Believing in Mathematics

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

In my experience, many students in K-12 and post-secondary mathematics courses believe that:

  • all math problems have known answers,
  • failure and misunderstanding are absent from successful mathematics,
  • their instructor can always find answers to problems, and
  • regardless of what instructors say, students will be judged and/or assessed based on whether or not they can obtain correct answers to problems they are given.

As long as students believe in this mythology, it is hard to motivate them to develop quality mathematical practices. In an effort to undercut these misunderstandings and unproductive beliefs about the nature of mathematics, over the past several years I’ve experimented with assignments and activities that purposefully range across the intellectual, behavioral, and emotional psychological domains. In this article, I provide a toolbox of activities for faculty interested in incorporating these or similar interventions in their courses.

Psychological Domains

A useful oversimplification frames the human psyche as a three-stranded model:

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A Framework for Integrating Conceptual and Procedural Understanding in the First Two Years of Undergraduate Mathematics

By Karen Keene and Nicholas Fortune, North Carolina State University

One common instructional approach during the first two years of undergraduate mathematics in courses such as calculus or differential equations is to teach primarily analytic techniques (procedures) to solve problems and find solutions. In differential equations, for example, this is true whether the course is strictly analytical or focuses on both analytic techniques and qualitative methods for analysis of solutions.

While these analytic techniques play a major part of the early undergraduate mathematics curriculum, there is significant discussion and research about the importance of learning the concepts of mathematics. Many researchers in mathematics education encourage teaching mathematics where students learn the concepts before the procedures and are guided through the process of reinventing traditional procedures themselves (e.g., Heibert, 2013). Additionally, educators who have developed mathematical learning theories often set up a dichotomy between the two kinds of learning (e.g., Skemp, 1975; Haapasalo & Kadijevich, 2000). At the collegiate level, we as professors may agree that these educational ideas hold merit, but also firmly believe that students have a significant amount of content to learn and may not always be able to spend the time necessary to allow students to participate fully in the development of conceptual understanding and the reinvention of the mathematics (including procedures).

However, some researchers, including ourselves, provide evidence that “teaching the procedures to solve problems and find solutions” and “providing ways for teaching concepts first so students will truly understand” can be integrated, and that the notion of learning procedures does not need to be shallow and merely a memorized list (Star, 2005; Hassenbrank & Hodgson, 2007). Our framework to merge these two ways of teaching is titled the Framework for Relational Understanding of Procedures. It was developed as part of Rasmussen and colleagues’ work in differential equations teaching and learning (Rasmussen et. al., 2006). Skemp coined the original definition; she defines relational understanding as “knowing both what to do and why” and contrasts it to instrumental understanding as “rules without reason” (1976, p. 21).

Following, we describe the six components of the Framework for Relational Understanding of Procedures. The idea is that each category can be used to consider and enhance students’ learning as they study a procedure.  For each one, we provide a brief explanation, questions about student thinking, and an example of an exam question related to each component taken from our work in differential equations.  Likely, each instructor could add other algorithms in differential equations as well as other courses.

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Creating a Classroom Culture

By Taylor Martin and Ken Smith, Sam Houston State University

A good educator must facilitate learning for a classroom full of students with different attitudes, personalities, and backgrounds. But how? This question was the starting point for a new Faculty Teaching Seminar in the math and statistics department at Sam Houston State University. In the conversation that transpired, we looked to identify the most important components of creating a class culture that best enables us to achieve learning outcomes. What are our goals? How do we get the ball rolling each semester? How do we get our students on board? Read on to find out…

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Recent Reports and Recommendations Related to Courses in the First Two Years of College Study

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

While one important component of successful teaching and learning is what happens inside the classroom, an equally important component involves decisions made at the administrative level that impact our classroom environments. A challenge that mathematics departments face is to make successful arguments for resources that support high-quality programs and courses for our students. Such arguments are often bolstered when the activities of a department are placed within the context of recommendations from professional societies.  

In this article we survey a selection of recent reports and recommendations related to courses in the first two years of college study, with the goal of providing an overview of these reports for faculty and department leaders. It is worth noting that most of these were created with grant support from the National Science Foundation (NSF). There are at least seventeen professional societies involved in mathematics education efforts, of which six are represented in these reports: American Mathematical Society (AMS), Mathematical Association of America (MAA), American Statistical Association (ASA), Society for Industrial and Applied Mathematics (SIAM), American Mathematical Association of Two-Year Colleges (AMATYC), and National Council of Teachers of Mathematics (NCTM). Continue reading

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A Skeptic’s Guide to Service Learning in Mathematics

By Priscilla Bremser, Contributing Editor

Many college and university students do volunteer work in local communities, and can learn valuable lessons in the process. The term “service learning” refers more specifically to service activities that are integral parts of academic courses. It can sometimes be difficult for mathematicians to envision how such projects could be included in their courses, especially courses focused on “pure” topics; for example, I have difficulty imagining how one would include such activities in Abstract Algebra. I have found myself, however, teaching courses in which service learning made sense, and I’ve implemented some service-learning projects with varying outcomes. Below I share some lessons I’ve learned in the process. Continue reading

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Wanted, Mathematicians for an Important but Difficult Task

By Art Duval, Contributing Editor, University of Texas at El Paso; Kristin Umland, Associate Professor, Department of Mathematics and Statistics, University of New Mexico (on leave), and Vice President for Content Development, Illustrative Mathematics; James J. Madden, The Patricia Hewlett Bodin Distinguished Professor, Department of Mathematics, Louisiana State University; and Dick Stanley, Professional Development Program, University of California at Berkeley

At the 2016 Joint Mathematics Meetings in Seattle this past January, an unusual mix of mathematicians and mathematics educators gathered for an AMS special session on Essential Mathematical Structures and Practices in K-12 Mathematics. This was the fourth consecutive special session at JMM organized by Bill McCallum and other folks at Illustrative Mathematics that focused on work in mathematics of mutual concern to mathematicians, mathematics educators, and K-12 teachers. The theme this year was inspired by a conversation between Dick Stanley and Kristin Umland about ratios and proportional relationships, and the talks were selected and ordered to highlight the development of mathematical ideas that are both upstream and downstream of this terrain.

Academic mathematicians are able to describe mathematical ideas in an efficient way. Across specialties, they share tools of language and habits of communication that have been shaped in order to facilitate the exchange of abstract knowledge. One purpose of the special session was to apply this cultural skill to selected topics in K-12 mathematics. The participants sought to create clearly expressed and easily understood descriptions of topics that are rarely developed clearly in the K-12 curriculum, such as measurement, number systems, proportional relationships, and linear and exponential functions. Although many people have been working in this area in recent years, much more needs to be done.

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Preparing the Next Generation of Students in the Mathematical Sciences to “Think with Data”

By Johanna Hardin, Pomona College, and Nicholas J. Horton, Amherst College

As statisticians in mathematics departments, we have both spent many department meetings, departmental reviews, and water-cooler conversations discussing the merits of various different curricular decisions with respect to the calculus sequence (“Why not take linear algebra before calculus III??”), upper division electives (“But those classes are needed for graduate school!”), and number and order of courses required for the mathematics major/minor. Recently, more of those discussions have related to critical components of the statistics curriculum, and how courses from mathematics ensure that statistics students have a solid quantitative foundation. These kinds of conversations reinforce the fact that there are strong connections between mathematics and statistics, and these connections can and do affect decisions about undergraduate curricula.

More generally, this is an exciting time to be in a quantitative field. The amount of data available is staggering and there is no end to the need for models that harness the deluge of information presented to us every day. Mathematicians, Statisticians, Data Scientists, and Computer Scientists will all play substantial roles in moving quantitative ideas forward in a new data driven age.   To be clear, there are challenges as well as opportunities in what lies ahead, and how we move forward – particularly with respect to training the next generation of mathematical, statistical, and computational scientists – requires deep and careful thought.

The goal of this blog post is to share some of the recent pedagogical ideas in statistics with our mathematician colleagues with whom we – as statisticians – are intimately engaged in building curricula. We hope that the description of the recent developments will open up larger conversations about modernizing both statistics and mathematics curricula.   Many of the ideas below on engaging students in and out of the classroom, connecting courses in sequence or in parallel, and assessing new programs are relevant to all of us as we work to better our own classrooms.

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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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