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

By Benjamin Braun, Editor-in-Chief, University of Kentucky; Priscilla Bremser, Contributing Editor, Middlebury College; Art Duval, Contributing Editor, University of Texas at El Paso; Elise Lockwood, Contributing Editor, Oregon State University; and Diana White, Contributing Editor, University of Colorado Denver.

Editor’s note: This is the first article in a series devoted to active learning in mathematics courses.  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,, 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, 2004

In 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) 39:183–199

A 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.


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
Proc. Natl. Acad. Sci. U.S.A. 2014 111 (23) 8410-8415

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

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

  1. Cam Wieczorek says:

    Hello – I am a senior studying math education at the University of Illinois. In my early field experience observations the past few semesters I have seen active learning implemented in the math classrooms through the form of CPM (College Preparatory Mathematics). Students are constantly working in small groups of 3-4, and the lessons are designed in accordance with the Common Core State Standards. As you mentioned, there is a lot less lecturing on the teacher’s part and a lot more discovery-based learning geared at getting students involved/engaged with the learning process. When I first was introduced to CPM, it took me a little bit of time to get used to it, considering I was used to mainly the lecture-style of teaching when I was in high school. However, after three semesters of observing teachers use this style and personally writing lesson plans that incorporate it, I have become more comfortable with it, and it is clear that when implemented properly, it definitely benefits students’ learning. I do have one concern with this approach though – when students go to college, they most likely will have lectures that do not use this active learning approach. How would you defend active learning when someone could argue, “If we’re trying to prepare students for college, why not have teachers lecture like they do at the university level? This active learning won’t even be present at the university level, so why introduce it if it will disappear from their education after graduation?”

    • Ben Braun says:

      I think there are two answers to your question. First, the university level teaching is changing significantly right now. I believe there will be much more active learning going on in university classes in 5-10 years than there are now, much like the changes that have taken place over the past decade. Second, if you do want to train students to take in a lecture, then the way to do this isn’t to have them sit through many lectures. Instead, you want to give them feedback on their process of evaluating ideas and engaging with content, so help them become independent thinkers and learners. This will prepare them for lecture-based courses by giving them the skills needed to listen to a lecture and evaluate the ideas.

      • Paul Anderson says:

        You and Cam make some interesting points. I have similar experiences as Cam. I have seen the shift in how mathematics is taught within middle and secondary schools. Like Cam, at first I was overwhelmed by the differentiation of teaching methods. However, I soon adapted.

        You mentioned that you believe that there will be more active learning in universities in the next 5-10 years, with this point I hope that you are right. Not only will students get more out of classes with the application of active learning, but I think students will end up enjoying the material more if they are actually engaged in the learning.

        I also strongly agree when you way that preparing students for college is not teaching them how to sit through lectures but
        “learning how to learn.” What I mean by this is simply that it is vital for students to be able to adapt to different teaching methods, different environments, and to become self-reliant and individual thinkers. Students must learn what to do once they are given factoids and how to break these pieces of information down into a more conceptual understanding.

        • John Carlson says:

          Did you attend conferences in which you learned how to get started with active learning strategies? Or have you just developed your teaching methods over time, without much help from “experts”? I am a post-secondary teacher who desperately wants to implement more active learning strategies in my classrooms. I try to do that, but I find myself moving back toward lecture mode when I realize that there’s only X weeks left in the semester and we still have to cover these Y objectives.

          • Ben Braun says:

            For me, I read a significant amount about teaching and learning and then experimented in my courses with those ideas, and had many discussions with other interested faculty in my department. I know other faculty who have primarily learned active learning techniques through attending conferences and workshops. I think it is very common for faculty and K-12 educators to do more active learning up front, then default a bit to lecture for the reasons you describe. In my experience, it is key to keep in mind that you can lecture with effective active learning strategies incorporated. Short 2-3 minute problem breaks or “discuss this with your partner” breaks can significantly increase the engagement of students during a lecture, even if it is not 100% small group work all the time. So, don’t think of it as either turning on active learning or turning it off — rather, think of it as a balance, some days you have 25% active learning, some days you have 90% active learning, etc. What is important is that throughout a course, there is a significant percent of time spent with students actively engaged, distributed reasonably through each week.

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