Learning Mathematics through Embodied Activities

By Hortensia Soto-Johnson, Professor, School of Mathematical Sciences, University of Northern Colorado

Those of us who teach mathematics know that students struggle writing the symbolism of mathematics even through they can articulate some of the concepts behind the symbolism. Those of us who interact with children know that they struggle articulating their thoughts even though they can convey their thoughts through gesture. For example, children point to indicate what they want and touch items or use their fingers as they learn to count. It is through such bodily action that children learn to recognize three objects as the quantity three without simultaneously touching and counting one, two, three. Athletes and musicians also apply bodily actions to master their sport or instrument respectively. For example, how many times have you have seen a basketball player shoot an imaginary ball into an imaginary hoop? Consider how a piano teacher places a student’s hand on top of the teacher’s hand as the teacher plays the piano. These are just a few ways in which we use our body to learn, so why not use it purposefully to promote the learning of mathematics? Continue reading

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What’s in Your Syllabus?

by Priscilla Bremser, Contributing Editor

I had what seemed the perfect first full-time teaching position, in that much of the planning for Calculus had already been done when I arrived.  The department chair handed me the textbook and the syllabus, essentially a day-by-day schedule of book sections and homework assignments.  This being the United States Naval Academy at Annapolis, where every student takes Calculus, a lot of wisdom had gone into the schedule.  I now look back at that syllabus with a mixture of gratitude for the jump start and recognition that much has changed.  What’s in your syllabus?  What does your institution require, and what is most important to you?  What is decidedly not in your syllabus? Do you hand out a paper copy on the first day, or is it all online?  How well does the syllabus reflect what you want your course to be? Continue reading

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From the Editors: Blog Update and 2016 Joint Meetings Highlights

by the Editorial Board

We want to begin this post with thanks to all of our readers and contributors — we appreciate your feedback and ideas through your writing, social media comments, and in-person conversations at mathematical meetings and events. In-person conversations have been on the minds of the editors recently because we had our first-ever in-person meeting as an editorial board at the 2016 Joint Meetings in Seattle.  This was great fun and gave us a chance to seriously reflect on our blog, its role in the mathematical community, and what we want to do over the next year or two.  In this post, we give a brief update about a change to the structure of our blog, followed by some highlights of our experiences attending the joint meetings. Continue reading

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Social Media as a Teaching Resource

By Drew Lewis, Assistant Professor, Department of Mathematics, The University of Alabama

Like many mathematicians, the only formal training I have received as a teacher was in graduate school.  After a one semester seminar on teaching, I was set loose on three recitation sections of unsuspecting calculus students and expected to improve my teaching primarily by trial and error, discussion with peers and mentors, and feedback from students and classroom observations.  While I still use all of these to improve my teaching, I have found that social media has become an indispensable tool to helping me improve as a teacher.  I use “social media” in a broad sense here — I would include any quasi-public interactive online discussion in my definition.  This includes platforms like Facebook and Twitter that most people associate with the term “social media”, but also things like a discussion in the comments section of a blog, or a discussion board-based online community.  Further, the key value of social media is not in the availability of information, but the interactions and discussions that are generated.  In conjunction with trial and error, I have learned more about teaching through social media than I have through any other method. Continue reading

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Don’t Count Them Out – Helping Students Successfully Solve Combinatorial Tasks

By Elise Lockwood, Contributing Editor, Oregon State University

Solving counting problems is one of my favorite things to do. I love the challenge of making sense of the problem, the work of correctly modeling what I am trying to count, and the fact that I get to reason about astonishingly large numbers. I did not always feel this way about solving counting problems, though. For much of my mathematical career, counting was a mystery – a jumble of poorly understood formulas and equations that just made me miserable. As an undergraduate, I struggled to grasp the difference between order mattering or not mattering, what the respective factorials represented in confusing-looking formulas, and why I should care about how many full houses could be chosen from a deck of cards. My teachers at the time may have shared the sentiment nicely captured by Annin and Lai: “Mathematics teachers are often asked, ‘What is the most difficult topic to teach?’ Our answer is teaching students to count” (2010, p. 403).

At some point during graduate school (thanks to an influential professor who loved counting), I turned the corner and became more interested in understanding counting. Through lots of practice, I began to improve in my ability to solve counting problems. Since that time I have committed my research interests to learning everything I can about undergraduate students’ counting – what they do when they approach counting problems, why they struggle, and how we might help them solve such problems more effectively. Continue reading

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Connections between Abstract Algebra and High School Algebra: A Few Connections Worth Exploring

by Erin Baldinger, University of Minnesota; Shawn Broderick, Keene State College; Eileen Murray, Montclair State University; Nick Wasserman, Columbia University; and Diana White, Contributing Editor, University of Colorado Denver.

Mathematicians often consider knowledge of how algebraic structure informs the nature of solving equations, simplifying expressions, and multiplying polynomials as crucial knowledge for a teacher to possess, and thus expect that all high school teachers have taken an introductory course in abstract algebra as part of a bachelor’s degree. This is far from reality, however, as many high school teachers do not have a degree in mathematics (or even mathematics education) and have pursued alternative pathways to meet content requirements of certification. Moreover, the mathematics education community knows that more mathematics preparation does not necessarily improve instruction (Darling-Hammond, 2000; Monk, 1994). In fact, some research has shown that more mathematics preparation may hinder a person’s ability to predict student difficulties with mathematics (Nathan & Petrosino, 2003; Nathan & Koedinger, 2000). Nevertheless, the requirements for traditional certification to teach secondary mathematics across the country continue to include an undergraduate major in the subject, and many mathematicians and mathematics educators still regard such advanced mathematics knowledge as potentially important for teachers.

Given this, it is important that, as a field, we investigate the nature of the present mathematics content courses offered (and required) of prospective secondary mathematics teachers to gain a better understanding of which concepts and topics positively impact teachers’ instructional practice. That is, we need to explore links not just between abstract algebra and the content of secondary mathematics, but also to the teaching of that content (e.g., see Wasserman, 2015). In November 2015, a group of mathematicians and mathematics educators met as a working group around this topic at the annual meeting of the North American Chapter of the Psychology of Mathematics Education. We began to probe the impact understanding connections such as those described above might have on teachers’ instructional choices.  For example, how does understanding the group axioms shift teacher instruction around solving equations? How does understanding integral domains shift teacher instruction around factoring?  Through answering questions such as these, mathematicians and mathematics educators can better support teachers to connect advanced mathematical understanding to school mathematics in meaningful ways that enhance the quality of instruction.

In the remainder of this blog post, we explain and discuss three frequently cited examples of connections between abstract algebra and high school mathematics.

Continue reading

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Shredding My (Calculus) Confidence

By A.K. Whitney, journalist.  In 2009, Whitney went back to school to find out, once and for all, if journalists really are as bad at math as they fear they are; her blog about the experience, Mathochism, runs on Medium three days a week.

This fall, the Mathematical Association of America released a five-year study on college calculus that showed that, no matter how elite their learning institution may be, far too many students lose confidence in their math abilities after Calculus 1. As someone who recently spent a lot of time in calculus classrooms, I understand how that can happen.

Between 2012 and 2013, I enrolled in four different Calculus I courses. This may seem excessive even to the math-loving crowd reading this blog, but let me explain. Of the four, I dropped two, failed one and passed one. Of the four, two were in a community college classroom (the dropped and the failed), while two were Massive Open Online Courses, or MOOCs (one dropped, one passed, the latter with an 89.3 percent).

To be honest, I never set out to take this many calculus courses. Ideally, it would have been one and done. Some quick context: I am a print journalist with 20 years of experience in print and online. While always interested in science, I gave up on math at age 12. I spent the next 26 years as an avowed word person and math phobe, until leaving my full-time newsroom job to go freelance. Suddenly having so much time to think (the freelance career took a while to get going) made me question my youthful decision, and since I was already taking a computer class, I gave a remedial pre-algebra class a try. This turned into the Mathochism Project, where I was determined to revisit high school math as an adult, and write a blog about the joys and terrors of the experience.

To my surprise, there were mostly joys. From pre-algebra to pre-calculus, I did very well, and became delighted not just with math as a subject but also with my ability to understand it, getting mostly As and high Bs. I finished pre-calculus with a high B, and a strong level of confidence.  Then the terror began, though I didn’t realize it at first. Continue reading

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A Beginner’s Guide to Standards Based Grading

By Kate Owens, Instructor, Department of Mathematics, College of Charleston

In the past, I was frustrated with grades. Usually they told me very little about what a student did or didn’t know. Also, my students didn’t always know what topics they understood and on what topics they needed more work. Aside from wanting to do well on a cumulative final exam, students had very little incentive to look back on older topics. Through many conversations on Twitter, I learned about Standards Based Grading (SBG) and I implemented an SBG system in several consecutive semesters of Calculus II.

The goal of SBG is to shift the focus of grades from a weighted average of scores earned on various assignments to a measure of mastery of individual learning targets related to the content of the course. Instead of informing a student of their grade on a particular assignment, a standards-based grade aims to reflect that student’s level of understanding of key concepts or standards. Additionally, students are invited to improve their course standing by demonstrating growth in their skills or understanding as they see fit. In this article I will explain the way I implemented SBG and describe some benefits and some drawbacks of this method of assessment. Continue reading

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What is Early Math and Why Should We Care?

By Jennifer S. McCray, Assistant Research Scientist and Director of the Early Math Collaborative at Erikson Institute

Effective early childhood math teaching is much more challenging than most people anticipate.  Because the math is foundational, many people assume it takes little understanding to teach it, and unfortunately this is distinctly not the case.  In fact, the most foundational math ideas — about what quantity is, about how hierarchical inclusion makes our number system work, about the things that all different shapes and sizes of triangles have in common — are highly abstract ones.  While we should not expect or encourage young children to formally recite these ideas, they are perfectly capable of grappling with them.  Further, they need to do so to develop the kind of robust understanding that will not crumble under the necessary memorization of number words and symbols that is to come in kindergarten.  In preschool, before there is really any opportunity for “procedural” math, it is important that we give children ample opportunity to think about math conceptually.  In this essay I will discuss several profound ideas from early childhood mathematics, including examples of effective early math classrooms.  Along the way I will share some of the resources that my colleagues and I have developed to help early childhood educators develop as skillful teachers of early mathematics.

Erikson.ChristopherHouse.178.scaled Continue reading

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Active Learning in Mathematics, Part VI: Mathematicians’ Training as Teachers

By Benjamin Braun, Editor-in-Chief, University of Kentucky; Priscilla Bremser, Contributing Editor, Middlebury College; Art Duval, Contributing Editor, University of Texas at El Paso; Elise Lockwood, Contributing Editor, Oregon State University; and Diana White, Contributing Editor, University of Colorado Denver.

Editor’s note: This is the sixth and final article in a series devoted to active learning in mathematics courses. The other articles in the series can be found here.

How are mathematicians trained as teachers, what are the effects of this training, and what can we do to improve the quality of this training? We feel these questions are particularly important at this time, as a clamor of recent calls for the dramatic improvement of postsecondary education, made from both inside and outside of the mathematical community, has not abated. From the outside, we hear this call in venues ranging from opinion pieces in major newspapers [1,2,3] to federal advisory reports to the President of the United States [4] and beyond. The message has also been clearly conveyed by leadership from professional societies in the mathematical community: in early 2014, an article titled “Meeting the Challenges of Improved Post-Secondary Education in the Mathematical Sciences” was published in the AMS Notices, MAA Focus, SIAM News, and AMSTAT News through a coordinated effort by the professional societies — we urge any readers who have not already done so to read this statement.

Yet in order to be effective and achieve meaningful change, any actions taken by our professional societies and other leadership in the mathematics community must get buy-in from individual mathematicians who are in the classroom daily, working face-to-face with students. From our training in both mathematics and the teaching of mathematics, we each carry disciplinary habits, ways of thinking, biases, and strengths, many of which occur subconsciously as part of our mathematical culture, and all of which impact our teaching. In order to improve mathematical teaching and learning on a large scale, we must all work to better understand how mathematicians grow and develop as teachers, so that we may more thoughtfully respond to the educational challenges of our time. In this article, we focus our discussion on the topic of pedagogical training and development for graduate students and early-career faculty, with a view toward active learning. Continue reading

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