underpin the technology. Findings show that pre-service teachers offer a wide array of ethical arguments related to access to data, that informs their effort to think critically on oppressive situations. However, there is also an indication that their reasoning can be limited by lacking understanding of the related data science methodology, implying that mathematics teacher education should encompass more of this.

The rise of Big Data Analytics (BDA) and related methodologies, have collectively been labeled the fourth industrial revolution.BDA is a new phenomenon that sits at the intersection of statistics, computer science, and domain knowledge.Using advanced statistical and computer science methods, data scientists are able to extract stories about human behavior and world phenomena to predict the future (Matthews, 2019).Notably, the ability to predict human behavior has reshaped the world economy into a data driven one in which leading entities are those that have the financial, technological, and human resources to be proactive instead of reactive (Agrawal, 2017).However, as analogous to previous industrial revolutions, rapid societal changes have led to negative ramifications for marginalized groups, including immigrants, the impoverished (O'Neil, 2016), women and non-binary gendered peoples (D'Ignazio & Klein, 2020), and people of color (Sabbagh, 2020).Relating to the theme of the special issue, Mathematics in Society: Exploring the mathematics that underpins social issues, the perspective taken in this study is grounded in the necessity to understand the potentially harmful and liberatory aspects of the mathematical components of this 4th industrial revolution in order to fully reap the benefits of the new technologies and BDA methodology.Since students will become both consumers and producers of these new technologies, we identify mathematics education as an arena for making connections between ethics, critical thinking and the mathematics of BDA.Thus, pre-and in-service teachers' understanding of such connections constitute important factors for consideration and connects to the question raised in the call for papers for the special issue, how can research in mathematics education support a more robust and critical understanding of the world?As such, this study seeks to investigate pre-service teachers' (PSTs) ethical and mathematical reasoning in the context of BDA, with particular emphasis on their considerations of accessibility to data and algorithms.
This study serves both as a phase of a collaborative, Design Based Research project initiated by the Charlotte Ethical Mathematics

Literature review
In the sections that follow, we will briefly describe the literature related to discrimination in machine learning, access to data in machine learning, and research-based arguments regarding the need for ethical mathematics education in the Age of Information.The literature review intends to illuminate the role of data in the data driven practices of BDA, how existing injustices can be reproduced by the new technology, and how ethical aspects can become pivotal when teaching and learning about mathematical aspects of the new technology.The literature review ends with identifying a gap in the related research that corresponds to the focus of this study on perspectives of access to data.

Discrimination and access to data in machine learning
The mathematical algorithms used for predictive models in BDA are developed on training data and tested for accuracy on a test data set.Data scientists typically choose the algorithm that predicts with the highest accuracy.Importantly, if the training data is biased towards particular groups, then the algorithm itself will produce biased outcomes.
Well known data sources used in BDA include a digital compilation of users' behaviors and characteristics.Such "digital traces" generate a user profile constituted by personal attributes and predicted traits in the absence of desired data.As a result, the demand for more data to improve convenience and efficiency has led to companies' increasing surveillance of peoples' behaviors (Zuboff, 2019).Today, organizations are able to access trends in our movement through mobile phone locations, internet traffic on company websites and third party cookies that track internet browsing.In addition, the data collected often goes beyond its initial purpose, as exemplified by the court conviction of the Swedish Police authority for gathering ethnicity data (which is sensitive and protected data in Sweden) in ways that went beyond the purpose of police work (Renfors & Justitieombudsmannen, 2015).Because of this, some scholars have stressed that access to such personal information can make individuals and groups more vulnerable (O'Neil, 2016;Zuboff, 2019), while others have countered that the precision in the statistical predictions are too low to pinpoint and exploit specific individual's characteristics and that only characteristics on a group level are predictable to any meaningful degree (Sumpter, 2018).Despite arguments about precision, the collection of sensitive and/or behavioral information on individuals (e.g.race, ethnicity, gender, etc.), whether accurate or not, has been shown to yield predatory and discriminatory ramifications for some groups (O'Neil, 2016).
An additional concern relates to the ability of machine learning methods to predict characteristics of individuals based on their available data.That is, if sensitive data, such as race or gender, is omitted from the training data, the algorithms can still adopt stereotypes from proxies in the data (O'Neil, 2016).To clarify, information is contained in that data that can be connected to the demographic characteristics of the person, regardless of whether or not those characteristics are explicit.For example, a person's zip code can be connected to their neighborhood of residence, and potentially their race or ethnicity.This constitutes a problem for democracy when such algorithms are used to make important automated decisions for citizens (Villani et al., 2018), creating tensions between different perspectives of fairness (e.g., the right not to be evaluated on the basis of one's race on the one hand, and not being discriminated against because of racially biased data on the other, (Corbett-Davies et al., 2017).Naturally, this begs consideration of the ethical implications of using personal data when making decisions in society.As a result, we argue an approach to mathematics education in which students are positioned to consider the ethical implications of their mathematical work on individuals and social groups.The literature related to this argument is discussed next.

Impetus for an ethical mathematics education
The rapid growth in our capacity to analyze data has catalyzed a shift towards data science in the job market and in the amount of data literacy required by the general public.As it stands, the BDA industry is populated by data scientists who come from predominantly privileged backgrounds (white and/or male; upper income) resulting in what D'Ignazio and Klein (2020) term a privilege hazard.That is, phenomena in which teams of data scientists are primarily composed of people from dominant groups in privileged positions.Designs created in these contexts often reflect the dominant perspectives, experiences and values of the privileged creators at the expense of nondominant identities and viewpoints (D'Ignazio & Klein, 2020;Noble, 2018).Having limited or no experience with social and/or financial struggle or the lived-experiences of the masses, privileged data scientists are often ill-equipped to identify oppressive situations in the world resulting in a lack of consideration for the impact of their models on societal groups (D'Ignazio & Klein, 2020).Consequently, a salient threat to democratic society today is the hard coding of discrimination in the processes used by world governing entities (D'Ignazio & Klein, 2020).
To combat the negative effects of BDA and the privilege hazard, there is an increasing amount of research that adopts a sociopolitical perspective on BDA congruent with the aim of this study.Such research stresses the relevance of power in relation to data in education (Rubel et al., 2021), proposing that power and responsibility may be hidden when people interact with mathematics, statistics, and technology (Straehler-Pohl, 2007).To clarify, the algorithms used in BDA to make decisions about users are either hidden behind a user interface and/or the code is uninterpretable by the common citizen, making it impossible for the user to critique the variables, methods, and/or claims made by the organization (O'Neil, 2016).This phenomenon is further reflected in mathematics curricula where, for instance, the societal and democratic aims for mathematics education are expressed in general terms and do not call for any of the specific mathematical understandings required to perceive how power can operate in BDA (Andersson, et al., 2022).
This article draws on an emerging body of research that calls for the explicit grounding of mathematics education in ethics (Atweh & Brady, 2009;Boylan, 2016;Ernest, 2018).Such scholars view mathematics as undeniably integrated into the technical, political, industrial, military, social, etc., facets of the world having real effects on individuals and groups in society (D'Ambrosio, 1998;Lengnink, 2005).Further, given the mathematical nature of data science and its prevalence in the social, economic, and political aspects of globalized society, it has become imperative that students become proficient in its techniques and algorithmic ways of thinking (Boaler & Levitt, 2019;Cobb, 1999;O'Neil, 2016;PISA, 2020).Initiatives to develop a standardized K-12 data science curriculum are taking shape across the globe.These include efforts to transform the current mathematics curriculum, or provide for the eventual shift to more relevant, modern schooling (Boaler & Levitt, 2019;Koh, 2020;Tong & Yong, 2015).K-12 data science programs, however, have yet to fully integrate ethics and social justice into their coursework.They typically focus on developing students' foundational understanding of the data life cycle and the basic statistical and/or computer science concepts necessary for navigating it (Gould et al., 2016;Heinemann et al., 2018;Tong & Yong, 2015).As such, we adopt methods from Critical Mathematics Education (CME) to guide our research in developing ethically grounded data scientists and citizens.However, there is a shortfall of research in CME related to peoples' personal data, their perspectives on access to data, and their consideration of, and responsibility toward ecologies and the other (Boylan, 2016;Andersson et al., 2022;Register et al., accepted).Given the expansive reach of BDA, its implications for marginalized groups, and its overall effect on society, we find it imperative to expand the scope of CME and to address the aforementioned gaps.

Theoretical orientation
The essence of what we consider an ethical mathematics education is tied to the transformative ideal for mathematical education.That is, we see mathematics education as a means to "create the world in a new way" (Atweh & Brady, 2009, p. 270), going beyond responding to injustice (typical of CME and Teaching Math for Social Justice) and focusing on responsible creation.Drawing from relational ethics, Boylan (2016) proposes four ethical dimensions that mediate the learning of mathematics and include relationships with others, the societal and cultural, the ecological, and the ethical self (Boylan, 2016).Notably, in Western cultures, considerations of the other are typically subordinated to neoliberal and individualistic attitudes concerned with individualism, power, success, and self-preservation (Wiggan, 2012).In contrast, Boylan (2016) suggests that ethical reasoning concerns issues of fairness and choice, but also must consider relationships outside of those that exist within our own communities and beyond human interactions.Thus, mathematics education should also develop a sense of social response-ability that includes "the ability to respond to the demands of our own well-being and the ability to respond to the demands of the other" in our current and future lives (Atweh & Brady, 2009, p. 269).Considering the ethical implications of the mathematically grounded sciences in the Digital Age, it is essential that mathematics is learned in the context of ethical design where students are encouraged to consider the micro and macro-level implications of their mathematical products.

Critical mathematics education (CME)
Given its explicit focus on identifying and dismantling oppressive relations and power structures, ethical mathematics education adopts the core tenets of Critical Mathematics Education (CME) as a basis for promoting socially response-able mathematicians and data scientists (Atweh & Brady, 2009).Skovsmose (1994) identified several core tenets of CME that we see as essential to the development of ethical mathematics education.Namely, schooling should serve as preparation for active participation in political life.Thus, mathematics education should center applied mathematics as it is used in the increasingly globalized world.Further, mathematics should be promoted as a tool for exploring and analyzing critical societal features, but that can also be used in unethical ways.Considering the potentially problematic function of mathematics in modern technology and society, mathematics itself should be rejected as objective and neutral since humans employ mathematic and should be critiqued with regard to the reproduction of societal inequalities and discrimination.Regarding these concerns, CME scholars (Gutiérrez, 2013;Gustein, 2006;Frankenstein, 1983;Rubel et al., 2016;Skovsmose, 1994) argue for a contextualized mathematics curriculum which is based on students' macro and micro-level realities for the purposes of developing a critical consciousness of, and empowerment to dismantle, the oppressive forces in one's life.

Critique of CME
Boylan (2016) suggests that within CME there is a lack of discourse related to consideration of the other as an ethical imperative for mathematics.Critical Mathematics pedagogies typically focus on liberation of the self or that person's social group.They adopt Freire's (1970Freire's ( )/(2018) ) viewpoint that people in oppressed positions cannot be set free by their oppressors, but must develop the literacy, reflection, and agency to free themselves.Such studies typically focus on groups of racially and/or ethnically homogenous and C.H. Andersson and J.T. Register historically marginalized populations of students who explore social justice issues relevant to their community through mathematics (Berry, 2003(Berry, , 2004;;Gutstein, 2006;Rubel et al., 2016;Rubel, 2017).However, while Freire certainly opposed the concept of teachers and/or oppressors as saviors, he suggested that to dismantle oppressive systems, the oppressed and the oppressors must work together to transform reality.That is, the oppressors must also free themselves from their oppressive role through their own development of critical consciousness.With that being said, Freire's work typically conceptualized oppression on the basis of economic class, receiving criticism for his lack of explicit attention to race and gender (Gutstein, 2006;Hooks, 2014).The present lens considers oppression in terms of unearned societal advantages based on a diversity of privileges including but not limited to race, gender, socioeconomic status, dis/ability status, education, and their complex intersections (Crenshaw, 1991).
The outlined ethical dimensions of mathematics illuminate gaps in mathematics education as it currently stands.While attempts to integrate the societal and cultural dimension into mathematics education have been gaining traction (Berry, 2003(Berry, , 2004;;Esmonde, 2014;Gutstein, 2006;Kokka, 2020;Rubel et al., 2016;Rubel, 2017), significant work needs to be done to promote students' consideration of the other, the ethical self, and the ecological impact of their mathematical products, and to understand how such reasoning can be supported in classrooms.For this work to be effective, it is necessary to understand the diverse ethical orientations of those who will implement such curricula (e.g., pre-service teachers) and the characteristics of ethical reasoning that promote the consideration of the ethical dimensions of mathematics (Boylan, 2016).Thus, the results of this study will serve to inform future frameworks used for analysis of individuals' ethical reasoning in mathematics and data science, for making curricular design decisions, and for preparing teachers to implement such curricula.As a starting point, the following CMC analytic framework is used to understand how individuals think critically in mathematics and data science contexts which have micro and macro-level ethical implications.

Critical mathematics consciousness (CMC)
The development of ethical mathematicians and data scientists may be aided by their development of critical mathematics consciousness (CMC) (Stephan et al., 2021).Derived from Freire's conception of critical consciousness, the definition for CMC used in this research attempts to critically analyze the mathematical action and/or product from an ethical and sociopolitical perspective, and refers to.
the awareness that human beings do mathematics; thus, there are potential ethical dilemmas and implications of mathematical work which may affect entities at the individual, group, societal, and/or environmental level.CMC includes sociopolitical, ecological, and communicative mathematical awarenesses (Table 1) and a willingness and commitment to act (i.e., critical mathematics agency) (Register et al., 2021;Stephan et al., 2021).

Assessing CMC
Importantly, people may exhibit different levels of consciousness at different times and in different contexts (Stephan et al., 2021).Their consciousness is not a part of their personality but is fluid, dynamic, and tied to the context of the issue and their relevant experiences.Within a given context, an individual's reasoning can be classified as one of six levels of consciousness (see Fig. 1.CMC Growth Framework).Shor (1993) defines critical transitivity (CT) as the actualization of full critical consciousness where the individual engages in critical reflection, recognizes the systemic influence on oppression, and is empowered to act by attacking the causes of oppression at the systems level.Freirean theory maintains this as the goal of a liberatory education.Semi-transitivity includes critical thought on the ethical and critical aspects, may see the systemic cause of the oppression, but does not attack it at its root.Rather, they may perceive and attack the cause as isolated incidents, isolated semi-transitive (IsT) or put their faith in other individuals or groups to change oppressive situations, systemic semi-transitive (SsT).Individuals who demonstrate disempowered (Di) consciousness exhibit some critical thought, but feel restricted in their ability to act, likely due to a lack of power.Similarly, those who exhibit intransitive consciousness, feel disempowered to act, but do not exhibit critical thought, perceiving their oppressive situation as a consequence of God's will or bad luck, rendering their agency as irrelevant.Finally, Stephan et al. (2021) incorporate King (1991) construct of dysconsciousness (Dy) referring to a lack of critical thought caused by a "distorted vision of oppression" where inequity is the "natural order of the world" (p.3).They believe that the oppressed are at fault for their current situation and are thus solely responsible for overcoming their oppression.As King explains, dysconsciousness may be taught through familial and educational influences, or by media effects.

Ethical reasoning in mathematics [ERiM] Principles
Stephan et al. (2021) and Register et al. (2021) argue that high levels of CMC imply an awareness of ethical implications to be

Sociopolitical
Mathematics is used to model and interpret the real world and can be used to make decisions (both at the individual and systemic levels) that may further disenfranchise (or liberate) marginalized groups.

Ecological
Mathematics has been socially constructed by human beings and thus has implications for humans, animals, the environment, and its interconnected ecologies.

Communicative
Mathematical communication has the power to educate and mis-educate society and encourage the masses to act in certain ways.
C.H. Andersson and J.T. Register considered in the process of making mathematical decisions and a sense of personal responsibility to do right by the affected parties.As such, the Ethical Reasoning in Mathematics [ERiM] Principles Framework is used to identify the specific ethical considerations that individuals' make in order to help classify their demonstrated levels of CMC in specific contexts (Stephan et al., 2021).Unlike CMC, ERiM does not identify individuals as having an exhibited level of consciousness, but rather identifies specific categories of ethical principles involved in data science contexts as identified by discipline experts.For instance, the ERiM framework contains ethical principles that speak to considerations made in the process of making data-based decisions.These include violations of privacy (privacy), fair access to benefits (fairness), the accuracy of data and/or predictive algorithms (accuracy), who is accountable for the effects of these algorithms (accountability), whose property is the data to sell (property), whether decisions made are loyal to specific individuals, groups, or entities (loyalty), whether algorithms are biased or objective (algorithm bias), if the algorithms are readily available for inspection (transparency), what ecological impact they may have (ecological), how they effect employment (employment), whether they are discriminatory (discrimination), and who has (or should have) access to the data, algorithm, or findings (accessibility).By identifying contexts in which individuals' express concern for the ethical impact of their mathematics products and decisions, we anticipate that the ERiM Principles may support the design of curricular materials that foster their development of critical consciousness.For this study, however, we explicitly focused on the accessibility element of the ERiM framework in hopes of speaking to the forms of reasoning that PSTs demonstrate with regard to who should have access to personal information, and in what forms.

Access to data
Accessibility is an element of the ERiM framework that we explicitly focused on in the development of the task-based interview questions in the study and in our analysis of PSTs reasoning.In the analysis, a new construct emerged which allowed us to better classify PSTs considerations of access to data.To encompass two perspectives on the role of data in PSTs answers, we consulted literature on BDA in oppressive situations and formulated two related concerns, dearth and excess, collectively called access to Data (AtD): • Dearth: The problem can be understood as being caused or amplified by limited possession of reliable and valid data, e.g.too few data points, biased data, or data that is impossible to analyze (Williams et al., 2018).• Excess: The problem can be understood as being caused or amplified by an excess of reliable data (or what is portrayed as reliable data) possessed by a potential adversary (Darragh, 2021;Zuboff, 2019).
Importantly, dearth and excess are not to be understood as mutually exclusive.It is possible to construe the role of data as satisfying both perspectives, for example O'Neil (2016) identifies the dangers of an adversary having too much data and using algorithms that are perceived to have the capability to predict risk and performance (excess), but simultaneously lacking sufficient data for these predictions to be fair for all groups (dearth).Like CMC, AtD does not necessarily represent an individual's general feelings towards access to data, but rather their reasoning in that specific context.

Framework interactions
The discussed analytic frameworks are used reflexively to identify an individual's level of CMC within a given context.Recall that our goal for mathematics education is to foster high levels of CMC in our students and teachers.Thus, it is important to understand  what constitutes and motivates high levels of CMC.When used together, the ERiM framework (including AtD) allows us to identify the ethical considerations that PSTs make in their reasoning, which speaks to the mathematical awarenesses that they possess in that context (potentially motivating the decisions they make).However, being aware that there is an issue does not necessarily imply recognition of the systemic factors that cause or uphold injustice, or the personal agency to dismantle such systems.Therefore, the CMC growth framework helps us to characterize these essential components, speaking to their CMC.

Methodology
This study employs a Design Based Research (DBR) approach which implies the development of learning theories that either do not exist or are in need of reform (Bakker & Van Eerde, 2014).Such theories speak to both the process and means of supporting learning in dynamic and diverse classrooms (Cobb et al., 2003).The findings of the wider DBR project at UNC Charlotte have illuminated that secondary students demonstrate a range of ethical concerns in their reasoning related to BDA which may serve as a starting point for the development of curricula intended to foster students' CMC in data-based contexts (Register et al., 2021;Stephan et al., 2021).Given that PSTs are positioned at the forefront of education in the digital age, we chose to focus our study on PSTs' ethical reasoning and CMC in hopes to provide more specific information about the preparedness of the future generation of teachers to support the emergence of ethical mathematicians, citizens and data scientists.As aligned with the tenets of CME, such teacher preparation includes the deconstruction of former identities to redevelop teacher identities that support diverse and innovative minds.This involves teachers' examination of privilege, personal biases, the history of opposing cultures and belief systems, and the development of political knowledge of both themselves and their prospective students (Gutiérrez, 2013).It also includes the development of a consistently reflective, adaptive, and dialogical practice that accommodates the changing world and needs of their students.Thus, our goal is to provide a means to support teachers in becoming revolutionary leaders, intent on "unveiling reality" and "coming to know it critically" while discovering themselves as "its permanent recreators" who are committed to participating alongside their students (Freire, 1970(Freire, /2018, p. 69), p. 69).Finally, we chose to focus on accessibility due its permeating influence on society and the lack of literature related to issues of access to data in mathematics education.

Author positionality
The authors of this paper serve as doctoral students at the PSTs' respective Universities in the Mathematics Education program.The first author grew up in a lower class, social-democratic area in Western Sweden.He moved to a segregated area of southern Sweden and formed a racially mixed family.The second author currently resides in the "deep South" after growing up in a lower-middle class family in a small, and highly-liberal state, in the Northeastern U.S.Both authors are White, middle class, first generation University students who moved from predominantly White areas to more urbanized and segregated communities to receive their education credentials and teach high school mathematics.This experience catalyzed their teaching and research towards social justice and ethics, where they see education as an arena for bridging the divide between privileged and unprivileged groups.Their overlapping research in the marginalizing effects of BDA and digital traces served as a catalyst for this study.

Participant selection methods
Participants were sought by asking for volunteers for interviews on ethics and BDA within the mathematics education teacher preparation programs from the researcher's respective universities.Both researchers recruited PSTs in the last two years of their program from courses in which they served as instructors.Available demographics of the participants are listed in Table 2 below.Note that participant demographics related to the race/ethnicity of Swedish (SWE) PSTs are unavailable due to Swedish policy restrictions and the sensitive aspects of collecting race-based data (as opposed to being a common practice in the United states (U.S.).

Interview task development
The semi-structured, task-based interviews consisted of a series of six tasks developed to reflect common and/or troubling discourses in the media and data science industry in order to trigger participants' critical/ethical reasoning.Specifically, they were intended to elicit PSTs notions of systemic injustices related to the mathematical classification of individuals according to social,  'Neil, 2016) and to consider such notions when making data-based decisions.To develop the tasks, interview questions from the pilot studies conducted by the C-EMDT with high school students were adapted to align with the mathematical knowledge of undergraduate mathematics PSTs (Stephan et al., 2021).Their purpose was not to "teach" but to gauge how PSTs reason through and make decisions in real world, BDA contexts.To achieve this, the tasks positioned PSTs as decision-makers in order to promote agency while drawing on each of the CMC   similar contexts outside of the University environment.Prior to implementation, the tasks were tested on community members and doctoral advisors from both countries to select those that promoted ethical discourse and elicited a diversity of ethical considerations from the participants.This study will report the findings of one of six tasks, called Mapping Crime.

Mapping crime
The Mapping Crime task was developed to draw on PSTs notions of historically biased data, police targeting, reinforced stereotypes, and discrimination.Heat maps were sourced from public data (see: https://docplayer.se/185788546-Var-begas-den-allvarligastebrottsligheten-i-malmo.html,© Persson & Hallström and https://crimegrade.org/safest-places-in-charlotte-nc-metro/,©Open-StreetMap contributors).Mapping Crime was created as a local task, in that the maps represented crime in their respective regions, and reflected dominant and/or historical discourses related to the perceived criminality of certain groups in each city (based on race in the U.S. and immigration in SWE).For instance, Black populations have been historically criminalized by society in the U.S., while Arab and immigrant populations have been so in Sweden (Anderson, 1988;Lainpelto, 2019).The goal for this task is not to reinforce common stereotypes, but draw attention to them so that PSTs may apply reasoning to critique and dismantle related discourses using data based representations that promote such misleading discourses (e.g. by drawing on their knowledge of the systemic contributions to crime rates including aggressive policing, police targeting and surveillance).Though we cannot speak to the participants' dispositions after the interviews, the testing and implementation phases indicated that participants reacted to the questions in the intended way, by questioning rather than upholding the illustrated stereotypes.
A salient goal of this task was for PSTs to recognize how data representations may influence both the perceptions and behaviors of individuals within a system.To probe their consideration of access to data, PSTs were asked who might be interested in having access to the application, should they have access, and how they feel the maps themselves might change over time given police and public use of the crime mapping application.Part 2 was intended to gauge PSTs sociopolitical awareness of the potential for negative consequences on marginalized populations and society.We drew on our knowledge of local stereotypes related to the perceived criminality and economic ability of social groups, positioning two hypothetical friends in locations on the map that would trigger the PSTs considerations of reinforced stereotypes, social immobility, discrimination, and ecological impact due to targeted policing.Here, we hoped that the participants may speak to the cyclical nature of BDA in which what the user observes in the data will influence their behavior, which then influences the data in an ongoing feedback loop (O'Neil, 2016).
Mapping Crime centered around a geo-localization application for data driven police work.It is thus related to the trend of predictive policing based on machine learning, by potentially having the same feedback mechanism between data use and data collection that also intersects with existing prejudices and inequalities.Such applications have been described as promising for reducing crime since they would enable the police to better plan where to use their resources, but also critiqued since they give little input on how to work and could therefore increase the risk of discriminatory practices, profiling, skewed depictions and stigmatization of minorities due to focusing on correlations rather than causality (Meijer & Wessels, 2019).Although Meijer and Wessels (2019) found little empirical evidence for the benefits and none for the drawbacks of this kind of data driven police work, we still see a utility of such applications for discussing ethical aspects of BDA since the use of such applications are contested in public debate (e.g.Heaven, 2020) and statistical and mathematical literacy is required to engage in profound discussion about them.Specifically in Mapping Crime, there are several concepts that participants can build their arguments on.This includes but are not limited to; area density, a general understanding of distributions and reading graphical representations of them, proportional reasoning for comparing across neighborhoods when both crime density and population density change, per capita versus a simple count, the role of bias in data collection (e.g. when arrests are being overrepresented for some minorities due to racism), and finally probability, since arrests and therefore, the consequences of arrests, are probabilistic in nature.
In addition to the concerns with data driven police work mentioned above, Mapping Crime addresses the exacerbated complexity of potential ramifications that is added by the dimension of who has access.An example of this, is an expressed concern within the predictive policing community that organized crime could potentially adopt the same kind of data driven methodology if suitable data is available to them (Pearsall, 2010).In addition, the follow up questions within the task (blue text in Figs. 2 and 3) were meant to draw attention to the different ethical and mathematical aspects mentioned above.We conjectured that such questions would encourage participants to develop rich and diverse answers, but also recognize that they may draw attention away from other possible answers, e. g. expressing the opinion that not even the police should have access to the data discussed in the task.From the viewpoint of the research questions, and that we wanted the task to portray the reality of the data industry (that by definition must chose to work with data and sell the outcome), we see these limitations as acceptable, as we believe that there is still ample of opportunities to express advanced CMC and perspectives on access to data.

Data collection methods
The approximately one hour interviews were recorded in person or through Zoom.The researchers began by building rapport, describing the expectations for the interview, and asking a series of pre-interview questions to gauge PST their dispositions towards the role of mathematics and mathematics education in society, controversial topics, and key understandings/skills that they feel their students need to succeed in the Digital Age.Interview expectations were to employ rough draft thinking and reason outloud.PSTs were shown each task, asked to interpret the information, and then respond to the designated questions.The role of the researcher was to create a safe space, provide directions and prompt PSTs to elaborate on their reasoning in cases where their perspective was unclear.PSTs were allowed to skip questions that they deemed uncomfortable.
Importantly, all of the PSTs knew their respective interviewer from coursework and the student-teacher relationships may have impacted their responses to the tasks.However, the contexts of the tasks had not been seen or discussed with participants prior to the interviews and their prior instruction did not include topics related to ethical reasoning or BDA.Our decision to interview our own PSTs was grounded in our socio-emotional understanding of our own PSTs and our desire to make them feel comfortable discussing controversial topics.Additionally, proponents of DBR argue that designs must respond to the needs of participants in a particular context.Thus, we were concerned only with the results of the participants we studied, recognizing that the specific results may not transfer to populations that are considerably different from our studied population.

Process of analysis
The coding process was guided by the CMC and ERiM Principles analytic frameworks.The two authors independently coded the interview transcripts from their respective PSTs.The Swedish interviews were first transcribed and coded in Swedish, then translated The participant's decision is integral to determining their CMC level.Though they may make ethical considerations at a high level of critical thought, the decision that they make speaks to their willingness to act on those considerations (i.e.agency).
Which maps to include; Who should have access to the application; Adaptations to be made to the application.
2. Identify any mathematical reasoning demonstrated in their response.
To understand how participants leverage mathematics to understand the issue at hand.This is integral to CMC as the pilot studies (Stephan et al., 2021;Register, 2021) showed that mathematical knowledge can either support or hinder their critical thought in specific contexts (e. g. proportional reasoning is essential for analyzing representation and identifying issues related to disproportionality, inequity, and discrimination).
Proportional reasoning when comparing the two maps; Statistical reasoning related to the effects of the application on society (distribution changes, probability of future crimes); Data science knowledge about filtering methods and the effects of real time data.

Identify ethical considerations made using the ERiM framework
To determine what ethical issues are at the forefront of PSTs reasoning as it relates to data.
General concern for an action being unfair for individuals → fairness.Concern about disparate outcomes between specific social groups, or marginalized populations → discrimination.
Concern for accuracy of the maps → accuracy.
4. If accessibility (AtD) was considered, determine whether they demonstrated an excess perspective, dearth perspective, or a combination of the two.Excess: Open access to the Mapping Crime app might be dangerous because it could be weaponized by criminals/police, to target specific neighborhoods and/or reinforce harmful stereotypes about the people who live in certain neighborhoods.
Both: People should have access to the information to make informed decisions, but there are potential repercussions of doing so on marginalized communities which should be resolved.

Identify demonstrated CMC awarenesses (sociopolitical, ecological, communicative).
To determine what types of awareness PSTs demonstrate with regard to communication, sociopolitical and ecological impact of data science.
Communicative: concern for how the information is communicated or miscommunicated (map accuracy).Sociopolitical: concern for marginalized groups or individuals (police targeting, reinforced criminal stereotypes).Ecological: concern for the effect on entities or ecologies (migration patterns).
6. Classify PTS's demonstrated reasoning according to its probable CMC level.
Holistic analysis of interview, including ethical considerations, CMC awareness, and decisions made-to determine their overall level of critical thought, on whom they place of onus of responsibility for change, their role in catalyzing change, and at what level that change needs to occur (individual, local, system, etc.).Note: a PST who demonstrated all three awarenesses, but either did not recognize the systemic influence on the issue, or did not see themselves as agents of change may still be classified as having lower a level of CMC for that task.
CT: Recognizes application's influence on marginalized groups (reinforced stereotypes, feedback loop), and migration patterns.Explains how they could reduce negative effects by employing safeguards related to who has access to the application, the amount and format of information represented, and how that information is used.
SsT: ^ but places responsibility of safeguards in others hands.
IsT: ^ but identifies a solution that either does not attack the system or translate outside of the local context (show only the simple count).Di: ^ no solution presented or solution does not protect marginalized communities.Dy: Identifies criminals as the issue.Does not address discrimination, targeting or stereotyping.
Int: Sees the entire problem as the way of the world.

C.H. Andersson and J.T. Register
into English in order to preserve connotations of the language.This enabled a sensibility to cultural and local contexts.After coding and classifying our respective PSTs, we coded the interviews from the other group, then met to determine a consensus on codes that differed.In a final step, coding patterns within answers were discerned and related to PSTs choices when making decisions, allowing us to determine their overall CMC level for the task.See Table 3 below.An excerpt of the coding of Cruz is found in Table 4 below (see the appendix for the full coding for Cruz).Notice that the codes and the coding comments in this excerpt partly overlap and support each other.This is a feature of the step by step procedure explained in Table 3; the next step typically builds on previous steps.For example, once the decision is coded and described, ERiM follows from the rationale of the decision, which in turn informs which kind of CMC awareness that is displayed.In this case, Cruz' exploration of the topic is related to different perspectives conveyed by the two maps (communicative awareness) and possible migrationary impacts

Mapping crime
The findings from Mapping Crime indicate that PSTs in both the US and SWE demonstrated a range of ethical considerations, CMC levels, and perspectives on accessibility that were influenced by their personal experience, mathematical conceptions, and understanding of the BDA industry.CMC levels ranged from dysconscious to critically transitive.There was also a range of responses related to the PSTs' belief in how much data should be accessible, who should be able to access it, and in what format.An overview of PSTs CMC levels and exhibited awarenesses as well as their reasoning related to accessibility can be seen in Table 5.Details of PSTs' ethical and mathematical reasoning specific to their CMC and considerations of access to data will follow.An acronym after a PST name refers to the CMC level of the task.

Access to data
PSTs pondered over who, outside of the police, should have access to the mapping application, what specific information the maps should include, and the format of the maps themselves.Their responses and proposed solutions were classified according to how they described the problem, that is, can it be understood as being caused by a limited possession of reliable and valid data (dearth) or as being caused by an excess of data in the hands of a potential predator or adversary (excess)?Significantly, those who were classified as demonstrating higher levels of CMC (6 PSTs) considered both the dearth and excess perspectives on AtD as well as discrimination and the ecological impact of police and public use of the mapping application.

Dearth perspective
PSTs who demonstrated the dearth perspective on accessibility were primarily concerned with ( 1) providing open access to the application, (2) providing more disaggregated data, and (3) using both maps to provide a more accurate representation of crime in the region.Regarding the first, several PSTs argued that who is granted access to the data is not a cause for concern, and in some cases, is beneficial.For instance, after noting that insurance companies would be interested in accessing the data, Erik (Di) suggested that not only should they be allowed to access it, but that the information should be accessible by all.Similarly, Erin and Rachel (IsT and Dy) commented on how people who are moving or those who work in real estate would also be interested in having access.Notably, both argued that if their access was granted, it may be useful to include more disaggregated information about the historic crime rates in the area.Whereas Rachel considers the effects of accessibility on the consumer, Daniel and Carl (IsT and CT) discuss the effects of accessibility to disaggregated information on police distribution for the purposes of more effective crime prevention (Table 6).
A final characteristic of AtD in Mapping Crime involves PSTs considerations of how the maps should be used.Those who demonstrated the dearth perspective argued that greater access to information provides a more accurate representation of crime in the area and thus argued that both maps should be used together.

Excess perspective
PSTs who demonstrated the excess perspective on AtD were primarily concerned with the negative effects of having open access to the mapping application, and limiting which maps should be accessible.For instance, Erin (IsT) argued against the police having access to both maps to guide patrols, stating that since the maps tell a different story, only the second map should be used.Erin believed that the second map should be considered more accurate for the purpose of police targeting because, in her opinion, population should not be the deciding factor over where the police patrol given similar crime counts.
PSTs who wanted to limit access to the application to specific groups argued that open access could invite potentially malevolent and/or ignorant users.For instance, Cruz (Di) argued that open access to the application could invite criminals to track where the police will monitor, potentially influencing them to commit their crimes elsewhere.Similarly, Alice (Di) argued that by providing access to the public, criminal organizations (terrorists) could use the information to recruit other terrorists within the high crime areas.Notably, those who were primarily concerned with criminality did not reach higher levels of CMC because they did not explicitly consider the potentially discriminatory effects of the application.(Table 7).

AtD as it applies to discrimination, and ecological impact
For those who displayed higher levels of CMC, ecological impact and discrimination were major concerns related to AtD.For instance, several PSTs reasoned that open access to the data could potentially contribute to biased targeting by police and/or the public.For Katherine, this bias came in the form of higher police presence in certain areas contributing to increased reports of crime, but not necessarily reflecting the criminality of the area.Similarly, Daniel argued that open access to the application could reinforce police and public discrimination due to preexisting social prejudices, especially for criminalized populations living in areas with historically high crime rates.(Table 8).

Both dearth and excess
Those like Daniel and Katherine, who discussed the discriminatory effects of open access to the application, seemed to consider both the dearth and excess perspectives in their reasoning.Of the six PSTs who considered both types of accessibility, all displayed levels of CMC that include critical thought related to the influence on oppressed groups, and some agency to dismantle oppressive systems.For instance, Erin (IsT) argued for more disaggregated data within the maps (dearth) but stated that she would only use the simple crime count map to show that crime is everywhere and to avoid profiling (excess).Similarly, Birgitta (SsT) argues that open access to the application could be useful (dearth), but too much accessible information could also have negative impacts on society (excess), like criminal use of the application, the potential for discrimination, and effects on the housing market.Both Erin and Birgitta demonstrated ethical considerations in their reasoning.However, Birgitta's proposed solution placed the onus of responsibility on another person or entity, stating that she did not "want to be the one to decide, whether it should be released or not."Similarly, Erin's solution, though aimed at reducing discrimination, did not necessarily address the oppressive system itself.Rather, she pinpointed an isolated characteristic of the system that she felt might reduce the potentially marginalizing effects of the application.That is, she would "rather give the public and the police the map that doesn't include the population or the residencies" because "it shows crime can happen anywhere".
PSTs who displayed critical transitivity in this task considered discrimination, ecological impact, and both the dearth and excess perspectives on AtD.In addition, they demonstrated critical thought, attention to oppressive systems, and proposed solutions at the systemic level (components of CMC).For instance, Katherine (CT) ponders both the benefits (dearth) and constraints (excess) of providing open access to the mapping application, but ultimately decides that its use by the police or public is not worth the risks of discrimination and de facto segregation through the housing market (excess).Specifically, she states that if the public has access to the application, people may flock to low-crime areas that are typically wealthy, resulting in more resources for those areas.In addition, she disagrees with the assumption that targeting "crime-ridden areas'' will lower their crime rates.Rather, the targeting of those areas specifically, may actually make the crime rates go up, and deter people from moving there.
Josephine (CT) further discusses the benefits and constraints of open access, including the negative effects on the housing market and racial profiling by police (excess).(Table 9).
Josephine ultimately determines that both maps should be used because she believes that more information is always better (dearth).Her considerations of racial profiling by police due to the nature of the application was also evident in Carl and Jenny's (both CT) reasoning.They argued that providing statistical information to a population that may not have the appropriate statistical

Rachel (Dy)
It'd be cool to look at the historic and the current rate of crime [.] Could we [also] have it separated based on the type of crime?[…] because if one car gets broken into [its not ideal] but if you have a bunch of murders, or like attempted murders, or assaults or whatever, like you might want to steer away from that area.Erik (Di) I think this can be an open information as I see it.So that, no I do not see that there is any major problem actually.

Daniel (IsT)
Does this include things like tax crimes and tax evasion which can also have a very high penalty?[.]If you are going to use it to distribute where the police should be, then maybe you should probably look at the type of crime that is solved by a patrol.So you would need some filter to sort out crimes.

Carl (CT)
If this is to be used to distribute police presence in the city, one would probably have needed to take more account of the frequency of crimes, and to what extent these crimes could have been prevented by police presence.
C.H. Andersson and J.T. Register knowledge to interpret them could have potentially negative effects.As intended, accessibility was a consideration made at every CMC level, albeit in slightly different ways.However, the differentiating factors between those who displayed higher levels of CMC were the ways in which the PSTs considered the ecological impact of using the maps and their potential for discrimination in the community.Importantly, such considerations were noticeably dependent on the PSTs' demonstrated proportional and statistical reasoning.

Mathematical reasoning
Up until this point, we have sparsely discussed the role that PSTs mathematical reasoning played in their demonstrated ethical reasoning.Recall the first map represented the density of crimes by population and/or severity of the crime, whereas the second map represented a simple frequency of crimes in the same area.In their reasoning, PSTs drew on their understanding of proportions, probabilities and distributions, and one, drew on his knowledge of data analytics to understand the maps and their effects on society.
As a general trend, all PSTs drew on their proportional reasoning to understand what was being represented in the two graphs.(Table 10).
Additionally, several PSTs used their statistical knowledge to reason that the mapping application is a probabilistic representation of where crime is likely to occur based on past data from the police.They drew on their understanding of distributions to discuss how the map may change according to real time data depending on how the public and police act on their interpretation of the results.For instance, people may move to low crime areas to feel safe and police may target high crime areas to either reduce crime or fill a quota, both of which will influence migration patterns within the area.As mentioned above, Katheriene (CT) described how the maps could make the police find and report more crimes in certain areas.In contrast, Daniel's expressed reasoning reflected his experience as a data analyst and his objective trust in statistics, a common theme that we observed with high school students in the pilot studies and a common notion in Western cultures.(Table 11).
As mentioned in the dearth perspective section, Daniel expressed his data science knowledge by arguing that filtering methods should be applied to the algorithm to break the information down according to which crimes could actually be addressed through police patrols.In addition, he discussed the function of real time data and its potential effects both on the mapping application itself, and on the community as a result.Specifically, Daniel argues that if people choose where to live based on the adaptive maps, you will not only see these changes in real time, but the changing populations may illustrate a faulty depiction of the criminality of some regions given that crime is typically more concentrated in densely populated areas.Such reasoning was shared by several other PSTs who discussed their concerns about the behavioral (ecological) effects of open access to real time data, a characteristic of BDA that is not a common consideration in the theoretical mathematics of the secondary school classroom.

Cruz (Di)
So in the end, if you were to give people the information, people would flock to safer places.And maybe someone who makes a living with cat burgling is going to be like, oh, people think they're safe.So they're not going to buy home security.I'm going to rob them now.

Alice (Di)
Yes, terrorist organizations would be one that I would keep it away from [accessing the data][.]Because of reduced recruitment.

Erin (IsT)
I just would rather give the public and the police the map that doesn't include the population or the residency's.It shows crime can happen anywhere.It doesn't happen in just one singular location.

Table 8
Example Responses (AtD, Discrimination and Ecological Impact).

Daniel (IsT)
People always have a lot of prejudices.And if you come from there [high crime area] and work at a good job, there is nothing to suggest that that person would do something bad.
[.] If this is public [.] and unfortunately, if there are any sort of racist ideas, then there are people who can draw such preconceived notions.

Katherine (CT)
I also was just thinking about the fact that if they're using these mapping features to identify [.] what areas they are going to patrol more, there's going to be more reported crime in areas where there's more police because there's [.] more police to catch things that are happening.

Table 9
Example Response (Both Dearth and Excess).

Josephine (CT)
I would definitely say for the [Black male living in a high crime, low income area], he's probably going to see a higher police presence [.].He may have a harder time putting his background behind him, because if he lives in an area where they already have the mindset that the people who live here are around more crime, then any small offense might be taken more seriously.Whereas for the girl living in the better part of town, she might be around police officials less often.And if she were to ever encounter any problems, like a speeding ticket or something, they might not look any further into it if she is in an overall better part of town.

Interviewer
And is that because of the area they live in or their identity?Josephine I think both [.] But, I think that if everybody saw a map more like the one on the right [count], it might challenge their perception on certain neighborhoods and the safety of them.
C.H. Andersson and J.T. Register

The role of mathematics in their ethical decision-making
Notably, proportional reasoning allowed PSTs to enter the task, whereas their knowledge of data science and statistical reasoning related to probabilities and distributions allowed them to elaborate on the marginalizing effects of the mapping application in society.Interestingly, their mathematical and statistical reasoning only partially served as a basis for their ethical decisions but was essential for reaching higher levels of CMC.That is, PSTs drew on their contextual knowledge of local crime, police targeting, and common stereotypes to guide their analysis, often contributing to a state of cognitive dissonance over which reasoning (contextual or mathematical) to trust.For instance, when asked which map(s) should be used by the police and in what capacity, several of the PSTs toggled between their typically positive orientation towards the use of proportional representations and their understanding of the context at hand.Many indicated that the use of both maps would provide a more well-rounded picture of crime in their respective cities.However, when asked which maps the police should use to target their patrols, they maintained that police should attend to the simple crime map because if crime occurs everywhere, it is unfair to target areas with high crime grades since those areas are often densely populated urban areas with a higher proportion of historically marginalized and/or criminalized people by society.(Table 12).
Partial mathematical and statistical conceptions may have also played a part in their decision making.While it is clear that many of the PSTs attempted to limit the negative effects of the application on marginalized communities, their lack of understanding related to the techniques of BDA restricted several PSTs from making efficient and effective decisions about how to improve the application.For instance, choosing to restrict the application to the simple crime map avoids discriminatory practices based on the map, but does not provide a solution for reducing crime in the city.Though not confirmed, we believe that such generalized responses may be due to a lack of knowledge related to the design and analytic possibilities within the data and computer science fields.Additionally, many mentioned that crime is typically concentrated in densely populated areas, but ignored this in their decision-making.In an interesting case, Birgitta chose not to make a decision about who should be allowed to use the mapping application because she did not want to be responsible for its outcomes.Leading up to this statement, she indicated a flawed understanding of probability, a key concept which lies at the heart of data science.Birgitta implies that since the events on the map have already happened, their probability of occurring again would be low, potentially contributing to her lack of agency.Thus, it seems that in order to make ethical decisions in BDA contexts that are effective, an understanding of the facets of BDA, including its design and analytic possibilities, its probabilistic nature, the effects of filtering and aggregating data, and among others, the effects of real time data on human behavior, are essential.

Discussion
The results indicated that most PSTs were able to draw on their mathematical and statistical knowledge to understand data-based representations and related claims that have ethical implications in society.Some of the PSTs further demonstrated their ability to draw on such knowledge to make ethical arguments.Thus, mathematics was used to both understand the issue and to suggest a solution.Additionally, PSTs made coherent mathematical connections between the role of the data in the mapping application and access to the data for different actors, (e.g., arguments for which map the Police should have access to based on mathematical interpretations of the maps).While we expected that PSTs at the end of their teacher preparation program could use mathematics in this way, less expected was the richness in perspectives when discussing access to data.
We explicitly asked the PSTs about access to data and expected that they would give reasons to either grant access or to deny it.As it

Map 1 Jenny (CT)
So when that says, like per 1000 residents, that means like they're basing off of like how many people out of a thousand for that area.
[.] Crime rates are a weight of type and severity of the crimes considered… The safest places in the Charlotte Metro area are in the green.The most dangerous areas in Charlotte are the red and moderate, the yellow.

Map 2 Jenny
So this isn't the ratio.This is just a count.Well, this kind of shows that….likeoverall there's a general pattern of not good.Alice (Di) The first picture says what type of crime it is and where in the city it is most common [.] and where to put specific efforts against specific crimes.The second picture rather says how common it is with crime in general.Whether there are certain regions where it is much more common, where you may have to have more police presence.
Daniel (IsT) There may not be as many who would like to move into the center city, or where the criminals are [.]But [the app] may make it a little clearer.At least for me, I trust the statistics more than someone saying that, or writing that you should not live there.So I could have been persuaded not to live here… If this data had been available.

Interviewer
So you're saying it's not that there's different amounts of crime.Is it that if they target those areas, they're going to be reported more often or be convicted more often?Katherine (CT) Potentially yes, or maybe there is less crime, but it's going to make it seem like those areas where it's already yellow might become red just because if there's more police around, there's more chances to get caught….When you're thinking of the African-American male, if he is living in that area that's red … and the police are going to be using this information to target more of the red areas, I guess, that it might be more difficult.
Like even if he is like, putting his past discretions behind him, he's still at risk of being targeted by police just because he is black and living in an area that is, (I might put this in quotes) "known for its crime".
C.H. Andersson and J.T. Register were, several PSTs gave elaborate descriptions that held both dearth and access perspectives concerning the same decision, giving detailed accounts of both the pros and cons.In this way, their answers clearly challenged the notion of general truths about access to data such as 'more data is always better' (dearth) or 'data should be protected' (excess).Instead, the PSTs drew from the specific context of the questions to guide their ethical reasoning thereby acknowledging that the complexity of real world situations require a more nuanced approach.Seeing such complexities and considering both dearth and excess perspectives were more prevalent among PSTs that demonstrated higher CMC levels while other PSTs demonstrated less advanced perspectives on AtD leading to scant ethical reasoning.This suggests that although advanced reasoning on AtD occurred among some of the PSTs, it cannot be expected to occur automatically and consistently among all PSTs.Rather we see that there is potential to develop PSTs reasoning as it relates to who should have access to data and in what format and quantity.
The long-term effects and consequences of data driven practices (referred to as ecological impact in the Findings) was another consideration we explicitly asked PSTs to make in their reasoning.Some responses here were also surprisingly advanced.Several PSTs described long-term interaction mechanisms between human behavior and the data.This is best illustrated by Katherine's reasoning, quoted in the findings section, on how the data can affect police behavior, which then affects future data collection.Understanding such feedback loops demonstrates insights into how access to data and mathematics can interact.In Mapping Crime however, PSTs mathematical concerns were primarily focused on the accuracy of the graphical representation of the crime data and how those representations would be used.A more robust analysis, however, is revealed in Katherine's reasoning.She realizes that the long-term effects of data driven practices are not only dependent on ( 1) what the data supposedly portray, or (2) who has access to it, she points at a phenomenon where (1) and ( 2) interact.This is because the color on the map changes when the police get access (i.e., the color no longer portrays only reported crime) it now also reflects the fact that the Police have access to the data and the concurrent change in patrolling as a result of this.This consequence analysis operates on another level than, for instance, saying that one of the maps is preferable over the other since it foregrounds more important perspectives, or that some persons should (not) have access to the data because of what they can do with it.Instead of drawing from only either mathematical understanding of the maps, or concerns for data access in this societal context, she combines them to produce a synthesis that goes beyond what either approach can achieve alone.While not all of the PSTs demonstrated this reflexive mathematical and AtD reasoning, we observed that PSTs have this capability, indicating a potential to foster such pluralistic reasoning.
One could perhaps argue that advanced reasoning on the interplay between mathematics and data access is too much for the needs of teachers, and that it would be sufficient with a lay citizen's caution about data in the hands of an adversary.This is reminiscent of how Skovsmose (1990) discusses reflective knowledge as important for citizens' ability to evaluate and critique, and as such, may occasionally be separated from technological knowledge.Instead, we argue that a general awareness that it might be a bad idea to give data/information about oneself to a potential adversary, may be enough to form a 'first line defense' of critical thinking without also understanding why or how, but it is not enough to develop rigorous ethical reasoning as it relates to BDA.The mathematics in Mapping Crime is comprehendible for PSTs.But in other applications of BDA, mathematics can go far beyond their current training.By tweaking Mapping Crime to involve predictions of future maps, we could for instance have introduced predictive statistics with machine learning.For such examples, knowing the involved mathematics is a requirement for doing the kind of insightful synthesis of AtD and mathematics that Katherine did.Such synthesis can uncover important perspectives.We therefore argue that a lay citizen's caution regarding data in the hands of an adversary, does not cover all needs for education.Rather, we suggest that this level of understanding, while potentially sufficient for consumers of BDA, is insufficient for future industry professionals and policymakers.Thus, a contribution of this study to the CME research field is that teachers need to understand the why and how behind common BDA practices, since they will teach students who will either become producers or consumers of such practices.

Implications
Our findings have implications for this special issue's aim to support a more robust and critical mathematical understanding of the world, in that they show the possibility and value of PSTs in discussing access to data in relation to ethical reasoning on BDA.As was described in the literature review, accessibility to data is an important topic for understanding how BDA can operate in society.Our results show that it is possible for PSTs to discuss this important concept in depth, and hence it ought to be possible to include AtD in teacher education.Our results also indicate that ethical reasoning can be enhanced if it is simultaneously informed by both understanding the mathematics involved, and AtD in the particular context.We therefore argue that an Ethical Mathematics Education in the digital age ought to include instruction on the key concepts and skills of data science and BDA.
This raises two challenges for mathematics education.First, teacher education must encompass more of the mathematical technologies used in BDA.This includes probability and statistics, that in the case of the USA have been situated in the periphery of mathematics education (Boaler & Levitt, 2019).Second, teacher education may have to reconsider how data is discussed in general.In

Cruz (Di)
If used separately, the first map that has a proportion will probably be somewhat misguiding because they'd probably be more focused on areas like this and might disregard areas like this, where crimes are still happening, but due to the population, it might not be as visible but it's still important because this is a group of people that you are simply leaving to suffer.Birgitta (SsT) It has already happened.So then the probability is not so high that it happens again.Yes, tricky.I do not want to be the one to decide, whether it should be released or not.
C.H. Andersson and J.T. Register mathematics education, we typically hold the perspective that more data and/or more information is better.We rarely discuss the benefits of regulating and restricting data from a mathematical or ethical standpoint.It is important to integrate the ethical underpinnings of this viewpoint as it relates to privacy, property, discrimination, ecological impact, etc.In addition, our findings indicate that an ability to consider multiple perspectives on a societal issue may be connected to higher levels of CMC.This is indicated by the combination of excess and dearth perspectives held by PSTs at the critical levels of CMC.Therefore, a context of a highly complex authentic societal issue, can become a resource to draw from when developing PSTs' CMC, equivalent with using wicked problems in mathematics education (e.g., Steffensen et al., 2018).
To summarize, an additional awareness of what outcomes are possible with BDA could draw attention to the value of personal data.This includes assessing the ramifications of providing open access to data, or providing access to specific groups.Teachers would then be able to focus their instruction on the nuances of BDA that promote the use of individuals' personal data, having significant consequences for individuals and groups in society.Such knowledge may promote new possibilities for ethical mathematics education in the Digital Age.
C.H.Andersson and J.T. Register
C.H.Andersson and J.T. Register

C
.H.Andersson and J.T. Register
C.H. Andersson and J.T. Register political, behavioral, and economic stereotypes which are often communicated through data-based representations (O

Table 3
Coding Process.

Table 4
An excerpt of the codes for Cruz.

Table 5
Mapping Crime Ethical Considerations and Demonstrated CMC.
Note: Rows indicate which PSTs were classified at each CMC level.Columns indicate the AtD reasoning and ethical awareness demonstrated by those PSTs within each CMC level.For instance, One SWE PST and three US PSTs were classified as Critically Transitive for this task.The SWE participants demonstrated an excess AtD perspective while the US PSTs demonstrated either an excess perspective or drew on both excess and dearth AtD perspectives.All PSTs (SWE and US) demonstrated ecological, communicative, and sociopolitical awareness in their reasoning.C.H. Andersson and J.T. Register

Table 12
Example Responses (Cognitive Dissonance between Mathematical and Contextual Knowledge).