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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A Call for More Context

By Art Duval, Contributing Editor, University of Texas at El Paso

One of the highlights of my summer was attending a research conference, Stanley@70, celebrating the 70th birthday of my Ph.D. advisor Richard Stanley. Because it was a birthday conference, many of the speakers went out of their way to say a little something about Richard Stanley, with mathematical or personal anecdotes. One talk in particular, by Lou Billera, did an especially good job giving the history and context of the study of face numbers of simplicial polytopes, in which Richard played an essential role. (The slides don’t totally convey the breadth of the talk, but at least give you some idea of the mathematical story he was telling.) I really appreciated Lou’s talk, and I know (from asking them) that other participants did too. This got me thinking that the mathematical community could do more of this sort of thing, not just at conferences, but more importantly in courses for our undergraduate majors and graduate students. In these courses, we rightfully focus on the truth of mathematical results. Let’s also spend some time sharing with our students why we care about the mathematical objects and ideas that show up.

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Why is Slope Hard to Teach?

By Sybilla Beckmann, Josiah Meigs Distinguished Teaching Professor in the Department of Mathematics at the University of Georgia, and Andrew Izsák, Professor of Mathematics Education in the Department of Mathematics and Science Education at the University of Georgia.

One of the challenges of teaching mathematics is understanding and appreciating students’ struggles with material that to the instructor, after years of thinking about it, may seem straight forward. Once we understand an idea, it may seem almost impossible not to understand if it is presented clearly enough. Yet experienced math teachers know that presenting mathematical ideas clearly, as important as that is, is generally not enough for students to learn the ideas well, even for dedicated and determined students. At the same time, students who struggle can have insightful and productive ways of solving problems and reasoning about mathematical ideas. Research into how people think about and learn mathematics reveals why this surprising mix of struggle and competence can coexist: learners can use what they do understand to make sense of new things, yet ideas that are tightly interconnected and readily available for an expert may be fragmented or inchoate for a learner.

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Call for Nominations: American Mathematical Society Awards

From The Editorial Board.

We thought our readers might be interested to know that nominations are now open for several American Mathematical Society awards related to teaching and learning.  The deadline for nominations for the following awards is September 15, 2014.

-  Award for Impact on the Teaching and Learning of Mathematics.
-  Award for an Exemplary Program or Achievement in a Mathematics Department.
-  Mathematics Programs that Make a Difference.

More information about these awards and the nomination process can be found here: http://www.ams.org/profession/prizes-awards/prizes

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Assessment in Postsecondary Mathematics Courses

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

Our understanding of the importance of processes and practices in student achievement has grown dramatically in recent years, both in mathematics education and education more broadly.  As a result, at the K-12 level explicit practice standards are given in the Common Core Mathematics Standards [1] and the Next Generation Science Standards [2] alongside content standards.  At the postsecondary level, studies regarding student learning and achievement have revealed the importance of many key practices, and accessible sources exist on this topic [3, 4, 5].  Further, we understand now that not all advanced postsecondary mathematics students are well-served by the same curriculum; for example, pre-service high school mathematics teachers need to develop unique ways of practicing mathematics compared to math majors with other emphases [6, 7].  As discussed by Elise Lockwood and Eric Weber in the previous post on this blog, mathematicians generally appreciate these issues; for readers unfamiliar with mathematical practice standards, their article is a nice introduction to this topic.

All of this leads us to the following question:

Given the breadth of both content and practices required for students to deeply learn and understand mathematics, what are effective techniques we can use at the postsecondary level to gauge student learning? Continue reading

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Some Thoughts on the Teaching and Learning of Mathematical Practices

By Elise Lockwood, Contributing Editor, Oregon State University and Eric Weber, assistant professor of mathematics education in the College of Education, Oregon State University.

As students’ mathematical thinking develops, and they encounter more advanced mathematical topics, they are often expected to “behave like mathematicians” and engage in a number of mathematical practices, ranging from modeling and conjecturing to justifying and generalizing. These mathematical practices are distinct from specific content students might learn because they are characteristics of broader behavior, rather than mastery of a single concept or idea. However, these practices represent indispensable components of what it takes to become a successful mathematician. Continue reading

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Mathematics Instruction, an Enthusiastic Activity

By William Yslas Vélez, Professor in the Department of Mathematics at the University of Arizona.

The best recruiting tool I have to convince students that they should continue in the study of mathematics is the mathematics that I am teaching, no matter the level. It is all fascinating. In almost every lower division course that I have taught I have convinced at least one student to add the mathematics major. The last time I taught second semester calculus, three students added the math major and one the math minor (and the student selecting the math minor simply could not fit in the last three mathematics courses for the major). One of those students is now a graduate student in biostatistics at Harvard. Continue reading

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