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

Posted in Assessment Practices, Classroom Practices | 2 Comments

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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The Role of Mathematics Departments in the Mathematical Preparation of Elementary Teachers

By Diana White, Contributing Editor, University of Colorado Denver

Mathematics departments have long provided the bulk of the mathematics content training for both practicing teachers and those studying to be teachers.  This is a tremendous responsibility, and one that presents a variety of challenges and opportunities.  In this post, we start early in the mathematical spectrum – with elementary teachers and how mathematics departments impact their mathematical preparation. Continue reading

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Do mathematicians need new journals about education?

By Priscilla Bremser, Contributing Editor, Middlebury College.

In the past nine months, I’ve heard colleagues at three different meetings—an AMS sectional meeting in Louisville, the Joint Mathematics Meetings in Baltimore, and the Contemporary Issues in Mathematics Education workshop at the Mathematical Sciences Research Institute—identify a need for journals focused on publishing useful refereed articles for mathematicians about mathematics education. This raises several questions that get at fundamental issues in the complicated and sometimes uneasy relationships among research mathematicians, mathematics education specialists, and those with interests in both areas (I put myself in the last category). Continue reading

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On being a friendly mathematician

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

Almost fifteen years later, Lucy Michal still remembers the exact words Phil Daro told the leaders of the El Paso Collaborative for Academic Excellence as they were preparing to launch the K-16 Mathematics Alignment Initiative, which Lucy would direct: “Find a friendly mathematician.”  The goal was to align mathematics in grades K-16, through regular meetings of a working group of a few dozen local teachers of all grade levels.  Phil had many contacts, including national authorities in K-12 mathematics, but, for a project like this, he stressed the need for local mathematics experts.  A “friendly mathematician” would be respected for mathematics, but would also understand the importance of working with both pre-service and in-service teachers.  I became one of those friendly mathematicians.  What did I do to live up to this billing? Continue reading

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Teaching practices between and beyond all lecture and all student discovery

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

This post is inspired by an article by Karen Marrongelle and Chris Rasmussen [1], in which they discuss the false dichotomy between all lecture and all student discovery as the two exclusive teaching strategies available for mathematics teachers.  I’ve noticed that many discussions among postsecondary mathematics teachers lead to a debate of the merits of these two classroom teaching strategies, with the result that interesting teaching practices are left undiscussed.  Below I describe three key teaching practices that I’ve learned about and used over the past several years that fit between and beyond these extremes.  I’ve observed that when I use these practices, students are generally more engaged in the course, e.g. attending office hours, asking questions in class, forming study groups, etc.  Though they appear simple, using these practices successfully has required perseverance and effort on my part, and a willingness to regularly revise their implementation. Continue reading

Posted in Assessment Practices, Classroom Practices | 2 Comments

Striking the Balance between Examples and Proof

By Elise Lockwood, Contributing Editor, Oregon State University.

As an undergraduate, it was easy for me to assume that as my professors conducted mathematical research, beautiful, complete proofs came to them in moments of epiphany. Their work was mysterious to me, and I believed that somehow their superior intelligence and vast mathematical knowledge gave them immediate access to all things abstract. Had I been asked then, I likely would have said that mathematicians didn’t need to think about examples in their own research – surely they had outgrown the need for concrete examples. Continue reading

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Transformation of a Math Professor’s Teaching

By Jerry Dwyer, professor in the Department of Mathematics & Statistics and Director of the Science, Technology, Engineering and Mathematics Center for Outreach, Research & Education (STEM-CORE) at Texas Tech University.

This essay describes the changes that have taken place in my teaching philosophy and practice over the past 30 years or so. I have always loved teaching and the satisfaction of explaining concepts to others. However, my understanding and love of the profession has greatly increased over the years, with some pivotal moments emphasizing when those changes occurred. I present these reflections as a possible encouragement to others who may wonder about their teaching and how to make it more satisfying and enjoyable. Continue reading

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Welcome to “On Teaching and Learning Mathematics”

By The Editorial Board.

There are major changes afoot in both K-12 and postsecondary mathematics education.  For example, the widespread adoption of the Common Core State Mathematics Standards [5] has been a focal point for everyone involved in K-12 education in the United States.  The 2012 report Engage to Excel [4] from the President’s Council of Advisors on Science and Technology (PCAST) included substantial recommendations for experimentation and change in mathematics education, including at the undergraduate level for preservice teacher training.  A large and growing body of research [6] in STEM disciplines (Science, Technology, Engineering, Mathematics) is demonstrating that undergraduate learning and achievement can be increased by implementing evidence-based teaching practices.  Funding agencies such as the National Science Foundation are responding to this seriously, for example through the Widening Implementation and Demonstration of Evidence Based Reforms (WIDER) program [7]. Continue reading

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