Learning Mathematics in Context with Modeling and Technology

Dr. Brian Winkel, Professor Emeritus, Mathematical Sciences, United States Military Academy, West Point NY USA and Director of SIMIODE.

I cannot accept that mathematics be taught in a vacuum. Yes, mathematics is beautiful, be it pure or applied. However, in our age of immediacy for students we need to move more of our efforts to teaching mathematics in context, in touch with the real world. We should incorporate more modeling and applications in our mathematics courses to richly support and motivate our students in their attempts to learn mathematics and we should support colleagues who seek to use this approach.

Over the course of time I have moved to this position. At first I used applications of mathematics in course lectures, e.g., error correcting codes in algebra, cryptology in number theory, life sciences in calculus, and engineering in differential equations. Then I assigned students to read articles in other disciplines and share these applications in class. Finally, I incorporated projects in which students could see and practice the application of mathematics.  Introducing a modeling scenario makes the mathematics immediate; what do I do right now?  Students desire to address the problem at hand, which is real to them, primarily because it intrigues them and piques their curiosity. Thus the mathematics becomes a necessary tool they are ready to learn. I eventually used the application to motivate the learning of the mathematics before introducing that mathematics.  This is a “flipping” of content.  

Some students are a bit shy, even resistant, to this approach. However, in an active and supportive learning environment in which students work in small groups and the teacher works the room by watching, visiting, listening, and assisting the groups, students do amazing things. Sometimes they get off a workable track, but colleagues and teachers bring them along. Students make mistakes, but as we know, learning from mistakes is an important part of learning [BrownEtAl2014]. Indeed, we do it all the time ourselves and call it conjecture and research. Continue reading

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Imagining Equity

By Priscilla Bremser, Contributing Editor, Middlebury College

In my Mathematics for Teachers course, students take a fresh look at foundational concepts, such as fractions and place value, from an advanced perspective.  For some of them, our work together exposes weaknesses in their backgrounds, and unsettling stories emerge regularly, but  B.’s story stands out.  B. was a senior Japanese Studies major who offered insightful observations during problem-solving sessions.  As the semester progressed, it became clear that there was a gap in his mathematical knowledge.  He explained that he moved to the U.S. speaking only Spanish, and missed out on the mathematics being taught while he was learning English. He soon moved to a different city, and never learned how to add fractions.  A significant chunk of the college curriculum was inaccessible to him because his middle school had no mechanism for accommodating his language transition.  B. has many strengths, and he will do well in the world, but he was shortchanged at a critical phase in his mathematics education.

We have all had students who arrive at college unprepared to do college-level mathematics.  While there are many contributing factors at play, it’s clear that inequities in pre-K-12 education systems play an important role.  It’s also clear that it is extremely difficult, if not impossible, to make up in four years for disparities experienced over fifteen years. Although we work in higher education, nevertheless we must advocate for greater equity in pre-college education.  If we don’t, we’re simply perpetuating injustice.

That injustice is reflected in persistent and significant differences in educational attainment among demographic groups in the United States.  It’s not just that students from some groups are less prepared for college. Those college students have too many peers who don’t have access to college at all, for reasons that are well beyond their, or their families’, control. Continue reading

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Thick Derivatives

by Tevian Dray, Professor, Department of Mathematics, Oregon State University

One of the iconic messages of the calculus reforms that took place in the 1990s is the “Rule of Four,” emphasizing the use of multiple representations: algebraic, geometric, numeric, and verbal. But what is a numerical representation of the derivative?

In a recent study [1], we asked faculty in mathematics, physics, and engineering to determine a derivative based on experimental data they had to collect themselves, using the apparatus shown in Figure 1. The physicists and engineers had no trouble doing so—but the mathematicians refused to acknowledge a computed average rate of change, however accurate, as a derivative. The physicists and engineers knew full well that their computation was an approximation, but they also knew how to ensure that it was a good one.

PDM Figure 1: The Partial Derivatives Machine, designed by David Roundy at Oregon State University. In this mechanical analog of a thermodynamic system, the variables are the two string positions (the flags) and the tensions in the strings (the weights). However, it is not obvious which variables are independent, nor even how many independent variables there are. For further details, see [1].

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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:

psyche Continue reading

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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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