By Brian Katz, Augustana College
I think that mathematics draws in some people and repels others in large part because of the distinctive role of authority in our discipline and teaching, especially when we act as content experts and discussion leaders in the classroom. For instance, consider the following phrases from students, distilled from my interactions with college students over the past 15 years.
I’m not a math person. I learn best when you show me a bunch of examples and then I practice them. It’s true, so why do I have to prove it? That’s just how my last teacher told me to do it. I always liked math because there was one right answer. I just want to teach high school; why do I have to learn this? Wait, what – you want me to ask my own question!? Do I have to simplify my fractions? Well, that’s what the computer said was the answer. The test was unfair because it had problems we didn’t discuss in class. ~silence~
I expect that these comments are also familiar and painful to the reader. I think that each of these comments is in part a symptom of ways students have internalized a relationship with authority from our teaching. In this post, I will illuminate the role of authority in mathematics teaching, argue that taking a more overt stance toward it can better support both the students we repel and the ones we attract, and offer a handful of strategies for taking such a stance.
Depending on whom you ask, the truth of a mathematical conclusion can stand independently of a human authority or based entirely on the word of an authority. Mathematicians will often claim (e.g., [8]) that we depend only on proof to develop reliable knowledge and will dismiss student efforts to use empirical evidence or the word of an expert, but researchers [13] are showing that mathematicians use these strategies as well. I certainly agree that deductive reasoning occupies a special place in our discipline, but the absence of methods, evidence, and theoretical frameworks in the discussion of mathematics quietly places the math itself in the position of perfect (Platonic) authority. I do not take issue with this perspective except when our silence about authority leaves students to grapple with it alone, often painfully. This post will focus on these implications for our students.
Models of Student Development
My thinking about authority begins with Williams Perry’s work from the 1950s and 60s on the epistemological development of college students [14]. In broad strokes, Perry’s scheme describes a sequence of positions from which the students he studied viewed truth or knowledge; I will use a condensed version of this scheme with three main positions. From Dualism, students view knowledge as binary, and Authorities know the difference between true and false. From this position, students equate learning with memorizing any information that these Authorities pass on to them. From Multiplicity, students notice that much of knowledge is context-dependent and come to believe that any perspective is a valid source of knowledge. As a result, while in this position, students become less interested in the perspectives of others, including authority figures, instead focusing on their own authority. From Relativism, students begin to demand that other perspectives be justified; as a result, knowledge becomes the result of argument and evidence. In this position, students initially focus on learning HOW authorities want them to think, rather than WHAT. Eventually, these students acknowledge that their arguments must be grounded on accepted assumptions, and they become concerned with establishing appropriate and personal precepts. Common elements of college programs, such as first-year writing seminars and introduction-to-proof courses, encourage our students to adopt a relativistic position in their collegiate work by helping them practice appropriate modes of argumentation. For the purposes of this post, I will group students as though they have a predominant or preferred position.
My dualist students often find enticing or comforting the idea that mathematics is a field in which truth is absolute and completely known because they feel that it allows them to avoid pesky ambiguity, especially in their course assignments. These same students regularly say dismissive things about other disciplines to me, most commonly about literature courses. They seem to believe that the work of engaging literature reduces to forming a personal opinion and that all opinions are equally valid; they will often go further to rail against grading in these courses because they see it as relying on whether their valid opinion happens to match the one held by the teacher. To be clear, these are deleterious conceptions of both disciplines; I think that one of the most important goals of a college (liberal arts) education is helping students change these stereotyped views of disciplines. While these dualist students sometimes come to college liking math, they have been set up for a painful bait-and-switch when their math work shifts suddenly from execution of provided algorithms to generation of original arguments, leading to their comments about the good old days of Calculus. Perry saw evidence in his data of students retreating to earlier positions when facing difficulty, which can explain student resistance in their first proof courses, not to mention the first time they are asked to do something original and creative in math. I’m not surprised that many of these students consider switching to engineering or economics because they believe the math used there is aligned with their earlier perceptions of mathematics.
The other large group of incoming students, those for whom multiplicity is the preferred epistemological position, curiously say essentially the same things about mathematics and literature to me, with exactly opposite emotional value attached to the descriptions. They seem to hate that math has no room for their individual perspective and prefer discussion-oriented courses in the humanities and social sciences because they offer spaces in which they are encouraged to consider their own perspective. The departure of this group from studying mathematics is one of the leaky joints in the mathematics pipeline discussed by CSPCC [2] and even the Obama administration [7]. Perry’s data suggested that some students in this position aligned with authority figures and others against and that this alignment impacted their trajectory. I think that the importance of this split path can be understood by considering research that responds to one of the major critiques of Perry’s work. The population on which he based his scheme, namely students at Harvard at the time, was overwhelmingly male and non-representative of today’s college students in many other ways.
Belenky, Clinchy, and Goldberger, using a population of female interviewees whose ages varied widely, developed a scheme called “Women’s Ways of Knowing” [1]. These two schemes share many common features, but WWK illuminates two important observations. First, some of the women in this study talked about ending educational experiences because they perceived educational environments as placing no value on their voices and offering no pathways to expertise. In contrast, it seems important that the students in Perry’s population were male and enrolled at an elite college in a time when going to college was a rarer and more intentional choice than today; in other words, Perry’s population was likely selected to contain participants who already envisioned themselves as future authorities, an attribute that may help a person persist in (collegiate mathematics) courses when they experience friction between their epistemology and their course work. Combining this with the under-representation of women in our discipline, it’s hardly surprising that female students switch out of math and STEM more than male students [6]; they must endure years of study in an environment that seems not to value any student voices, and when students begin to find their voices, their male peers may have more support for identifying as future authorities themselves. Second, the WWK scheme also adds an important, fifth position (before the others) called Silence, from which interviewees experienced a world in which they had no access to knowledge or truth. I think the multiplicitous and silent students are driven from math similarly, though the “myth of genius”, which correlates to heavily with gender representation in STEM fields [10], might be a silencing factor well before college.
There is a third, much smaller group of students who view mathematics as relativists. I believe I was in this group; I liked how I could avoid memorizing almost anything in math because I knew that everything could be derived from the definitions. Importantly, this perspective seemed to maintain but reframe the dualist and multiplicitous things I liked about math. Yes, our knowledge and truth seemed to be of a different sort than in other disciplines, but this characteristic came from the kinds of arguments that are possible about abstract objects, not by declaration. And yes, there wasn’t much room to disagree with a theorem, but I felt empowered to make my own arguments given our egalitarian access to definitions, to try to extend algorithms or make them more efficient, and to have my own personal way of thinking about problems. I conjecture that most of us who have persisted in mathematics made similar transformations of our love for math that helped us persist and that many who did not persist did not make. Significantly, I believe that math classrooms can be set up to be welcoming to students in all of these positions while also helping them move forward.
Using these Models to Inform Pedagogy
Fortunately, Perry’s research also illuminates at least three kinds of experiences that impel students to move to later positions: encountering questions without known answers or about which Authorities disagree, engaging a pluralism of ideas among peers, and rigorously justifying claims and questioning assumptions. To me, this feels like a description of a classroom organized by inquiry. I further claim that mathematics makes it particularly easy to include these experiences in our classrooms: each new definition gives us instant access to an unlimited supply of perfect copies of objects into which we can explore, often very quickly and cheaply, while entertaining open conjectures that require justification. In contrast, work in the natural and social sciences seems to require data collection that can be slow and expensive, work in the arts and humanities can require nuanced analysis that uses a holistic perspective that is not immediately accessible, and work in applied fields often interacts with a great deal of information from its context. I’m not suggesting that these aspects of other disciplines are negative; in fact, I think each discipline has opportunities to highlight different facets of authority and epistemology more easily and that contrasting these themes across disciplines is a key mechanism for supporting student growth. I am saying that I see a way that our disciplinary context allows me to put the epistemological work of education front and center in the math classroom, in contrast to placing it in an invisible role.
Building a classroom that supports these epistemological themes can be challenging because it requires course materials and teaching skills that are consistent with this goal. A complete discussion of such course materials and general teaching skills is beyond the scope of this post, but I would suggest the reader start with the writings coming from these two threads [17,4]. Instead, I would like to offer two suggestions for strategies that engage authority in your classrooms overtly but don’t require an immediate reimagining of your teaching practice. First, I suggest that you read and discuss some mathematics education literature with your students — here are some of my favorites. Harel and Sowder [9] describe categories of students’ perspectives on mathematical justification called “proof schemes”; these categories align extremely well with Perry’s described above, and discussing this paper (or another that uses it) can help students understand professors’ expectations in advanced mathematics courses. Weber and Alcock [18] discuss student and expert behaviors when validating proofs; this paper always leads to discussion of subtle but important points including the role of communal expectations for proofs. If your students don’t yet have experience with proof, then articles or videos about Carol Dweck’s work on “growth mindsets” [15,5] or Paul Lockhart’s provocative (and informal) “A Mathematician’s Lament” [3] are both strong choices; these papers are not overtly about authority and epistemology, but they help students talk about the discipline in a way that allows an instructor to engage their perspectives. And while it’s not about mathematics, I also enjoy discussing summaries of Perry’s scheme with students in many courses, most commonly using Chapter 1 of this reference [11]. You may worry that these activities would take away time from other learning objectives, but the opposite has been my experience; through these reflection opportunities, students are able to see course activities in a new light that deepens understanding of the past work and makes future work more effective and efficient.
Second, I suggest that you humanize the mathematical practices in your classroom. I think that small changes in our language can help students adapt a more productive stance toward authority. Using “we” and “us” communicates that the people in the room are engaged in mathematics and have some local authority. Tagging ideas, questions, and conjectures with student’s names when we reference them highlights the fact that individuals are impacting the development of the mathematics; asking students to use their peers’ names suggests that these people have perspectives that we will consider. This language makes space for the psychological and social aspects of our work that we may crush with our silent authority otherwise. In addition to investing students with some authority, I think it is important to humanize ourselves to resist becoming the abstract authority. As a first step, we can talk about moments inside and outside of the course during which we are or were uncertain mathematically. However, I think that telling students that they can only know mathematical things about us as people communicates a preference to be seen as a distant authority. Personally, I feel an ethical obligation to go further and be a multifaceted person with my students. For example, I end up coming out in some of my courses because doing otherwise feels hypocritical given the level of openness I’ve asked of students about their lives and minds [12]; I also appreciate that they are now aware of at least one queer mathematician as a potential role model.
Conclusion
I would like to reiterate the importance of mathematics educators taking a more overt stance toward authority. I think we are compelled to use the distinctive interaction of mathematics with authority to help students mature generally in their lives, and I think the lenses discussed here help me see opportunities for this work. For example, Perry noticed changes in the most common starting position of first-year students in the 1960s, which he connected to the marked changes in the national dialog and the role of Authority therein. Analogously, I believe that these schemes are going to be key tools for understanding the waning public faith in higher education and for responding to the changing needs, perspectives, and skills of students entering college during the next decade, especially regarding the authority of hands-on parents and of information technology. In the context of our classrooms, I have argued above that when we vest all authority with mathematics in the abstract, we seem to create a vacuum that leads students to engage with us as Authorities instead, in ways that attract some students while setting them up for struggle and make other students feel unwelcome. I have made some conjectures about how this might perpetuate the under-representation of women in mathematics, and I would make similar connections and conjectures about under-representation of people of color (e.g, [16]). In summary, I think that being intentional and overt about the role of authority in our teaching can transform our classrooms into more inclusive and equitable spaces, and I hope you feel empowered to use the ideas in this post to do this work.
References:
[1] Belenky, M. F. (1986). Women’s ways of knowing: The development of self, voice, and mind. Basic books.
[2] Characteristics of Successful Propgrams in College Calculus: Publication & Reports – Retreived from http://www.maa.org/programs/faculty-and-departments/curriculum-development-resources/national-studies-college-calculus/cspcc-publications
[3] Devlin, K. (2008, March) Lockhart’s Lament. Retreived from https://www.maa.org/external_archive/devlin/devlin_03_08.html
[4] Discovering the Art of Mathematics – Retreived from https://www.artofmathematics.org/
[5] Dweck, C. (2014, November). The Power of Believing that You Can Improve (TED talk). Retreived from http://www.ted.com/talks/carol_dweck_the_power_of_believing_that_you_can_improve
[6] Ellis, J., Fosdick, B.K., and Rasmussen, C. (2016). Women 1.5 times more likely to leave STEM pipeline after calculus compared to men: Lack of mathematical confidence a potential culprit. PLoS ONE 11(7): e0157447. doi10.1371/journal.pone.0157447
[7] Feder, M. (2012, December 18) One Decade, One Million more STEM Graduates (based on a statement by the President’s Council of Advisors on Science and Technology). Retreived from https://www.whitehouse.gov/blog/2012/12/18/one-decade-one-million-more-stem-graduates
[8] Fischbein, E. (1982). Intuition and proof. For the learning of mathematics, 3(2), 9-24.
[9] Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. Research in collegiate mathematics education III, 234-283.
[10] Leslie, S. J., Cimpian, A., Meyer, M., & Freeland, E. (2015). Expectations of brilliance underlie gender distributions across academic disciplines. Science, 347(6219), 262-265.
[11] Love, P. G., & Guthrie, V. L. (1999). Understanding and Applying Cognitive Development Theory: New Directions for Student Services, Number 88 (Vol. 27). John Wiley & Sons. (Chapter 1)
[12] Lundquist, J. & Misra, A. (2016, October 18) Establishing Rapport in the Classroom. Retreived from https://www.insidehighered.com/advice/2016/10/18/how-engage-students-classroom-essay
[13] Mejia-Ramos, J. P., & Weber, K. (2014). Why and how mathematicians read proofs: Further evidence from a survey study. Educational Studies in Mathematics, 85(2), 161-173.
[14] Perry Jr, W. G. (1999). Forms of Intellectual and Ethical Development in the College Years: A Scheme. Jossey-Bass Higher and Adult Education Series. Jossey-Bass Publishers, 350 Sansome St., San Francisco, CA 94104.
[15] Popova, M. (2014, January 29) Fixed vs. Growth: Two Basic Mindset that Shape Our Lives. Retreived from https://www.brainpickings.org/2014/01/29/carol-dweck-mindset/
[16] Robinson, M. (2011, Spring). Student Development Theory Overview. Retrieved from https://studentdevelopmenttheory.wordpress.com/racial-identity-development/
[17] Teaching Inquiry-Oriented Mathematics: Establishing Supports – Retreived from http://times.math.vt.edu/
[18] Weber, K., & Alcock, L. (2005). Using warranted implications to understand and validate proofs. For the Learning of Mathematics, 25(1), 34-51.