By Benjamin Braun, Editor-in-Chief, University of Kentucky; Priscilla Bremser, Contributing Editor, Middlebury College; Art Duval, Contributing Editor, University of Texas at El Paso; Elise Lockwood, Contributing Editor, Oregon State University; and Diana White, Contributing Editor, University of Colorado Denver.
Editor’s note: This is the third article in a series devoted to active learning in mathematics courses. The 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 , 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.  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  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 , ; 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  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.  that is discussed by Bressoud  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  (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 .
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 .
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 . 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 , , calculus , , math courses for pre-service elementary school teachers , statistics , and mathematical biology . We refer the interested reader to these references for in-depth discussions regarding the benefits and drawbacks of inverted classroom environments.
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  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 .
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 , and for this (among other factors) the math emporium model has attracted attention from national news organizations . 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  and a recent study by Webel et al. , are available for readers interested in learning more about the math emporium model.
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 .
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 , 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  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.
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 . 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 . 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 , , , 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  and algebraic geometry .
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.
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