When Small Changes Lead to Big Impact: Hysteresis in Mathematics Teaching
Communicated by Notices Associate Editor William McCallum
The purpose of this article is to highlight some insights gleaned from the emerging body of research that focuses on undergraduate mathematics teaching. Unlike the domain of research that examines student reasoning and what it means to understand particular content, it is only within the last dozen or so years that the field has closely examined instructional practices at the undergraduate level. Indeed, Speer and colleagues in their 2010 comprehensive review referred to collegiate mathematics teaching as “an unexamined practice” (Speer et al., 2010, p. 99). The situation today, however, is quite different and reflects a growing interest in undergraduate mathematics teaching. For comprehensive reviews of this literature we point readers to Rasmussen and Wawro (2017) and Melhuish et al. (2022).
Our goal in this report is to present these insights in a way that is accessible to a wide range of readers (often mathematics education research reports are steeped in terminology and theory because they are aimed at an educational research audience). In pulling out these research-based insights, our aim is to contextualize, motivate, and provide some accessible instructional recommendations and resources that are relevant for a wide range of teaching approaches.
We structure our discussion of research on undergraduate mathematics teaching by first highlighting three articles that examine the benefits and limitations of lecture followed by a brief review of the value of instruction that actively engages students. We then shift to an overview of research-based instructional strategies that reflect easy entry points for actively engaging students in class. We conclude this brief review with a collection of resources where readers can learn more.
There Are Good Reasons to Lecture
Lectures have a long history in undergraduate mathematics instruction. Advantages of lectures include efficient delivery of a large amount of material, opportunities to motivate the topic, inspire students, and provide the necessary background for study outside of class, and the chance for instructors to model mathematical ways of thinking and doing mathematics. Indeed, mathematicians conversant with the culture and practice of mathematics are well-positioned to motivate, inspire, and be models for how mathematics is done (e.g., Byers, 2007). Research that illuminates the practice of lecture, therefore, can highlight its value and nature, as well as its limitations.
For example, Weber and Fukawa-Connelly (2023) investigated what mathematicians learn from attending other mathematician’s lectures. These researchers interviewed 13 mathematicians who participated in a two-week workshop on inner model theory, a topic in set theory. Central findings include an appreciation of the big picture of the domain and how experts in this area think about the topic, the highlighting of important results and techniques, and the feeling of being well-equipped to embark on their own subsequent study of the material. These findings illustrate how lectures differ from textbooks or research papers that are necessarily concerned with rigor and correctness, and often mask the intuition and big-picture ideas that lectures can convey. Instead, findings from this study illustrate the potential for undergraduate mathematics lectures to model doing mathematics, provide a road map of the domain, and motivate students and hence facilitate independent study. We conjecture that because of the distance in mathematical maturity between where undergraduate students are and where mathematicians are, specific pedagogical actions are needed so that students can benefit from lectures in the same ways that mathematicians and graduate students do.
Another relevant and illuminating study that examined the practice of lecturing is the international study by Artemeva and Fox (2011). Based on lecture recordings and interviews with over four dozen lecturers from Australia, Canada, Israel, Spain, Sweden, the United Kingdom, and the United States, these researchers identified a collection of common practices, which they refer to as “chalk talk.” Chalk talk practices include verbalizing everything written on the board along with metacommentary about what was written, using rhetorical questions to signal transitions, the use of pointing to highlight key ideas, verbal references to the text, assignments, and notes, and checks for understanding. It is precisely these chalk talk practices that enable lecturers to convey the practice of how experts do and think about mathematics. Some differences across the cultural-historical educational contexts include the acceptability of making mistakes, the willingness of students to ask questions, and instructor use of notes during the lecture. Studies like those by Weber and Fukawa-Connelly and Artemeva and Fox are valuable because they highlight qualities of good lectures and how they might contribute to student learning.
In addition to some of the positive aspects of lecture-oriented instruction, the research literature has also revealed nuanced ways that account for why students do not always take away with them what the instructor intends. For example, Lew et al. (2016) studied what advanced undergraduate students learned from a real analysis lecture and compared this to what the professor thought to be the central points of the lecture. These researchers conducted a case study of one professor, Dr. A, who had an excellent reputation as a real analysis instructor with over 30 years of experience. As a case in point, they report on one 11-minute proof that a sequence with the property that for some is convergent. They first interviewed Dr. A and asked him about the goals of his lecture and why he presented this proof to students. They then showed him a video recording of the lecture and asked him to stop the recording at every point he thought he was trying to convey mathematical content and to specify the content. They then conducted interviews with three pairs of students with the following four “passes” through the lecture:
Pass 1: Students recalled what they learned from the proof by reviewing their notes.
Pass 2: Students watched the lecture again in its entirety, took notes, and were asked what they learned and what they thought the instructor was trying to convey.
Pass 3: Students were shown short specific clips of the video and asked what they thought the professor was trying to convey.
Pass 4: Students were asked whether particular content highlighted by Dr. A in his interview could be gleaned from the proof they just watched.
When students recognized the intended content conveyed by Dr. A.
Content conveyed by professor | Pair | Pair | Pair |
#1 | #2 | #3 | |
To show sequence is convergent without a | Pass 3 | Pass 3 | Never |
limit candidate, show it is Cauchy | |||
Triangle inequality is important for proofs in | Pass 2 | Pass 3 | Pass 3 |
real analysis | |||
Geometric series in one’s “toolbox” for working | Never | Never | Never |
with bounds and keeping quantities small | |||
How to set up proofs to show a sequence is | Pass 4 | Pass 2 | Pass 4 |
Cauchy | |||
Cauchy sequences can be thought of as | Pass 3 | Pass 3 | Pass 3 |
“bunching up” | |||
As shown in Figure 1, even strong mathematics students with an experienced and accomplished instructor miss many of the intended points made by the instructor. In fact, none of the students picked up on Dr. A’s intended content during the first pass, which is typically the only chance they get to glean insights from a lecture. A primary reason for this was that the instructor conveyed most of the important points orally, but the students’ focus was what was written on the board. This finding resonates with the description of chalk talk where metacommentary is a prominent feature of lecture. The authors conclude with two important implications. First, as educators we can help students become better note takers, attending to what is verbalized as well to what is written on the board. Second, as lecturers, we can be more attuned to the possibility that important ideas which we convey via metacommentary may not be remembered by students. This suggests the need for heightened awareness of what is and what is not written on the board.
Lecture has a long history in undergraduate mathematics teaching and is not without merit. Research, however, is increasingly spotlighting how alternative forms of instruction that actively engages students can improve student success, especially in introductory mathematics courses required of STEM-intended students. Informed by this growing body of literature, professional societies are calling on the field to look toward other forms of instruction. For example, the AMATYC, AMS, ASA, MAA, and SIAM came together to produce the Common Vision document, which states the following:
We see a general call to move away from the use of traditional lecture as the sole instructional delivery method in undergraduate mathematics courses … Even within the traditional lecture setting, we should seek to more actively engage students than we have in the past. (Saxe & Brady, 2015, p. 19)
In the next section we provide a few highlights from the literature on active learning as well as easy to use strategies that can, as Saxe and Brady urge, be used in more traditional lecture settings.
Engaging Students During Class Time is Good
Across undergraduate Science Technology Engineering and Mathematics (STEM) courses, the educational research literature fairly consistently finds that teaching practices that encourage student engagement during in-class time are beneficial to student learning. Most comprehensively, Freeman et al. (2014) conducted a meta-analysis of 225 research articles that compared outcome variables between classes that incorporated some sort of “active learning” techniques to classes that did not. In their meta-analysis, they found that student exam scores in classes that incorporated active learning techniques were 6% higher and failure rates were 12% lower, on average, than in classes that did not incorporate active learning techniques.
Research studies conducted specifically in undergraduate mathematics classes have found increasing student engagement in class, through the use of nonlecture teaching techniques, has a positive impact on learning outcomes, persistence, and success. For instance, across multiple studies Laursen and colleagues have found higher student reports of confidence and learning gains in inquiry-based learning courses than in more traditional lecture-based courses (e.g., Kogan & Laursen, 2014).
A retrospective account of two different threads of inquiry as well as a definition of inquiry-based mathematics education is detailed in the overview paper by Laursen and Rasmussen (2019). Laursen and Rasmussen (2019) define inquiry in terms of four aspirational pillars. Two of these pillars emphasize what students do (students engage deeply with coherent and meaningful mathematical tasks and students collaboratively process mathematical ideas) and two emphasize what teachers do (teachers inquire into student thinking and teachers foster equity in their design and facilitation choices). Importantly, this definition does not define inquiry in terms of prescriptive behaviors, but rather lays out four aspirational goals that can be achieved in different ways that fit the style of each instructor. For an overview of how inquiry-based mathematics education has been taken up in many European countries we refer readers to the Platinum ProjectFootnote^{1} (Gomez-Chaco et al., 2021).
While inquiry-based learning is a rather large-scale shift from traditional lecture courses, with the classes often devoting more than half of each class session to small-group work, student presentation of problems at the board, and whole-class discussion, there are also ways to improve student learning outcomes with much more incremental changes. In the next section we highlight several of these easy-to-implement strategies.
It Doesn’t Have to Be All or Nothing, Here Are Some Ideas About How to Get Started
Trying to characterize instruction as strictly “lecture-oriented” or “active learning” is often an unnecessary division as most of our teaching is not one or the other. For instance, when we look at “lecture-oriented” classes there are lots of examples where instructors frequently ask questions to increase student engagement (e.g., Artemeva & Fox, 2011) and a recent study identified a sizable subgroup of inquiry-based learning instructors whose reported instruction looked almost exactly like the reported instruction of people who had never heard of inquiry-based learning (Vishnubhotla et al., 2022). In fact, there are many student engagement techniques that could be incorporated into a predominantly lecture-based class with only minor alterations. In fact, going back to the Freeman et al. (2014) meta-analysis, they defined “active learning” classrooms as ones that included “vaguely defined ‘cooperative group activities in class,’ in-class worksheets, [or] clickers” (p. 8414)—with as little as 10% of the class time devoted to active learning. This suggests that minor active learning modifications to lecture may still go a long way in terms of improving student engagement and learning. Here we discuss some of these lecture-compatible, student engagement techniques that can easily be fit into an existing class.
Exit Cards: Exit cards (aka one-minute papers) are a quick way to gather information at the end of class on how students are thinking about the class material. Exit cards can be asked quite generally, (e.g., what is one thing you learned today and one question) or they can be specific to one of the main ideas from class (e.g., draw a diagram to illustrate Rolle’s theorem). Given their versatile nature, they can be used to support students in reflecting on their own learning or as a source of information for the instructor about how their students are thinking about the material and what could use more attention in either the homework or the next class session.
Think-Pair-Share: Whereas exit cards, at most, require just a few minutes at the end of class, the think-pair-share method builds in more structured and intentional pauses during class time. For this method, you pose a quick question or task to the students and ask them to first reflect on this privately (think), they then turn to a neighbor to discuss (pair), before sharing within a small group or to the whole class (share). This is a highly versatile technique, as there are numerous opportunities during any class in which students could use just a few minutes to consider the material being presented. This includes: completing a computation, thinking about how they would approach a problem, generating a (counter)example, or considering the validity of a conjecture. As discussed by Braun et al. (2018), “giving students time to think about and discuss mathematics mid-lecture encourages their active participation in the class… and serves as an effective comprehension check in which students are able to refocus their attention during a lecture” (p. 125).
Individual Response Systems (Clickers): There are certainly times when we want to provide students time to think about a question or task, but the “pair” and “share” aspects of the previous method do not match the teacher’s goals or instructional environment (e.g., high-enrollment lecture hall). Using individual response systems (or “clickers”) still provide students time to reflect and engage in class—and still provide the instructor with timely information about the extent to which the students are following along with the material. While some instructors prefer to use some technology-delivered individual response system (e.g., clickers), classroom voting can also be done with a simple show of hands, with colored index cards (red = a, blue = b, etc.), or by having students hold fingers in front of their chests to indicate the option number for which they are voting. The polling questions used can range from quick homework/reading checks at the beginning of class, to checks for understanding or assessments of prior knowledge, to end-of-class reflections. Given the real-time data that is collected through the use of clickers and other means, they not only serve as a way to help students stay engaged with the material in real time, they can also serve as a real-time formative assessment, which differs from exams in that these assessments provide information that can be used by teachers to make better-founded decisions about the next steps in instruction.
Guided Notes: We know that, in addition to what they write on the board, most math instructors also provide a lot of commentary and insights that they do not write down—and that are virtually never written down by students (Artemeva & Fox, 2011; Lew et al., 2016). As an alternative to having the students write down for themselves every bit of the lecture, guided notes provide students with “a print out of the notes which contain gaps and they can fill the gaps in as the lecture proceeds” (Iannone & Miller, 2019, p.6). The idea behind the use of guided notes is that it frees up time and mental capacity for students to focus on the larger ideas of the lecture and the rich verbal commentary that they may miss otherwise. As explained by one of the students in a study on the use of guided notes “I think I am probably more engaged in the ones [lecturers] that do give us gappy notes because you can kind of stop writing and actually listen to what the lecturer is saying. Rather than just having to copy down everything for an hour.” (Iannone & Miller, 2019, p.14)
A fairly recent census survey of all mathematics departments that offer a graduate degree found both a readiness and willingness to implement more active learning strategies, but also a need for guidance on how to more effectively implement active learning (Rasmussen et al., 2019). In the following section we point readers to a number of resources on ways to actively engage students in class and how to begin shifting department culture so that active learning, while perhaps not yet the norm, is at least not an outlier.
Where to Learn More
An excellent comprehensive resource for actively engaging students both inside and outside the classroom is the MAA Instructional Practices (IP) GuideFootnote^{2} (Abell et al., 2018). The IP guide is a “how to” guide, full of concrete examples. The edited volume contains three foundational types of practices: classroom practices, assessment practices, and course design practices. The book is intended for all mathematics instructors, from those with years of experience to those just starting out in the profession. The classroom practices section contains a wealth of research-based instructional strategies that range from quick and easy to implement to more elaborate strategies that require greater preparation. The assessment practices section contains guiding principles for both summative and formative assessments. Summative assessments include things like quizzes and exams where the purpose is to evaluate student proficiency. Formative assessments, on the other hand, are intended to provide evidence about student progress to help instructors determine appropriate and needed next steps. In the previous section we gave the example of “exit cards” as one example of a formative assessment technique. The IP guide provides other examples as well as effective methods for designing summative assessments. Lastly, the course design practices section discusses and provides engaging vignettes focused on the plans and choices instructors make prior to teaching and what they do after teaching to modify and revise for the future.
For those instructors that are interested in using classroom voting to actively engage students, one can now find many tried and true good questions for a variety of content areas. One good source of information is the edited volume, Teaching Mathematics with Classroom Voting: With and Without ClickersFootnote^{3} by Cline and Zullo (2011). Another excellent source of inspiration for good classroom voting questions can be found at this website.Footnote^{4} Our goal here is not to provide a comprehensive list of resources, but rather to offer interested instructors a starting point for locating resources.
Another useful resource is the Academy of Inquiry Based Learning,Footnote^{5} which provides a variety of professional development and learning community opportunities for faculty to learn more about inquiry-based mathematics education. A good place to start at their website is the IBL Blog Articles, which include articles on getting started, learning about what IBL is, nuts-and-bolts topics, classroom organization and management, productive failure and growth mindsets, and more.
In addition to classroom-level resources, we also want to point readers to department-level resources that can help shift department culture to be more supportive of engaged student learning. While transforming practice can and often does happen one instructor at a time, research has pointed to the power and sustainability of change when the department, rather than the individual, is the focus of change. The recently edited book, Transformational Change Efforts: Student Engagement in Mathematics through an Institutional Network for Active LearningFootnote^{6} (Smith et al., 2021a), reports on a national study of departmental change and change mechanisms that allow math departments to incorporate and sustain active learning in Precalculus to Calculus 2. Another excellent resource is a triple special issue of PRIMUSFootnote^{7} that features 26 mathematics departments in the process of transforming their introductory mathematics courses typically required for all STEM majors (Smith et al., 2021b). We hope that these stories of on-the-ground change efforts will inspire readers to develop their own change efforts, ones that fit their departmental and institutional context.
Thinking about making a change can be overwhelming, whether it be at the individual instructor level or the department level. For those interested in making use of some different active learning strategies, whether this is in a primarily lecture class or not, we hope the research insights and resources provided here, while limited, make your life easier. For those that are already familiar with some of the instructional approaches highlighted in this article, we hope that the numerous references are useful for you to dive into and expand your repertoire. For those interested in better understanding the process of departmental change, especially change that is aimed at infusing more active learning into the introductory mathematics courses, we hope that the department change references provide you with inspiration and lessons learned.
References
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Credits
Photo of Chris Rasmussen is courtesy of Chris Rasmussen.
Photo of Estrella Johnson is courtesy of Estrella Johnson.