*Editor’s note: This is the fourth article in a series devoted to active learning in mathematics courses. The other articles in the series can be found **here**.*

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. Below, 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. As 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. The path to active learning is not always a smooth one, and is almost always a winding road.

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. *We invite all our readers to share their own stories in the comments at the end of this post.* We 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.

There is one more implicit message contained in the reflections below that we want to highlight. All mathematics teachers, even those using the most ambitious student-centered methods, use a range of teaching techniques combined in different ways. In our next post, we will dig deeper into the idea of instructor “telling” 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.

**Priscilla Bremser:**

I began using active learning methods for several reasons, but two interconnected ones come to mind. First, 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. As I have developed and taught my seminars over the years, I’ve become convinced that students learn better when they are required to express themselves clearly and precisely, rather than simply listening or reading. At 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’d been hearing about at American Mathematical Society meetings and reading about in journals.

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. Thinking 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 “Any questions?” for guidance, which didn’t 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. Because I’m joining each conversation for a while, I get a more accurate perception of students’ comprehension levels.

This semester I’m teaching Mathematics for Teachers, using an IBL textbook by Matthew Jones. I’ve already seen several students throw fists up in the air, saying “I get it now! That’s so cool!” How well I remember having that response to my first Number Theory course; it’s why I went into teaching at this level in the first place. On the other hand, a Linear Algebra student who insists that “I learn better from reading a traditional textbook” leaves me feeling rather deflated. It seems that I’ve failed to convey why I direct the course the way that I do, or at least I haven’t yet succeeded. The truth is, though, that I used to feel the same way. I regarded mathematics as a solitary pursuit, in which checking in with classmates was a sign of weakness. Had 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. Most of them come around eventually.

**Elise Lockwood:**

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’t until much later that I took a combinatorics course as a master’s 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.

For example, we once were discussing a counting problem in class (I can’t 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.

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.

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’ 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’ 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.

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.

**Diana White:**

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 “lecturer” to all out inquiry-based learning. I was enthusiastic and ambitious, but lacking in the skills to genuinely teach in the manner in which I wanted.

As a junior faculty member, I was already sold on the value of inquiry-based learning and student-centered teaching. I 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. Then, the summer before I started my current position, I attended both the annual Legacy of R.L. Moore conference and a weeklong workshop on teaching number theory with IBL through the MAA PREP program. The enthusiasm and passion at both of these was contagious.

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. In hindsight, it was too much for me to try to both switch from being a good “lecturer” to doing full out IBL and running an intensely student centered classroom, all while teaching new courses in a new place. I 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.

Figuring out specifically what was going wrong was a challenge, though. Those 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.

I thus backtracked to more in the center of the spectrum, using an interactive lecture Things smoothed out and students became happier. What I am not at all convinced of, though, is that this decision was best for student learning. Despite 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. To 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. Yet 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 “hide” with my earlier classes, due to the natural accountability built into the process, so struggling students had to confront their weaknesses much sooner.)

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. However, I would offer some thoughts, aimed primarily at junior faculty.

Don’t be afraid to start slow. Even if it’s not where you want to end up, just getting started is still an important first step. Negative perceptions from students and colleagues are incredibly hard to overcome.

Don’t underestimate the importance of student buy-in, or of faculty buy-in. I 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.

Don’t be afraid to politely request to decline teaching assignments. When I was asked to teach the history of mathematics, a course for which I had no knowledge of or background in, I wasn’t comfortable asking to teach something else instead. While it has proved really beneficial to my career (I’m 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.

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.

**Art Duval:**

My first experience with active learning in mathematics was as a student at the Hampshire College Summer Studies in Mathematics program during high school. Although 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. The 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.

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 *do* something. Early in my teaching career, I got a big push towards using active learning course structures from teaching “reform calculus” and courses for future elementary school teachers. In each case, this was greatly facilitated by my sitting in on another instructor’s section that already incorporated these structures. Later I learned, through my participation in a K-16 mathematics alignment initiative, the importance of conceptual understanding among the levels of cognitive demand, and this helped me find the language to describe what I was trying to achieve.

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. This sort of engagement, in addition to being good for the students, is very addictive to me. My 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. As with many habits, after I’d done this for a while, it became hard *not* 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.

Of course, there are always challenges to overcome. The biggest difficulty I face with including any sort of active learning is how much more time it takes to get students to realize something than it takes to simply tell them. I 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. Still, I keep including as much active learning as I can in each course. The 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. Along the same lines, I occasionally run into former students who took my courses many years ago, and it’s 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.

**Other Essays and Reflections:**

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

David Bressoud, Personal Thoughts on Mature Teaching, in *How to Teach Mathematics, 2nd Edition*, by Steven Krantz, American Mathematical Society, 1999. Google books preview

Jerry Dwyer, Transformation of a Math Professor’s Teaching, http://blogs.ams.org/matheducation/2014/06/01/transformation-of-a-math-professors-teaching/

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

Ellie Kennedy, A First-timer’s Experience With IBL, http://maamathedmatters.blogspot.com/2014/09/a-first-timers-experience-with-ibl.html

Bob Klein, Knowing What to Do is not Doing, http://maamathedmatters.blogspot.com/2015/07/knowing-what-to-do-is-not-doing.html

Evelyn Lamb, Blogs for an IBL Novice, http://blogs.ams.org/blogonmathblogs/2015/09/21/blogs-for-an-ibl-novice/

Carl Lee, The Place of Mathematics and the Mathematics of Place, http://blogs.ams.org/matheducation/2014/10/01/the-place-of-mathematics-and-the-mathematics-of-place/

Steven Strogatz, Teaching Through Inquiry: A Beginner’s Perspectives, Parts I and II, http://www.artofmathematics.org/blogs/cvonrenesse/steven-strogatz-reflection-part-1, http://www.artofmathematics.org/blogs/cvonrenesse/steven-strogatz-reflection-part-2

Francis Su, The Lesson of Grace in Teaching, http://mathyawp.blogspot.com/2013/01/the-lesson-of-grace-in-teaching.html

]]>*Editor’s note: This is the third article in a series devoted to active learning in mathematics courses. The other articles in the series can be found 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.

**Student-Centered Classroom Environments**

In his decade-long study of highly-effective college teachers [1], Ken Bain found that such teachers establish in their courses a *natural critical learning environment* in which…

…people learn by confronting intriguing, beautiful, or important problems, authentic tasks that will challenge them to grapple with ideas, rethink their assumptions, and examine their mental models of reality. These are challenging yet supportive conditions in which learners feel a sense of control over their education; work collaboratively with others; believe that their work will be considered fairly and honestly; and try, fail, and receive feedback from expert learners in advance of and separate from any summative judgment of their effort. — Ken Bain,

What the Best College Teachers Do

From this description is it clear that these environments engage students with tasks at all levels of cognitive demand, a concept described in Part II of this series. While these environments require effort and diligence to establish and maintain, Bain makes it clear that these environments arise in every conceivable teaching environment, including small discussion-focused courses in the humanities, large-lecture style courses in the sciences, practicum-based courses in medical fields, field-based courses in the social sciences, and more. These diverse teaching environments have been the catalyst for the development of many successful models of active learning that support student engagement; this is one source of the challenge behind defining the phrase “active learning,” as we discussed in Part I of this series. Bain’s work complements and reinforces the explicit consideration in the meta-analysis by Freeman et al. [2] of varied active learning techniques.

Acknowledging the effectiveness of a range of active learning techniques across diverse settings is particularly important in the context of postsecondary mathematics teaching and learning. In contemporary college and university courses, lecturing remains the dominant teaching technique used by mathematics faculty. Many faculty view the use of either active learning (with the stereotypical interpretation mentioned previously) or lecture as an exclusive choice with two diametrically opposed options, yet nothing could be farther from the truth. Marrongelle and Rasmussen [6] have described a spectrum of teaching that ranges from “all telling” to “all student discovery.” Mathematics education researchers have invested significant effort toward understanding teaching and learning across this spectrum, including recent efforts to better understand the pedagogical moves of mathematicians who use traditional lecture as their instructional practice [4], [5]; we will investigate this topic in more depth in a forthcoming article in this series. The most important aspect of this for mathematics teaching is that there are opportunities to use active learning techniques at all points on this spectrum, the single exception being the extreme end of “instructor lecture only — no questions or comments allowed by students,” which we believe is extremely rare in practice.

In the rest of this article we will describe techniques and environments that we include as active learning, using our definition from Part I of this series. We will begin with techniques that fall closer to the “all telling” end of the spectrum and end with techniques closer to the “all student discovery” end. It is important to discuss techniques that can be used across this spectrum because there are many high-quality, concerned teachers who, while not wanting to make the jump to all-student-discovery, are deeply interested in increasing student learning and engagement. These responsible, committed teachers are valuable members of the mathematical community. Indeed, in the case of Calculus, a recent report by the MAA regarding successful calculus programs [3] found that the most important aspect regarding student persistence from first- to second-semester calculus is the presence of three factors: classroom interactions that acknowledge students; encouragement and availability on the part of the instructor; and the use of fair assessments. These are among the qualities that the report uses to describe “good” teaching, and these qualities afford ample opportunities for the introduction of effective active learning techniques. The MAA report uses the term “ambitious teaching” to describe the use of more sophisticated and complex active learning techniques by teachers moving beyond the qualities of good teaching, which represents a shift further along the spectrum discussed by Marrongelle and Rasmussen.

An important observation is that the words “active” and “interactive” are not synonymous in our discussion. For example, a lecture in which an instructor tells jokes that elicit laughter from students, or asks students to fact-check an elementary arithmetic calculation with their calculator, is interactive. These actions acknowledge the presence of students, hence fall under “good” teaching. However, those techniques do not provide tasks in which students directly engage with content during class, thus aren’t within the boundaries of active learning. Similarly, active learning must go beyond asking students to “think hard.” For example, after a particularly complicated example in a calculus class, or upon completion of a proof in an advanced course, an instructor might tell students to “think about what we just did for a minute or two,” then ask if there are any questions. While again this act would fall within the bounds of “good” teaching, the absence of a specific task given to students, with a specific goal, prevents this from being considered an active learning technique.

**Active Learning Techniques for Lectures**

One of the best examples of an active learning technique suitable for use in lectures is “think-pair-share.” In this technique, the instructor provides students with a short task — perhaps a short computational problem, or a step in a proof to complete, or an example for them to create a hypothesis based on. After providing the students with 2-5 minutes of time to independently consider the task, the students are asked to compare their answers with the people sitting around them, or with their partner if they have been placed into explicit work pairs. Finally, some or all of the students are asked to share their answers in some manner, either with the groups next to them or with the entire class. The benefits of using this technique for students are that they have a chance to be energized during lecture, have a chance to pause and consider the content the lecturer has been presenting, and have to explain their thinking to peers. In classes with large numbers of students for whom English is not their first language, students also can discuss the content with peers in a language they might understand more clearly. Instructors benefit as well, as they can gather feedback from students to determine how well content is understood. The main drawback instructors report for this technique is that in the time it takes to complete a think-pair-share, the instructor could have covered more examples or moved on to other content topics more quickly. According to research on activities such as this, for example a study of physics students by Deslauriers et al. [7] that is discussed by Bressoud [8] in the context of mathematics, these benefits far outweigh the drawbacks.

Because this technique is relatively simple, it is applicable in almost every conceivable teaching environment. Even in medium- and large-lecture settings, instructors have used both low- and high-tech feedback response systems for the “share” stage of the technique. For example, many instructors use multiple-choice problems as think-pair-share prompts in conjunction with classroom response systems, i.e. “clickers.” These systems typically come with additional data analysis features that allow instructors to more carefully review student responses over time to detect problematic content areas. Even at institutions where faculty do not have access to sophisticated systems of this type or do not want to deal with the technology, many instructors have successfully had students share their answers by holding up colored pieces of paper, providing a visual representation of their responses. This technique is introduced in Prather and Brissenden [9] (p. 10) as a small part of a larger article about a very focused form of think-pair-share applicable to all disciplines; for a more practical introduction to these “A-B-C-D cards”, with examples from a statistics classroom, see Lesser [26].

In addition to think-pair-share, there are many related examples of “classroom voting” techniques that can be used to increase student engagement during a lecture-based course. An in-depth description of these techniques can be found in the MAA volume *Teaching Mathematics with Classroom Voting: With and Without Clickers* [10].

**Inverted (or “Flipped”) Classes**

In an inverted (or “flipped”) classroom environment, instructor presentations of basic definitions, examples, proofs, and heuristics are provided to students in videos or in assigned readings that are completed prior to attending class. As a result, class time becomes available for tasks that directly engage students. The type of task that instructors use during this time ranges from using complicated think-pair-shares, with complex problems or examples, to having students work in small groups on a sequenced activity worksheet with occasional instructor or teaching assistant feedback. The inverted model of teaching has been used as the structure for entire courses, as an occasional event for handling topics that are less amenable to lecture presentations, as the basis for review sessions or problem solving sessions, and more. While the mere act of inverting a classroom is not inherently active, the structure of the inverted classroom environment is typically used to support in-class tasks with higher levels of cognitive demand, hence our inclusion of this as an active learning environment.

Compared to implementing think-pair-share and classroom voting techniques, creating inverted classroom environments requires both more effort and time on the part of the instructor and significantly more institutional support, especially in the areas of technology and data storage support. Having said that, the inverted classroom model is being explored in many disciplines, and many colleges and universities have experience with this technique even if mathematics faculty do not. This breadth of use across disciplines is reflected in a recent volume on best practices for flipped classrooms [11]. In mathematics, faculty have used combinations of video- and readings-based assignments to invert classes across a surprising range of content areas, including linear algebra [12], [15], calculus [13], [16], math courses for pre-service elementary school teachers [14], statistics [17], and mathematical biology [18]. We refer the interested reader to these references for in-depth discussions regarding the benefits and drawbacks of inverted classroom environments.

**Math Emporium**

The emporium model of teaching, like inverted classrooms, is not a technique but a learning environment that supports active learning techniques. The typical math emporium [23] is centered around a large room filled with computer workstations, in which students progress through self-paced online courses. Unlike inverted classes, many emporium models do not include a lecture component at all. Also unlike inverted classes, most math emporiums have been developed to handle remediation issues and low-level courses such as developmental mathematics and college algebra. An emporium usually has tables at which students can work collaboratively and is staffed by a large number of teaching assistants and tutors. Because the work of students is self-paced, and is driven in some emporium models by adaptive learning systems such as Aleks, students spend most of their time actively engaging with course content, providing opportunities for engagement with a range of tasks. In the emporium environment it is important that tasks be designed with levels of cognitive demand in mind, as there is evidence that some students who are successful in emporium programs are not engaging in high-cognitive work that promotes deep learning [19].

An interesting aspect of the math emporium model is that it was developed and is promoted as a means of both helping students learn and managing the economic reality that many institutions face of increased student enrollment with flat or decreasing instructional resources. The operating costs of an emporium can be lower than that of traditional teaching environments [23], and for this (among other factors) the math emporium model has attracted attention from national news organizations [21]. With a teaching environment that combines significant infrastructure investment at the institutional level and a shift from the traditional economic model on which college classes are built, it is not surprising that the emporium model has been more controversial in the mathematical community than techniques like classroom voting or less comprehensive changes such as inverted classes. Thoughtful discussions and methodological studies, for example Bressoud’s *Launchings* column on this topic [20] and a recent study by Webel et al. [19], are available for readers interested in learning more about the math emporium model.

**Laboratory Courses**

The use of computer technology in math courses does not have to be as dramatic as in emporium models. Since the 1990’s, many mathematics courses have included exercises and computer lab activities using programs such as Mathematica, Maple, and MATLAB. The use of computer algebra systems in postsecondary mathematics courses is now widespread, with a wide range of benefits reported by mathematicians teaching with technological tools, often representing students engagement at higher levels of cognitive demand [22].

The use of technology to teach mathematics can go far beyond simple augmentation of traditional courses, serving as the basis for an environment focused on active learning. For example, in 1989 the mathematics and statistics department at Mount Holyoke College created a new sophomore-level course for their majors that they called the Laboratory in Mathematical Experimentation, or, for short, “the Lab”. The course consisted of six to seven mathematical labs in which students were given a problem to explore, usually with a computer (or calculator) and programs already written by the instructors. Students would use the results of their experiments to make and test conjectures, and then ultimately write arguments to justify some of their conjectures. The course succeeded “beyond any of [the faculty’s] expectations.” Students became more likely to engage with mathematics actively, and did better in their upper-division analysis and algebra courses than students who did not take the course. The labs for this course were eventually distilled into a book, *Laboratories in Mathematical Experimentation* [24], from which the above historical summary was taken. While the original computer code was written in BASIC, mathematicians have adapted the code to other languages such as *Mathematica* (and even improved it on the way). Students typically write up the results of each lab, and this is where they get to practice writing mathematics. In order for students to succeed in this type of course, they are forced to abandon the common misconception that mathematics consists of nothing more than applying formulas the teacher gives you. Another example of a laboratory-style course, influenced by the Mount Holyoke approach, is given by Brown [25] in an article regarding the recent development of a course on experimental mathematics suitable for both mathematics majors and students fulfilling a general education requirement.

**Inquiry-Based Learning**

Arguably the most well-known example of active learning in mathematics is Inquiry-Based Learning (IBL). Recent research studies have found that IBL courses have a positive effect on students, with particularly strong benefits for low-achieving students [31]. In mathematical culture, IBL (sometimes incorrectly identified as synonymous with the “Moore-method”) has its roots in the teaching methods of R.L. Moore, whose teaching methods were extremely beneficial for some students. However, his overt racism and bias in his classroom precluded many students from participating in his classes [27]. This tension has led some mathematicians to be caught between a desire to use and promote IBL methods and a desire to remove any suggestion of acceptance of the negative aspects of Moore’s teaching [28], [29], [30], a situation the mathematical community needs to resolve.

One of the main organizations promoting IBL is the Educational Advancement Foundation (EAF), which holds an annual “Legacy of R.L. Moore Conference” each summer. Despite the tension surrounding this naming, the EAF has been by far the largest promoter of IBL (which is now much more broadly construed) in mathematics. In addition to their summer conference, now a vibrant meeting full of early-career faculty eager to learn and share best-practices related to IBL, they sponsor both large grants and small grant programs through the Academy of Inquiry Based Learning (AIBL). As an example of a classroom environment that falls close to the aforementioned “all student discovery”, AIBL describes a “typical” day in an IBL class:

Class starts. The instructor passes out a signup sheet for students willing to present upcoming problems. The bulk of the time is spent on student presentations of solutions/proofs to problems. Students, who have been selected previously or at the beginning of class, write proofs/solutions on the board. One by one, students present their solutions/proofs to their class. The class as a group (perhaps in pairs) reviews and validates the proofs. Questions are asked and are either dealt with there or the presenter can opt to return with a fix at the next class period. If the solution is approved as correct by the class, then the next student presents his/her solution. This cycle continues until all students have presented. If the class cannot arrive at a consensus on a particular problem or issue, then the instructor and the class devise a plan to settle the issue. Perhaps new problems or subproblems are written on the board, and the class is asked to solve these. Teaching choices include pair work immediately or asking students to work on the new tasks outside of class, with the intention of restarting the discussion the next time. If a new unit of material is started, then a mini lecture and/or some hands-on activities to explore new ideas and definitions could be deployed. If no one has anything to present OR if everyone is stuck on a problem, pair work or group work can be used to help students break down a problem and generate strategies or ways into solving a particularly hard problem.

Note that while this clearly falls toward one end of the active learning spectrum discussed previously, this does not describe a classroom consisting of pure, unguided student discovery. Rather, students are provided direction through a scaffolded series of activities, some independent, some in pairs, some in small groups, and some with the whole class, including mini-lectures as appropriate. Faculty teaching in an IBL environment need to develop facility with a range of teaching strategies, and need to develop familiarity with many “teaching moves” that are not typically used in lecture environments. Opportunities exist for faculty to receive training in these areas, for example through workshops and minicourses at the Joint Mathematics Meetings and MathFest, or through workshops sponsored by the Academy for Inquiry-Based Learning and other organizations.

The other aspect of IBL that requires attention from faculty is the scaffolding of content. Fortunately, many existing resources are available for faculty interested in teaching an IBL course. The Journal of Inquiry-Based Learning in Mathematics contains refereed course notes on a variety of topics, ranging from first-semester calculus to modern algebra to real analysis to mathematics for elementary school teachers. These notes contain sequences of tasks carefully designed to guide students through an area or topic of mathematics. There are also many excellent freely available texts that are suitable for IBL use contained on independent websites, such as the Active Calculus textbook series and Ken Bogart’s guided inquiry combinatorics text. Published textbooks also exist to support IBL courses, e.g. in number theory [32] and algebraic geometry [33].

**Conclusion**

Active learning is hard to define, but at its core is having students work on mathematical tasks of varying levels of cognitive demand during class. As we have seen in this survey, there are multiple teaching environments in which active learning can be used, and multiple active learning techniques through which student tasks can be provided. However, thus far in our series on active learning we have avoided discussion of a fundamental truth: learning to effectively design and use active learning techniques is challenging, and the process of integrating these activities into one’s “teaching toolbox” requires both patience and a willingness to persist through setbacks. In this way, the process of developing and implementing new pedagogical tools is akin to the process of learning and discovering mathematics.

In the three remaining articles in this series on active learning, we will direct our attention to the ways in which personal experiences can shape and affect our development and choices as teachers. In Part IV of this series, we, the authors, will reflect on aspects of our personal experiences as teachers who have struggled to find effective ways to engage students. In Part V, we will explore the role of “telling” in the mathematics classroom and gain a better understanding of the subtle ways in which instructor lecture, student activities, and constructivist educational philosophies can support each other. In Part VI, our sixth and final article on this topic, we will consider the ways in which professional training as a mathematician can be both a benefit and a hindrance to broadening and developing as a teacher of mathematics.

**References**

[1] Scott Freeman, Sarah L. Eddy, Miles McDonough, Michelle K. Smith, Nnadozie Okoroafor, Hannah Jordt, and Mary Pat Wenderoth. Active learning increases student performance in science, engineering, and mathematics. *Proc. Natl. Acad. Sci. U.S.A. *2014, 111, (23) 8410-8415

[2] Bain, Ken. *What the Best College Teachers Do*. Harvard University Press, 2004.

[3] Bressoud, D., V. Mesa, C. Rasmussen. Insights and Recommendations from the MAA National Study of College Calculus. MAA Press, 2015.

[4] Artemeva, N., & Fox, J. (2011). The writing’s on the board: the global and the local in teaching undergraduate mathematics through chalk talk. *Written Communication*, 28(4), 345-379.

[5] Fukawa-Connelly, T. P. (2012). A case study of one instructor’s lecture-based teaching of proof in abstract algebra: making sense of her pedagogical moves. *Educational Studies in Mathematics*, 81(3), 325-345.

[6] Marrongelle, Karen and Rasmussen, Chris. Meeting New Teaching Challenges: Teaching Strategies that Mediate between All Lecture and All Student Discovery. *Making the Connection: Research and Teaching in Undergraduate Mathematics Education.* Carlson, Marilyn P. and Rasmussen, Chris, eds. MAA Notes #73, 2008. pp 167-177.

[7] Deslauriers, L., E. Schelew, and C. Wieman. Improved Learning in a Large-Enrollment Physics Class. *Science*. Vol. 332, 13 May, 2011, 862-864.

[8] Bressoud, David. The Worst Way to Teach. *MAA Launchings Column*, July 2011. https://www.maa.org/external_archive/columns/launchings/launchings_07_11.html

[9] Prather, E., & Brissenden, G. (2008). Development and application of a situated apprenticeship approach to professional development of astronomy instructors, Astronomy Education Review, 7(2), 1-17. http://astronomy101.jpl.nasa.gov/files/Situated%20Apprentice_AER.pdf

[10] Cline, Kelly Slater Cline and Zullo, Holly, (eds). *Teaching Mathematics with Classroom Voting: With and Without Clickers*. MAA Notes #79, 2011

[11] Julee B. Waldrop, Melody A. Bowdon, (eds). *Best Practices for Flipping the College Classroom*. Routledge, 2015.

[12] Robert Talbert (2014) Inverting the Linear Algebra Classroom, *PRIMUS*, 24:5, 361-374.

[13] Jean McGivney-Burelle and Fei Xue (2013) Flipping Calculus, *PRIMUS*, 23:5, 477-486.

[14] Pari Ford (2015) Flipping a Math Content Course for Pre-Service Elementary School Teachers, *PRIMUS*, 25:4, 369-380.

[15] Betty Love, Angie Hodge, Neal Grandgenett & Andrew W. Swift (2014) Student learning and perceptions in a flipped linear algebra course, International Journal of Mathematical Education in Science and Technology, 45:3, 317-324.

[16] Veselin Jungić, Harpreet Kaur, Jamie Mulholland & Cindy Xin. On flipping the classroom in large first year calculus courses. *International Journal of Mathematical Education in Science and Technology*. Volume 46, Issue 4, May 2015, pages 508-520.

[17] Jennifer R. Winquist and Kieth A. Carlson. Flipped Statistics Class Results: Better Performance Than Lecture Over One Year Later. *Journal of Statistics Education*. Volume 22, Number 3 (2014).

[18] Eric Alan Eager, James Peirce & Patrick Barlow (2014) Math Bio or Biomath? Flipping the mathematical biology classroom. *Letters in Biomathematics.* 1:2, 139-155

[19] Corey Webel, Erin Krupa, Jason McManus. Benny goes to college: Is the “Math Emporium” reinventing Individually Prescribed Instruction? *Math*AMATYC* Educator*, May 2015, Vol. 6 Number 3.

[20] Bressoud, David. The Emporium. *MAA Launchings Column*, March 2015. http://launchings.blogspot.com/2015/03/the-emporium.html

[21] Daniel de Vise. “At Virginia Tech, computers help solve a math class problem.” The Washignton Post. April 22, 2012. https://www.washingtonpost.com/local/education/at-virginia-tech-computers-help-solve-a-math-class-problem/2012/04/22/gIQAmAOmaT_story.html

[22] Neil Marshall, Chantal Buteau, Daniel H. Jarvis, Zsolt Lavicza. Do mathematicians integrate computer algebra systems in university teaching? Comparing a literature review to an international survey study. *Computers & Education*, Volume 58, Issue 1, January 2012, Pages 423-434

[23] Barbara L. Robinson and Anne H. Moore. The Math Emporium: Virginia Tech, in *Learning Spaces*, Oblinger, Diana G. (ed). Educause, 2006.

[24] Cobb, G., G. Davidoff, A. Durfee, J. Gifford, D. O’Shea, M. Peterson, Pollatsek, M. Robinson, L. Senechal, R. Weaver, and J. W. Bruce. 1997. *Laboratories in Mathematical Experimentation: A Bridge to Higher Mathematics*. Key College Publishing.

[25] David Brown (2014) Experimental Mathematics for the First Year Student, *PRIMUS*, 24:4, 281-293. http://faculty.ithaca.edu/dabrown/docs/scholar/experimental.pdf

[26] Lesser, L. (2011). Low-Tech, Low-Cost, High-Gain, Real-Time Assessment? It’s all in the cards, easy as ABCD! *Texas Mathematics Teacher*, 58(2), 18-22. http://www.math.utep.edu/Faculty/lesser/LesserABCDcardsTMTpaper.pdf

[27] Reuben Hersh & Vera John-Steiner. *Loving and Hating Mathematics: Challenging the Myths of Mathematical Life*. Princeton University Press, 2011.

[28] Kung, David. Empowering Who? The Challenge of Diversifying the Mathematical Community. Presentation at June 2015 Legacy of R.L. Moore — IBL Conference, Austin, Texas. https://www.youtube.com/watch?v=V03scHu_OJE

[29] Lamb, Evelyn. Promoting Diversity and Respect in the Classroom. *AMS Blog on Math Blogs,* 17 August 2015. http://blogs.ams.org/blogonmathblogs/2015/08/17/promoting-diversity-and-respect-in-the-classroom/

[30] Salerno, Adriana. Talkin’ Bout a Teaching Revolution. AMS PhD+Epsilon Blog, 3 August 2015. http://blogs.ams.org/phdplus/2015/08/03/talkin-bout-a-teaching-revolution/

[31] Kogan, M., & Laursen, S. L. (2014). Assessing long-term effects of inquiry-based learning: A case study from college mathematics. *Innovative Higher Education,* *39*(3), 183-199

[32] David Marshall, Edward Odell, and Michael Starbird, *Number Theory Through Inquiry*. The Mathematical Association of America, 2007.

[33] Thomas Garrity, Richard Belshoff, Lynette Boos, Ryan Brown, Carl Lienert, David Murphy, Junalyn Navarra-Madsen, Pedro Poitevin, Shawn Robinson, Brian Snyder, Caryn Werner. *Algebraic Geometry: A Problem Solving Approach*. American Mathematical Society, Student Mathematical Library, 2013.

*Editor’s note: This is the second article in a series devoted to active learning in mathematics courses. The other articles in the series can be found 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.

**Bloom’s Taxonomy and its Variants**

A well-known and long-established framework in educational psychology is Bloom’s taxonomy [2]. In 1956, Benjamin Bloom and a team of educational psychologists outlined multiple levels of skills in the cognitive domain of learning, increasing from simple to complex. These are often simplified into six skill levels: knowledge, comprehension, application, analysis, synthesis, evaluation. By associating verbal cue words with each level, they categorized test questions over a variety of topics at the college level, and found that *over 95%* of these questions were at the very lowest level, “recall of knowledge” [11, p. 1]. Since these original findings, which were further developed in a second volume published in 1964, the core ideas of Bloom’s taxonomy have been widely used in education across disciplines.

The original taxonomy has been extended and adapted by many researchers in educational psychology. For example, Anderson et al. [1] developed a two-dimensional extension of Bloom’s taxonomy with a cognitive process dimension (remember, understand, apply, analyze, evaluate, create) similar to Bloom’s taxonomy, but also with a knowledge dimension (factual knowledge, conceptual knowledge, procedural knowledge, and metacognitive knowledge) — a taxonomy table encoding this appears below. When categorizing a task by this taxonomy, the cognitive process is represented by the verb used when specifying the task (what the student is doing) and the knowledge process dimension corresponds to the noun (what kind of knowledge the student is working with). In 2002, a special volume of the journal *Theory Into Practice* was devoted to this revised taxonomy; examples of applications of this taxonomy can be found throughout the volume.

Remember | Understand | Apply | Analyze | Evaluate | Create | |

Factual Knowledge | ||||||

Conceptual Knowledge | ||||||

Procedural Knowledge | ||||||

Metacognitive Knowledge |

An important shortcoming of each of these taxonomies for mathematicians is that the specific descriptors used for the different levels aren’t always appropriate for mathematics. For instance, in Bloom’s taxonomy, application comes after comprehension, which does make sense in a general context. But trying to apply this to mathematics, it is too easy to put routine word problems in the “application” category. The idea of “application” in the general sense is to take ideas presented in one context and be able to use them in a somewhat new setting, but in mathematics the word “application” can be used to represent both the development of a mathematical model to fit a situation or data set and the “cookbook” application of a previously-established mathematical model; most word problems in textbooks fit into the latter category.

**Specialized Cognitive Taxonomies and General Student Intellectual Development**

Around the same time as [1], several papers appeared that used taxonomies specialized to mathematics, e.g., [15, 19, 20, 21]. These have the two-dimensional nature of [1], with the columns or verbs replaced by labels that are specific to mathematics, while the rows or nouns simply correspond to different topics in mathematics. In 2006, Andrew Porter [14] explained it this way:

Unfortunately, defining content in terms of topics has proven to be insufficient at least if explaining variance in student achievement is the goal [9]. For example, knowing whether or not a teacher has taught linear equations, while providing some useful information, is insufficient. What about linear equations was taught? Were students taught to distinguish a linear equation from a non-linear equation? Were students taught that a linear equation represents a unique line in a two space and how to graph the line? For every topic, content can further be defined according to categories of cognitive demand. In mathematics cognitive demand might distinguish memorize; perform procedures; communicate understanding; solve non-routine problems; conjecture, generalize, prove.

More details about this taxonomy of levels of cognitive demand can be found in [15]. A comparison of various such taxonomies can be found in [15].

Similarly, several papers of Mary Kay Stein and various co-authors [19, 20, 21] analyze mathematical tasks and how they are implemented, focusing on middle school, using four levels of cognitive demand: Memorization; procedures without connections; procedures with connections; and “doing mathematics”. They identify the first two levels as “low-level”, matching the first two levels of [15]; and they identify the last two levels as “high-level”, matching the last three levels of [15].

There are also broad models for student intellectual development across not only individual topics but their entire college experience. One of the first such models is due to William Perry, and it can be (overly) simplified into the following description. Most college students will begin with the belief that there are right and wrong answers to questions, and that professors hold the knowledge of which these are. As students progress through their studies, they realize that sometimes their teachers are not always aware of the answers to questions, and also that answers can be more subtle than merely “right” or “wrong.” After this realization, students often enter a phase of relativism, where everyone’s opinions are equally valid. In the final stages of intellectual development, students recognize that different areas of intellectual inquiry have different standards and (some students) develop a balance between intellectual independence and commitment to the discipline. The Perry model has been refined and revised by many psychologists to account for diverse student experiences with respect to gender and other factors; an excellent survey of these developments, with pedagogical implications, has been given by Felder and Brent [6,7].

As Thomas Rishel points out [16], students in the early stages of the Perry model or one of its variants often enjoy mathematics precisely because all the answers are perceived as known, and they frequently value mathematical problems that focus on verification of these truths. As these students begin to encounter complicated modeling problems, or as they are first asked to seriously participate in proof-based mathematical reasoning, the cognitive load of such tasks can be much higher than for students who have developed further along this model. Thus, the intellectual stage of development for a given student can impact the level of cognitive demand for various tasks and problems they will encounter in mathematics courses.

**Practical Issues: Level Identification and Task Assessment**

Given these theoretical frameworks for both cognitive engagement and intellectual development, a practical challenge for instructors is to use these frameworks effectively to increase the quality of teaching and learning in the classroom.** **With any of the cognitive taxonomies, it can be hard to assess precisely which level(s) a given student task is hitting. The taxonomy tables discussed in previous sections provide instructors with tools to produce reasonable cognitive demand analysis of the tasks they give students. Engagement with all cognitive levels is necessary for deep learning to take place, so it is important that mathematics faculty identify and provide students with tasks representing a range of levels. Since lower-level tasks are typically already most prevalent, and easiest to assess both in terms of time and resources, faculty have to make the effort to bring in the higher levels. As a result, three challenges for instructors are to identify high-quality mathematical activities for students at higher levels of cognitive demand, to develop methods for assessing student work on such activities, and to create or make use of institutional programs, culture, and resources to support the use of high-quality activities. We will comment on the third issue in our next article in this series.

Some mathematics problems afford a wide range of cognitive engagement. For example, in the K-12 setting Jo Boaler and others have promoted activities described as “low-floor, high-ceiling” (LFHC) [23]. These are activities that can give students practice in lower levels of cognitive demand, but also are open-ended enough to eventually lead to (grade-appropriate) mathematical investigations with high-cognitive demand. Good examples of problems that students can engage with all the way from elementary school procedures to the highest levels of cognitive demand, leading to college-level abstract topics, can be found on the youcubed website, on sites for Math Circles, and on sites for Math Teachers’ Circles. When students are working on LFHC problems, they have flexibility in how they navigate through the problem. Unless explicit guidance is given regarding how students should investigate a LFHC problem, it is possible for them to spend most of their time working inside a small range of cognitive demand. Consequently, it is important for instructors to provide some pathways or scaffolding for students to use when first engaging with such problems.

Though they are not as common as they deserve to be, mathematicians have developed a wide range of techniques for assessing high-cognitive demand tasks, including written assignments, group work, projects, portfolios, presentations, and more [3, 4, 10, 12, 13]. However, task-appropriate techniques for assessing a given high-cognitive demand task can be challenging to identify and put in practice. It is important that the method for assessing specific tasks be selected in the context of overall course assessment. Some mathematicians have been experimenting with grading schemes that more directly support high-cognitive demand assignments, such as specifications grading and standards-based grading. Unfortunately, the fact remains that there is much to be learned about the efficacy of different methods of assessment [17].

**Active Learning and Theories of Learning**

Implicit in our discussion has been an important point that should be made explicit: as Stein et al. state [19], “…cognitive demands are analyzed as the cognitive processes in which students actually engage as they go about working on the task” as opposed to what students witness others doing. Thus, it is not possible to discuss the cognitive level of a mathematical proof itself, though proofs certainly vary in level of sophistication. Rather, one focuses on the cognitive level of what a student is asked to do with the proof: memorize the proof verbatim? construct a concrete example illustrating the proof method? derive a similar result using the same technique? analyze the proof in order to identify the key steps? compare the proof to a different proof of the same result? These tasks are all different from the perspective of cognitive demand, hence they are not interchangeable from the perspective of student learning, yet they exhibit superficial similarities and would each generally be considered valuable for students to complete. It is worth remarking that the verbs used in describing each of the tasks are helpful indicators of level of cognitive demand, as the taxonomies suggest.

This observation brings us back to active learning, which by definition has as a primary goal to engage students through explicit mathematical tasks in the classroom, in view of peers, instructors, and teaching assistants. One major effect of active learning techniques is that the mathematical processes and practices of students, which are tightly interwoven with high-cognitive achievement, are brought into direct confluence with peer and instructor feedback. Thus, active learning techniques complement the shift in emphasis described by Schoenfeld, from received knowledge to committed engagement, of the primary goal of student learning. Active learning techniques are also well-aligned with contemporary theories of learning, for example constructivism, behaviorism, sociocultural theory, and others [18, 22].

As one example of this alignment, constructivism is based on the idea that people construct their own understanding and knowledge through their experiences rather than through the passive transfer of knowledge from one individual to another. This is a prominent theory of learning among mathematics education researchers with many refinements, e.g. radical constructivism remains agnostic about whether there actually is any objective truth/reality, while social constructivism views individual thought and social interaction as inseparable with no model for a socially isolated mind. Generally, constructivism’s emphasis on the actions of the learner reinforces the need to emphasize consideration of the cognitive demands placed on students.

**Conclusion**

As research regarding the teaching and learning of postsecondary mathematics and science matures and becomes more well-known, both inside the mathematics community and beyond, significant evidence is building that active learning techniques have a strong impact on student achievement on high-cognitive demand tasks. We began this article with recognition that mathematicians are fully aware of students’ difficulties with mathematical tasks at all levels, and the observation that the mathematical community has developed language such as “mathematical maturity” and “elegance” which is often applied to distinguish successful from unsuccessful student work. Our main purpose in writing this survey of concepts related to levels of cognitive demand is to introduce mathematicians to the rich and complex set of ideas that have been developed in an attempt to distinguish different types of student activities and actions related to learning. Given the current evidence supporting the positive impact of active learning techniques, mathematics faculty will have an increased need for a refined language with which to discuss both the successes and failures of our students and the efficacy of the large variety of active learning techniques that are available. In our next post in this series, we will discuss the most prominent of these active learning techniques and environments with an eye toward both institutional constraints (as discussed in Part I of this series) and student learning in the context of levels of cognitive demand.

**References**

[1] Anderson, L.W. (Ed.), Krathwohl, D.R. (Ed.), Airasian, P.W., Cruikshank, K.A., Mayer, R.E., Pintrich, P.R., Raths, J., & Wittrock, M.C. *A taxonomy for learning, teaching, and assessing: A revision of Bloom’s Taxonomy of Educational Objectives (Complete edition)*. New York: Longman. 2001

[2] Bloom, Benjamin, et al., eds. *Taxonomy of Educational Objectives: the classification of educational goals. Handbook I: Cognitive domain.* New York: Longmans, Green. 1956

[3] Benjamin Braun. Personal, Expository, Critical, and Creative: Using Writing in Mathematics Courses. *PRIMUS*, 24 (6), 2014, 447-464.

[4] A. Crannell, G. LaRose, and T. Ratliff. *Writing Projects for Mathematics Courses: Crushed Clowns, Cars, and Coffee to Go.* Mathematical Association of America, 2004.

[5] J. Epstein. 2013. The Calculus Concept Inventory—Measurement of the Effect of Teaching Methodology in Mathematics. *Notices of the American Mathematical Society.* 60 (8), 1018–1026.

[6] Felder, Richard M. and Brent, Rebecca. The Intellectual Development of Science and Engineering Students. Part 1: Models and Challenges,* Journal of Engineering Education*, Volume 93, Issue 4, October 2004, 269–277.

[7] Felder, Richard M. and Brent, Rebecca. The Intellectual Development of Science and Engineering Students. Part 2: Teaching to Promote Growth,* Journal of Engineering Education*, Volume 93, Issue 4, October 2004, 279–291.

[8] Scott Freeman, Sarah L. Eddy, Miles McDonough, Michelle K. Smith, Nnadozie Okoroafor, Hannah Jordt, and Mary Pat Wenderoth. Active learning increases student performance in science, engineering, and mathematics. *Proc. Natl. Acad. Sci. U.S.A. *2014, 111, (23) 8410-8415

[9] Gamoran, A., Porter, A.C., Smithson, J., & White, P.A. (1997, Winter). Upgrading high school mathematics instruction: Improving learning opportunities for low-achieving, low-income youth. *Educational Evaluation and Policy Analysis, 19*(4), 325-338.

[10] Bonnie Gold, Sandra Z. Keith, William A. Marion, (eds). *Assessment Practices in Undergraduate Mathematics*. Mathematical Association of America Notes #49, 1999.

[11] Karin K. Hess. Exploring Cognitive Demand in Instruction and Assessment. National Center for Assessment, Dover, NH 2008. http://www.nciea.org/publications/DOK_ApplyingWebb_KH08.pdf

[12] Reva Kasman. Critique That! Analytic writing assignments in advanced mathematics courses. *PRIMUS* XVI (2006) 1–15.

[13] John Meier and Thomas Rishel. *Writing in the Teaching and Learning of Mathematics*. MAA Note #48, 1998.

[14] Porter, Andrew. Curriculum Assessment, In J. L. Green, G. Camilli, & P. B. Elmore (Eds.), *Complementary methods for research in education (3rd edition)*. Washington, DC: American Educational Research Association, 2006. http://www.andyporter.org/sites/andyporter.org/files/papers/CurriculumAssessment.pdf

[15] Andrew C. Porter and John L. Smithson. Defining, Developing, and Using Curriculum Indicators. CPRE Research Report Series RR-048, December 2001. Consortium for Policy Research in Education University of Pennsylvania Graduate School of Education. https://secure.wceruw.org/seconline/Reference/rr48.pdf

[16] Rishel, Thomas. *Teaching First: A Guide for New Mathematicians*. MAA Notes #54, 2000

[17] Schoenfeld, Alan H., ed. *Assessing Mathematical Proficiency*. MSRI Book Series, Volume 53, 2007.

[18] B. Sriraman, & L. English (Eds.). *Theories of mathematics education.* New York: Springer, 2010.

[19] Mary Kay Stein, Barbara W. Grover, and Marjorie Henningsen. Building Student Capacity for Mathematical Thinking and Reasoning: An Analysis of Mathematical Tasks Used in Reform Classrooms. *American Educational Research Journal*, Vol. 33, No. 2 (Summer, 1996), pp. 455-488

[20] Stein, Mary Kay and Smith, Margaret Schwan. “Mathematical Tasks as a Framework for Reflection: From Research to Practice.” *Mathematics Teaching in the Middle School,* Vol. 3, No. 4 (January 1998), pp. 268-275

[21] Stein, Mary Kay and Smith, Margaret Schwan. “Reflections on Practice: Selecting and Creating mathematical Tasks: From Research to Practice.” *Mathematics Teaching in the Middle School*, Vol. 3, No. 5 (February 1998), pp. 344- 350

[22] T. Rowland & P. Andrews (Eds.). *Master class in mathematics education: International perspectives on teaching and learning*. London: Continuum Publishers, 2014.

*Editor’s note: This is the first article in a series devoted to active learning in mathematics courses. The other articles in the series can be found here.*

“…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, et.al, 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.

**What is Active Learning?**

The core tenet of active learning is that providing students with opportunities to actively engage with content during their classes leads to positive learning outcomes. In mathematics, the phrases “active learning” and “inquiry-based learning” (IBL) are closely related, though opinions vary regarding the extent to which they are related or overlap. Here are some particularly insightful descriptions of active learning and IBL from the literature.

Active learning is generally defined as any instructional method that engages students in the learning process. In short, active learning requires students to do meaningful learning activities and think about what they are doing. While this definition could include traditional activities such as homework, in practice active learning refers to activities that are introduced into the classroom. The core elements of active learning are student activity and engagement in the learning process. Active learning is often contrasted to the traditional lecture where students passively receive information from the instructor.

— Does Active Learning Work? A Review of the Research, Michael Prince, J. Engr. Education, 93(3), 223-231, 2004In the context of mathematics, IBL approaches engage students in exploring mathematical problems, proposing and testing conjectures, developing proofs or solutions, and explaining their ideas. As students learn new concepts through argumentation, they also come to see mathematics as a creative human endeavor to which they can contribute. Consistent with current socio-constructivist views of learning, IBL methods emphasize individual knowledge construction supported by peer social interactions.

— Assessing Long-Term Effects of Inquiry-Based Learning: A Case Study from College Mathematics, Marina Kogan & Sandra L. Laursen, Innov High Educ (2014) 39:183–199A student-centered instructional approach places less emphasis on transmitting factual information from the instructor, and is consistent with the shift in models of learning from information acquisition (mid-1900s) to knowledge construction (late 1900s). This approach includes

- more time spent engaging students in active learning during class;
- frequent formative assessment to provide feedback to students and the instructor on students’ levels of conceptual understanding; and
- in some cases, attention to students’ metacognitive strategies as they strive to master the course material.

— Discipline-Based Education Research: Understanding and Improving Learning in Undergraduate Science and Engineering, S. R. Singer, N. R. Nielsen, and H. A. Schweingruber (eds.), National Research Council, The National Academies Press, 2012

The PNAS authors do not attempt to define active learning, but instead consider in their analysis “papers representing a wide array of active learning activities, including vaguely defined ‘cooperative group activities in class,’ in-class worksheets, clickers, problem-based learning (PBL), and studio classrooms, with intensities ranging from 10% to 100% of class time.”

In alignment with the broad picture painted by the descriptions just given, *our approach is to use the phrase “active learning” to represent any classroom strategy that provides students with opportunities to directly engage with content during class, whether individually or collaboratively with peers.* Avoiding a singular definition of active learning increases the risk of faculty, administrators, and other stakeholders “speaking past” one another. However, as we discuss next, we believe it is important to emphasize the multiplicity of approaches to increased student engagement and to emphasize the need for clear language when discussing different classroom environments.

It is important to observe that mathematics education researchers have investigated the impact of active learning techniques on mathematical learners for decades, especially at the K-12 level. See the paper “Active Learning in a Constructivist Framework” by Anthony, listed in the references, for an example from the mid-1990’s that contains a nice exposition of different interpretations of “active learning.” We are choosing to emphasize in this article that the effects of active learning transcend disciplines, and that student-centered pedagogical techniques are currently the subject of a broad discussion across the sciences.

**Important Considerations**

The following three fundamental issues must be considered when implementing or supporting active learning strategies. These issues complicate our ability to have a coherent national dialogue regarding postsecondary mathematics teaching, and are a frequent source of confusion among different stakeholders in higher education at the national, regional, and local level.

*Classroom Environment*: Often as a result of factors beyond the control of individual faculty (or even departments), classroom environments vary wildly from institution to institution. “Typical” class sizes can run from fifteen to six hundred, with varying levels of grading support. In environments where courses are often taught in a hybrid fashion, meaning a mix of in-person contact time and online modules, contact time is structured differently than in a traditional three-to-five hour per week class meeting structure. All of these considerations and more impact the choices of active learning strategies available for a course or institution, through both restrictions on the type of direct interactions available and enrichments of the type of technology-driven interactions available.

*Teaching Environment*: Mathematics faculty experience an incredibly diverse range of employment conditions. In contrast to public stereotypes of tenure-stream faculty at research-intensive institutions, postsecondary mathematics teachers include both long-term faculty and part-time or adjunct faculty, tenure-stream and non-tenure stream, with many different administrative job requirements and varying levels of support for pedagogical innovation. This range of faculty profiles creates an equally broad range of needs regarding how pedagogical training and mentoring is delivered, and raises questions such as: how much preparation time do faculty have available? Are the courses under consideration being taught by experienced faculty or those teaching for the first or second time? Do faculty performance evaluations and/or job renewals depend on consistently satisfactory student evaluations? Has the institution in question had any historical focus regarding the teacher-training of new faculty hires?

*Course/Student Goals*: Learning outcomes for courses, and the student expectations accompanying them, vary dramatically among faculty, courses, and institutions. For example, courses that primarily serve as part of a general education or quantitative literacy component will typically have fundamentally different goals and expectations for students than courses that primarily serve as a pathway to STEM majors. Individual faculty often have distinct models of student learning, ranging from a view of teaching/learning as the transfer of knowledge and facts to the view of developing students’ ability to solve new problems and/or grapple with and develop understanding of new ideas. Many faculty have different expectations for students regarding the level of cognitive tasks that they are expected to carry out, and often these expectations are implicit in the way they structure their course rather than explicitly communicated to students and peers.

**Aspects Related to Efficacy and Public Policy**

Several additional issues are directly brought up in the PNAS article. The strength of their results led the PNAS authors to suggest that “STEM instructors may begin to question the continued use of traditional lecturing in everyday practice.” Having said that, they point out that to date, active learning has been implemented primarily by faculty interested in experimenting with new pedagogical strategies. What is not clear is if the high efficacy levels observed so far for active learning techniques would be seen if implemented by almost all mathematics faculty. We feel this is a key question, especially if the use of active learning strategies are mandated without a robust support/reward system and full recognition that transitioning to new pedagogical techniques is never a smooth or effortless process. It would also be worthwhile to compare the efficacy of different active learning techniques in mathematics, in analogy to the work of Prince provided in the references.

Finally, the PNAS authors point out that increased student learning as a result of active learning techniques will lead to increased student success rates, thus resulting in fewer repeats of mathematics and science courses. This has the potential to save students significant amounts of time and tuition. Active learning also has been found to have a disproportionately beneficial effect on members of minoritized groups in STEM fields, revealing that fundamental issues regarding equity are at hand. In addition to the ethical and moral questions these points raise for the mathematics teaching community, these qualities of active learning are drawing the attention of individuals involved in student advocacy, public policy, grant and scholarship funding, and related fields. We believe that increased support and attention to the success of our students from people outside the mathematical teaching community should be welcomed, and that an inclusive discussion of how to best help students learn mathematics at a deep level will lead to a richer teaching and learning experience for all.

**A Preview**

In the remaining articles in this series, we will explore how different tasks carry varying levels of cognitive demand on the part of students, and we will provide examples of active learning techniques that address these levels in various teaching and classroom environments. We will share some of our personal experiences as teachers who have experimented with pedagogical strategies, and consider the difficult issue of finding balance between providing students with time for exploration and providing students with direct feedback and instruction. Finally, we will discuss potential points of tension between scholarly training in research-level mathematics and scholarly development of pedagogical strategies and techniques.

**References**

Glenda Anthony

*Active Learning in a Constructivist Framework
*Educational Studies in Mathematics, Vol. 31, No. 4 (Dec., 1996), pp. 349-369

Scott Freeman, Sarah L. Eddy, Miles McDonough, Michelle K. Smith, Nnadozie Okoroafor, Hannah Jordt, and Mary Pat Wenderoth

*Active learning increases student performance in science, engineering, and mathematics
*

Marina Kogan & Sandra L. Laursen

*Assessing Long-Term Effects of Inquiry-Based Learning: A Case Study from College Mathematics
*Innov High Educ (2014) 39:183–199

Michael Prince

*Does Active Learning Work? A Review of the Research
*Engr. Education, 93(3), 223-231, 2004

R. Singer, N. R. Nielsen, and H. A. Schweingruber (eds.)

*Discipline-Based Education Research: Understanding and Improving Learning in Undergraduate Science and Engineering
*National Research Council, The National Academies Press, 2012

“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.

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I first discovered the work of Carol Dweck from a link on Terry Tao’s blog. Dweck is a psychologist at Stanford whose studies on the relationship between self-beliefs and achievement have had a tremendous impact in education and beyond. Her message is simple: when people, whether students or otherwise, believe that they are capable of improving their abilities through hard work and sustained effort, then they achieve more than when they believe they have innate abilities that will at some point be reached. In other words, if you believe that failure is a natural part of growth and development, then you are more likely to persist through failure and setbacks. On the other hand, if you believe you succeed because you are smart, then when you experience a failure, even a small one, you likely conclude that you are not actually smart and give up as a result. The former belief is referred to as a “growth” mindset, while the latter is called a “fixed” mindset.

Dweck’s work fit naturally in the context of math education research I had been reading regarding the use of active learning, and it also fit with my personal experience. As an undergraduate, I majored in English composition and Mathematics, originally intending to be a high school teacher. While I was interested and active in the math club and math competitions, I wasn’t a particularly strong math student, earning a mixture of A’s, B’s and a C in math major courses. After completing my degree, I took a job at a planetarium as a low-level manager. I had previously considered going on to graduate school in mathematics, so in my spare time I read about math and science. Slowly, through a series of fortunate moments, I came to understand the depth of my lack of mathematical knowledge. I found out that \(x^2+y^2=1\) is the equation for the unit circle because of the Pythagorean theorem; in high school, this had simply been presented as a fact. I was cleaning up the science demo room in the planetarium one day when a NASA video about trigonometric functions came on — I quietly closed the doors and watched for 45 minutes, taken aback by the beautiful connection between sine, cosine, tangent, their graphs, and the unit circle, which I had never seen.

The next year I started graduate school. On a regular basis I told my wife that I was definitely the dumbest person in my complex analysis course, but that I was doing my best anyway. On more than one occasion I sat on the living room floor and burst into tears, overwhelmed by the stress of trying to understand the barrage of ideas one encounters in graduate mathematics courses. I spent a lot of time in the math library that first year in graduate school, reading books like Serge Lang’s *Basic Mathematics* (a high-school text) and Liping Ma’s *Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States* (about elementary school mathematics); it was at this time that I came to understand the importance of the distributive law and its role in the multi-digit multiplication algorithm. It was also at this time that I became deeply aware of how I had been doing math without real understanding, demonstrating high-level mathematical knowledge without substance. Only as time passed did I realize how many students of mathematics, even successful students, operate in this way.

**************

When I first read Dweck’s work on mindsets, I had already begun using various pedagogical and assessment techniques: active learning through small group work, reflective essays as homework, semester-long individual projects. While students responded well to this, I was never satisfied at the end of the course. There were too many students who didn’t develop their understanding of mathematics, who were hesitant to fully engage in the course. I decided to directly intervene, using Dweck’s work as the basis for an explicit discussion of the role of beliefs in learning and achievement.

Because our first-year calculus courses are taught using a large-lecture/recitation, highly coordinated structure, I only felt free to experiment in my smaller upper-level courses for math majors and minors. On the first day of class, I assigned Dweck’s survey article “Is Math A Gift?: Beliefs That Put Females at Risk”; recently I have instead used her Scientific American article “The Secret to Raising Smart Kids.” I also had students write a one-page autobiographical statement about their previous experiences in math courses. The second day of class was devoted entirely to psychological aspects of mathematics: How do you feel about mathematics? Do you actually like it? Do you feel you are good at it? What are the reasons you have succeeded to this point? At first I tried to have discussions with the entire class sitting in a circle, but found that it is much more effective to assign students to groups and have them talk with their peers — I don’t have to hear everything they say in order for the discussion to be meaningful. I found that starting the class this way completely changed the tone of my courses for the better. It was surprising and refreshing to the students for a math course to start in this manner, and it set the stage for our classroom discussions about mathematics to include both technical and psychological aspects.

The biggest surprise I had, and a challenge I still struggle with as a teacher, is the remarkable ability of students to argue in favor of the dominance of innate talent in mathematics. Cultural conditioning regarding the myth of genius is strong and embedded; for many of my students, this had developed into the false belief that the goal of doing math is to be brilliant, rather than to gain reasonable mastery and improve one’s understanding. Students frequently compared doing math to training to be an elite athlete. In more than one small group, in more than one class, I heard statements such as “no amount of hard work will make someone play basketball like Michael Jordan.” The fact that these same students enjoyed playing basketball for the purpose of honing their skills and enjoying the company of friends, rather than becoming a legendary athlete, usually didn’t occur to them until I raised the point explicitly.

**************

Ten months ago, I was grading homework that had asked students to work on a particularly challenging open-ended problem. I was struck by the fact that almost a third of the students made a negative remark such as “this is wrong, I’m an idiot” in their solutions. This type of language is so culturally embedded in our mathematical discourse that we often don’t realize the level of negative self-talk we use. I frequently hear mathematicians make comments like “Oh, I see that now, I should have realized that before” when the reality is that we earn our realizations through effort and persistence, because real mathematical understanding requires time. The day after I graded that homework assignment, I implemented the following course policy, which is now part of every course I teach.

Students are not allowed to make disparaging comments about themselves or their mathematical ability, at any time, for any reason. Here are example statements that are now banned, along with acceptable replacement phrases.

- I can’t do this –> I am still learning how to do this
- That was stupid –> That was a productive mistake
- This is impossible –> There is something interesting and subtle in this problem
- I’m an idiot –> This is going to take careful thought
- I’ll never understand this –> This might take me a long time and a lot of work to figure out
- This is terrible –> I think I’ve done something incorrectly, let me check it again
Please keep in mind the article we read by Carol Dweck. The banned phrases represent having a fixed view of your own intelligence, which does not reflect the reality that you are all capable of dynamic, continued learning. The suggested replacement phrases support and represent having a growth mindset regarding your abilities and your capacity for improvement.

In my most recent courses, I introduced this policy on the second day of class, following our small group discussions of Dweck’s article, and I subsequently enforced it vigorously. Doing so has revealed even further for me the depth of the challenge math teachers face — everything operates against our goal of student learning, even the words and phrases we are subconsciously trained to use. How can I hope to have my students believe in their own abilities, when their default descriptions of their work are derogatory?

I frequently teach courses for pre-service teachers, and one remarkable aspect of building a classroom environment around growth mindsets is the connection to the Standards for Mathematical Practice in the Common Core State Standards for Mathematics. At some point in time during every course that serves preservice teachers, I show students these standards — their typical response is to be shocked that these are required of K-12 students, and also to feel uncertain of how to interpret some of them. These standards both implicitly and explicitly reflect the fact that authentically doing mathematics involves trying, failing, trying again, making mistakes, correcting, and shifting perspective. I have students work on a tough problem in small groups, one I don’t expect them to solve during class or at all, and stop every few minutes to reflect on which of the practice standards they have used, and whether or not there were missed opportunities to bring others into play. I insist that the students not criticize themselves for their missed opportunities, simply acknowledge them and, from that recognition, improve.

**************

It is reasonable to wonder if any of these activities have a meaningful effect on students. Mindset interventions, as they are often called, are being included as part of academic orientations at various universities. I often reflect on the ability my students have demonstrated to resist these messages regarding mathematics, even though various studies provide strong evidence that such interventions improve academic achievement. I wonder: while students’ performance improves after a brief mindset intervention, how much does it change what they believe about the nature of mathematical ability?

I’ve found that the hardest questions to ask students are the ones I most want to know the answer to: *“What are you really thinking? What do you truly understand? What do you believe you are capable of accomplishing?”* To obtain reasonably deep answers to these questions, I decided early in my teaching career that I need to have students write reflective essays in my courses. Here are excerpts from end-of-semester essays that four of my students have allowed me to share.

**Student 1:** I had always conceived of mathematics — and, by extension, science and engineering — as a field advanced by sheer brilliance. Yes, I realized that these fields were more parts failure than success, but nothing has contributed to cementing in my mind that anyone can succeed in any field through hard work and dedication than the Dweck article presented near the beginning of the semester. I have made this an integral message in my private chemistry tutoring; no regular client of mine this semester has managed to escape my spiel about how they can’t allow their fears and lack of confidence to hold them back from working hard to succeed.

**Student 2: **Speaking of teaching, the [Dweck] article that we read at the very beginning of the semester has stuck with me this whole time and is something I want to be sure I keep in mind when I have a classroom of my own. While I would still say that most of the mathematicians we have read about are super geniuses, they did work diligently towards what they wanted to achieve. It stood out to me that when you present these geniuses as people who worked really hard it influenced the students’ thought processes in a positive way, making them more likely to try, whereas when presented with material that said they were geniuses the students took a more negative approach of “I don’t have the gift therefore I can’t do the math.” It will definitely influence the way that I present material in my own future classroom, making sure to focus on working hard rather than “just being good at math.”

**Student 3:** One of the misconceptions I held when I came into this class at the beginning of the semester was that if I had to spend a large amount of time on a problem that meant I was dumb. I don’t believe that I ever voiced that opinion to anyone, but I now know that it was there. And it was because of this and my other math courses this semester, that I only slightly have that thought. I learned the very hard way that good mathematics takes time. I can no longer just plug and chug like I could with calculus or matrix; now I actually have to think about what I’m doing. It was, and still is, a very frustrating feeling, but underneath that feeling is the understanding that sometimes this is just what solving problems is: it’s time, and frustration, and sometimes having a tantrum before the problem can be solved.

**Student 4:** I had always thought that mathematics was a gift. You were either good at math or you were not. It was this reasoning that caused me to believe that some people were born to be mathematicians, while others were doomed to always struggle in mathematics. However, I have seen several people (including myself) in this class go from struggling in math class to having an impressive mathematical skill set. I now see mathematics as more like athletics. While some people are more naturally gifted than others, hard work will pay off in the end. This does not mean that I will not have more challenges. It does mean that I can face those challenges, and that in most cases I can learn the mathematics in order to do what is required.

**************

For reasons that I don’t fully understand, our mathematical culture encourages us to define our mathematical ability by what we don’t know, what we aren’t able to do, rather than by what we do know and have learned how to do. The power of culture is strong, with deep roots — I don’t truly believe that the ripple effect from my teaching will spread very far. Yet I cannot help but think of all the students who persist in mathematics. In spite of so many unspoken doubts, so many negative influences, these students have made their way through the doors to our classrooms. And I cannot help but think of the many thoughtful, capable students who turn away from mathematics and give up hope. We are surrounded by potential, by possibility, by self-inspiration yearning for a spark. I believe that the brightest sparks come from people rather than mathematics. That our thoughts, emotions, and beliefs are the gateway toward a more diverse, equitable, proficient, and beautiful mathematical culture. The key is allowing time for these alongside technical mathematics in our classrooms; real mathematical understanding requires time.

]]>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.

**About Queensborough Community College**

A member of The City University of New York (CUNY), the largest urban university system in the nation, Queensborough Community College truly serves its community. About 84% of Queensborough students live in Queens. Our students come from 139 countries and speak 87 different languages; 31% are Hispanic, 26% are Asian or Pacific Islander, 26% are Black, and 18% are White, reflecting the diversity of Queens and the larger metropolitan area. In 2013-14, 76% of our first-time, full-time freshmen qualified for some form of financial aid. In Fall 2014, 70% of freshmen needed remediation in mathematics, 27.4% in writing, and 23.1% in reading, with some students needing remediation in two (19%) or all three (12%) subjects (2014-2015 Factbook).

As stated in its mission statement, Queensborough is “committed equally to open-admissions access for all learners and to academic excellence within an environment of diversity.” In order to achieve its goals, the institution pays a “focused attention to pedagogy,” creating many opportunities for faculty to get involved in innovative teaching practices.

For several years now, Queensborough has been working on integrating High-Impact Educational Practices into the fabric of the college, a process that started even before George Kuh’s landmark book “High-Impact Educational Practices: What They Are, Who Has Access to Them, and Why They Matter” (AAC&U, 2008).

**High-Impact Educational Practices (HIPs)**

HIPs are pedagogical practices proven to promote student engagement, satisfaction, acquisition of desired knowledge, skills and competencies, persistence, and attainment of educational goals. Well-implemented, HIPs demand considerable time and effort from both instructor and students. They facilitate learning outside of the classroom as well as collaborations and meaningful interactions among participants (e.g. student-student, student-faculty, student-community), and they provide frequent and substantive feedback (Kuh, 2008).

Queensborough has institutionalized seven of ten practices recognized as HIPs: Writing Intensive Courses (WI), Academic Service-Learning (ASL), Learning Communities, Collaborative Assignments and Projects, Common Intellectual Experience (CIE) or “common read,” Undergraduate Research (UR), and Global and Diversity Learning. I have received professional development for and implemented four of these: WI, ASL, CIE, and UR. I have used these HIPs in varied courses: College Algebra, Pre-Calculus, Intro to Probability and Statistics for non-majors; Number Systems for early childhood/elementary education majors; Discrete Mathematics and Probability courses for STEM majors.

In courses where I integrate a HIP (or a combination of HIPs), coursework requires that students use the lens of mathematics to examine real-life, real-time problems (e.g. health care, public education, oil consumption, police-community relations, human rights) and that they educate others and/or report their findings to a real audience of peers, the community, or at a professional conference. I often have the students working in groups. In part, this is done to make my workload more manageable, but group work also allows students to serve as cultural, academic and linguistic resources for one another, and it increases the potential for learning in the affective domain. I always embed the HIP into the course so that all students have to participate, as opposed to making it optional or offering it just to the honors students in a class. Many challenges come with that choice, especially in classrooms as academically diverse as ours, and –every time– I tell myself that next time I am just going to stay on the safe side, using less involved teaching strategies. But when I look back and see how students benefited and how exciting the work was – not just for them but for me – I cannot envision it in any other way and I just consider my next HIP.

**Benefits to students**

Students’ responses to HIPs in my math classes have been positive. More than half of the students report that the HIP made them take more responsibility for their own learning and made them more aware of their strengths and weaknesses, and about 60% report that the HIP component of the course made them understand course material better than from my lectures and readings. About three-fourths of the students report that the HIP made them see how course material can be used in everyday life, and close to two-thirds said that they are more aware of the role of mathematics in disciplines and majors besides their own. A few students report that the experience helped them to clarify their career or specialization choices.

When I ask my students what is “the most valuable lesson” learned while participating in the HIP, their responses are as diverse as they are and have led to memorable reflections and aha! moments. Many students point to lessons learned about teamwork, time management, the use of technology, or the mathematics involved. But others point to areas of student learning and student life that one would hardly ever speak of in a more traditional class, especially in a mathematics class. Here are some examples:

- students who come to class, talk to no one, and leave (common among community college students) start to realize how much they are missing out by not interacting with peers and faculty and discover how rich and rewarding academic learning and college life can be;
- some students emerge as natural leaders, more often than not to their own surprise;
- students start to see mathematics differently and some start talking more comfortably about mathematics, including talking to their children at home;
- students learn that
*they can*. They learn that they can be thrown into totally unfamiliar territory, and that despite their many competing obligations and daunting deadlines, they can deliver. I, too, learn that lesson every time that I work on a new project with a class. While I start the semester thinking that what I am asking students to do is attainable, once a project unfolds I am just another learner, dealing with the uncertainties and unanticipated challenges, so absent from traditional textbook exercises, of working on something*new*and*real*.

I have developed relationships with students that I know would have not occurred in a more traditional setting. I have nominated students for awards and written letters of recommendations for scholarship, transfer, graduate school, or job applications for students who would have never spoken to me had it not have been because of those HIPs. Even if they had approached me, I would have had very little to say about them without those HIPs.

I must concede that I haven’t observed any differences in terms of grades and, to be honest, I am no longer disappointed by that fact. My college’s Office of Strategic Planning, Assessment, and Institutional Effectiveness is currently engaged in a multiyear assessment of the impact of HIPs on student learning outcomes and only time can tell us if there are any significant gains there. In all, my classroom experiences point beyond traditional measures of student academic success and peak into gains in social capital, and that alone makes these practices worth exploring!

What others have found is that HIPs lead to greater engagement and retention among undergraduate students and that they have a profound impact on the experiences of traditionally underserved students (__Finley, A. & McNair, T. 2013;__ Brownell and Swaner, 2010; Kuh, 2008). These studies have, almost entirely, looked at baccalaureate granting institutions and relied on self-reported benefits. Kuh’s work, in particular, drew from datasets of the National Survey of Student Engagement NSSE. As pointed out by Finley and McNair (Finley, 2012; Finley, A. & McNair, T. 2013), there is a gap between what students think they have learned and what students can do, and the authors have called for large scale assessments of learning outcomes and students’ competencies.

It appears that community colleges have much to contribute to the conversation on High-Impact Educational Practices, and much to gain, as a large number of community college students match the profile of those most likely to benefit.

*References:*

Brownell, J. E., & Swaner, L. E. (2010). Five high-impact practices: Research on learning outcomes, completion and quality. Association of American Colleges and Universities.

Finley, A. (2012). *Making progress?: What we know about the achievement of liberal education outcomes.* Association of American Colleges and Universities.

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.

**Practical notes – so what does this look like? **

I have experimented some with peer grading, but I still have more to fine tune in terms of the practical details. There are several possibilities for how it might best work, and I outline some guiding practical principles:

1) *Be transparent about the process*. Before my students begin the peer grading process, we have explicit classroom discussions about the purpose of the activity and what is expected in terms of the nature and the amount of feedback they should give. I understand that learning to give valuable, constructive feedback may take time, and that there will be a range of quality, especially initially. I give students a handout with some examples of what helpful feedback looks like (probing or challenging questions, thoughtful comments with reasons and justifications, catching serious mathematical errors) and what unhelpful feedback looks like (suggestions for changes without explanation or justification, positive comments with no reasoning, missing mathematical mistakes), in order to exemplify an effective critical analysis.

I also check over the peer graders’ comments to ensure that they are putting effort into the process, and this is a small part of their grade. I will typically offer comments and use a check/check plus/check minus system to give feedback to the peer graders (so far this has been sufficient), but one could also come up with a more rigorous feedback system. Also, the peer grader’s comments should have no actual bearing on the other students’ grades, as I view the activity of peer grading as a formative assessment, not a summative one.

2) *Tailor assignments for peer grading*. Not every assignment is appropriate for peer grading, nor should students necessarily grade most or all of the assignments your students complete. I designate a handful of assignments that are explicitly geared toward peer grading. One such assignment might consists of a few proofs (perhaps in the midst of or at the end of a unit on proving) or some carefully selected counting problems that have the potential for being particularly illustrative of important ideas. I intentionally pick tasks that make a nice “peer grading assignment” so as to make the process efficient and effective.

3) *Facilitate variety*. I think there is benefit to letting students see a variety of proofs or problems from a variety of their peers. It is not necessarily helpful if a student grades only one other student’s proofs – rather, the power in the exercise is in being able to see what a handful of other students did. There are many options for how to implement this – a student could grade few problems from many students, or many problems from few students, or several problems from several students. I prefer the third option, which I feel gives students the most opportunity to see a variety of responses. Practically, this may mean that students take shifts in peer grading. In a class of 30, I may have six students grade one peer grading assignment (which would consist of a few problems), so they each grade five classmates’ work. Then, in the next assignment, six different students would do the grading. Over the course of the term, students would get to see a variety of problems, proofs, and topics from many of their peers.

**Different topics, different courses. **

There are a number of different types of courses for which this could be particularly useful. One type of course is proof-based, upper division courses for math majors. These would include courses like discrete mathematics, introductory analysis/advanced calculus, and abstract algebra. It is in courses like these that students are make formative steps in their development as proof writers, and they are also encountering advanced, challenging topics. Students are learning important ideas both about what is convincing (to themselves, to a professor, or to the mathematical community) and also what a proof must include to be convincing and also rigorous. In these courses, students encounter difficult topics and must learn, in a relatively short amount of time, how to solve increasingly abstract problems and how to write proofs that are clear, concise, mathematically correct, convincing, and rigorous.

Another type of course for which I think peer grading would be particularly useful is in preservice teacher courses, whether geared toward elementary, middle, or high school teachers. Using peer grading in such courses not only gains the affordances outlined below, but there is also the pedagogical benefit that future teachers could uniquely appreciate. Peer grading simulates the exact kind of activity teachers will eventually need to do. Thus, gaining experience making sense of, interpreting, and assessing student’s arguments is great practice for what will become a prominent part of their jobs. They have to think hard about what other people are thinking, which is an essential part of being a teacher. Sometimes we try to simulate this kind of activity by providing examples of hypothetical student work that we ask preservice teachers to evaluate, but peer grading could offer a more authentic kind of experience.

In addition, peer grading might also be particularly beneficial for students who will go on to do technical work in industry (such as data scientists, engineers, actuaries, etc.). Experiences with peer grading could help them hone critical analysis skills that such work will require.

**What might be gained from allowing students to see and evaluate their peer’s proofs?**

I offer a few specific benefits that students might experience if they interact meaningfully with other students’ problem solutions and proofs.

1) *Improved conceptual understanding*. Students can gain mathematical insights from their classmates. They may learn a more elegant or clean way of formulating an argument (perhaps demonstrating an efficient total-minus-bad solution to a counting problem), or they may actually be introduced to a brand new mathematical technique or idea (such as a multiplicative expression for the sum of the first *n *natural numbers). Additionally, the student who is grading may have key mathematical ideas reinforced through the process reading through many different proofs or solutions to the same problem. That is, ideas that might have been only partially developed for students as they wrote a proof may now be solidified through revisiting and evaluating many different formulations of the same argument. The point is that mathematical ideas can be strengthened and even developed by immersing oneself in others’ mathematical work.

2) *Enhanced mathematical communication*. In addition to actually learning some mathematics, students can learn valuable lessons about communicating mathematically. By seeing what others say and how they say it, students can appreciate how they themselves communicate. Perhaps they thought they were being clear, but then they read another more articulate response and realize how to say something more clearly. Seeing a variety of proofs or problem solutions can highlight the distinction between an effective bit of communication and a poorly articulated argument.

3) *Renewed empathy for others and increased confidence*. Finally, the experience of grading can help students better understand and appreciate their own struggles and success. For students toward the bottom of the class, they benefit from the mathematical insights and seeing well-formulated proofs that their classmates produce. For students at the top, they can gain empathy for their classmates, appreciating that others might actually be struggling with the material. Also, in my experience, students can tend to lack confidence about their own abilities, and many students are hesitant to raise questions for fear of appearing unintelligent or lost (this is especially true for female math students). However, it is often the case that many students in a class will find an idea challenging, and the activity of peer grading could help to bring this reality to light. Students can realize that others and can be empowered by it – not in a “I can’t believe how stupid so and so is” kind of way, but rather in a “Wow, so it seems that other people are struggling with these topics, too” kind of way. This realization can, in a sense, level the playing field, and can perhaps even boost confidence among students who need it.

I have framed these benefits in terms of how they will help the grader, but note that the student whose work is being graded also stands to benefit from peer feedback. They may realize that ideas they thought they were communicating clearly are actually not easily understood, or they might come to learn particular mathematical topics that they need to work on.

*References:*

Freeman, S. & Parks, J. W. (2010). How accurate is peer grading? CBE Life Sciences Education, 9(4), 482-488. http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2995766/

Peer Assessment. Retrieved from http://ctl.utexas.edu/teaching/assess-learning/feedback/peer-assessment.

Self and Peer Assessment Resources. Retrieved from

http://primas.mathshell.org/pd/modules/7_Self_and_Peer_Assessment/html/index.htm

Sivan, A. (2000). The implementation of peer assessment: an action research approach. Assessment in Education, 7, 2: 193-213.

]]>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 8^{th} 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.

I consider K-12 Outreach to be a partnership with local school districts to improve education, to provide unique learning experiences for everyone involved, and to work collaboratively towards building a future generation of problem solvers. Although this is a broad definition, it allows for a wide range of activities that can help achieve those objectives. The key component for K-12 Outreach is the *partnership*, making genuine connections with superintendents, teachers, principals and students. Approaching the partnership with an understanding that each person has a critical expertise can make a K-12 program a success. I have had some efforts succeed and some that were epic fails, and both relied on trust and appreciation of all the people involved.

The first thing I always do when forming a new partnership is admit that I am by no means an expert (or even qualified) at teaching school-aged children. I rely on teachers to help me understand the appropriate level of material, identify challenging topics that could use more relevance and motivation, and to communicate in a language students can grasp. I remember leading a session about drawing a scaled roller coaster blue print to a group of 7^{th} and 8^{th} graders at our summer camp one year and thinking it was going well. Then I was met with blank stares and nobody knew how to get started. Luckily there was an 8^{th} grade teacher in the room who reworded all my directions for them and they immediately got to work.

Likewise, teachers often do not have the resources or time to learn about how mathematics is being used to solve real world problems. Math modeling, open-ended questions, and interdisciplinary problems within the mathematics classroom are rare, yet emerging, scenarios in K-12 schools. From my experiences, teachers need to be able to trust and feel comfortable asking questions with faculty. When running teacher professional development workshops, I usually have undergraduate and graduate student helpers. They make the setting more comfortable and bridge the gap in terms of technical expertise. In general, I found that using college students can strengthen any level of K-12 Outreach. They usually have great energy and insight and are closer in age to the students we are serving. Participating also is a way to strengthen their resumes and instill an understanding in future mathematicians that K-12 Outreach is valuable.

I have piloted some small-scale efforts as well as participated in both state and federally funded student driven and teacher professional development programs. One effort simply involved organizing an essay contest for local middle school students to celebrate Math Awareness Month (April). Our math club spearheaded the whole thing. We got local businesses to donate prizes (camping equipment, gift certificate to a music shop, sporting equipment) and then students had to relate their essay to how mathematics is used in that arena (for example, why is mathematics important if you are planning a camping trip or how is mathematics used in baseball?) All we really had to do was circulate an announcement to superintendents and then enjoy reading the essays and choosing winners.

There are a variety of pre-existing national STEM programs that provide ways for faculty to make connections with K-12 teachers and students. MATHCOUNTS is a middle school competition that I have been the local director of for the last 11 years. We provide the facilities to hold the annual competition and Clarkson Student volunteers have even worked with teachers throughout the school year to coach teams. In the past, we have also provided one-day workshops for teachers to help them develop coaching activities.

Another opportunity that requires no funding (and actually provides an honorarium!) is to become a judge (or problem author) for the SIAM (Society for Industrial and Applied Mathematics) Moody’s Mega Math (M3) Challenge. The M3 Challenge is a free mathematical modeling competition for high school juniors and seniors held annually in March. Judges have a week to read through roughly forty solution papers online and score them based on a given rubric. See __http://m3challenge.siam.org/__ for more information.

A much more ambitious program is our NYSED STEP (Science Technology Entry Program) after-school and summer camp program, called IMPETUS for Career Success (Integrated Math and Physics for Entry to Undergraduate STEM). This program connects Clarkson faculty and graduate/undergraduate students with 11 school districts and 150+ 7-12 grade students. Highlights are a week-long Summer Roller Coaster Engineering Camp, weekly after-school STEM enrichment activities which include research projects, a model roller coaster design competition, tutoring and mentoring services, and monthly on-campus STEM workshops centered around research and STEM careers, providing a variety of STEM experiences. At camp, students apply math, physics, and simulation with hands-on lab activities. Students predict the behavior of a roller coaster traveling along a wall-mounted track whose shape can be adjusted to accommodate multiple hills, loops, and jumps. Activities include designing a roller coaster from a scaled drawing and wire model that undergoes a complete energy and safety analysis and is simulated via software so students experience their ride virtually and cross-check their velocity and acceleration computations. We also use the VR2002 Virtual Roller Coaster to teach students about accelerations, model predictions, and data analysis. Students visit a Six Flags to collect acceleration data wearing kinematic vest. The highlight is a workshop with a roller coaster engineer who built The Comet. To see more about the scope of this program, see __http://web2.clarkson.edu/projects/impetus/__.

Getting started in K-12 outreach activities seems intimidating but the pay-off is huge. A simple web search will reveal numerous programs, examples, curriculum samples, and funding opportunities that may seem overwhelming. The most important step is getting started and then to keep trying and learning. For my roller coaster camp program, we were denied funding twice before finally being awarded a NYSED grant. Seeing the evolution from the birth of the idea to where we are now is one of the most rewarding experiences of my career. Better yet is seeing what our graduating seniors go on to do. We are in our tenth year and the program is continuously changing and improving. My advice is to start small but think big.

]]>*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.

*Applications*

I wish that I had been introduced earlier and more often to applications, as they would have provided me with a better idea of potential areas of specialization after graduation. For example, in linear algebra we could have learned about the role eigenvectors play in Google’s PageRank algorithm, and in number theory we could have learned about how encryption facilitates e-commerce. My textbooks and courses were mostly filled with theorems, definitions, and proofs, and relatively few examples of applications. With more such examples, I believe that students would think more about the value of a math degree and the growing demand for graduates with a math major. Vassar has recently begun inviting graduates back to talk about their career paths. I wish this program had existed when I was there. I would have also liked to learn more about fields where we are only just starting to discover the prominence of math, such as web development and social networks. Additionally, there are many industries in which math’s long-existing role continues to expand, such as movie animation and national security. Incorporating examples like these into the curriculum shows students that mathematical theories influence new applications, and in turn, new applications drive theoretical research by uncovering additional problems. Perhaps an introductory course focusing on real-world applications (with each unit dedicated to a different field where math is used) could show students more of these connections.

*Other STEM fields*

The mathematical sciences continue to be the foundation for exciting research and development in the other STEM fields. Yet I’m sure there are other math graduates like me who didn’t take classes on these subjects and were surprised at the extent to which these fields are used in their lines of work. Much of the work in rapidly evolving areas such as compressed sensing or drug design is being done by people with a foundation in multiple STEM disciplines. To keep up with the broadening of the mathematical sciences and to equip students for a wider range of careers, I think a requirement to take a course in at least one other math-related discipline would be an asset to majors. Students would also benefit from improved interdepartmental collaboration, which could include joint courses that count for credit in more than one discipline or classes co-taught by professors from different departments. For example, a class using computer science skills to analyze large data sets could be applied towards either a computer science or math major. Freshmen and sophomores would take comfort in knowing that, regardless of which subject(s) they pursue further, their math departments offer many worthwhile options beyond core math classes. I might have double majored, or at least taken more science and technology-based courses, had an environment more like this existed.

*Statistics*

I enjoyed the variety of math courses that I took (e.g., linear algebra, modern algebra, multivariable calculus, number theory, probability, and real analysis), yet I wish I had selected my courses with more regard to post-college interests. If I could redo my undergraduate years, I would take more statistics courses. I think a department requirement would help students recognize how important a statistics background is in increasing their mathematical value, and by extension, their employability in data-driven careers. Without taking statistics, students who end up in mathematical jobs would likely have to teach themselves key concepts and tools, such as modeling via simulation or statistical inference, in the workplace. More data than ever is generated, collected, and used for research in today’s world. Because of this, fields from neuroscience to advertising are looking for employees with statistical and computational expertise. The Mathematical Sciences in 2025 says that by 2018, U.S. businesses will need another 140,000-190,000 employees with advanced quantitative skills and deep analytical talent, and who are adept in working with big data. Courses where students work with large data sets and form their own conclusions would be beneficial. I don’t use high-level statistics in my job, but I use many tools that are honed in a statistics class. My work revolves around organizing, analyzing, interpreting, and presenting data. If I had studied some more advanced statistics concepts, I might have been able to find ways to apply them to my job. For example, reading about hierarchical models in The Mathematical Sciences in 2025 made me wonder if my company uses them, especially since it seems like they might be valuable in determining magazines’ sales potential. With the print magazine industry struggling, we analyze data to try to find ways to cut costs without sacrificing revenue. I sift through the numbers to find anomalies, opportunities, and trends, and I use the data to generate ideas for tests and measure results. I also have to make the best of messy data. For instance, our second largest wholesaler recently went out of business, so the twenty thousand stores serviced by that wholesaler were suddenly without magazines until the chains made deals with new wholesalers. Unsurprisingly, the sales data from that transitional period is one big anomaly.

*Computer Science*

I also wasn’t aware of the extent to which math is used in computer science, and of how vital computer science is to ongoing developments in countless fields. I didn’t even consider enrolling in a computer science course. Looking back, I wish that I had been required to take courses in that department. Having knowledge in areas that combine math and computer science skills (e.g., math modeling, simulations, programming, and coding) has become more essential for mathematical careers. Although I don’t use advanced computer science concepts in my job, I’ve had to learn certain data analysis and computer systems skills that I wish I had gotten a head start on in college. I was initially surprised to see that Time Inc. has its own systems built by in-house programmers that hold data on each magazine-selling store in the country and suggest a number of copies to put in each store. I have taught myself rudimentary coding to better understand the logic, structure, and language, but I would have loved to have had a jumpstart in college on querying data and forming conclusions. I communicate frequently with programmers, testing systems and making suggestions for enhancements. Recently, they were having trouble getting a report to display necessary results, and I gave them a query I had written which helped them finish building the report. I was able to figure out the logic in this instance, but building queries for other reports would be beyond my understanding. There have been numerous situations like this where I’ve felt that having even a basic computer science foundation could have led to faster progress and a stronger group effort.

Much of 21st century research will have a foundation in math, and there will be surprising connections to other fields, as well as jobs that haven’t even been conceived yet. Building on core math concepts through the incorporation of more real-world applications and further linking these concepts to statistics, computer science, and other STEM disciplines will help broaden the perspective of potential math majors, and better prepare them for the rest of college and their subsequent careers. This will not only create more well-rounded students, but will also strengthen math’s relationship with the other disciplines in the real world.

]]>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.

My first years of teaching, I would prepare for each class by writing notes of just about every word I would write on the board. (Like a low-tech Powerpoint presentation.) However, this greatly facilitated the sort of class where, as the saying goes, the ideas travel from my notes to my students’ notes, without having to pass through the brains of any of us. Because the other trait I imagined in my movie hero teacher was making the students truly understand (and not just memorize) the material, over time I brought in activities to encourage more active learning and interaction, which also made my classes less tightly-scripted.

I also started loosening up and allowing more of my personality and sense of humor to show, for instance slipping in more clever cultural references or ironic asides. This is part of who I am, how I communicate outside the classroom, even when discussing serious mathematics with colleagues. Sometimes it’s just hard to avoid. One of the small ways I have of making class more interactive is to ask students to help me with a proof or equation. After watching countless episodes of Blue’s Clues when my son was little, I find it almost impossible to do this without saying “You *will* help me, won’t you?“, just the way Steve, the show’s host, said it.

If I thought of anything amusing related to what we were discussing in class, I would share it. The payoff for this sort of thing is immediate, in the smiling or laughing faces of students. I could justify it by noting it made class more enjoyable, and maybe helped students remember ideas better. And it made me feel more like the movie hero teacher.

Now, shortly after I started teaching, I served on the jury for a trial. The lead attorney for one side had very much the attitude in court I was trying to cultivate in the classroom. He seemed to want to be friendly with us on the jury, and, while I don’t think he introduced any actual humor, he was certainly very relaxed and smiled a lot. The attorney for the other side was more down to business. I definitely liked the first attorney more, but I found myself sometimes a little irritated at him for being less serious.

I would occasionally remember this trial as I grew more comfortable injecting humor in the classroom. And I eventually started questioning my motives. Was I doing this because it helped students learn mathematics, or because I wanted them to like me? Here’s the final note of cognitive dissonance that made me confront myself. I would tell students we didn’t have time in class for some things, such as a review for an exam. But if we don’t have time for a review, how do we have time for a joke?

And here’s another concern. The clever wordplay and cultural references that I love so much have a special risk when dealing with students from another culture, or who are still mastering the language. I am especially aware of this, being on the border with another country. (It is literally true that I can see Mexico from my office window.) A number of years ago, teaching Game Theory, the textbook referred to the two players of some game as “Norm” and “Cliff“. I asked my students who understood this reference (click on the links to see the answer);So what did I change? I’ve kept in my cheerful outlook and my sense of wonder at mathematics (for instance overreacting for dramatic effect when some calculation yields a surprising result). But for anything beyond that, I now have two criteria for including anything entertaining:

**Does it take away precious class time?**Class time together is one of our most limited resources, and so I want to reserve it for the most important things, the ones that cannot be done individually or outside of class.**Does it unnecessarily distract the class from important mathematics?**Does what I am thinking of*reinforce*the mathematics, or is it just funny? Very rarely, it is worthwhile to take a short mental break to give everyone a chance to catch their breath. But I also recall Gian-Carlo Rota’s observation that a valuable trait for doing mathematics is Sitzfleisch, the ability to sit and concentrate for long periods of time.

And, with some practice and careful attention, I have now pretty much trained myself to avoid anything that fails either of these tests. I can now think of a funny idea, and consciously choose to not share it. For instance, here is a joke I used to tell when the idea for uniqueness came up:

How do you catch a unique rabbit? You “neak” up on it.

How do you catch a tame rabbit? “Tame” way, u-nique up on it.

The problem is, students may remember this joke, and so may remember the word unique, but does it help them understand the idea, or remember how to show something is unique, or anything mathematical at all?

On the other hand, some jokes help make a point.

Two campers are in their tent in the woods when they hear a bear. The first camper starts putting on shoes, and the other camper says, “You don’t have time for that, shoes aren’t going to help you outrun a bear.” The first camper replies, “I don’t have to outrun the bear, I only have to outrun

you.”

I tell this in Calculus I as we are starting global optimization, to make the point that sometimes it’s not enough to just have a very good solution or almost the best solution, but you have to have the best solution, second to none. Here, the joke may help to drive home this larger point, which some students may overlook in the thicket of algebra and derivatives that will shortly arise when we get to practice problems.

One final anecdote: In Linear Algebra, I was trying to make the point that even though we can’t put all matrices into diagonal form, we *can* at least put all matrices into Jordan canonical form (of which diagonal form is a more special case). I reminded students that not all matrices are diagonalizable, and wrote on the board, “You can’t always get what you want.” I realized only as I finished writing it that, without intending to, I was quoting the Rolling Stones. But in this case, it worked out perfectly because the next line (as realized by some students, who then practically sang it to everyone else) is “But if you try sometime, you just might find you get what you need,” which reinforced exactly the point I was about to make about Jordan canonical form.

And, come to think of it, this is a pretty good summary of where I am now. There will always be a part of me that still *wants* the adoration that comes from being the movie hero teacher. But what I *need*, what my students need, is for them to learn mathematics. And, as the song says, if I “try sometime,” I can give my students that opportunity while still sharing my enthusiasm for mathematics.

Oh, and if you want to hear the joke about the mathematician in the balloon, you can stay after class.

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