{"id":999,"date":"2015-10-10T00:01:07","date_gmt":"2015-10-10T04:01:07","guid":{"rendered":"http:\/\/blogs.ams.org\/matheducation\/?p=999"},"modified":"2015-12-28T07:17:04","modified_gmt":"2015-12-28T12:17:04","slug":"active-learning-in-mathematics-part-iv-personal-reflections","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/matheducation\/2015\/10\/10\/active-learning-in-mathematics-part-iv-personal-reflections\/","title":{"rendered":"Active Learning in Mathematics, Part IV: Personal Reflections"},"content":{"rendered":"

By Benjamin Braun, Editor-in-Chief<\/a>, University of Kentucky; Priscilla Bremser, Contributing Editor<\/a>, Middlebury College; Art Duval,\u00a0Contributing Editor<\/a><\/span><\/i>, University of Texas at El Paso; Elise Lockwood,\u00a0Contributing Editor<\/a><\/span><\/i>, Oregon State University; and Diana White,\u00a0Contributing Editor<\/a><\/span><\/i>, University of Colorado Denver.<\/span><\/i><\/p>\n

Editor\u2019s note: This is the fourth article in a series devoted to active learning in mathematics courses. \u00a0The other articles in the series can be found <\/span><\/i>here<\/span><\/i><\/a>.<\/span><\/i><\/p>\n

In contrast to our first three articles in this series on active learning, in this article we take a more personal approach to the subject. \u00a0Below, the contributing editors for this blog share aspects of our journeys into active learning, including the fundamental reasons we began using active learning methods, why we have persisted in using them, and some of our most visceral responses to our own experiences with these methods, both positive and negative. \u00a0As is clear from these reflections, mathematicians begin using active learning techniques for many different reasons, from personal experiences as students (both good and bad) to the influence of colleagues, conferences, and workshops. \u00a0The path to active learning is not always a smooth one, and is almost always a winding road.<\/span><\/p>\n

Because of this, we believe it is important for mathematics teachers to share their own experiences, both positive and negative, in the search for more meaningful student engagement and learning. \u00a0<\/span>We invite all our readers to share their own stories in the comments at the end of this post.<\/span><\/i> \u00a0We also recognize that many other mathematicians have shared their experiences in other venues, so at the end of this article we provide a collection of links to essays, blog posts, and book chapters that we have found inspirational.<\/span><\/p>\n

There is one more implicit message contained in the reflections below that we want to highlight. \u00a0All mathematics teachers, even those using the most ambitious student-centered methods, use a range of teaching techniques combined in different ways. \u00a0In our next post, we will dig deeper into the idea of instructor \u201ctelling\u201d to gain a better understanding of how an effective balance can be found between the process of student discovery and the act of faculty sharing their expertise and experience.<\/span><\/p>\n

Priscilla Bremser:<\/b><\/p>\n

I began using active learning methods for several reasons, but two interconnected ones come to mind. \u00a0First, Middlebury College requires all departments to contribute to the First-Year Seminar program, which places every incoming student into a small writing-intensive class. The topic is chosen by the instructor, while guidelines for writing instruction apply to all seminars. \u00a0As I have developed and taught my seminars over the years, I\u2019ve become convinced that students learn better when they are required to express themselves clearly and precisely, rather than simply listening or reading. \u00a0At some point it became obvious that the same principle applies in my other courses as well, and hence I was ready to try some of the active learning approaches I\u2019d been hearing about at American Mathematical Society meetings and reading about in <\/span>journals<\/span><\/a>.<\/span><\/p>\n

Second, I got a few student comments on course evaluations, especially for Calculus courses, that suggested I was more helpful in office hours than in lecture. \u00a0Thinking it through, I realized that in office hours, I routinely and repeatedly ask students about their own thinking, whereas in lecture, I was constantly making assumptions about student thinking, and relying on their responses to \u201cAny questions?\u201d for guidance, which didn\u2019t elicit enough information to address the misunderstandings around the room. One way to make class more like office hours is to put students into small groups. I then set ground rules for participation and ask for a single set of problem solutions from each group. This encourages everyone to speak some mathematics in each class session, and to ask for clarity and precision from classmates. \u00a0Because I\u2019m joining each conversation for a while, I get a more accurate perception of students\u2019 comprehension levels.<\/span><\/p>\n

This semester I\u2019m teaching Mathematics for Teachers, using an <\/span>IBL textbook by Matthew Jones<\/span><\/a>. I\u2019ve already seen several students throw fists up in the air, saying \u201cI get it now! \u00a0That\u2019s so cool!\u201d How well I remember having that response to my first Number Theory course; it\u2019s why I went into teaching at this level in the first place. \u00a0On the other hand, a Linear Algebra student who insists that \u00a0\u201cI learn better from reading a traditional textbook\u201d leaves me feeling rather deflated. It seems that I\u2019ve failed to convey why I direct the course the way that I do, or at least I haven\u2019t yet succeeded. \u00a0The truth is, though, that I used to feel the same way. \u00a0I regarded mathematics as a solitary pursuit, in which checking in with classmates was a sign of weakness. \u00a0Had I been required to discuss my thinking regularly during class and encouraged to do so between sessions, I would have developed a more solid foundation for my later learning. Remembering this inspires me to be intentional with students, and explain repeatedly why I direct my courses the way that I do. \u00a0Most of them come around eventually.<\/span><\/p>\n

Elise Lockwood:<\/b><\/p>\n

I have a strong memory of being an undergraduate in a discrete mathematics course, trying desperately to understand the formulas for permutations, combinations, and the differences between the two. The instructor had presented the material, perhaps providing an example or two, but she had not provided an opportunity for us to actively explore and understand why the formulas might make sense. By the time I was working on homework, I simply tried (and often failed) to apply the formulas I had been given. I strongly disliked and feared counting problems for years after that experience. It wasn\u2019t until much later that I took a combinatorics course as a master\u2019s student. Here, the counting material was brought to life as we were given opportunities to work through problems during class, to unpack formulas, and to come to understand the subtlety and wonder of counting. The teacher did not simply present a formula and move on, assuming we understood it. Rather, he persisted by challenging us to make sense of what was going on in the problems we solved.<\/span><\/p>\n

For example, we once were discussing a counting problem in class (I can\u2019t recall if it was an in-class problem or a problem that had been assigned for homework). During this discussion, it became clear that students had answered the problem in two different ways — both of them seemed to make sense logically, but they did not yield the same numerical result. The instructor did not just tell us which answer was right, but he used the opportunity to have us consider both answers, facilitating a (friendly) debate among the class about which approach was correct. We had to defend whichever answer we thought was correct and critique the one we thought was incorrect. This had the effect not only of engaging us and piquing our curiosity about a correct solution, but it made us think more carefully and deeply about the subtleties of the problem. <\/span><\/p>\n

Now, studying how students solve counting problems is the primary focus of my research in mathematics education. My passion for the teaching and learning of counting was probably in large part formed by the frustrations I felt as an undergraduate and the elation I later experienced when I actually understood some of the fundamental ideas.<\/span><\/p>\n

When I have been given the opportunity to teach counting over the years (in discrete mathematics or combinatorics classes, or in courses for pre-service teachers), I have tried my hardest to facilitate my students\u2019 active engagement with the material during class. This has not taken an inordinate amount of time or effort: instead of just giving students the formulas off the bat, I give them a series of counting problems that both introduce counting as a problem solving activity and motivate (and build up to) some key counting formulas. For example, students are given problems in which they list some outcomes and appreciate the difference between permutations and combinations firsthand. I have found that a number of important issues and ideas (concerns about order, errors of overcounting, key binomial identities) can emerge on their own through the students\u2019 activity, making any subsequent discussion or lecture much more meaningful for students. When I incorporate these kinds of activities for my students, I am consistently impressed at the meaning they are able to make of complex and notoriously tricky ideas.<\/span><\/p>\n

More broadly, these pedagogical decisions I make are also based on my belief about the nature of mathematics and the nature of what it means to learn mathematics. Through my own experiences as a student, a teacher, and a researcher, I have become convinced that providing students with opportunities to actively engage with and think about mathematical concepts — during class, and not just on their own time — is a beneficial practice. My experience with the topic of counting (something near and dear to my heart) is but one example of the powerful ways in which student engagement can be leverage for deep and meaningful mathematical understanding. <\/span><\/p>\n

Diana White:<\/b><\/p>\n

What stands out most to me as I reflect upon my journey into active learning is not so much how or why I got involved, but the struggles that I faced during my first few years as a tenure-track faculty member as I tried to switch from being a good \u201clecturer\u201d to all out inquiry-based learning. \u00a0I was enthusiastic and ambitious, but lacking in the skills to genuinely teach in the manner in which I wanted.<\/span><\/p>\n

As a junior faculty member, I was already sold on the value of inquiry-based learning and student-centered teaching. \u00a0I had worked in various ways with teachers as a graduate student at the University of Nebraska and as a post-doc at the University of South Carolina, including teaching math content courses for elementary teachers and assisting with summer professional development courses for teachers. \u00a0Then, the summer before I started my current position, I attended both the annual <\/span>Legacy of R.L. Moore conference<\/span><\/a> and a weeklong workshop on teaching number theory with IBL through the MAA PREP program. \u00a0The enthusiasm and passion at both of these was contagious. \u00a0<\/span><\/p>\n

However, upon starting my tenure track position, I jumped straight in, with extremely ambitious goals for my courses and my students, ones for which I did not have the skills to implement yet. \u00a0In hindsight, it was too much for me to try to both switch from being a good \u201clecturer\u201d to doing full out IBL and running an intensely student centered classroom, all while teaching new courses in a new place. \u00a0I tried to do way too much too soon, and in many ways that was not healthy for either me or the students, as evidenced by low student evaluations and frustrations on both sides.<\/span><\/p>\n

Figuring out specifically what was going wrong was a challenge, though. \u00a0Those who came to observe, both from my department and our Center for Faculty Development, did not find anything specific that was major, and student comments were somewhat generic – frustration that they felt the class was disorganized and that they were having to teach themselves the material. \u00a0<\/span><\/p>\n

I thus backtracked to more in the center of the spectrum, using an interactive lecture \u00a0Things smoothed out and students became happier. \u00a0What I am not at all convinced of, though, is that this decision was best for student learning. \u00a0Despite the unhappiness on both our ends when I was at the far end of the active learning spectrum, I had ample evidence (both from assessments and from direct observation of their thought processes in class) that students were both learning how to think mathematically and building a sense of community outside the classroom. \u00a0To this day, I feel torn, like I made a decision that was best for student satisfaction, as well as for how my colleagues within my department perceive me. \u00a0Yet I remain convinced that my students are now learning less, and that there are students who are not passing my classes who would have passed had I taught using more active learning. (It was impossible to \u201chide\u201d with my earlier classes, due to the natural accountability built into the process, so struggling students had to confront their weaknesses much sooner.)<\/span><\/p>\n

It is hard for me to look back with regrets, as the lessons learned have been quite powerful and no doubt shaped who I am today. \u00a0However, I would offer some thoughts, aimed primarily at junior faculty. \u00a0<\/span><\/p>\n

Don\u2019t be afraid to start slow. \u00a0Even if it\u2019s not where you want to end up, just getting started is still an important first step. \u00a0Negative perceptions from students and colleagues are incredibly hard to overcome.<\/span><\/p>\n

Don\u2019t underestimate the importance of student buy-in, or of faculty buy-in. \u00a0I found many faculty feel like coverage and exposure are essential, and believe strongly that performance on traditional exams is an indicator of depth of knowledge or ability to think mathematically.<\/span><\/p>\n

Don\u2019t be afraid to politely request to decline teaching assignments. \u00a0When I was asked to teach the history of mathematics, a course for which I had no knowledge of or background in, I wasn\u2019t comfortable asking to teach something else instead. \u00a0While it has proved really beneficial to my career (I\u2019m now part of an NSF grant related to the use of primary source projects in the undergraduate mathematics classroom), I was in no way qualified to take that on as a first course at a new university.<\/span><\/p>\n

I have personally gained a tremendous amount from my participation in the IBL community, perhaps most importantly a sense of community with others who believe strongly in active learning. \u00a0<\/span><\/p>\n

Art Duval:<\/b><\/p>\n

My first experience with active learning in mathematics was as a student at the <\/span>Hampshire College Summer Studies in Mathematics<\/span><\/a> program during high school. \u00a0Although I’d had good math teachers in junior high and high school, this was nothing like I’d seen before: The first day of class, we spent several hours discussing one problem (the number of regions formed in 3-dimensional space by drawing \\(n\\) planes), drawing pictures and making conjectures; the rest of the summer was similar. \u00a0The six-week experience made such an impression on me, that (as I realized some years later) most of the educational innovations I have tried as a teacher have been an attempt to recreate that experience in some way for my own students.<\/span><\/p>\n

When I was an undergraduate, I noticed that classes where all I did was furiously take notes to try to keep up with the instructor were not nearly as successful for me as those where I had to <\/span>do<\/span><\/i> something. \u00a0Early in my teaching career, I got a big push towards using active learning course structures from teaching \u201c<\/span>reform calculus<\/span><\/a>\u201d and courses for future elementary school teachers. \u00a0In each case, this was greatly facilitated by my sitting in on another instructor\u2019s section that already incorporated these structures. \u00a0Later I learned, through <\/span>my participation in a K-16 mathematics alignment initiative<\/span><\/a>, the importance of conceptual understanding among the <\/span>levels of cognitive demand<\/span><\/a>, and this helped me find the language to describe what I was trying to achieve.<\/span><\/p>\n

Over time, I noticed that students in my courses with more active learning seemed to stay after class more often to discuss mathematics with me or with their peers, and to provide me with more feedback about the course. \u00a0This sort of engagement, in addition to being good for the students, is very addictive to me. \u00a0My end-of-semester course ratings didn’t seem to be noticeably different, but the written comments students submitted were more in-depth, and indicated the course was more rewarding in fundamental ways. \u00a0As with many habits, after I’d done this for a while, it became hard <\/span>not<\/span><\/i> to incorporate at least little bits of interactivity (think-pair-share, student presentation of homework problems), even in courses where external forces keep me from incorporating more radical active learning structures.<\/span><\/p>\n

Of course, there are always challenges to overcome. \u00a0The biggest difficulty I face with including any sort of active learning is <\/span>how much more time<\/span><\/a> it takes to get students to realize something than it takes to simply tell them. \u00a0I also still find it hard to figure out the right sort of scaffolding to help students see their way to a new concept or the solution to a problem. \u00a0Still, I keep including as much active learning as I can in each course. \u00a0The parts of classes I took as a student (going back to junior high school) that I remember most vividly, and the lessons I learned most thoroughly, whether in mathematics or in other subjects, were the activities, not the lectures. \u00a0Along the same lines, I occasionally run into former students who took my courses many years ago, and it\u2019s the students who took the courses with extensive active learning, much more than those who took more traditional courses, who still remember all these years later details of the course and how much they learned from it. <\/span><\/p>\n

Other Essays and Reflections:<\/b><\/p>\n

Benjamin Braun, The Secret Question (Are We Actually Good at Math?), <\/span>http:\/\/blogs.ams.org\/matheducation\/2015\/09\/01\/the-secret-question-are-we-actually-good-at-math\/<\/span><\/a><\/p>\n

David Bressoud, Personal Thoughts on Mature Teaching, in <\/span>How to Teach Mathematics, 2nd Edition<\/span><\/i>, by Steven Krantz, American Mathematical Society, 1999. \u00a0<\/span>Google books preview<\/span><\/a><\/p>\n

Jerry Dwyer, Transformation of a Math Professor\u2019s Teaching, <\/span>http:\/\/blogs.ams.org\/matheducation\/2014\/06\/01\/transformation-of-a-math-professors-teaching\/<\/span><\/a><\/p>\n

Oscar E. Fernandez, Helping All Students Experience the Magic of Mathematics, <\/span>http:\/\/blogs.ams.org\/matheducation\/2014\/10\/10\/helping-all-students-experience-the-magic-of-mathematics\/<\/span><\/a><\/p>\n

Ellie Kennedy, A First-timer\u2019s Experience With IBL, <\/span>http:\/\/maamathedmatters.blogspot.com\/2014\/09\/a-first-timers-experience-with-ibl.html<\/span><\/a><\/p>\n

Bob Klein, Knowing What to Do is not Doing, <\/span>http:\/\/maamathedmatters.blogspot.com\/2015\/07\/knowing-what-to-do-is-not-doing.html<\/span><\/a><\/p>\n

Evelyn Lamb, Blogs for an IBL Novice, <\/span>http:\/\/blogs.ams.org\/blogonmathblogs\/2015\/09\/21\/blogs-for-an-ibl-novice\/<\/span><\/a><\/p>\n

Carl Lee, The Place of Mathematics and the Mathematics of Place, <\/span>http:\/\/blogs.ams.org\/matheducation\/2014\/10\/01\/the-place-of-mathematics-and-the-mathematics-of-place\/<\/span><\/a><\/p>\n

Steven Strogatz, Teaching Through Inquiry: A Beginner\u2019s Perspectives, Parts I and II,\u00a0<\/span>http:\/\/www.artofmathematics.org\/blogs\/cvonrenesse\/steven-strogatz-reflection-part-1,\u00a0<\/span><\/a>http:\/\/www.artofmathematics.org\/blogs\/cvonrenesse\/steven-strogatz-reflection-part-2<\/span><\/a><\/p>\n

Francis Su, The Lesson of Grace in Teaching, <\/span>http:\/\/mathyawp.blogspot.com\/2013\/01\/the-lesson-of-grace-in-teaching.html<\/span><\/a><\/p>\n

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By Benjamin Braun, Editor-in-Chief, University of Kentucky; Priscilla Bremser, Contributing Editor, Middlebury College; Art Duval,\u00a0Contributing Editor, University of Texas at El Paso; Elise Lockwood,\u00a0Contributing Editor, Oregon State University; and Diana White,\u00a0Contributing Editor, University of Colorado Denver. Editor\u2019s note: This is … Continue reading →<\/span><\/a><\/p>\n

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