Good Times, Bad Times

Learning School Science and Mathematics through Faith: An argument for critical thinking rather than compulsion. Three Papers.

Abstract

This paper is a personal sense-making exploration of the concepts of epistemology and authority and their importance for educational research. Perceived parallels between religious, scientific, and mathematical knowledge construction are examined. Drawing from personal experience and the literature, I argue that mathematical and scientific knowledge must be grounded in experience and rationality rather than faith if meaningful learning is to occur.

Keywords: Authority, Anthropogogy, Epistemology, Religion, Mathematics, Science

Introduction

Our probing of difficulties in the reconception of valid authority relations in contemporary education sets a broad direction for the reconstruction, sometimes fundamental, of our institutions of formal education. I don’t know whether we as educators will prove equal to the task of reconstruction. I do believe that the burden is upon us. (Benne, 1970, p. 410)

I began this paper as an assignment for a course on collaborative learning completed during my second semester as a doctoral student in mathematics and science education at a southeastern university. My instructors stressed that my classmates and I should complete our assignments with publication in mind. Originally, I intended the paper to be a thorough survey of the literature related to authority in mathematics education. But being new to the field (as my previous studies were of mathematics, philosophy, and physics) I realized that my lack of experience and the limited time I had to work on the assignment would most likely inhibit me from creating a paper worthy of publication. So in the spirit of autonomy (at the risk of being in err) I diverged from what I perceived as the traditional expectations for a literature review. I decided that rather than agonizing over methodology and details of scope I would write a literature review that was reflective in nature so that I could gain a better grasp on the construct of authority and its importance to education.

This paper is the result of that assignment, now revised in light of my professors’ feedback. It is comprised of two parts. The first is a reflection on epistemology—in which scientific, mathematical, and religious knowledge are discussed. My personal religious and educational experiences guide this reflection and provide insight into my conclusions. The second is a review of the literature related to authority relationships in mathematics education. Benne’s (1970) conception of anthropogogical authority provides the framework for analysis.

Theoretical Framework

I view this paper as an opportunity to engage in legitimate peripheral participation (Lave & Wenger, 1991). I recognize that the feedback I get from instructors, reviewers, and other readers, will contribute to my assimilation into the community of practicing educational researchers.

Methodology

The paper was constructed largely through free-form writing. This consisted of brainstorming, reflecting on literature, and working out the ideas that were buried in my psyche. I then crafted my thoughts and organized them so that they would be comprehensible to a reader. Although my purpose was to make sense of the literature and the related constructs to the best of my abilities, I have attempted to craft the paper so that it is readable and that others may understand the conclusions I have reached about the topics of epistemology, authority, and education. In this paper, I will argue that an anthropogogical approach to mathematics teaching and learning will ensure that mathematical knowledge is grounded in experience and rationality rather than faith.

Epistemology

Where I Began: My Original Motivation and Research Lens

The idea for this paper began with an idea that mathematics and science learning is based on faith. I became whole-heartedly consumed with the notion that neither critical thinking nor meaningful knowledge is obtained through the memorization of factswithout thoughtful considerations of the evidence. I viewed the United States educational system as authoritarian: to comply is to accept without question the claims made by the educational authorities. I saw this as an affront to human intelligence and likened it to religious indoctrination. A quote by Norris (1997) summarizes my sentiment to some degree: “To ask of other human beings that they accept and memorize what the science teacher says, without any concern for the meaning and justification of what is said, is to treat those human beings with disrespect and is to show insufficient care for their welfare” (p. 252).

I reflected that the way science and mathematics have been traditionally taught is similar to religious indoctrination. As a child in a religious family, I was expected by my family members and other religious authorities to accept religious doctrines without question. I assume this is the case with other young children, many who have not learned or are not capable of questioning the legitimacy of such authorities. Similarly, school pupils who have not yet learned how to appropriately question the legitimacy of an expert authority are expected to accept and memorize what is presented as scientific and mathematical fact. I contend that if students merely accept claims as presented by their teacher or the textbook without examining the justification for such claims then the students will not develop meaningful understandings of the concepts involved. The majority of education should not consist of appeal to authority but rather to reasoning, sense-making, and justification.

The Epistemologies of Religion and Science

In a paper titled Authority in Science and Religion with Implications for Science Teaching and Learning (2013), Mike U.Smith provided a thorough review of literature in order to outline the differences between the epistemologies of religion and science and the role that authority plays for each. Based on his analysis, Smith listed 14 implications for high school and undergraduate science instruction. The overarching theme of these implications is that explicit instruction on epistemology should be incorporated into science courses. Some of these are that teachers should, “Address what counts as evidence in science (compared to other fields, e.g., religion) … [and] explicitly address when it might be appropriate to trust teachers and textbooks and when it is not” (p. 620).

Smith’s (2013) implications were directed for secondary/undergraduate science education. Do they apply for primary school? At what age-level can children be expected to effectively and appropriately question the legitimacy of authority? Most young children would not question the religious ideas taught to them by their parents or church leaders nor the scientific claims presented to them by their school teachers. Harris and Koenig (2006) reported that children unquestionably accept the testimony of adults as fact.

After reading Smith’s (2013) work, I hypothesized that engaging primary students in an examination of the evidence of the natural world, along with the formation and testing of conjectures, would encourage and support critical thinking while at the same time allowing students to be actively engaged in the scientific process. I was interested in how the critical thinking of elementary school students could be developed if the students were engaged in a purely evidence-based science curriculum devoid of appeal to authority.

Clashes between religion and science.

There is a concern, especially in the south, that students do not accept the theory of evolution because of their religious beliefs. Smith (2013) wrote that “the teacher can be sensitive to these student concerns by promoting open-mindedness (as opposed to indoctrination or scientism), rationality, and tolerance for differing views—not speaking in judgmental terms about non-scientific views but acknowledging that science takes no position about their validity” (p. 625). Smith argued that discussions about the fundamental differences in the ways knowledge is acquired in the fields of religion and science must take place for meaningful progress to be made on this issue. However, even though the way knowledge is acquired in science is fundamentally different from the way it is acquired in religion, school children often do not learn science in a way that is different from the way they learn about religion. The following section is a personal reflection in which I use my religious background and educational experiences to demonstrate how I have been indoctrinated into science in much the same way as I was indoctrinated into religion.

Personal epistemological concerns.

In some Christian circles as well as some books of the Bible, goodness and truth are associated with God, and evil and lies are associated with the devil. Evil is sometimes seen as synonymous with the world. In the first epistle of John it is written, “We know that we are of God, and the whole world lies under the swayof the wicked one” (1 John 5:19 New King James Version). If one believes in the inerrancy of the Bible, and another worldly source of information (such as a science textbook) appears to contradict the Bible, then it is logical to conclude that the worldly source is evil and full of lies. Although I do not have convictions that the Bible is inerrant, I believe in God and I take the testimony of the Bible seriously. These concerns about good, evil, truth and falsity sometimes affect the lens through which I view the world. I recently came to the realization that most of the scientific knowledge I had acquired was based on the testimony of others. I realized that I didn’t really know if anything I had been taught about science was truth.

Testimony.

Harris and Koenig (2006) reviewed a substantial amount of literature on the ways that children obtain scientific and religious knowledge through testimony. In regards to scientific knowledge, the authors found that children were not at all skeptical of testimonial claims, sometimes accepted counterintuitive claims, but also actively assimilated the knowledge gained through testimony. The authors claimed that in some instances children ontologically differentiate between beings such as god and germs. One possible explanation offered was that children notice that belief in god is something that many adults are not sure of, whereas a belief in germs is taken as fact by most adults. Thus,children may make an ontological distinction between god and germs, and between religious and scientific knowledge. An alternative explanation provided by the authors is that children make the distinction between the two types of knowledge because religious knowledge does not always align with experiential observations.

Harris and Koenig (2006) also reported that some cognitive anthropologists have argued that it is precisely because religious claims are counterintuitive that they are so easily remembered. I reflect this may also be true of scientific claims. To a young child, the counterintuitive idea that the world is spherical may result in feelings of wonder and awe. The child may find joy and amazement in reflecting on a spherical earth despite the fact that it appears flat. I recall that this is precisely what happened to me when I first encountered the ideas of quantum mechanics. The excitement I felt about these counter-intuitive notions led me to study university physics for an entire year, eventually leading to my acquisition of a physics minor.

I recall that my time studying physics was spent reflecting on testimonial claims about quantum mechanics and special relativity, and actively assimilating these ideas into my every day thoughts. When I watched the leaves fall, I thought about their motion in terms of the curvature of space time. Yet, I never once had the opportunity to experimentally verify any of the claims of my teachers or textbooks. I learnedthat particles such as electrons sometimes appear to move from one side of an energy barrier to another. This phenomenon is called quantum tunneling. The situation is analogous to a person walking through a wall. According to the definitions of perturbation, disequilibrium, and accommodation as explained by Bartlo (2013), I gather that the assimilation of the idea of quantum tunneling was a perturbation which resulted in my awestruck disequilibrium. According to Bartlo, Piaget hypothesized that learning occurs during the accommodation phase in which the perturbation is eliminated as a result of the persons incorporating the new information into their existing knowledge structures. I accepted the idea of quantum tunneling, but it left me in a state of sustained disequilibrium. Because I was never able to empirically verify this knowledge, I may not have been able to accommodate the knowledge. I have an idea about the notion of quantum tunneling. I have been told it exists, and I probably examined some equations related to it. But I do not really have any meaningful knowledge about it. Additionally, I can conceive that the phenomenon might not actually exist.

Science as faith.

I have recently had discussions with a colleague who said that scientific claims can be traced back to empirical observations which can then be verified by others. He also maintained that religious claims are ultimately rooted in subjective experiences which cannot be empirically verified. Certainly I could recreate the experiments that led scientists to believe in the phenomenon called quantum tunneling, but why did my education not consist of this exercise? Why do students learn science through the testimony of their teachers and textbooks? I realize that because of my lack of empirical investigation of scientific claims, I perceive my belief in scientific theories to be a kind of faith. Just as I may choose to have belief in the miracles of Jesus, I can choose to believe in quantum tunneling. Both notions are counterintuitive, and both based on the testimony of others. A person cannot empirically verify that Jesus walked on water. However, one could perform a test to confirm the existence of quantum tunneling. Learning about science does not have to be based on faith, but for me it has been.

Choice and Legitimate Appeal to Authority.

I am worried that because of an overreliance on testimony, the epistemology of school science has become identical to the epistemology of religion. At some point, probably during a study of evolution, the student may feel that they have to make a choice between faith in science and faith in religion. Up until that point, it was not problematic to have faith in both. The way that these students came to gain scientific and religious understanding was the same. Faced with a contradiction, it is logical to retain one body of knowledge and reject the other.

Smith (2013) emphasized that the ways knowledge is acquired in the fields of religion and science are fundamentally different and argued that instruction must address these differences. Smith noted that scientists sometimes accept the claims made by other trusted scientists because it may not always be possible or desirable to recreate scientific experiments or examine the empirical evidence for scientific claims. There is a chain of authority from the researcher that initially looked at the empirical evidence, the scientists in the field who accepted the work, and the outlets in which the scientific findings are made known to the public. Thus an important skill for scientists, secondary and tertiary students, and a scientifically literate public is the ability to properly evaluate scientific claims. I reflect that it is important for school science students to learn how to rationally assess and critically judge the knowledge claims of authorities. This may alleviate some of my concerns with school science instruction, but I still worry about the overreliance on appeals to authority and students’ ability to acquire meaningful knowledge.

Conviction and Understanding in Mathematics

In a recent survey on epistemic cognition and the ways mathematicians gain conviction, Weber, Inglis, and Mejia-Ramos (2014) found that in addition to being convinced by deductive arguments, mathematicians sometimes gain conviction empirically or by appeals to expert authority. Working from the premise that mathematics students should become convinced of mathematical claims in a manner similar to the way mathematicians are, they argued that it is unreasonable to expect students to become convinced of mathematical truths solely through deductive reasoning. Furthermore, it is not problematic that students gain conviction about mathematical claims through appeals to authority or empirical evidence since some mathematicians do the same. The authors maintained that deductive arguments are useful for developing an understanding of mathematics and that low mathematics achievement is not a result of students’ beliefs that knowledge can be gained through authoritarian or empirical evidence, but rather “the absence of the belief that using one’s own reasoning is also a valuable source of knowledge” (p. 62). They argued that if students are to be expected to gain conviction in the same ways that mathematicians do, then it is overly simplistic to insist that students only be encouraged to gain mathematical knowledge through deductive reasoning. Instead, students should be given instruction on when it is appropriate to gain conviction by empirical arguments or through appeals to authority.

Conclusions

Weber et al. (2014) and Smith (2013) argued that students need more training in judging expert authorities. The fact is that we live in an age when a wealth of knowledge has already been acquired by the human community. While we struggle to understand more, it is important that we stand on the shoulders of giants and appeal to the authority of experts—but not in a naïve manner. Knowledge claims should be judged critically. The fact that some knowledge claims have repeatedly been judged and accepted by the experts may be enough evidence for conviction. However, for those human beings who will succeed in discovering the previously undiscovered, they must come to understand the processes by which knowledge is acquired.

In light of these reflections, I recall the literature related to the nature of science (Lederman et al., 2002), and the nature of proof in mathematics (de Villiers, 1990). A theme of this literature is that nothing is certain. Scientific theories are based on inductive reasoning, and mathematical proof is dependent on unproven axioms. According to lay epistemic theory, “knowledge is itself a form of belief ... one’s knowledge develops continually through an ongoing cycle of calls to evidence and subsequent conclusion-making” (Fulmer, 2014, p. 199). It appears that one may consider conviction as a type of faith (regardless if that conviction was obtained through deduction or appeal to authority). It is imperative for educators to conceptualize the nature of knowledge if we are to make rational decisions about teaching and learning. Kenneth Benne (1970) claimed that if educational institutions are to maintain relevancy and legitimacy then we must also begin the task of reconceptualizing another important construct—authority.

Kenneth Benne’s Conceptualization of Authority

At one point I intended to make Kenneth Benne’s conception of anthropogogical authority the foundational lens through which I conducted this literature review. Benne’s conception is idealistic and yet relevant. But, when I tried to describe the conception to my colleagues at dinner, one of them said, He seems out there. I believe that I had attempted to describe Benne’s conception of authority along these lines:

Anthropogogical authority is a view of authority as a source of social growth. The basic idea is that rather than focusing on pedagogy and the teaching of the child by the more learned teacher, anthropogogical means the teaching of humans by other humans. Everyone should be in the continual process of reeducation. It is not that one person is placed above another person, but rather people are equals and all have something to contribute to the human community.

I had just been introduced to the idea, had not yet had a significant amount of time to adequately conceptualize its meaning, and yet I was singing its praises.

Benne (1970) called for educators to reconceptualize authority in order to meet central human needs in a changing world. Rather than focusing on education as a purely intellectual enterprise, Benne maintained that education should be a means of finding one’s place in the community of human beings. Students must see themselves as having an impact on the world they live in. Benne wrote that the teacher, “becomes a surrogate bearer of anthropogogical authority only by virtue of responsible efforts to mediate between the present community involvements of those being educated and their expanding and deepening affiliations in the common life of a wider community” (p. 402). Anthropos means human; peda means child, and agogosmeans leader (http//www.etymonline.com). Anthropogogy should bring to mind the notion of a human leader, rather than a mere child leader. It points to the human community rather than just the classroom or specialized communities. However, as I make sense of Benne’s work I realize that anthropogogical authority is closely related to the theory of situated learning (Lave and Wenger, 1991). Amit and Fried (2005) first made this comparison in a paper on authority relationships in mathematics education. A student may obtain valid membership into a community of practice through valid anthropogogical relationships. An anthropogogical relationship does not consist of a mere transmission of knowledge. By way of one’s actions and thoughts, the student comes to view herself as a contributing member of the field. However, Benne took this a step further by shifting the focus from specialized fields to the larger human community.

My analysis of Benne’s conceptualization is based on Authority in Education (1970) which was in part a restatement and revision of some of the views expressed in a 1943 work titled A Conception of Authority. Benne (1970) wrote that the impetus for his 1943 work was a desire to form a person-centered/democratic conception of authority. This was in response to the fact that many educators subscribed to the view of education as a process of transmission of moral and intellectual culture, tradition, and knowledge. Another impetus for the 1970 work was that the role of authority in education had been widely ignored.

Personal Maturity and a Context for a New Conception of Authority

Benne felt that traditional views of authority were not sufficient to support human needs in our rapidly changing modern culture. Benne’s ideas of personal maturity can shed light on his concerns. Benne maintained that personal maturity has traditionally been thought of as autonomy and independence. Maturity is associated with freedom, rationality, and not being under authority. But Benne wished to shift the focus from independence to interdependence so that maturation can be seen as autonomy within interdependence. Authority is then a positive force rather than an oppressive one. The goal of educators must be to help students develop autonomy within interdependence.

Positive and Negative Freedom

The human species has been plagued by power struggles. Independence from oppressive authorities may be a necessary condition for human happiness. Even today’s society is plagued by oppressive authorities who wish to maintain power and wealth. In an undergraduate course on Marxism, my classmates and I discussed positive and negative freedom. Negative freedom can be viewed as the absence of an oppressive force, whereas positive freedom is the ability to act willfully. Negative freedom may be a necessary precursor to positive freedom, but it is not sufficient. Traditional notions of authority have focused on negative freedom. From this vantage point authority is viewed as something to be shunned and done away with. However, I reflect that Benne’s conception of authority emphasizes the need for positive freedom. Positive freedom allows us to willfully cooperate interdependently to achieve our goals.

Benne argued that an individual may desire to disrupt the traditional culture if it does not meet that individual’s needs. In such a situation, I reflect that the individual sees the culture as an impediment to his or her independence. An alternative view of maturation as becoming an autonomous, interdependent individual can lead to the realization that one can play a meaningful role in working to rejuvenate society, culture, and the world. Humans must depend on other trusted humans. In doing so, we enter into authority relationships.

Defining the Authority Relationship

As authority relationships are necessary for human existence, Benne wished to define authority relations in a way that it would make it possible to distinguish between relationships that were rooted in power versus those rooted in mutual human benefit. Benne (1970) defined an authority relationship as “a triadic relation between subject (s), bearer (s), and field (s)” (p. 393). The field is the medium through which the bearer and the subject interact. A bearer bears authority because he has a specialized knowledge or skill related to a particular field. The subject who wishes to advance his or her knowledge or skill in a field stands to benefit from accepting the authority of the bearer. The authority of the bearer is directly correlated with the bearer’s legitimacy.

Valid and Invalid Relationships

By introducing the concept of the field, Benne (1970) desired to introduce the notion of rationality into the conceptualization of authority. The bearer can rationally assess his or her own degree of competence within a field, and can also be judged as legitimate by the subject. Benne wrote that the legitimacy of the bearer is subjective. It does not exist in the abstract, but in the minds of other human beings. I reflect that the subject subjectively views the legitimacy of the bearer in relation to a certain field, and he voluntarily enters into a relationship with the bearer (who also enters voluntarily). Through this relationship the bearer and subject interact in a way beneficial to both parties—a valid authority relationship. I wish to operationalize the concept of valid authority relationship as follows: A valid authority relationship is a mutually beneficial, voluntary relationship between a subject, a legitimate bearer, and a field.I define an invalid relationship as one that does not meet all the definitions of the valid authority relationship. In order to illuminate the differences between actual and ideal authority relationships, Benne described three types of authority relationships: the authority of expertise, authority of rules, and anthropogogical authority.

Expert Authority

Benne described two characteristics of expert authority relationships: 1) The specialized expertise of the bearer is dependent upon his understanding or ability within the field. 2) The subject is not as knowledgeable about the field as the bearer is. Benne provided the example of a doctor and patient to describe how expert authority relationships can be mutually beneficial. But Benne implied that an educator determining the curriculum without attending to the needs of the student is like a doctor prescribing treatment for a patient whom they have not seen. He claimed that in such a situation an authority relationship does not exist. According to my operational definition of valid authority relationships, the relationship is not valid because the students have not voluntarily agreed to participate. The teacher may not appear to have legitimacy because the instruction does not meet the needs of the students.

Rule Authority

Humans often submit themselves to a set of rules. In contrast to expert authority in which the field is independent of the bearer, under rule authority the field is created by the rules. The subjects are those who comply with the rules. Benne (1970) borrowed a line from Dewey and claimed that the bearer is “the moving spirit of the whole group” (p. 397). If the rule authority relationship is valid then it must be understood that the rules are subject to change and negotiation (in this case the rules can be considered as the bearer). To judge the legitimacy of the rules one cannot merely appeal to traditions. A valid authority relationship does not exist when a school system imposes rules that the students do not value.

Anthropogogical Authority

Benne (1970) offered a rebuke of traditional authority relations. Benne argued that when one sees education as a process of enculturation, it reinforces the view that authority is a one-way street in which the more cultured masters are responsible for teaching the uncultured. I reflect that Benne must have been influenced by the cultural changes in the U.S. that were related to the civil rights movement. Benne (1970) wrote that “the model of education as cultural transmission [should be] replaced by a model of education as basically a process of personal and cultural renewal” (p. 387). The teacher and student must co-participate in the construction of a just society. The culture that the teachers would attempt to transmit may not be relevant to the developing human community. In fact, a child may have a better feel for the pulse of the worldwide community than the learned professor. Benne (1970) wrote that “the ultimate bearer of educational authority is a community life in which its subjects are seeking fuller and more valid membership ... [And] the teacher becomes a surrogate bearer of anthropogogical authority only by virtue of responsible efforts to mediate between the present community involvements of those being educated and their expanding and deepening affiliations in the common life of a wider community” (pp. 401-402). I do not think that our educational system is supporting students to become autonomous, interdependent, contributing members of the wider community. Many students lose their voice, and feel alienated from a culture that is being forced upon them through schooling. In April of 2014, a sixteen-year old student stabbed 21 of his classmates in a Pennsylvanian school (Daley, 2014). In a note found in his locker, the student wrote, "I can't wait to see the priceless and helpless looks on the faces of the students of one of the 'best schools in Pennsylvania' realize their precious lives are going to be taken by the only one among them that isn't a plebeian” (Daley, 2014). Benne (1970) wrote that, “Just as the broad tasks of pedagogy are set by the induction of the chronologically immature into viable relationships and participation within a human culture, the reduction of the alienation of deviant persons and groups within a culture sets the broad tasks of anthropogogy” (p. 391). If we continue to view education as a means to disseminate knowledge rather than develop community members, then educators run the risk of losing legitimacy and alienating more and more students.

Democracy and Education

There have been recent attempts by educators to incorporate democracy into mathematics and science classrooms (e.g. Vithal, 1999; Basu & Barton 2010). Basu and Barton (2010) wrote that the “key to democracy is the notion of constituents having rights. Traditional science classrooms limit students to being consumers of knowledge who are expected to memorize facts selected as important by their teacher ... However, science can be viewed as a tentative discipline in which different evidence-based opinions compete for status. If exposing students to the process of doing science is indeed given worth in science education, free speech becomes an important tool through which students can engage in debate about competing evidence-based ideas, rather than accepting facts as they are provided” (p. 74). I reflect that we need more classrooms focused on the nature of specialized communities of practice (rather than a focus on the accepted body of knowledge of such communities). In such democratic classrooms students will have a voice. They will realize that they have the opportunity to become contributing members of human society. Benne intended for anthropogogy to capture the essence of these ideas. Benne believed that education could play an important role in the rejuvenation of culture by which both young and old view themselves as valid contributors to the renewal of society.

Analysis of Authority Relationships in Mathematics Education

I will now present an analysis of the authority relationships discussed in the mathematics education literature. I will analyze these in terms of my reflections on epistemology and Benne’s (1970) conceptualization of authority.

Deborah Ball’s Third Grade Class.

“The things that children wonder about, think, and invent are deep and tough. Learning to hear them is, I think, at the heart of being a teacher.”

(Deborah Ball, 1993, p. 374).

Deborah Ball successfully taught a group of third graders how to reason mathematically, engage in mathematical discourse, and understand what counts as mathematical proof. With her students in mind and her eye on the mathematical horizon, she wrote, “How do I value [the students’] interests and also connect them to ideas and traditions growing out of centuries of mathematical exploration and invention?” (p. 374). Ball’s role as teacher was not to disseminate knowledge, but build a bridge between the students’ ideas and the discipline of mathematics. She supported her students’ active engagement in mathematical practices in a way that honored the discipline of mathematics and respected her students as learners.

Discovery and contribution.

In one documented classroom episode, a student named Sean had claimed that some numbers can be partitioned into an “odd number of groups of two” (Ball, 1990, p. 387). In Ball’s reflections she wrote that she struggled with the notion that Sean’s idea wasn’t part of established mathematical knowledge. In the end, she decided to allow the class to formulate and prove conjectures related to Sean numbersfor several days. Ball wrote that, “it seemed defensible to give the class firsthand experience in seeing themselves capable of plausible mathematical creations” (p. 387). This episode highlights the way Ball’s instruction supported students so that they could see themselves as becoming more valid members of the mathematical community. I reflect that Ball’s instruction was consistent with the spirit of anthropogogy.

Magdalene Lampert’s Fourth Grade Class

Magdalene Lampert is another researcher and teacher who I believe respected her students in the spirit of anthropogogy. In Lampert’s classic (1990) work on mathematical knowing she described her students practices and claimed “that the students had learned to regard themselves as a mathematical community of discourse, capable of ascertaining the legitimacy of any member’s assertions using a mathematical form of argument” (p. 42). Lampert’s role was not to disseminate mathematical knowledge, but to listen to her student’s arguments and guide instruction so that student thinking and reasoning was supported. Lampert was concerned with the epistemology of mathematics, and her work highlighted the importance of learning mathematics in a way that is consistent with how knowledge is acquired in the community of practicing mathematicians. Drawing on the ideas of Poyla and Lakatos, Lampert (1990) described mathematics as a social process of conscious guessing, conjecturing, refuting, and generalizing where “reasoning and mathematical argument—not the teacher or the textbook—are the primary source of an idea’s legitimacy” (p. 34). I reflect that the instructional styles of Lampert and Ball lead to student opportunities for develop critical thinking skills and meaningful knowledge. Furthermore, a student experiencing mathematics through this type of instruction would be able to see themselves as becoming a member of the mathematical community.

Making Sense and Sharing Mathematical Authority

There are frequent calls in the mathematics education literature to share mathematical authority with students (e.g. Webel, Evitts, & Heinz, 2002; Wilson, Melvin, & Lloyd, 2000). In their analysis of a college calculus classroom Gerson & Bateman (2010) observed four types of authority relationships: hierarchical, mathematical, expertise, and performative. They defined mathematical authority as “authority whose legitimization is based in mathematics” (p. 202). Bearers of mathematical authority gain legitimacy by providing mathematical arguments or drawing from communally-accepted mathematical knowledge. Ball and Lampert’s students did not rely on the teacher as the final arbiter of mathematical truth. The teachers shared the mathematical authority in ways that promoted sense-making amongst the students. Ball (1993) wrote “I am trying to model my classroom as a community of mathematical discourse, in which the validity for ideas rests on reason and mathematical argument, rather than on the authority of the teacher or the answer key. … I am searching for ways to construct classroom discourse such that the students learn to rely on themselves and on mathematical argument for making mathematical sense” (p. 388). Thus the mathematical authority is with the students rather than the teacher.

When did the mathematics education community first begin to recognize the need for teachers to share mathematical authority with students? I gather that Piaget’s ideas may have played an important role. Cobb, Yackel, and Wood (1989) described a teaching project in which a second-grade teacher shifted the authority for mathematical justification entirely to the students. The authors found that the students exhibited positive emotional acts in the classroom. They jumped for joy or hugged a partner after solving a tough problem. The authors claimed that one of the instructional goals of the project classroom was for students to develop “intellectual and moral autonomy” (Cobb et al., 1989, p. 117). Constance Kamii, one of Piaget’s students, wrote that “For Piaget, the aim of education was intellectual and moral autonomy” (Kamii, 1984, p. 410). The notion of anthropogogical authority also provides a lens for interpreting the sharing of mathematical authority. Recall that Benne (1970) felt that the task of educators was to aid in the development of student’s autonomy within interdependence. Intellectual authority should be shared with students so that they may become more capable of contributing to the classroom society as well as the society at large.

Social norms and collaborative learning.

Yackel, Cobb, and Wood (1991) described how a teacher in one of the project classrooms used her authority to guide the renegotiation of classroom norms. The teacher would use paradigm cases or whole class instruction to encourage cooperation and perseverance. Similarly Ball (1993) and Lampert (1990) used their institutional authority (Gerson and Batemen, 2010) to guide the renegotiation of classroom norms beneficial for mathematical discourse. Additionally these teachers resisted taking the mathematical authority upon themselves and shifted it to the students. How do these authority relationships impact collaborative learning?

In addition to learning about mathematical content, Ball writes that “the students seemed to be developing a sense for what they could learn from one another” (p. 393). One of Ball’s students Riba commented that, “discussions are helpful because one person may have a good idea when it is taking a long time to figure it out all by yourself” (p. 393). In addition to supporting autonomy, Yackel et al. (1991) wrote that the problem-solving atmosphere and negotiation of norms that supported collaborative learning resulted in student interactions that naturally led to learning because of the opportunities students had to “verbalize their thinking, explain or justify their solutions, and ask for clarifications” (p. 401). Students seldom have opportunities for this type of learning in traditional classrooms.

Amit and Fried (2005) documented expert authority relationships that existed in two eighth-grade mathematics classrooms. They found that while the teacher encouraged the students to work cooperatively, they seldom did, and “rather, they seemed to ignore their co-workers and turn to the teacher, or sometimes to a ‘knowing member of the class” (p. 153). The students viewed their teachers as infallible sources of mathematical expertise, and the degree of legitimacy they attributed to others was based on the perceived likelihood that the person would possess needed information (to solve a math problem). Students did not feel any need to question the answers given to them by the authorities they sought out. (I am reminded of Harris and Koenig’s (2006) findings that children accept the claims of authorities without question). The authors expressed the sentiment that establishing expert authority relationships in collaborative small groups does not lead to dialogue or “true collaborative learning” (p. 162). They wrote that “by turning always from one figure to another, and never to themselves, the students not only fail to develop their own mathematical thinking but they also perpetuate this failure by always defining themselves as outsiderswith respect to mathematical discourse” (p. 165). The student cannot become a participant in the community of practice if they are merely vehicles for the transmission of knowledge.

Amit and Fried (2005) were the first to mention anthropogogical authority in the mathematics education literature. They described Benne’s conceptualization as follows: “The exercise of authority (rather than mere power) as negotiation and the bringing of a group into accord turns out to be the key to ... anthropogogical authority—and this kind of authority is the kind that best describes that of the discipline of mathematics” (p. 163). For Amit and Fried, anthropogogical authority is the means by which the student gains fuller membership into the mathematical community. The authors did notice one group of students, Yana and Ronit, who seemed to be working collaboratively. The authors wrote, “They consulted with one another, raised possibilities on their own, revised opinions, and seemed to arrive at common conclusions. In other words, rather than treating one another as possible sources of answers, Yana and Ronit treated one another as intelligent interlocutors who could work together to make progress on the question at hand” (p. 159). Fried and Amit (2008) documented Yana and Ronit’s collaboration in another work on the interrelation of authority and proof.

How is the traditional classroom different?

Ball (1993) recognized the problems inherent with explaining mathematical ideas to her students. She recalled that she sometimes noticed that students held incorrect mathematical beliefs that were likely due to the fact that they had attempted to memorize the explanations of a previous teacher rather than make sense of the mathematics on their own. Throughout Ball and Lampert’s (1993) work there is an emphasis on listening to the students. Yackel et al. (1991) wrote that the students increasing intellectual autonomy allowed the [project] teacher to have more time to listen and observe students rather than “the chore of constantly monitoring and supervising the children’s activity” (p. 401). Often in typical classrooms students must only listen to their teachers. Their role in authority relationships is always the subject. When thinking of the classrooms of the reform-oriented teachers discussed here, I am again reminded of Benne (1990) and his notion of anthropogogical authority. Sometimes it is necessary to be the bearer, and other times it is necessary to be a subject if one is to be an interdependent member of society. If children are to become valid members of the world community then they must have the opportunities to be bearers of authority in the classroom. Instructional models that shift the mathematical authority to the students thus fulfill an important human need.

Other Reflections on the work of Ball, and Lampert, and Anthropogogy

Benne (1990) argued that education must be relevant to the needs of students. To be relevant, education cannot be a mere transmission of culture and knowledge. Reform-oriented classrooms attend to students’ human needs more than traditional drill-and-practice sessions can. If students are to become mature, autonomous and interdependent members of society they must learn to work with others. Ball (1990) wrote, “I aim to develop each individual child’s mathematical power through the use of the group. I am to develop the children’s appreciation for and engagement with others different from themselves” (p. 388). In addition to developing their own power, autonomy, and cooperation skills, the students need to learn that meaningful work is difficult yet rewarding. Ball also wanted her students to know that “understanding and sensible conclusions often do not come without work and some frustration and pain—but that they can do it, and that it can be immensely satisfying” (p. 394). Although Ball possess a great deal of content knowledge, she recognized that she could learn more by attempting to see mathematics as her students saw it. I have attempted to use these three quotes of Ball’s to highlight how her instruction met not only the students’ intellectual needs but also human needs necessary for functioning autonomously and interdependently in the human community.

Final Reflections

The idea that’s really been tugging at my heart/intellectual self: In K-12 (and onward) education, students should not be expected to accept any scientific or mathematical (or intellectual) notion merely because of appeal to authority. An educational system grounded in the memorization of an authority’s statements of “fact” will not produce a body of critically thinking individuals. Learning has become a matter of faith in intellectual authorities.

I would like to emphasize the word expected. Weber et al. (2014) argued that it was not problematic for students to gain conviction from their teacher or textbook as mathematicians gain conviction from trusted experts. Smith (2013) claimed scientists also obtain acquire knowledge through appeal to authority. Furthermore Smith maintained that students need to be taught that knowledge can be gained from authorities and how to appropriately judge the legitimacy of sources. The results of Harris and Koenig (2006) imply that students naturally gain conviction through the testimony of authorities. But the problem I see with education is that students are expectedto be convinced without justification. We ask them to pass tests based on the knowledge of information they gained solely by the word of their instructor. As Benne (1970) argued, we can no longer view education as a means by which the teacher transmits knowledge to the student.

This transmission of culture is not necessarily the same as that wrote about by Stigler and Hiebert (2009) in which teaching is viewed as a cultural practice. Rather the view is that culture becomes outdated quickly in the modern era, and education can no longer be viewed as enculturation into a process that students may not feel they have a need to be enculturated into. Ball (1993) struggled to build “bridges between the experiences of the child and the knowledge of the expert” (p. 374). Unfortunately what so often happens is that the knowledge of the expert is expected to be taken as fact by the student without the student having the experiences to construct knowledge on their own.

I gather that it is a goal of many science teachers that their students become familiar with scientific theories. Should the theories be presented in their final form to be memorized? It seems as though we are denying the students the opportunity to actively construct the knowledge of those theories. It could be argued that it is infeasible to recreate the experiments used to gather the data that provides evidence to support these theories. I hypothesize that when children learn science in a rote manner, the meaning, significance and applicability is often lost. By examining the empirical evidence for a theory, students can come to knowledge of a theory in a way that it is meaningful. Then the significance and applications may be properly understood. In mathematics, theorem is often presented without proof. If an educational goal is to develop critically thinking members of society, then it should not be standard practice to accept theorems without proof or theories without evidence.

Reflection on the Course

Writing this literature review has allowed me to think about what interests me, and what I can contribute to human knowledge. As I work on completing my PhD in Mathematics and Science Education, I reflect that the research I do does not have to be limited by my concentration title: Mathematics Education. There is plenty of science education research that I find very interesting, and I think I will study it now and again. Also, just as Mathematics Education researchers often draw on psychology or sociology (Lerman, 2000), I may want to draw on my background in philosophy and incorporate philosophical literature into education research. I have found that Kenneth Benne was an Educational Philosopher (Feinberg, 1993). Before writing this paper I did not know that such a title existed.

But then again, I will need to focus my research efforts in order to write a meaningful authoritative dissertation. At this point I am not sure if I would like my dissertation work to be focused on proof, authority, or epistemology. I feel that this literature review gave me the opportunity to write about a lot of things that I have in the past I may not have thought had anything to do with education research.

Perhaps a discussion of religion must take place if progress is to be made in biology education. I have learned that my faith influences the way I interpret literature. I am not sure if I should make this explicit in academic writing. I realize that I still have a soft spot for philosophy, and that my philosophical background gives me a unique perspective that I should take advantage of in my writing. I have learned that I am extremely impractical, and that I will need to collaborate with others on research projects. I read through my reflections and notice all the places where I need citations but I do not have any. Are these gaps in the literature, or does research exist that would support my claims? Probably the latter. How to become familiar with it all? I reflect that if I really focus my efforts on one research topic then I should be able to better pull from the literature when writing. I notice that I am better adept at this in mathematical research on proof than in reflections on epistemology, authority or testimony.

I am not sure if my work has been legitimate peripheral participation. Hopefully, it has contributed to my becoming a participant in the field of education research in the future. I have learned that I need to really focus on understanding pieces of literature. I do not just need to write summarizes, I need to talk about research with my colleagues, and write about articles reflectively.

I recall when Dr. Blaylock (pseudonym), one of the instructors who assigned this work, asked me how my literature review was coming along. Specifically she asked me if I thought it might be something that I could publish. I told her about my project, and she was excited. She was excited about autonomy. I feel that if I had not been studying authority I would not have made the decision to write the literature review in a way that I thought would develop myself, and instead I would have been focusing on creating a literature review that matched the form I imagined my instructors expected. Dr. Blaylock said that she hoped the students in my class would be more autonomous, and that the goal of the PhD program is for students to develop into autonomous members of the field. To echo Benne’s ideas, I would suggest that graduate students should all strive to become autonomous and interdependent members of the field. In fact, I reflect that I feel I am engaged in anthropogogical relationships with my professors! I would elaborate further, but this literature review is due in seven minutes.

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References

Amit, M., & Fried, M. N. (2005). Authority and authority relations in mathematics education: A view from an 8^thgrade classroom. Educational Studies in Mathematics, 58, 145-168.

Ball, D. L. (1993) With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. The Elementary School Journal, 93, 373-397.

Bartlo, J. R. (2013). Why ask why: An exploration of the role of proof in the mathematics classroom. (Doctoral dissertation, Portland State University). Retrieved from http://pdxscholar.library.pdx.edu/open_access_etds/1075/

Basu, S. J., & Barton, A. C. (2010). A researcher-student-teacher model for democratic science pedagogy: Connections to community, shared authority, and critical science agency. Equity & Excellence in Education, 43, 72-87.

Benne, K. D. (1970). Authority in education. Harvard Educational Review, 40, 385-410.

Cobb, P., Yackel, E., & Wood, T. (1989). Young children’s emotional acts while engaged in mathematical problem solving. In D. B. Mcleod, & V. M. Adams (Eds.), Affect and mathematical problem solving: A new perspective(pp. 117-148). New York, NY: Springer-Verlag.

Daley, E. (2014, April 28). Boy in Pennsylvania school stabbing spree left note about plans. Reuters. Retrieved from http://www.reuters.com/article/2014/04/25/us-usa-pennsylvania-stabbing-idUSBREA3O20K20140425

De Villiers, M. D. (1990). The role and function of proof in mathematics. Pythagoras, 24, 17–24.

De Villiers, M. (2004). The role and function of quasi-empirical methods in mathematics. Canadian Journal of Science, Mathematics, and Technology Education, 4, 397-418.

Feinberg, W. (1993). Kenneth D. Benne. Proceedings and Addresses of the American Philosophical Association, Vol.66 No. 5, pp. 78-80.

Fried, M. N., & Amit, M. (2008). The co-development and interrelation of proof and authority: The case of Yana and Ronit. Mathematics Education Research Journal, 20(3), 54-77.

Fulmer, G. W. (2014). Undergraduates’ attitudes toward science and their epistemological beliefs: Positive effects of certainty and authority beliefs. Journal of Science Education and Technology, 23, 198-206.

Gerson, H., Bateman, E. (2010). Authority in an agency-centered, inquiry-based university calculus classroom. The Journal of Mathematical Behavior, 29, 195-206.

Harris, P. L., & Koenig, M. A. (2006). Trust in testimony: How students learn about science and religion. Child Development, 77, 505-524.

Kamii, C. (1984). Autonomy: The aim of education envisioned by Piaget. Phi Delta Kappan, 65,p. 410-415.

Lampert, M. (1990). When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. American Educational Research Journal, 27, 29-63.

Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. New York: Cambridge University Press.

Lederman, N. G., Abd-El-Khalic, F., Bell, R. L., Schwartz, R. S. (2002). Views of Nature of Science Questionnaire: Toward Valid and Meaningful Assessment of Learners’ Conceptions of Nature of Science. Journal of Science Teaching, 39¸497-521.

Lerman, S. (2000).The social turn in mathematics education research. In (ed.) J. Boaeler, Multiple Perspectives on Mathematics Teaching and Learning (pp. 19-44). Ablex: Westport, CT.

Norris, S. P. (1997). Intellectual independence for nonscientists and other content-transcendent goals of science education. Science Education, 81, 239-258.

Smith, M. U. (2013). The role of authority in science and religion with implications for science teaching and learning. Science and Education, 22, 605-634.

Stigler, J. W., Hiebert, J. (2009). The teaching gap: Best ideas from teachers for improving education in the classroom. New York, NY: Free Press.

Vithal, R. (1999). Democracy and authority: A complementarity in mathematics education? Zentrablatt für Didaktik der Mathematik, 31(1), 27-36.

Webel, C., Evitts, T. A., & Heinz, K. (2010). Shifting mathematical authority from teacher to community. The Mathematics Teacher, 104, 315-318.

Weber, K., Inglis, M., & Mejia-Ramos, J. P. (2014). How mathematicians obtain conviction: Implications for mathematics instruction and research on epistemic cognition. Educational Psychologist, 49, 36-58. Retrieved from http://homepages.lboro.ac.uk/~mamji/files/EP-2014.pdf

Wilson, M., & Lloyd, G. M. (2000). Sharing mathematical authority with students: The challenge for high school teachers. Journal of Curriculum and Supervision, 15, 146-169.

Yackel, E., Cobb, P., & Wood, T. (1991). Small-group interactions as a source of learning opportunities in second-grade mathematics. Journal for Research in Mathematics Education, 22, 390-408.

THE JUSTIFICATION PRINCIPLE

In this paper I articulate and defend the justification principle: a subject should not be taught merely through appeal to authority, but justification appropriate to the subject should always accompany the introduction of new content. Often researchers do not reveal their motivation for the principles they advocate. These principles spring from a belief system at the core of the scholar’s being. This wellspring, the soul of the academic, is often hidden so that the merits of argument alone may be considered and judged. In my advocation of the justification principle, I draw on a personal narrative in the form of an autoethnography. This personal narrative, infused with supporting literature, describes how I came to the realization that I have been indoctrinated into science in a manner similar to the way I was indoctrinated into religion as a child.If my teachers had adhered to the justification principle, then I certainly would not have come to such a conclusion. I also describe how my own violations of the justification principle as a teacher have led to its articulation and this defense.

THE EPISTEMOLOGIES OF RELIGION AND SCIENCE

Scientific claims can be traced back to empirical observations which can then be verified by others, whereas most religious claims cannot be empirically verified; instead they must be acquired through subjective experience or testimony. Smith outlined the differences between the epistemologies of the two disciplines, particularly with regards to authority¹. He noted that in science empirical evidence is the ultimate authority, whereas in religion the ultimate authority lies with God. However, in each discipline there is a chain of authority by which knowledge is transmitted. An important skill for scientists is the ability to judge whether the results presented by another researcher are reliable. A scientist does not have the expertise required to confirm every scientific result. There is a certain degree of trust in science as there is in religion also. Smith notes that many Christians adopt a critical stance, and engage in debate about interpretations of scripture and divine revelation. However other Christians adopt beliefs through the testimony of a pastor or the Bible without thoughtful consideration.

Smith recommended that teachers, “Address what counts as evidence in science (compared to other fields, e.g., religion) … [and] explicitly address when it might be appropriate to trust teachers and textbooks and when it is not” (p. 620). This recommendation is clearly a violation of the justification principle. However, is it reasonable to expect students to perform a science experiment every time they learn about a new scientific theory? Smith makes the point that our students need to become critical consumers of scientific research if they are to thrive in the modern world. Not everyone can be a scientist, and a valid appeal to a scientific authority can be useful for meeting the needs of today’s citizenry.

However, Smith’s recommendations were directed at secondary/undergraduate science instruction. Do they apply to primary school? At what age-level can children be expected to effectively and appropriately question the legitimacy of authority? Most young children would not question the religious ideas taught to them by their parents or church leaders nor the scientific claims presented to them by their school teachers. Harris and Koenig reported that children unquestionably accept the testimony of adults as fact.²As a child in a religious family, I was expected by my family members and other religious authorities to accept religious doctrines without question. Similarly, school pupils who have not yet learned how to appropriately question the legitimacy of an expert authority, are expected to accept and memorize what is presented as scientific fact. In such a scenario, scientific learning becomes strictly a process of rote memorization based on faith in one’s teacher. Norris wrote, “To ask of other human beings that they accept and memorize what the science teacher says, without any concern for the meaning and justification of what is said, is to treat those human beings with disrespect and is to show insufficient care for their welfare” (p. 252).³When such a situation exists in the classroom, the epistemologies of school science and religious indoctrination become identical.

In some Christian circles as well as some books of the Bible, goodness and truth are associated with God, and evil and lies are associated with the devil. Evil is sometimes seen as synonymous with the world. In the first epistle of John it is written, “We know that we are of God, and the whole world lies under the sway of the wicked one” (1 John 5:19 New King James Version). If one believes in the inerrancy of the Bible, and another worldly source of information (such as a science textbook) appears to contradict the Bible, then it is logical to conclude that the worldly source is evil or false. Although I do not have convictions that the Bible is inerrant, I believe in God and I take the testimony of the Bible seriously. Concerns about good, evil, truth and falsity sometimes affect the lens through which I view the world. I recently realized that I didn’t really know if anything I had been taught about science was true. I realized that most of my scientific knowledge had been acquired in the same way I acquired religious knowledge, through testimony.

KNOWLEDGE ACQUIRED THROUGH TESTIMONY

Harris and Koenig found that children were not at all skeptical of testimonial claims, sometimes accepted counterintuitive claims, but also actively assimilated the knowledge gained through testimony.⁴ The authors claimed that in some instances children ontologically differentiate between beings such as god and germs. One possible explanation offered was that children notice that belief in god is something that many adults are not sure of, whereas a belief in germs is taken as fact by most adults. Thus children may make an ontological distinction between god and germs, and between religious and scientific knowledge. An alternative explanation provided by the authors is that children make the distinction between the two types of knowledge because religious knowledge does not always align with experiential observations--perhaps it is precisely because religious claims are counterintuitive that they are so easily remembered. I reflect this may also be true of scientific claims.

As a young child, the counterintuitive idea that the world was spherical inspired me with feelings of wonder and awe. While glancing at the distant horizon I found joy and amazement in reflecting on the notion of a spherical earth that appears to be flat. I recall that this is precisely what happened to me when I first encountered the ideas of quantum mechanics. The excitement I felt about these counter-intuitive notions led me to study university physics for an entire year.

Just as I may choose to have belief in the miracles of Jesus, I can choose to believe in quantum tunneling. Both notions are counterintuitive, and both based on the testimony of others. A person cannot empirically verify that Jesus walked on water. However, one could perform a test to confirm the existence of quantum tunneling--an act of justification. Learning about science does not have to be based on faith, but for me it has been. Certainly I could recreate the experiments that led scientists to believe in the phenomenon called quantum tunneling, but why did my education not consist of this exercise? Why do students learn science primarily through the testimony of their teachers and textbooks?

I realize that because of my lack of empirical investigation of scientific claims, I perceive my belief in scientific theories to be a kind of faith. Furthermore, the scientific facts that I do believe in I have at best a surface-level understanding of. The reader should note that the violation of the justification principle in my physics education led to a dearth rather than a depth of knowledge. I do not possess a meaningful understanding of the knowledge I accepted through appeal to authority. Furthermore my lack of critical skepticism as a physics student runs contrary to the skepticism necessary for the development of scientific knowledge as practiced in the field. Certainly if I had been given the opportunity to empirically investigate the microscopic world and the phenomena of quantum tunneling I would have developed a deeper more meaningful understanding of the concept. Lack of adherence to the justification principle resulted in my lack of any real understanding of scientific knowledge. I only possess a rote knowledge of some scientific facts acquired through faith in my instructors and textbooks.

JUSTIFICATION IN MATHEMATICS

In this section I document how my own violation of the justification principle as a teacher of mathematics led me to articulate the principle. The setting is a tutoring session for pre-calculus. After spending more than our allotted time working on a pre-semester review, the student had one more problem left to complete. As it was past time to end our session, so I quickly explained how to complete the problem using Pascal’s triangle and the binomial theorem. The student admitted that while she believed that the binomial theorem was true, she didn’t understand it. While she accepted what I told her about the binomial theorem, it was given without the least bit of justification. Even without a formal proof, justification in the form of basic examples could have provided the student with a deeper understanding than she walked away with that day.

After the session was over, I rushed to my office to type the following (slightly edited but for the most part intact to capture my initial excitement and fundamental biases and motivation ):

Mathematics should never be taught without justification because

#1 – Sense-making. Meaningful mathematical knowledge;

#2 – Skepticism is a) Philosophically valuable, and b) Necessary for justice.

I realize that I am overly skeptical. I can realistically consider that the binomial theorem of mathematics may not hold because I haven’t proven the theorem (or read a proof) myself. Would it take a formal proof to convince myself? No. I would be convinced if I examined the problem long enough to find a reason for the underlying pattern (any mathematicians who are reading probably think that I am a complete buffoon as the reason for the pattern has naturally emerged in their minds while reading this), I am not fully convinced in the truth of an assertion unless I understand why it is true or complete a deductive proof. (Hence it is clear that while I do believe in the binomial theorem, I can conceive that it isn’t true because I haven’t struggled with the mathematics myself). For my student, the binomial theorem doesn’t make sense. This is strong support for my principal that mathematics should never be taught without justification. I am a skeptical person. Not everybody is. Not everybody questions their religion, their government, or their way of life. “The unexamined life is not worth living”. Mathematics is not worth studying without understanding. The aesthetic quality of mathematics is only apparent when one has insight. Our students cannot have this insight if the mathematics is handed to them pre-packaged ready for the microwave. Skepticism is important because the human way of life is full of injustice. It is worth questioning. It is worth questioning our endless wars. It is worth questioning our government structures. It is worth questioning the influence that textbook companies have on political decisions. In a populace that is not skeptical, tyranny reigns.

Proof is regarded as the ultimate form of justification in mathematics. A deductive proof not only serves to convince the skeptic, but also serves other roles such as illumination into why a theorem is true⁵. In a recent survey on epistemic cognition and the ways mathematicians gain conviction, Weber, Inglis, and Mejia-Ramos found that in addition to being convinced by deductive arguments, mathematicians sometimes gain conviction empirically or by appeals to expert authority.⁶ Working from the premise that mathematics students should become convinced of mathematical claims in a manner similar to the way mathematicians do, they argued that it is unreasonable to expect students to become convinced of mathematical truths solely through deductive reasoning. Their position is that it is not problematic that students gain conviction about mathematical claims through appeals to authority or empirical evidence since some mathematicians do the same.

Weber et al.’s position on appeal to authority is a violation of the justification principle. For example, the authors maintain that as most students will not be able to understand the fundamental theorem of and since the theorem is necessary for computing integrals, teachers should require that students accept the theorem strictly through appeal to authority as it is needed to compute integrals. Is it wise to sacrifice an in-depth understanding of arguably the most important theorem in calculus for the sake of enabling students to compute by hand what could just as easily be done with a computer? Each time the justification principle is violated in the mathematics classroom, the instructor removes the opportunity for students to engage deeply with the mathematical content.

Mathematicians often speak of the beauty of mathematics, something that students of mathematics seldom experience. How can something so beautiful inspire so much anxiety and disdain? The mathematics that students learn in school does not resemble the mathematics that a pure mathematician studies and loves. But the issue is not that the mathematical content is different, but rather the mathematical experience is different. Mathematical structures that could produce awe in the student are stripped of their true essence when they are presented as facts that students are forced to consume and reproduce on an examination.

CONCLUDING THOUGHTS

We live in an age when a wealth of knowledge has already been acquired by the human community. While we struggle to understand more, it is important that we stand on the shoulders of giants and appeal to the authority of experts—but not in a naïve manner. Knowledge claims should be judged critically. The fact that some knowledge claims have repeatedly been judged and accepted by the experts may be enough evidence for conviction. However, for those human beings who will succeed in discovering the previously undiscovered, they must come to engage more deeply with content through other forms of justification.

It is a goal that students become familiar with scientific theories. Should the theories be presented in their final form to be memorized? It seems as though we are denying the students the opportunity to actively construct the knowledge of those theories as we neglect the justification principle. It could be argued that it is infeasible to recreate the experiments used to gather the data that provides evidence to support these theories. I hypothesize that when children learn science in a rote manner, the meaning, significance and applicability is often lost. By examining the empirical evidence for a theory, students can come to a meaningful understanding of the theory with which the significance and applications may be properly understood. In mathematics, theorem is often presented without proof. If an educational goal is to develop critically thinking members of society, then it should not be standard practice to accept theorems without proof or theories without evidence.

Kenneth Benne, a founding member of the philosophy of education society, argued that more collaboration between philosophers and education researchers is sorely needed to address the problems of education.⁷To what extent can a teacher actually adhere to the justification principle? Under what circumstances is it more beneficial to abandon the principle than hold fast to it? Further consideration is needed to determine how the justification principle can foster healthy skepticism and guard against authoritarianism in society and the classroom. An educational system grounded in the forced memorization of an authority’s statements of “fact” will not produce a body of critically thinking individuals. Learning as faith in intellectual and institutional authorities should not be tolerated. It is my hope that philosophers of education as well as education researchers can weigh in on the justification principle, and consider its worth in education.

_________________________________________

1. Michael U. Smith, “The Role of Authority in Religion and Science with Implications for Science Teaching and Learning,” Science and Education22, (2014): 605-634.

2. Harris, Paul L., and Melissa A. Koenig. "Trust in testimony: How children learn about science and religion." Child development77, no. 3 (2006): 505-524.

3. Stephen P. Morris, “Intellectual Independence for Nonscientists and Other Content-Transcendent Goals of Science Education,” Science Education81, no. 2 (1997): 239-258.

4. Harris and Koenig. “Trust in Testimony”.

5. Michael de Villiers. "The Role and Function of Proof in Mathematics." Pythagoras24, no. 1 (1990): 17-24.

6. Keith Weber, Matthew Inglis, and Juan Pablo Mejia-Ramos. "How Mathematicians Obtain Conviction: Implications for Mathematics Instruction and Research on Epistemic Cognition." Educational Psychologist49, no. 1 (2014): 36-58.

7. Kenneth D. Benne. "The Philosopher and the Scientific Researcher in the Study of Education," Journal of Social Issues21, no. 2 (1965): 71-84.

THE CONTINUAL RENEWAL OF THE COMMUNITY OF MATHEMATICAL THINKERS

Middle Tennessee State University

A goal of mathematics education is to support students in developing their ability to think mathematically, i.e. to become members of the community of mathematical thinkers (Amit and Fried, 2005). To achieve this goal, it may be beneficial for researchers to do further work in reconceptualizing the role of authority in mathematics education. In this paper, I describe Kenneth Benne’s (1970) idea of anthropogogical authority. I then discuss mathematics instruction as described in the literature that best captures the spirit of anthropogogy.

ANTHROPOGOGICAL AUTHORITY

Educational philosopher Kenneth D. Benne (1970) desired to reconceptualize authority in education to distinguish between human relationships that were rooted in power versus those working for mutual human benefit. He argued that when one views education as a process of enculturation, it reinforces the view that authority is a one-way street in which the more cultured masters are responsible for teaching the uncultured. “[T]he model of education as cultural transmission [should be] replaced by a model of education as basically a process of personal and cultural renewal” (p. 387). Benne coined the term “anthropogogical authority” to describe a type of authority relationship in which culture is renewed rather than transmitted. While pedagogy brings to mind the instruction of the child by the more learned teacher, anthropogogy signals the teaching of humans by other humans, all involved in a continual process of reeducation. Benne wrote,

[T]he ultimate bearer of educational authority is a community life in which its subjects are seeking fuller and more valid membership ... [The teacher] becomes a surrogate bearer of anthropogogical authority only by virtue of responsible efforts to mediate between the present community involvements of those being educated and their expanding and deepening affiliations in the common life of a wider community” (pp. 401-402).

Benne was speaking not only of the broad human community, but also disciplinary communities. As an example of anthropogogical authority, he described the relationship between a doctor and a medical student. The student gradually bears more and more authority until she is a full member of the medical community. Amit and Fried (2005) related the notion of anthropogogical authority to mathematics education. They noted that the wider community into which students are becoming fuller members is the community of mathematical thinkers. But in mathematics classrooms, students often find themselves as outsiders to this community because of their overreliance on appeal to the expert authority of their teachers and classmates rather relying on than their own capabilities for mathematical reasoning (Amit and Fried, 2005). If students are to become fuller members of the community of mathematical thinkers, it is necessary that they bear more responsibility for sense-making and justification of mathematical ideas and rely less on appeal to authority. The teacher is not the only source of mathematical knowledge, but the students should also contribute. This sharing of the mathematical authority is necessary if students are to see themselves as contributing members of the community of mathematical thinkers. In the pages that follow, I further elaborate on the ideas of anthropogogical authority and the community of mathematical thinkers by highlighting some instruction described in classic mathematics education literature that I believe best captures the spirit of anthropogogy.

IN THE SPIRIT OF ANTHROPOGOGY

A mathematics classroom that honors the spirit of anthropogogy will support students’ development as mathematical thinkers and contribute to the renewal of the community of mathematical thinkers. I believe Magdalene Lampert’s fourth-grade mathematics instruction captures the spirit of anthropogogy well. In her (1990) work she claimed that the students in her class “had learned to regard themselves as a mathematical community of discourse, capable of ascertaining the legitimacy of any member’s assertions using a mathematical form of argument” (p. 42). Lampert’s role was not to disseminate mathematical knowledge as an expert authority, but to listen to her student’s arguments and support student thinking and reasoning. Her work highlighted the importance of learning mathematics in a manner consistent with the way knowledge is acquired in the highest rung of the community of mathematical thinkers--the community of practicing mathematicians. Drawing on the ideas of Poyla and Lakatos, Lampert described mathematics as a social process of conscious guessing, conjecturing, refuting, and generalizing where “reasoning and mathematical argument—not the teacher or the textbook—are the primary source of an idea’s legitimacy” (p. 34). For the community of mathematical thinkers to be continually renewed, students must have the opportunity to engage in this social process, not merely memorize the definitions and processes introduced by the teacher.

One of Lampert’s doctoral students, Deborah Ball, also documented her own teaching in a third-grade mathematics classroom. Ball (1993) supported her students’ active engagement in mathematical practices in a way that honored the discipline of mathematics and respected her students as learners. In one documented classroom episode, a student named Sean claimed that the number six is both even and odd because it can be partitioned into an odd number (three) of groups of two. After a class discussion he concluded that other numbers such as ten were also both even and odd. Rather than dismiss Sean’s ideas, Ball recognized that they were significant. In her reflections she wrote that she struggled with the notion that Sean’s idea was not part of established mathematical knowledge. She worried that continued discussion about these ideas might just confuse the students. But she ultimately decided to allow the class to formulate and prove conjectures related to Sean numbers (which Sean provided the definition for) for several days. Ball wrote, “it seemed defensible to give the class firsthand experience in seeing themselves capable of plausible mathematical creations” (p. 387). This episode highlights the way Ball’s instruction supported students so that they could see themselves as becoming more valid members of the community of mathematical thinkers, rather than “outsiders” (Amit and Fried, 2005) to that community. It is worth mentioning that Ball had to make sense of Sean’s mathematical ideas. Ball was not the only bearer of mathematical authority in the classroom. Authority was shared so that students could bear it as well and share in the creation of mathematical knowledge.

Yackel, Cobb, and Wood (1991) described a second-grade classroom that was part of a design project. The authors noted how the teacher in the classroom used her institutional authority to guide the renegotiation of classroom norms. She continually encouraged cooperation and perseverance, both key to developing and sustaining the community of mathematical thinkers. Students worked together on problems, and then shared their work during whole-class discussions in which the teacher summarized student ideas. The teacher in the study did not position herself as an expert authority to be relied upon, and did not verify correct answers. Like Ball and Lampert, she shifted the responsibility for mathematical explanation and justification to the students. Shifting authority is not valuable in its own right. It is necessary to support students in becoming fuller members of the community of mathematical thinkers.

CONCLUDING IDEAS ON ANTHROPOGOGY AND MATHEMATICS EDUCATION

All of the teachers described above valued and honored student thinking, collaboration, student explanation, and student justification. Rather than develop an overreliance on appeal to authority, students in their classrooms had the opportunity to refine their ability to think mathematically within a classroom community. Authority was shifted so that the teacher’s duty was to listen, make sense of, and appreciate student ideas. These teachers and the students in their classrooms embody the spirit of anthropogogy. Such classrooms promote the development and renewal of the community of mathematical thinkers.

Through an analysis of middle grades classrooms, Amit and Fried (2005) described the immense authority that mathematics teachers have in the eyes of students. One student said of the teacher, “I don’t check if what the teacher says is correct, I take it as obvious that it’s correct, especially if it has to do with mathematics” (p. 158 emphasis in original). The authors noted that reflection on anthropogogical authority is important not only for researchers, but also for teachers. They wrote that teachers “must learn to see themselves, like Benne’s teaching doctor, working to make their students into colleagues who finally will completely share authority with them” (p. 165). But this goal will not be easily realized as traditional authority relationships are dominant in mathematics teaching. It is noteworthy that the teacher in Amit and Fried’s study “tried to emphasize [to the students] that things in mathematics were not true because she said so, or because the book said so, or because anyone else simply said so” (p. 161). Yet, the students in the class often accepted statements as true merely because the teacher said so.

Weber, Inglis, and Mejia-Ramos (2014) found that in addition to being convinced by deductive and empirical arguments, mathematicians sometimes gain conviction by appeals to expert authority (especially if the mathematics involved is in a domain that is not their specialization). Working from the premise that mathematics students should become convinced of mathematical claims in a manner similar to the way mathematicians do, they argued that students are not behaving irrationally when they obtain conviction in mathematical claims through appeals to authority (since some mathematicians do the same). They wrote, “If we want students to know that π is irrational or to use the Fundamental Theorem of Calculus, we require them to rely on authoritarian evidence” (p. 32). I agree it is not irrational for students to become convinced solely through appeal to authority, but it is questionable whether teachers should expect and require students to do so. It may be a stretch to label such teaching practices as fascist, but it is not a stretch to claim that such practices do little to support students in becoming fuller members of the community of mathematical thinkers capable of sustaining and renewing the community. Is it wise to sacrifice an in-depth understanding of arguably the most important theorem in calculus for the sake of enabling students to compute an integral by hand that could just as easily be done with a computer? If the goal is to develop, sustain, and renew the community of mathematical thinkers, then what value is there to teach students that π is irrational if they are not capable of understanding what this means?

As Benne (1970) maintained, we can no longer view the purpose of education as the transmission of cultural knowledge to the student. Instead, more care must be taken to ensure that students develop their ability to think mathematically so that the community of mathematical thinkers, and thus the discipline of mathematics, may be continually renewed. If we are to value students’ development and growth as mathematical thinkers then students should not be required to accept any mathematical notion merely because a textbook or teacher said so. All too often such practices become the dominant form of learning in classrooms. If the community of mathematical thinkers is to be sustained and renewed, it is desirable that students rely more on their own reasoning abilities than the expertise of others. Such issues are not easily resolved as there is a deep cultural tradition of teaching mathematics in an authoritarian matter. Even teachers using alternative pedagogies in which authority is shifted sometimes expect students to accept a mathematical claim merely because the teacher said it is true. Authoritarian teaching is often seen as necessary for utilitarian (or perhaps economic) purposes, or to meet content standards in preparation for state or department examinations. Discussions on issues of authority and the goals of mathematics education must continue if we are to make progress on these issues.

References

Amit, M., & Fried, M. N. (2005). Authority and authority relations in mathematics education: A view from an 8^thgrade classroom. Educational Studies in Mathematics, 58, 145-168.

Ball, D. L. (1993) With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. The Elementary School Journal, 93, 373-397.

Benne, K. D. (1970). Authority in education. Harvard Educational Review, 40, 385-410.

Lampert, M. (1990). When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. American Educational Research Journal, 27, 29-63.

Yackel, E., Cobb, P., & Wood, T. (1991). Small-group interactions as a source of learning opportunities in second-grade mathematics. Journal for Research in Mathematics Education, 22, 390-408.

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