More Linear Algebra, Please

By Drew Armstrong, Associate Professor of Mathematics, University of Miami

Anyone who teaches mathematics in the US knows that the quality of education could be better, but we also know that the problems are complicated and defy easy solutions. I grew up in Ontario, Canada, where I attended high school and completed an undergraduate degree in mathematics. Afterwards I completed a Ph.D. in the United States and I have now been teaching undergraduate mathematics here for over ten years. These experiences suggest to me a change that would improve college mathematics education in the US. It won’t solve every problem, but it is something concrete that we can do right now.

Suggestion: Replace the typical one-semester “introduction to linear algebra” course with a two-semester linear algebra sequence. This would be taken in the first year of college, in parallel with calculus. It would not have calculus as a pre-requisite.

In effect, this would place linear algebra and calculus side-by-side as the twin pillars of undergraduate mathematics. I believe this would have several immediate benefits for the curriculum. In this blog post I’ll describe three of these benefits and then I’ll explain how my experience as a student in Canada and as a professor in the US has brought me to this position. Continue reading

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The Inefficiency of Teaching

By Gavin LaRose, University of Michigan

It could be the punchline of a joke that at any given college or university, at some point, the administration will lean on departments to be more “efficient” by teaching classes in larger sections, or online, or with some technology or another. By the metric of student credit hour to faculty work hour, of course, large lectures are tremendously efficient, and scale admirably. One may argue that there is little difference between an instructor lecturing to 100 or to 200 students, and little difference between an instructor rendered small by the distance to the front of a large lecture hall and one rendered small in the pixels of a video screen. This is the Massive Open Online Course (MOOC) model, which extends this efficiency of scale from 200 to 20,000. Anecdotally, the MOOC tide seems to be receding, but the pressures that argue for this efficiency are not going away. Many departments are being asked to teach, with fewer resources and greater accountability, more students whose mathematical preparation is weaker than in the past [2].

The difficulty here is that the student credit hour metric is easy to measure, while student learning is not. Research says that our efficient passive lecture does not result in the student learning gains we can see with more active teaching techniques [6,7,8]. Indeed, through the Conference Board of the Mathematical Sciences, the presidents of fifteen professional societies in the mathematical sciences have recognized this conclusion and endorsed the use of active teaching methods [4]. But neither the research nor the endorsement provide us with a simple, usable measure by which to demonstrate the effectiveness of these techniques. So our endeavor of teaching remains by easily applied metrics an inefficient one, and I increasingly think one that is inevitably inefficient by even more measures.

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Announcement Regarding White House Office of Science and Technology Policy Call to Action on Active Learning in STEM Education

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

This is an announcement to the mathematical community that the White House Office of Science and Technology Policy (OSTP) has issued a Call to Action for incorporating Active Learning in K-12 and higher education STEM courses. Their call includes a submission form where they ask: “What new (i.e., not yet public) activities or actions is your organization undertaking to respond to the Call to Action to improve STEM teaching and learning through the use of active learning strategies?” The deadline for submission of responses is Sept 23, 2016.  It would be excellent for the mathematics community to be well-represented among the submissions, so I encourage our readers to submit their activities and to share this information with others.

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An Inclusive Maths Conference: ECCO 2016

By Viviane Pons, maitre de conferences (Assistant Professor), Université Paris-Sud

Editor’s Note: An expanded version of this article previously appeared at

Being one of the few women in the men’s world of mathematics and computer science has led me to look around and spot our flaws when inclusivity is concerned. Let’s not fool ourselves: even though we think of us as being purely objective beyond bias, the maths world is not an inclusive paradise.  The academic world I personally live in is made of mostly white men, mostly from western countries (Europe, US, Canada, and just a little bit of Asia). If you look even closer, you’ll see that most of us come from well-off educated families. Except for the fact that I am a woman, I check all the other boxes myself and I am well-aware of it. Considering the multiple causes of this situation, what can be done? What can I do as a single individual in this world, when I’m busy fighting my own fights earning my right to stay around? Well, I’m not going to answer that just now, but I will share a very good experience I just had. I went to a CIMPA maths summer school in Colombia that was different: ECCO 2016. For the first time, I felt it was indeed inclusive in the best possible way. And, it was excellent maths too, so I was really happy.

First, a little bit of context.  As opposed to a classical conference where most presentations are short ones to announce new results, a summer school is usually made of mini-courses on a certain topic. At ECCO 2016, the main audience was made of students (masters students, PhD students, and undergrads) but some postdocs and even professors participated as well, as we are always keen on learning new things. It was in Colombia and the topic was combinatorics, which happens to be my field. ECCO runs every two years, and began in 2003 as a small event organized by Federico Ardila. He is a Colombian mathematician based in the US and we (the academic world) owe him thanks for many great researchers in combinatorics. I had noticed before that the number of Colombian people among researchers in combinatorics was astonishingly high, but before I met Federico I had no idea why. Most of this very active Colombian community is now organizing the conference. Over the years, ECCO has become quite a big event in combinatorics with a very positive, well-earned, reputation. This year, for the first time, it was a CIMPA school and there were over 100 participants.  So why was this conference so good?


Photo by Federico Ardila

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From the Editors: Changes for the Editorial Board

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

To start, I want to thank all of our readers, subscribers, and contributors — we appreciate your feedback and ideas through your writing, social media comments, and in-person conversations at mathematical meetings and events.  Since launching our blog in June of 2014, our articles have received over 189,000 unique page views!  We will continue to strive to provide high-quality articles on a broad range of topics related to post-secondary mathematics, and we welcome your feedback and suggestions.

I have a few changes to announce regarding the editorial board for On Teaching and Learning Mathematics.  Following two years of service as a founding Contributing Editor for our blog, Elise Lockwood is leaving our board to join the editorial board of the new International Journal on Research in Undergraduate Mathematics Education (IJRUME).  Many thanks to Elise for her excellent contributions that helped the blog have a great start over the past two years!  I know that we can look forward to hearing more from Elise in the future as a contributing author.

For 2016-2017, I am happy to welcome three new Contributing Editors to the board:

  • Luis David García Puente, Sam Houston State University
  • Jess Ellis, Colorado State University
  • Steven Klee, Seattle University

Luis, Jess, and Steve bring with them a wealth of expertise in teaching, research, and mentoring, and I am excited that they will be sharing their expertise with our readers.

For those of you who are regular readers, we will continue to publish articles roughly every two weeks, with a target goal of publishing 24 articles per year. Our next post is scheduled for August 22, 2016, where we will hear from Viviane Pons about her experience at a math summer school that was “inclusive in the best possible way.” Stay tuned!

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Does the Calculus Concept Inventory Really Measure Conceptual Understanding of Calculus?

By Spencer Bagley, University of Northern Colorado; Jim Gleason, University of Alabama; Lisa Rice, Arkansas State University; Matt Thomas, Ithaca College, Diana White, Contributing Editor, University of Colorado Denver

(Note: Authors are listed alphabetically; all authors contributed equally to the preparation of this blog entry.)

Concept inventories have emerged over the past two decades as one way to measure conceptual understanding in STEM disciplines, with the Calculus Concept Inventory (CCI), developed by Epstein and colleagues (Epstein, 2007, 2013), being one of the primary instruments developed in the area of differential calculus.  The CCI is a criterion-referenced instrument, measuring classroom normalized gains, which specifically is the change in the class average divided by the possible change in the class average.  Its goal was to evaluate the impact of teaching techniques on conceptual learning of differential calculus.  

While the CCI represents a good start toward measuring calculus understanding, recent studies point out some significant issues with the instrument.  This is concerning, given that there seems to be an increased use of the instrument in formal and informal studies and assessment.  For example, in a recent special issue of PRIMUS (Maxson & Szaniszlo, 2015a, 2015b) related to flipped classrooms in mathematics, three of the five papers dealing with calculus cited and used the CCI.  In this blog we provide an overview of concept inventories, discuss the CCI, outline some problems we found, and suggest future needs for high-quality conceptual measures of calculus understanding.

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The Second Year of “On Teaching and Learning Mathematics”

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

Another year has flown by, and so it is once again a good time to collect and reflect on all the articles we have been able to share with you since our last annual review.  I enjoyed the chance to re-read all the articles, and I was also surprised at the interesting variety of themes that emerged when I sorted them out.  It was not easy to put each article in a unique box, and I will point out the blurring between categories.  I hope you enjoy the chance to revisit these articles, and perhaps find new meaning from the juxtapositions here.

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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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