# Authority and Mathematics Education

**DOI:**https://doi.org/10.1007/978-3-319-77487-9_14-3

## Keywords

Authority of mathematics Democratic values Epistemic authority Cooperative learning Sociological perspectives## Definition

The nature and role of authority relations in mathematics education.

## Characteristics

The investigation of authority enters into mathematics education research by way of two main arguments. The first is that to the extent sociology, anthropology, and politics are relevant to understanding mathematics education (e.g. Skovsmose 1994; Dowling 1998; Lerman 2000), authority must be relevant as well since it is a central construct in all those attendant fields; indeed, any treatment of power, hierarchy, social regulation, and social relations must refer to the idea of authority in some way (see Krieger 1973 for a broad discussion).

The second argument refers to mathematics *per se*. It is that the perception of mathematics as universal, certain, and final gives the discipline itself an authoritarian character. Whether or not authority can be attributed to mathematics, *strictly speaking*, owing to this *perception* of the subject, authority may be transferred to those considered mathematical experts, often in matters having little to do with mathematics. From this it is easily seen how the issue of authority might arise in classroom situations: the authority of the mathematics teacher, for example, may trump the authority of the discipline, however that is understood, or discourage students from pursuing their own ideas.

In the social sciences generally, the *locus classicus* for the treatment of authority is Max Weber’s *The Theory of Social and Economic Organization* (Weber 1947). There, Weber describes “authority” (*Herrschaft*) as “…the probability that a command with a given specific content will be obeyed by a given group of persons” (p. 139). Authority, in this view, involves power, but it is not power alone, mere coercion: Weber points out that a relationship of authority always involves “…a certain minimum of voluntary submission” on the part of the controlled and an interest in obedience on the part of the authority (p. 247). Power (*Macht*) alone, by contrast, is only “the probability that one actor within a social relationship will be in a position to carry out his own will despite resistance, *regardless of the basis on which this probability rests* [emphasis added]” (p. 139).

The crucial point is that whatever power is associated with authority is recognized as *legitimate* by those who submit to it. Weber identifies three grounds of legitimacy and three concomitant “ideal types” of authority: traditional, charismatic, and legal authority. *Traditional authority* is the authority of parents or of village elders; c*harismatic authority* is the authority of one endowed with superhuman powers, a shaman for example. *Legal authority* is authority within an “established impersonal order,” a legal or bureaucratic system; the system within the legal authority acts is considered rational, and, accordingly, so too are the grounds of authority and the obedience it commands. These “ideal types” are not necessarily descriptions of individual authority figures. Weber’s claim is that authority can be analyzed *into* these types: the authority of any given individual is almost always an amalgam of various types.

*Expert or epistemic authority*, which is an essential aspect of teachers’ authority (Welker 1992), does not appear in Weber’s writings; but it is clear that because the grounds of such authority are rational and sanctioned by official actions, for example, the bestowing of an academic degree or a license, Weber could have deemed it a form of “legal authority.” Still, it is different enough and important enough for educational purposes to distinguish expert authority as a distinct authority type whose legitimacy founded on the possession of knowledge by the authority figure (regardless of whether the knowledge is true or truly possessed).

Students’ lives are influenced by a broad web of authorities, but the teachers’ authority is the most immediate of these and arguably the most important. It has been suggested too that teachers’ authority manifests elements not only of expert authority, but also traditional, legal, and even charismatic authority (Amit and Fried 2005). It is not by accident, then, that early sociological studies of education such as Waller’s classic study of education (Waller 1932) and Durkheim’s works on education (Durkeim 1961) underlined the authority of teachers.

Because of its pervasiveness and dominance, teachers’ authority can conflict with modes of teaching and learning which mathematics education has come to value. Such a conflict arises naturally between teachers’ authority and democratic values. This was studied by Renuka Vithal (1999) with respect to three domains: whole class interactions, group work, and the teacher-student-teacher-research complex. What was important for Vithal was that the teachers’ authority, although opposed to democracy, could actually live with democracy in a relationship of complementarity. She suggests that the very fact of the teacher’s authority, if treated appropriately, could provide an opportunity for students to develop a critical attitude towards authority (see also Skovsmose 1994).

To take full advantage of authority as Vithal suggests, or in any other way, it is essential to understand the mechanisms by which relations of authority are established and reproduced. Indeed, these may be embedded not only in social structures already in place when students enter a classroom, but in subtle aspects of classroom discourse. Herbel-Eisenmann and Wagner (2010), for example, have looked at lexical-bundles, small segments of spoken text, reflecting one’s position in an authority relationship. These lexical-bundles are as much a part of the students’ discourse as the teacher’s, recalling how authority relations are always a two-way street, as Weber was at pains to stress.

Since the overlapping role of teachers as expert authorities, as task controllers, to use Ernest’s term, has to do with mathematical content and how it is passed on to students, we are brought to the second argument concerning authority and mathematics education, for theThis analysis reflects that the teacher has two overlapping roles – namely as director of the social organisation and interactions in the classroom (i.e., social controller) and as director of the mathematical tasks and work activity of the classroom (i.e., task controller). This distinction corresponds to the traditional separation between being ‘in authority’ (social regulator) and being ‘an authority’ (knowledge expert) (p. 42)

*degree*of the overlap Ernest refers to is very much related to the authoritarian nature of mathematics. This is not a new phenomenon. For example, Judith Grabiner (2004), writing about Colin Maclaurin (1698–1746), pointed out that mathematics in the eighteenth century attained an authority greater even than that of religion since mathematics was thought to be able to achieve agreement with a universality and finality unavailable to religion.

This overwhelming authority of mathematics combined with the authority subsequently transferred to practitioners and teachers of mathematics creates a tension arising directly from the nature of mathematical authority. This is because what is essential about mathematical authority is precisely its independence from any human authority: mathematicians, no matter how great, must yield to a child pointing out a flaw in their work. Yet, Keith Weber and Juan Mejia-Ramos (2013) have shown that even professional mathematicians are influenced by human authorities or by authoritarian institutions, however counter that may be to the spirit of mathematics. This of course is all the more so with students.

How this plays out in a specific mathematical context can be seen in Harel and Sowder’s (1998) category of proof schemes based on external conviction, which includes a subcategory called “authoritarian proofs.” Typical behavior associated with this proof scheme is that students “...expect to be told the proof rather than take part in its construction” (Harel and Sowder 1998, p. 247). The authority of a teacher presenting a proof can thus take precedence over the internal logic behind the authority of the discipline: the whole notion of “proof” is vitiated when this happens since the truth of a claim becomes established not through argument but through a teacher’s authoritative voice.

But it is not only the authority of a teacher that may be operative here. Fried and Amit (2008) discussed authoritarian behavior *among students themselves* when they are working together on proof. The more general importance of this for cooperative learning was noted in Amit and Fried’s earlier paper on authority (2005), but considerable progress on that front can be seen in recent research such as that of Jennifer Langer-Osuna and others (e.g., Langer-Osuna 2016, 2017; Engle et al. 2014) who have studied closely the relationship between cooperative learning behavior and authority relations and, more importantly, the mechanisms by which authority structures become established among students.

Since the authority of mathematics as a discipline becomes ultimately the possession of the student as the teacher deflects her own authority, the problem of authority in mathematics education may be chiefly how to devolve authority (see also Benne 1970). The problem of authority, in this way, becomes the mirror problem of agency.There is a common perception that the authority in reform mathematics classrooms shifts from the teacher to the students. This is partly true, the students in Ms Conceptual’s class did have more authority than those in the traditional classes we followed. But another important source of authority in here classroom was the domain of mathematics itself. Ms Conceptual employed an important teaching practice–that of

deflectingher authoritytothe discipline. (p. 8)

## Cross-References

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