I am teaching Linear Algebra this semester, with pre-recorded classes and weekly meetings, and the students have a few short and simple exercises to solve every week, which count a bit (20%) to their final course grade.

This was designed in order to make sure that the students study a bit every week, and do not postpone studying the subject to the very last minute, as experience tells me and my colleagues that this is extremelly inefficient, specially in a (relatively) abstract course as Linear Algebra (this course focuses on abstract finite-dimensional Linear Algebra, with reduced use of matrices and linear systems, which are studied in a previous course).

However, I noticed that most student just leave their weekly assignments to the very last minute. Out of 51 students, only 2 have even opened this week's assignment assignments, which are due on Sunday, as of Friday.

As expected, the previous assignments grades are low, bordering on 60% mark. I should also add that they can take the weekly assignments as many times as they want to improve their grades, and the assignments are quite easy.

So I am looking for references (articles published in reputable peer-reviewed journals) which discuss the different practices of studying; e.g. comparing studying a bit every day vs accumulating a lot of material to study all at once, and how those different approaches affect mathematics learning.

I'm not even sure of what terms I should be looking for, as this is not my are of expertise, but I found that "cramming" seems to be related to the second approach mentioned above. All I could find were a few articles mentioning that cramming can be effective for students who already have good study habits, or for areas other than mathematics.


I'm looking for reputable articles comparing different ways of studying mathematics (spacing out the material or condensing it).

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    $\begingroup$ Bjork and Bjork are a good resource (bjorklab.psych.ucla.edu/research) on things related to the spacing effect (ncbi.nlm.nih.gov/pmc/articles/PMC3399982). The Learning Scientists like to blog about spacing and interleaving practice (learningscientists.org/blog/2016/7/21-1). Basically, cramming helps with short term performance, but misses out on most of the things that lead to long term benefits, i.e., the importance of forgetting and recall. $\endgroup$ – Carser Feb 19 at 21:51
  • $\begingroup$ What happens in the weekly meetings for this class? (Granted that there are "pre-recorded classes"). $\endgroup$ – Daniel R. Collins Feb 24 at 1:26
  • $\begingroup$ @DanielR.Collins There is the one synchronous meeting per week. I answer questions regarding the material (there are always one or two; usually from the same set of 4 students and related to the weekly exercises); I show a simple application or interpretation of some concept; and I solve one or two 'discursive/proof-type' question (e.g. "Prove that every linear function on a subspace extends linearly to the whole space", or something like that depending on the week's topic), to give the students some idea of how these ought to be solved in the exam. $\endgroup$ – Luiz Cordeiro Feb 24 at 3:41
  • $\begingroup$ So it sounds like what some call a "flipped" classroom. This is broadly in line with what I've heard elsewhere -- that it's unlikely to get more than a small number of students making time to actually watch the pre-taped lectures, and thus be prepared for exercise work. $\endgroup$ – Daniel R. Collins Feb 24 at 14:51
  • $\begingroup$ I wonder if there's any way to confirm/get data on when students are watching the lectures? I'm guessing that's occurring immediately before they submit exercises, and therefore that's the real bottleneck that needs fixing. $\endgroup$ – Daniel R. Collins Feb 24 at 14:52

If you can ignore the specific UBC context, this might be the type of paper the OP seeks:

Eric Eich. "The Cognitive Science of Learning Enhancement: Optimizing Long-Term Retention." (2011). Univ. British Columbia. HTML link.

Here's an excerpt comparing "massed" vs. "spaced" learning:

"The advantage of spacing over massing in inductive learning seems even more surprising in view of Figure 3 [...]. As is apparent in the figure, most participants believed they learned more from massed than from spaced presentations, even though their performance on the name-selection test proved otherwise."

"Massed" and "spaced" learning/practice are key search terms in this literature.

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    $\begingroup$ This is such an important point about learning that I see often: how we think learning happens best is not necessarily how it happens best, even when we are assessing our own learning. We often feel that we have learned something when we haven't, and feel that we haven't when we have! Teaching and learning are so complex :) $\endgroup$ – Carser Feb 23 at 18:55
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    $\begingroup$ This is a great reference! I am happy that the results indicate that our (instructors) general experience are in accordance with the linked results. I will keep the question open for the time being, not to dissuade other people answering. $\endgroup$ – Luiz Cordeiro Feb 23 at 19:13

To add to the excellent suggestions already made, I intersperse findings from this article (https://www.aft.org/sites/default/files/periodicals/dunlosky.pdf) throughout the semester in all my classes. The article reviews the results of a 2013 deep dive by educational psychologists into the effectiveness of 10 of the most popular study strategies students use (e.g, flash cards, highlighting the textbook). The psychologists found that practice testing and distributed practice were the two most effective of the 10 techniques they looked at. But that's only a tiny part of the story. I highly recommend taking a look at Table 4 in their main article (https://journals.sagepub.com/doi/pdf/10.1177/1529100612453266), which details the effectiveness of study strategies based on their intended purpose (e.g., memorization).

Another suggestion for Luiz -- since I use flipped classrooms too -- that follows up on Daniel's comment (wondering if there's a way to tell that your students are watching your videos) is to have your students submit written reflections on your videos. My reflections are Google Forms that ask them to: (1) summarize the main mathematical takeaways from the lesson; (2) identify the most interesting part of the lesson; (3) tell me what, if anything, they still find confusing; (4) share anything else that's on their mind; and (5) let me know if there are any issues happening now or on the horizon that might impact their learning. I read over these reflections before each class, use them to customize the class sessions, and construct lesson debriefs using them that answer all their individual questions in red text. I then upload this document (appropriately anonymized) to the LMS. Oh, and I encourage them to send themselves a copy of their Form submission. I sell this by telling them that the more care they take now with their reflections the easier their studying will be later when they go back and review their excellent past reflections; works every time and helps them see the value and payoff of good work now. I'm happy to share my Form with anyone interested.


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