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IMPACT Plus - Active Learning

By Robert Cappetta posted 11-18-2021 15:25:32


The traditional mathematics class is a “teacher-telling” environment where the students listen and watch the instructor perform mathematics.  The student’s role is to take notes in class, review them outside class, and complete homework assignments.  People taught in this manner can succeed in their courses by simply memorizing and “mimicking” the behavior of the instructor, yet these strategies will rarely succeed at the collegiate level in science, technology, engineering, and mathematics courses.   A great challenge with first-year college students is helping them realize that mathematics is not about formulas and procedures, but rather it should be focused on conceptual understanding and problem solving. Ganem (2009) writes, “Steps can be memorized but it will be a long time, if ever, before the concept and motivation for the process is understood. That raises the question of what exactly is being accomplished with such curricula? Learning techniques without understanding them does no good in preparing students for college. At the college level emphasis is on understanding, not memorization and computational prowess.” 


Might there a better way than the traditional lecture-based, teacher-telling STEM courses? In the 2014 Proceedings of the National Academy of Science (PNAS), Freeman, et al. (2014), contrasted student performance in traditional lecture-based STEM courses to performance in classes that used active learning strategies such as group problem solving, worksheets/tutorials completed in class, personal response systems, workshop course design etc.  In a meta-analysis of 225 studies, they found that students in active classes scored significantly better on examinations, and they were much less likely to fail.  Hake (1998) found that those taught using active strategies learned twice as much as students taught using a direct instruction approach.  There are countless studies that show the benefits of active learning, so it should come as no surprise that the educational focus has shifted from students passively receiving content information to diligently participating in learning activities (Frey, 2018).  Yet, many instructors are reticent to use an active learning approach.  Bonwell and Eison (1991) list several reasons for this reluctance.

  1. Educational tradition has a powerful influence on how people choose to teach.
  2. Faculty have fixed self-perceptions of the role of the teacher.
  3. Change creates discomfort and anxiety.
  4. There are limited incentives for faculty to change.
  5. There is a concern that there will not be sufficient time to cover all the course content.
  6. Active learning lessons may require much more teacher preparation time.
  7. It may be difficult to use these strategies in very large classes.
  8. Instructors may lack the training, materials, equipment, and resources.
  9. The greatest risk is the fear that students will not participate.

In addition, faculty worry about a loss of control, and they fear that they will be criticized for teaching in unorthodox ways.  Overcoming these fears is a substantial challenge, and instructors will need continual support if they are to be successful.  Without that assistance, these methods are less likely to yield positive results.  For example, in one study, it was found that 75% of instructors who have attempted specific types of active learning abandoned the practice altogether (Froyd, Borrego, Cutler, Henderson, & Prince, 2013). The evidence is clear.  Active learning results in improved student performance, yet instructors are resistant to implement it.  Organizations like AMATYC must provide the professional development needed to address this concern.

The “flipped classroom” model is a prevalent active learning method in collegiate mathematics.  Students in flipped classes are introduced to novel information outside class through videos, readings etc., and students spend time in class solving problems, asking questions, and completing activities with teacher support, but this approach will not work if the students refuse to complete the preparatory activities.  A key to a successful flipped classroom is to ensure that students engage with the material presented outside class (Prust et al, 2015). 


The following suggestions may increase that likelihood.

  1. Keep the videos short and focused on one topic. The University of Colorado recommends that videos be no more than ten minutes.  The total amount of video time per week should not exceed the traditional weekly lecture time of 150 minutes for a three-hour course.

  1. Prepare students to learn. Pose essential questions that will focus attention on key concepts. 
  2. Create accountability. For example, require students to complete guided notes, assess students with embedded quizzes and polls, ask students to summarize what they have learned, and monitor the students’ viewing times.
  3. Encourage multiple views. On subsequent viewings, ask students to assess their understanding by pausing the video and completing the problem before seeing it solved.  With each subsequent viewing, the instructor may introduce a new essential question to help students think about the problem in different ways. 


What should happen during class time in a flipped classroom?  There are several possible options.  The University of Michigan has compiled an extensive list.

Some examples follow:

  1. Think-Pair-Share: Have students work individually on a problem or reflect on a passage. Students then compare their responses with a partner and synthesize a joint solution to share with the entire class.
  2. Cooperative Groups in Class (Informal Groups, Triad Groups, etc.): Pose a question for each cooperative group while you circulate around the room answering questions, asking further questions, an
  3. Inquiry Learning: Students use an investigative process to discover concepts for themselves. After the instructor identifies an idea or concept for mastery, a question is posed that asks students to make observations, pose hypotheses, and speculate on conclusions. Then students share their thoughts and tie the activity back to the main idea/concept.


One method for engaging active learning mathematics classes is The Five Practices (Smith and Stein, 2011).  They are as follows:

  1. Anticipation: Determine the goal of the lesson. Create rich activities that achieve that goal.  Consider the multiple approaches/solution strategies that students may implement.
  2. Monitoring: Identify the strategies that are being used by groups of students.  Clarify the activity for students who may have difficulty getting started. 
  3. Selecting: Identify the groups who will share their work as part of a whole class discussion.
  4. Sequencing: Determine the order in which you want to steer the discussion.  Nabb (2020) identifies the following possible sequencing strategies:  informal to formal; common to unusual; simple to complex; incorrect to correct.
  5. Connect: Help the students see the connections among the various approaches.  Hopefully those connections lead to a deeper understanding of the relevant concept.


In 2016 the leaders of the mathematical professional organizations published a position paper advocating the use of active learning in collegiate mathematics.


The following is an excerpt from that document.

Post-secondary faculty and P-12 educators have successfully used active learning methods in a diverse set of institutions and across a broad range of teaching environments.  These methods have been shown to strengthen student learning and achievement in mathematics, to foster students’ confidence in their ability to do mathematics, and to increase the diversity of the mathematical community.  In recognition of this, we call on institutions of higher education, mathematics departments and the mathematics faculty, public policy-makers, and funding agencies to invest time and resources to ensure that effective active learning is incorporated into post-secondary mathematics classrooms.


AMATYC has an important role to play.  Particularly, we must extol the benefits of active learning and assist instructors overcome the many concerns.  This will only happen with good professional development, and AMATYC must be committed to providing those opportunities.




Bonwell, C. and Eison J. (1991). Active Learning; Creating Excitement in the Classroom. ASHE-ERIC Higher Education Report No. 1. Washington, D.C.: The George Washington University, School of Education and Human Development


Conference Board of the Mathematical Sciences (CBMS). (2016, July 15). Active learning in post-secondary mathematics education (Meeting Report).


Freeman, S. et al. (2014), Active learning increases student performance in science, engineering, and mathematics. Proc. Natl. Acad. Sci. U.S.A. 111, 8410–8415.


Frey, B. (Ed.) (2018). The SAGE encyclopedia of educational research, measurement, and evaluation. (Vols. 1-4). SAGE Publications, Inc.


Froyd, J. E., Borrego, M., Cutler, S., Henderson, C., & Prince, M. J. (2013). Estimates of use of research-based instructional strategies in core electrical or computer engineering courses. IEEE Transactions on Education, 56(4), 393– 399.


Ganem, J. (2009). A math paradox: The widening gap between high school and college math. American Physical Society. Retrieved6.


Hake R. (1998).  Interactive-engagement versus traditional methods: a six-thousand-student survey of mechanics test data for introductory physics courses. Am J Phys;16:64–74. 


Nabb, K. (2020). Active Learning in Undergraduate Mathematics: The Five Practices and the Five Dimensions of TRU Math. MathAMATYC Educator 12, 30-35, 68-69


Nguyen, K. A., Borrego, M., Finelli, C. J., DeMonbrun, M., Crockett, C., Tharayil, S.,  & Rosenberg, R. (2021). Instructor strategies to aid implementation of active learning: a systematic literature review. International Journal of STEM Education, 8(1), 1-18.


Prust, C. J., Kelnhofer, R. W., & Petersen, O. G. (2015, June). The flipped classroom: It's (still) all about engagement. In 2015 ASEE Annual Conference & Exposition (pp. 26-1534).


Smith, M.S., & Stein, M.K. (2011). Five practices for orchestrating productive mathematics discussions. Thousand Oaks, CA: Corwin Press