Learning Trajectories in Online Mathematics Courses

Published on August 16, 2018

Modified on December 11, 2020


Present research has devoted attention to a long-standing problem: how to better serve students who take K-12 online mathematics courses by investigating learner subgroups based on their semester-long learning trajectories. Mixture growth modeling was used to examine month-by-month scores students earned by completing assignments. The best-fitting model suggested four distinct subgroups representing (1) nearly linear growth, (2) exponential growth, (3) hardly any growth, (4) and early rapid growth. Follow-up analyses demonstrated that two different types of successful trajectories were more likely associated with advanced level courses, such as AP or Calculus courses, and foundation courses, such as Algebra and Geometry, were with the unpromising trajectory. Given those results, implications for practitioners and researchers were discussed from the perspective of self-regulated online learning and evidence-based mathematics instructional practices.

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Jemma Bae Kwon, Michigan Virtual


  • Parallel to K-12 online education’s phenomenal growth, online mathematics learning has continued to grow.
  • Student learning outcomes appear to be disappointing, and online mathematics learning is still under-explored.
  • It turned out that the longer students stayed in a learning management system (LMS), the greater their course grades.
  • Some counterintuitive results were also found when learning behaviors were measured by the total numbers of sessions and clicks in LMS.
  • Aggregate variables used by the previous research cannot adequately capture patterns of learning progress. Also, research on predictors that consistently account for success or failure in online mathematics learning is still in its early stages, which leads to uncertainty about dimension identity, i.e., how a set of variables defines and confirms a trait of behavior or development.


  • To address issues stated in the last bullet point above, this study suggested an alternative analytic approach – a person-oriented approach (to address the uncertainty about dimension identity) to growth modeling (to account for both inter- and intra-variability of learning behaviors) using growth mixture modeling.
  • The study found the largest group (73% of study sample) showing nearly linear growth and its more likelihood of coming from AP courses.
  • The second largest group showed an exponential growth profile (14%), and there was no difference from the largest group in final grades.
  • Students who enrolled for foundation courses, including Algebra and Geometry, and who indicate their enrollment reasons were credit recovery or personal learning preferences are more likely to show unpromising trajectories.


  • Explicit and implicit instructional practices that are closely aligned with established pacing guides could boost an individual’s odds of sustaining a steady pace, and in turn, his/her success in AP mathematics courses.
  • Students could succeed academically despite marked variation in their self-determined pathways and pacing, even in advanced courses such as Calculus.
  • An interim benchmark may help students pace themselves a little better in the first four months and increase their likelihood of achieving their end goals.
  • For at-risk learner groups, course design features, instructional practices, and learning support structures need to align with characteristics of those learner populations from the perspective of content mastery and self-regulated learning. Future studies will have to further our understanding of how to make such changes.
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With many students across the state 100% remote, demand for our online courses is greater than ever before. Because every course we offer is taught by a Michigan-certified teacher, this high volume of enrollments has created capacity issues for our teachers who provide each and every student with individual feedback.

While the Michigan Virtual team anticipated and planned for significant increases in student enrollments this Fall, the increased demand we’ve experienced has been unprecedented. As a result, we are taking steps to hire even more part-and full-time teachers to support larger numbers of student enrollments for Semester 2 as well as for Trimester 2 and 3. 

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