The Place of Mathematics and the Mathematics of Place

By Carl Lee, Professor of Mathematics at the University of Kentucky and Chellgren Endowed Professor at the Chellgren Center for Undergraduate Excellence.

Editor’s Note: Carl Lee is a recipient of the 2014 Deborah and Franklin Tepper Haimo Award from the Mathematical Association of America.  This essay is based on his acceptance speech at the 2014 Joint Mathematics Meetings.

My place.  I was born into a family littered with academics, teachers, and Ph.D.s, including a grandfather who was an educational psychologist at Brown serving on one of the committees to create the SAT.  My early interest in things mathematical was nurtured in a home stocked with books by Gardner, Ball and Coxeter, Steinhaus, and the like.  With almost no exception my public school teachers were outstanding.  I was raised in a faith community, Bahá’í, that explicitly acknowledges the presence of tremendous human capacity and the high station of the teacher who nurtures it.  I played and experimented with, and learned, mathematics in both formal and informal settings.  Thus I grew up in a place in which I was able both to feed my mathematical hunger as well as to have a clear idea of what it was like to teach as a profession.  I thrived.

I recount this not to present a pedigree to justify personal worthiness, but rather to emphasize that I enjoyed a perfect match between my personal mathematical inclination and my learning environments.  Because of this background, it took me a while to understand the sometimes profound gap between others’ mathematical place, and the consequent care required to pay attention to that place, when designing an effective realm for learning.  As a K–12 student I often engaged in math classes at a high cognitive level merely as a result of a teacher’s direct instruction (“lecture”).  As a teacher I quickly learned that I engaged few of my students by this process.  Not all developed their “mathematical habits of mind” or “mathematical practices” through my in-class lectures and out-of-class homework (often worked on individually).  I now better appreciate the significant role of personal context and informal education in the development of students’ capacity.

The student’s place.  There is an entire discipline of “place-based” teaching and learning, focused on recognizing and making explicit connections with the student’s physical location and social community (an “outer place”).  Mirroring and linked with this is a student’s personal cognitive place (or “inner place”)—here, I recall Vygotsky’s writings on the “ZPD,” the zone of proximal development, which he describes as “the distance between the actual developmental level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance, or in collaboration with more capable peers.”  That is to say, learning can be promoted when the material is above the student’s current state, but not so far above to be unattainable even with scaffolding and assistance.  Identifying these outer and inner student places, and making wise and deliberate instructive choices, are major challenges of the teacher.

With respect to the student’s outer place we are all well aware of the encouragement to teach mathematics through “real-world” problems.  The Common Core State Standards for Mathematical Practice encourage modeling with mathematics:  “Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace.”  My work with teachers in central Appalachia has convinced me authentic and locally placed problems can provide powerful stimulus and support for mathematical learning.

On the other hand, with respect to the student’s inner place, another mathematical practice advocates that students must “make sense of problems and persevere in solving them.”  Research by the psychologist Carol Dweck, for example, confirms that praise focused on developing a growth mindset positively affects subsequent student achievement, while praise that reinforces fixed intelligence beliefs has the opposite effect.  Further, fostering a growth mindset rather than a fixed mindset in the classroom with the explicit knowledge and understanding of the students appears to lead to increased academic achievement when students are aware of the value of the struggle.  Polyá was an early advocate of the deliberate shift toward raising the explicit awareness of and cultivating mathematical practices among students.  Seeing his film “Let us Teach Guessing” while in high school left a lifelong impression upon me.  As a result, I feel I must promote and observe struggle in my classroom—deliberately create opportunities in the classroom in which students grapple with mathematics and communicate with each other; carefully listen and use what I learn to shape what is to come; and provide an environment in which mistakes are opportunities for learning and not censure.

The student’s outer and inner places are, of course, deeply connected—where a student is mathematically is not isolated from his or her background and environment.  And in Appalachia (as in many other places), struggle is a part of life.  The preeminent Appalachian poet and writer Wendell Berry beautifully captures this notion in his essay “Poetry and Marriage” from Standing by Words.

There are, it seems, two muses: the Muse of Inspiration, who gives us inarticulate visions and desires, and the Muse of Realization, who returns again and again to say “It is yet more difficult than you thought.”  This is the muse of form.  It may be then that form serves us best when it works as an obstruction, to baffle us and deflect our intended course.  It may be that when we no longer know what to do, we have come to our real work and when we no longer know which way to go, we have begun our real journey.  The mind that is not baffled is not employed.  The impeded stream is the one that sings.

The place of community.  There is a continuum of participants and stakeholders in STEM education, including:  P-12 students, school teachers, counselors, principals, superintendents, parents, community members, college students taking math and science courses, majors in STEM fields, aspiring STEM teachers, higher education faculty in content departments teaching all of these types of students, higher education faculty in education departments teaching courses for future teachers and engaging in teacher training programs, practicing teachers including those who supervise student teachers or are enrolled in graduate programs, higher education faculty engaging in STEM education research or in outreach to schools, and various local, state, regional, and national agencies and organizations, public and private, commercial and non-profit.  There is a natural tendency for each of the diverse participants to operate within a somewhat limited sphere of activity.  If we wish to build institutional and regional capacity, there is an imperative need for mathematicians to lend their expertise to this continuum, and for institutions to appropriately reward their contributions.  The Mathematical Education of Teachers II (CBMS), for example, offers a call to action with explicit guidance and suggestions.

Reflecting on my work with others in these many roles in Appalachia, it is very clear to me that an appropriate understanding of place is essential.  Many regard rural Appalachia with “deficit vision” and wish to come in and “fix things.”  Yet Wendell Berry’s view is completely opposite—read his poem “The Wild Geese.”

Berry articulates his vision in “The Loss of the Future” from The Long-Legged House:  “A community is the mental and spiritual condition of knowing that the place is shared, and that the people who share the place define and limit the possibilities of each other’s lives.  It is the knowledge that people have of each other, their concern for each other, their trust in each other, the freedom with which they come and go among themselves.”

Bob Wells recalls Berry’s 2007 speech at Duke Divinity School.

“Whatever doesn’t fit a place is wrong,” Berry said.  “It doesn’t matter if it is true or false.  If it doesn’t belong, it is wrong.”  Without a standard of “place” as a measure of real prosperity, Berry said, we will never know what to make of development, technology, research, education, modernization, religion and the environment, or ecosphere.

I have learned that to be more effective I must view place from the perspective of a partner rather than as a knowledgeable outsider (however well-intentioned).  Consideration of place must be approached with an authentic attitude of partnership, setting aside such common barriers as “outsider-insider,” “knowledgeable-ignorant,” and “wealthy-poor.”  The wealth and strength of Appalachia include rich experience and an abiding sense of community, both of which can significantly contribute to sustainable approaches to educational challenges.

Sentiments such as these were central principles in two recent large-scale NSF funded projects in Appalachia that I had the privilege to work on.  ACCLAIM, an NSF Center for Learning and Teaching, focused on “the cultivation of indigenous leadership capacity for the improvement of school mathematics in rural places.”  A highlight of this project was the creation of an interinstitutional doctoral program in mathematics education built around issues in mathematics, mathematics education, and rural sociology.  Students in this program demonstrated a commitment to rural place and earned their degrees without having to quit their jobs.  Their desire to remain in their communities helped sustain ACCLAIM’s impact on future teachers.  The resulting dissertations were not required to address rural topics, but often did.  I encourage perusal of for uncovered understandings at the intersection of mathematics education and rural education.

AMSP, an NSF Mathematics and Science Partnership, was an ambitious Appalachian enterprise involving nine institutions of higher education and about 60 school districts.  Lessons learned during the earlier years led later to community-based Partnership Enhancement Projects generated by groups of stakeholder partners based on local concerns.  The place of the work (e.g., the school, district, or county) provided the explicit context in which participants evaluated challenges, assessed resources, planned, executed projects, and evaluated outcomes.

The place of mathematics and the mathematics of place.  On the one hand many (including mathematicians) value mathematics precisely because it transcends place, even though it may be initially motivated by a particular context (mathematical, physical, or otherwise).  On the other hand, the value of place (including rural or urban place, and personal place) offers a rich and meaningful setting in which to nurture the understanding of mathematics and make important connections that can promote mathematical learning and more effective teaching.  My present understanding is that the latter view is important to support the former.  In teaching and professional development I therefore try to work with others in a spirit of partnership — there are things that I know, and there are things that my partners know.  If we abandon a sense of superiority as we approach classroom teaching, professional development, or community capacity building, striving to understand our place, we can dramatically increase the efficacy of our work together.

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