Active Learning in Mathematics, Part III: Teaching Techniques and Environments

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 third article in a series devoted to active learning in mathematics courses.  The second article in the series is here.

It is common in the mathematical community for the phrases “active learning” and “inquiry-based learning” (IBL) to be associated with a particular teaching technique that emphasizes having students independently work and present to their peers in a classroom environment with little-to-no lecturing done on the part of the instructor.  Yet it is counterproductive for this method to be a dominant cultural interpretation of “active learning,” as it does not represent the range of teaching styles and techniques that fall along the active learning and IBL spectrums as considered by mathematicians who use these pedagogies, mathematics education researchers, federal and private funding agencies, and professional societies such as the AMS, MAA, SIAM, ASA, AMATYC, and NCTM.  In this article we will provide multiple examples of active learning techniques and environments that arise at institutions with different needs and constraints.  We begin by reflecting on general qualities of classroom environments that support student learning.  Continue reading

Posted in Assessment Practices, Classroom Practices | Tagged , , , , , , | 2 Comments

Active Learning in Mathematics, Part II: Levels of Cognitive Demand

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 second article in a series devoted to active learning in mathematics courses.  The first article in the series is here.

Mathematics faculty are well-aware that students face challenges when encountering difficult problems, and it is common to hear instructors remark that successful students have high levels of “mathematical maturity,” or are particularly “creative,” or write “elegant” solutions to problems.  To appreciate research results regarding active learning, it is useful to make these ideas more precise.  Motivated by research in education, psychology, and sociology, language has been developed that can help mathematicians clarify what we mean when we talk about difficulty levels of problems, and the types of difficulty levels problems can have. This expanded vocabulary is in large part motivated by…

…the “cognitive revolution” [of the 1970’s and 1980’s]… [which] produced a significant reconceptualization of what it means to understand subject matter in different domains. There was a fundamental shift from an exclusive emphasis on knowledge — what does the student know? — to a focus on what students know and can do with their knowledge. The idea was not that knowledge is unimportant. Clearly, the more one knows, the greater the potential for that knowledge to be used. Rather, the idea was that having the knowledge was not enough; being able to use it in the appropriate circumstances is an essential component of proficiency.

— Alan Schoenfeld, Assessing Mathematical Proficiency [17]

In this article, we will explore the concept and language of “level of cognitive demand” for tasks that students encounter.  A primary motivation for our discussion is the important observation in the 2014 Proceedings of the National Academy of Science (PNAS) article “Active learning increases student performance in science, engineering, and mathematics” by Freeman, et al. [8], that active learning has a greater impact on student performance on concept inventories than on instructor-written examinations.  Concept inventories are “tests of the most basic conceptual comprehension of foundations of a subject and not of computation skill” and are “quite different from final exams and make no pretense of testing everything in a course” [5].  The Calculus Concept Inventory is the most well-known inventory in mathematics, though compared to disciplines such as physics these inventories are less robust since they are in relatively early stages of development.  Freeman et al. state:

Although student achievement was higher under active learning for both [instructor-written course examinations and concept inventories], we hypothesize that the difference in gains for examinations versus concept inventories may be due to the two types of assessments testing qualitatively different cognitive skills.  This is consistent with previous research indicating that active learning has a greater impact on student mastery of higher- versus lower-level cognitive skills…

After introducing levels of cognitive demand in this article, our next article in this series will directly connect this topic to active learning techniques that are frequently used and promoted for postsecondary mathematics courses.

Continue reading

Posted in Classroom Practices | Tagged , , | Leave a comment

Active Learning in Mathematics, Part I: The Challenge of Defining Active Learning

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 first article in a series devoted to active learning in mathematics courses.

“…if the experiments analyzed here had been conducted as randomized controlled trials of medical interventions, they may have been stopped for benefit.”

So strong is the evidence supporting the positive effects of active learning techniques in postsecondary mathematics and science courses that Freeman,, made the statement above in their 2014 Proceedings of the National Academy of Science (PNAS) article Active learning increases student performance in science, engineering, and mathematics.  Yet faculty adoption of active learning strategies has become a bottleneck in post-secondary mathematics teaching advancement.  Inspired by the aforementioned PNAS article, a landmark meta-analysis of 225 studies regarding the positive effects of active learning, we will devote a series of posts to the topic of active learning in mathematics courses.  

An immediate challenge that arises when discussing active learning in mathematics is that the phrase “active learning” is not well-defined.  Interpretations by mathematics faculty of this phrase range broadly, from completely unstructured small group work to the occasional use of student response systems (e.g., clicker) in large lectures.  In this article we discuss several descriptions from the literature, including what we will take as our working understanding throughout this series of posts, discuss important considerations in the adaptation of such methods, and highlight some important aspects of the PNAS article. Continue reading

Posted in Classroom Practices, Education Policy, Online Education | Tagged , , | Leave a comment

The Secret Question (Are We Actually Good at Math?)

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

“How many of you feel, deep down in your most private thoughts, that you aren’t actually any good at math? That by some miracle, you’ve managed to fake your way to this point, but you’re always at least a little worried that your secret will be revealed? That you’ll be found out?”

Over half of my students’ hands went into the air in response to my question, some shooting up decisively from eagerness, others hesitantly, gingerly, eyes glancing around to check the responses of their peers before fully extending their reach.  Self-conscious chuckling darted through the room from some students, the laughter of relief, while other students whose hands weren’t raised looked around in surprised confusion at the general response.  

“I want you to discuss the following question with your groups,” I said.  “How is it that so many of you have developed negative feelings about your own abilities, despite the fact that you are all in a mathematics course at a well-respected university?”

If this interaction took place in a math course satisfying a general education requirement, I don’t think anyone would be surprised.  Yet this discussion repeats itself semester after semester in my upper-level undergraduate courses, for which the prerequisites are at least two semesters of calculus and in which almost every student is either a mathematics major or minor.  I’ve had similar interactions with students taking first-semester calculus, with experienced elementary school teachers in professional development workshops, with doctoral students in pure mathematics research seminars, and with fellow research mathematicians over drinks after dinner.  These conversations are about a secret we rarely discuss, an invisible undercurrent of embarrassment and self-doubt that flows through American mathematical culture, shared by many but revealed by few.  At every level of achievement, no matter what we’ve done, no matter how much we’ve accomplished, many of us believe that we’re simply not good at math. Continue reading

Posted in Assessment Practices, Classroom Practices, Student Experiences | Tagged , , , | 13 Comments

Why High-Impact Educational Practices (Despite Being So Labor–Intensive) Keep Me Coming For More

By Maria Mercedes Franco, Coordinator for Undergraduate Research & Associate Professor, Mathematics & Computer Science, Queensborough Community College-The City University of New York (CUNY)

By the time I was finishing graduate school, I had done much soul-searching and had come to realize that I have a passion for teaching and a strong commitment to the mission of public education. With my new awareness came the opportunity to interview for (and soon after accept) a position at Queensborough Community College, where I was encouraged early on to incorporate innovative pedagogies into my teaching. Now on my tenth year at the college, I look back and say without hesitation that High-Impact Educational Practices have brought me closer to larger and more diverse groups of learners – and closer to my ideals for higher education – than any other practice. Continue reading

Posted in Classroom Practices | 5 Comments

Let Your Students Do Some Grading? Using Peer Assessment to Help Students Understand Key Concepts

By Elise Lockwood, Contributing Editor, Oregon State University

On many occasions when I grade my students’ proofs, or when I read their solution to a particularly interesting problem, I am surprised by something I read. Sometimes I am surprised because I am disappointed with a given argument or a hand-wavy proof, but often I am surprised because I am impressed by a clever insight or an eloquent way of expressing an argument. Indeed, there have been occasions when I have learned something through the experience of grading my students’ work. Also, seeing the sheer variety of solution strategies that my students offer helps me to appreciate various mathematical approaches and makes me more attuned to their respective mathematical ways of thinking.

In this post I will discuss an activity that I call peer grading, by which I mean having students provide formative, written feedback on their classmates’ assignments. This involves giving students the opportunity to engage with and analyze work that their classmates have done. Peer grading has been used by other teachers (see the references at the end of this post), and my personal reflections on the value of engaging in the process of grading have convinced me that students can similarly benefit from grading other students’ work.

Continue reading

Posted in Assessment Practices, Classroom Practices | Tagged | 2 Comments

Start Small, Think Big: Making a Difference Through K-12 Mathematics Outreach

By Kathleen Fowler, Professor of Mathematics, Department of Mathematics and Computer Science, Clarkson University

Since starting my career as a faculty member in 2003, I jumped right in to K-12 Outreach and have never looked back. I was motivated by my strong connection to my community, which is located in St. Lawrence County, a geographically isolated, rural part of upstate New York. All K-12 districts in this county share the same problems of limited resources, significant poverty rates, and a “high needs” population. My choice to become involved in K-12 Outreach was a personal one. I had a very nonlinear path to becoming a mathematician. I was raised by a single mom who sold cars and told me I could do anything I wanted to if I hunkered down and worked hard. I went to three different colleges, changed majors three times, and took five years to get my undergraduate degree—waitressing for the last three years to support myself. I only had one female math teacher in 8th grade and one female math professor—but not until graduate school. My point is, I didn’t have many female STEM role models, but honestly not much of this occurred to me until I started to get involved in K-12 Outreach. However, I quickly understood that these experiences are not the norm and that not every child has an encouraging support system to motivate them. Even for students who do have strong family support, a lack of opportunities for resume building activities or enrichment such as Robotics or Science Olympiad or even an AP Physics class means they are not even competitive when they apply to colleges. I am raising two daughters in this community—they and their peers deserve the same opportunities as students in affluent suburbs scattered across “downstate” New York and elsewhere.

Feedback I’ve received from faculty from a variety of Universities that do K-12 Outreach imply that a common thread is a feeling of wanting to “give back” or to honor a K-12 teacher that made a difference in their lives. The bottom line is that this sort of service to the broader community is a win-win situation. In times of major budget cuts in education, new curriculum and assessments, exhausted teachers, overworked parents, and a new generation of students who need STEM problem solving skills more than ever, it feels great to help out in any possible way. In this article, I’ll describe what K-12 Outreach is and share examples about how mathematics faculty can get involved on a variety of levels. My hope is that, as mathematicians, we can share our expertise with and also learn from the K-12 community to strengthen STEM education through collaboration. Continue reading

Posted in Multidisciplinary Education, Outreach, Summer Programs | Leave a comment

What I Wish I Had Learned More About in College Mathematics

By Sabrina Schmidt, Data Manager at Time, Inc. and former undergraduate mathematics major at Vassar College

Editor’s note: The editorial board believes that in our discussion of teaching and learning, it is important to include the authentic voices of current and former undergraduate students reflecting on their experiences with mathematics.

When I graduated from Vassar College in 2010 with degrees in math and Italian, I wasn’t sure what was next for me. I applied for math-related jobs at my favorite media companies. Ultimately, Time Inc. offered me a position as a Data Analyst, a job which has been an ideal blend of my mathematical and entertainment interests. I manage store-level distributions for three magazines, Us Weekly, Rolling Stone, and Men’s Journal, all published by Wenner, a primary client. I determine how many copies of every issue go into each store by using formulas based on the store’s available checkout pockets and average sales. At Time Inc., I have been impressed and surprised by the variety of math-related projects. There is a Shopper Insights group that has developed an eye-tracking system that follows the movement of a consumer’s pupils while shopping and helps optimize magazine placement in stores. The Research divisions work on projects that include using subscriber data to help expand the reach of our brands and analyzing historical data to create new pricing strategies. They are doing a zone- pricing test for People magazine, where they are removing the cover price and setting different prices for different regions. In this blog post, I use examples from my work experience over the last five years to suggest ways in which undergraduate mathematics majors can be better prepared for math-related positions in companies. I discuss how I wish I had learned more about applications, computer science, statistics, and connections to other STEM fields.

Continue reading

Posted in Student Experiences | Tagged , , , | 8 Comments

(Don’t?) Make ’em Laugh

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

When I started teaching, I wanted to be the very best teacher.  Not just “the best teacher I could be”, but the very best teacher, the one students would tell their friends about and remember fondly years later, the kind of teacher they might imagine being the hero in a movie.  I don’t know what your movie hero teacher looks like, but mine is beloved by all the students (more Robin Williams than John Houseman).  So naturally, I wanted all the students to like me.  I also wanted them to share my love of mathematics, and see it as a joyful endeavor, not just a requirement to be checked off.  As a result, I started including more humor in my classes.  What I eventually realized, and had to confront, was that at least some of what I was doing was more about making me look like that movie hero teacher, or about making the class fun, than about helping my students learn mathematics.

Continue reading

Posted in Classroom Practices | Tagged | Leave a comment

Number Theory and Cryptography: A Distance Learning Course for High School Students

By Matt Baker, Professor of Mathematics, Georgia Institute of Technology

Last year, I began offering an online Number Theory and Cryptography course for gifted high school students through Georgia Tech.  Fourteen high school seniors from metro Atlanta took the course in Fall 2014, and overall I would say it was a big success.  We will be offering the course again in Fall 2015 and are expecting roughly double the number of students.  After describing the structure of the course, I will relate some of my experiences and describe some of the things I learned along the way.  I hope this article stimulates others to think outside the box about using technology in education without necessarily following the standard “MOOC” model. Continue reading

Posted in Classroom Practices, Online Education | Tagged , , , | 7 Comments