The Liberal Art of Mathematics

By Priscilla Bremser, Contributing Editor, Middlebury College

Somehow, over the last 600 years or so, mathematics has moved from the core of the liberal arts disciplines to entirely outside. We’re all used to this; a “liberal arts math” course is understood to serve non-STEM majors, for example. The reasons for this shift are interesting to ponder (see [1] and [2]), but in this post I suggest that we consider some of its unfortunate present-day implications. It’s also worth considering the broader aim of a liberal arts approach, which transcends disciplinary boundaries. Continue reading

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Reading Articles in Mathematics Education – It’s Not Just for Prospective Teachers!

By Elise Lockwood, Contributing Editor, Oregon State University.

When I teach classes for pre-service teachers, I typically have the students read and discuss a math education article about the teaching or learning of content they may eventually teach. This may include research articles (in journals such as Journal for Research in Mathematics Education, which typically report on research studies), or practitioner articles (in journals such as Mathematics Teacher, which offer practical insights without necessarily being rooted in rigorously conducted research).

Recently, however, I have also started to have students in more traditional postsecondary mathematics classes (not just those designed for pre-service teachers) read math education articles. Last term, for instance, after discussing counting problems in an advanced mathematics course, I had my students read an article by Batanero, Navarro-Pelayo, and Godino (1997) about effects of implicit combinatorial models on students’ solving of counting problems. Through such readings, my students can be exposed to research on students’ thinking about the very postsecondary content they are learning. I am always pleasantly surprised by the rich discussion such readings stimulate, and this made me reflect on the value of having students read such articles, even in their “pure” mathematics classes.

Both research and practitioner papers about math education can elicit valuable ideas and points of discussion from which math students can benefit. In this post, I make a case for three potential benefits of having students occasionally read math education articles in their math courses. Continue reading

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We Did the Math! Student Perspectives on Inquiry-Based Learning

By Sarah E. Andrews and Justin R. Crum, undergraduate Mathematics majors at Northern Arizona University, and Taryn M. Laird, graduate student in Mathematics at, and 2014 graduate of, Northern Arizona University.

Editor’s note: The editorial board believes that in our discussion of teaching and learning, it is important to include the authentic voices of undergraduate students reflecting on their experiences with mathematics. We thank Ms. Andrews, Mr. Crum, and Ms. Laird for contributing their essay. More information regarding inquiry based learning can be found at http://www.inquirybasedlearning.org/.

Inquiry based learning (IBL) classes inspired each of us to believe that we could go into mathematics. That we belonged. We may be able to prove something important or make an impact in the lives of other budding mathematicians. IBL classes have given us this confidence to believe in ourselves, and to have fun trying to discover for ourselves what math is and where it will lead us. It was not only this sense of being able to discover, however, it was also learning how to collaborate with others. Mathematics is not an isolated endeavor, but rather a concentrated attempt by groups of people working toward their common goal. In normal lecture-based classes, we would talk to our friends, and if we got stuck, we might ask one another what to do next. In the IBL classes, we would talk to each person in the class. Students would ask each other questions willingly. We would make new friends, and ask more questions, until each of us decided we were satisfied — we understood the material now. Continue reading

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Proportionality Confusion

By Dick Stanley, Professional Development Program, University of California at Berkeley

The notion of one quantity being proportional to another is certainly a very basic part of an understanding of mathematics and of its applications, from middle school through calculus and beyond. Unfortunately, the picture of proportionality that tends to emerge in school mathematics in this country is narrow and confused. Everyone learns the procedure of setting up and solving a proportion, but the connection of this to the idea of one quantity being proportional to another is tenuous.

In support of this statement, I summarize below the results of participant responses given in a workshop attended by teachers, mathematics educators, and mathematicians. The surprisingly shallow responses show a striking lack of a common, mathematically coherent understanding in this audience of the subject of proportionality. Continue reading

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“The Time Has Come”: Highlights of the 2014 AMS Committee on Education Meeting

By Benjamin Braun, Editor-in-Chief, University of Kentucky

The 2014 American Mathematical Society (AMS) Committee on Education (CoE) meeting took place on October 16-18 in Washington, D.C.  I attended as a member of the AMS CoE.  In addition to the committee members, there were many attendees from academic institutions, government, other professional societies, and the private sector.  Like the recent CBMS forum that Diana White discussed in a blog post earlier this month, the focus of the CoE meeting this year was the first two years of postsecondary mathematics education.  In this post, I will reflect on some of the key themes that stood out to me during the CoE meeting. Continue reading

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The First Two Years of College Mathematics: Reflections and Highlights from the CBMS Forum

By Diana White, Contributing Editor, University of Colorado Denver

In early October, approximately 150 educators and policy makers gathered together in Reston, Virginia for the fifth Conference Board of the Mathematical Sciences (CBMS) Forum entitled The First Two Years of College Mathematics: Building for Student Success. Participants came from almost every state in the country and represented higher education institutions ranging from two-year colleges to top-ranked research universities. We spent two days reflecting, learning, and in some cases planning how to improve the last year of high school mathematics and the first two years of college mathematics.

As is often my reaction at these types of conferences, I found the two days both sobering and energizing — sobering because of the sometimes harsh realities and challenges we face, energizing because of the good work participants report on and the many people gathered together who care so passionately and who dedicate so much of their time and energy to moving us forward. For those who could not join us in Virginia, this blog post will present a few key highlights from the Forum, in an effort to open a broader conversation about the future of the first two years of collegiate mathematics instruction. Continue reading

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Transformation of a Math Student’s Learning

By Morgan Mattingly, undergraduate double-major in STEM Education and Mathematics at the University of Kentucky.

Comment from the Editorial Board: We believe that in our discussion of teaching and learning, it is important to include the authentic voices of undergraduate students reflecting on their experiences with mathematics. This article is our first such contribution. We feel it provides a window into many of the subtle challenges students face as they transition to advanced postsecondary mathematics courses, and that it mirrors many of the themes discussed in previous posts. We thank Ms. Mattingly for being the first student to contribute an essay to our blog.

In previous math classes, I was the quiet worker who kept to herself and didn’t know when or how to ask questions. After improving my skills in a problem solving class, that has changed. The group work we did each day allowed me to be around other people who think significantly differently than I do. Being in this environment was difficult at first because I actually had to work through problems with other people, which was somewhat unfamiliar to me. My classmates and I were not just sitting down and reading information about specific math problems. We had to analyze and make sense of the best methods and strategies to use and present our ideas to each other. Confusion would set in when other students introduced different approaches. The only way I could understand their ways of thinking was to ask them to explain. Asking questions in math initially intimidated me, especially because my questions had to be directed to my peers. I did not want them to think that I could not keep up with the material or that I did not belong in the class. But I also did not want to misunderstand major mathematical concepts as a consequence of not asking questions. So I started asking my group members each week what strategies they used in their solutions. Although it may have seemed repetitive to them or obnoxious to have to explain their approaches, it helped me immensely. Through my question asking, I was able to talk and think about math in a unique way. I could compare my peers’ techniques to my own, which further stimulated my interest in the particular subjects that were covered in the class. This skill has been and will continue to be essential in my future relationship with mathematics. Continue reading

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Helping All Students Experience the Magic of Mathematics

By Oscar E. Fernandez, Assistant Professor in the Mathematics Department at Wellesley College.

Mathematics is a beautiful subject, and that’s something that every math teacher can agree on. But that’s exactly the problem. We math teachers can appreciate the subject’s beauty because we all have an interest in it, have adequate training in the subject, and have had positive experiences with it (at the very least, we understand a good chunk of it). The vast majority of students, on the other hand, often lack all of these characteristics (not that this is their fault). This explains why if I’d start talking to a student about how exciting the Poincare-Hopf theorem is, I probably wouldn’t see anywhere near the same excitement as if I were to, say, let them play with the new iPhone. This may seem like a silly hypothetical, but I believe it brings up all sorts of important points. For one, what does it say about our culture (and our future) when young people would rather be playing games on iPhones (or watching Youtube, or being on Facebook, etc.) than studying math or science? What causes our culture to be the way it is? How did companies like Apple and Facebook get students so interested in these activities? What are they doing that we math teachers aren’t? Continue reading

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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. Continue reading

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Teaching Mathematics Through Immersion

By Priscilla Bremser, Contributing Editor, Middlebury College.

Chapter 1 of Make It Stick:  The Science of Successful Learning [2] is called “Learning is Misunderstood.”  That is an understatement, as demonstrated by the remaining chapters.  The book has received several strong reviews ([3], [5], [8]), so rather than providing a critique, my aim here is to explore the ways in which its account of cognitive science research has validated some decisions I have made about my teaching and gotten me to reconsider others.

Since the early 1990’s, I have been using a form of what we now call Inquiry-Based Learning (IBL) in my Abstract Algebra course; more recently I’ve been doing so in Number Theory as well (using [6]).  This all started when Professor Bill Barker of Bowdoin College described an Algebra course built around small-group work, and I was hooked.  Surrounded here at Middlebury College by excellent immersion language programs, I realized that Bill was describing a mathematics immersion program.  I modeled my course on his so that my students would learn mathematics by speaking mathematics with each other, while I roamed the room as consultant.  That first post-conversion semester, there were numerous classes that went overtime before any of us noticed, so engaged were the students.   Continue reading

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