Assessing Just-in-Time Implementation in Differential Equations

Assessing Just-in-Time Implementation in Differential Equations | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Assessing Just-in-Time Implementation in Differential Equations Heather Lippert This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9020528/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract This study examined how differential equations students: (a) perceived Just-in-Time (JIT) learning to lessen prior knowledge gaps, (b) benefited from the JIT learning handouts, and (c) improved student learning outcomes. Results indicated that students positively benefited from the JITs, gained confidence in learning differential equations, and spent less time researching or studying prior knowledge. Students who did not complete the JITs results indicated that they already knew the material, did not have time to complete the JITs, or suggested that the handouts could have been better integrated into the course. The differential equations students provided suggestions for improvement and, overall, recommended keeping the JITs as a part of the course. Algebra Differential equations Just-in-Time learning Prior knowledge gaps in math Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Introduction Differential equations require extensive knowledge of Intermediate Algebra, Calculus II, and problem-solving strategies. When students reach differential equations, it is expected that they may have prior knowledge gaps because several classes have passed since Intermediate Algebra. Topics in Intermediate Algebra include solving systems of equations, completing the square, factoring, the quadratic formula, and exponent rules. These topics serve as a foundation for differential equations. Foster ( 2021 ) noted that prior knowledge gaps are not detrimental as long as the student can contribute something to the table, like relevant learned techniques. Differential equations employ various algorithmic solving techniques based on the classification of the equations (Camacho, Perdomo, & Santos-Trigo, 2009 ; Foster, 2021 ). How can students improve their recall of prior knowledge when these algorithmic equations require extensive knowledge of Intermediate Algebra or Calculus? Just-in-Time learning (JIT) is one tool designed to help students strengthen their foundational knowledge. The JIT handouts aim to motivate students both intrinsically and extrinsically, foster active learning, encourage classroom engagement, and provide remediation and support for students at all levels (Killi & Morrison, 2015 ; Novak, Patterson, Gavrin, & Christian, 1999 ). The goal is to alleviate the frustration associated with learning new and old material simultaneously in differential equations and to create a more meaningful learning experience. Literature Review: Just-in-Time Learning and Differential Equations Instruction Differential equations is essential for modeling dynamic systems in engineering, physics, and applied mathematics. However, differential equations is often regarded as one of the most challenging subjects in undergraduate mathematics due to their abstract nature, various problem-solving strategies, and reliance on prerequisite knowledge such as calculus, algebra, and linear algebra (Foster, 2021 ; Ganter & Barker, 2004 ). Students frequently enter differential equations courses without sufficient mastery of these foundational topics, resulting in frustration, disengagement, and poor academic performance. Challenges in Traditional Instruction Although widely used, traditional lecture-based instruction tends to be passive and does not adequately meet students' individual learning needs. Freeman et al. ( 2014 ) conducted a meta-analysis of STEM education and found that traditional lectures lead to significantly lower student performance compared to active learning methods. Arslan ( 2010 ) discovered that procedural learning (i.e., how to do something) supports conceptual learning (i.e., why something happens), but in a traditional teaching setting, procedural learning did not result in students applying that knowledge to new situations. Additionally, Bibi et al. ( 2017 ) noted that professors rely on algebraic, graphical, and numerical techniques to teach differential equations. In the context of differential equations, where cognitive load is already high due to complex concepts and calculations, lecture-based delivery often fails to reinforce prerequisite knowledge when it is most needed. The Just-in-Time Learning Approach JIT learning is an approach that delivers educational content based on students’ immediate needs, often identified through formative assessments conducted before class. Novak et al. ( 1999 ), pioneers of Just-in-Time Teaching, demonstrated that providing instructors with pre-class feedback enables them to modify lectures in real time, making instruction more responsive and learner-centered. JIT learning emphasizes timeliness, adaptability, and addressing misconceptions before they are reinforced. Simkins and Maier ( 2010 ) provided Just-in-Time Teaching suggestions across academic disciplines and found that it improved engagement, accountability, and performance by encouraging students to come prepared and actively participate in class. This is especially relevant for differential equations, where a lack of foundational understanding of calculus or algebra can significantly impede new learning. JIT strategies can assist instructors in identifying and addressing these deficiencies early. Application to Differential Equations Although JIT learning has been successfully applied in physics, biology, and economics, its implementation in teaching differential equations remains limited. Juter and Olsen ( 2014 ) investigated Just-in-Time Teaching implementation in undergraduate calculus courses and reported improvements in student engagement, problem-solving ability, and work habits. Their findings support the hypothesis that JIT can provide the necessary scaffolding to help students connect new concepts with prior knowledge. Mayer’s ( 2005 ) Cognitive Theory of Multimedia Learning supports the pedagogical rationale behind JIT strategies. According to Mayer, students learn more effectively when instructional content is aligned with their cognitive readiness and delivered in manageable chunks that reduce cognitive overload. In a subject like differential equations, where concepts are hierarchical and complex, JIT learning helps minimize extraneous load and focus attention on essential material. Limitations and Gaps Despite promising findings, the integration of JIT into mathematics instruction, particularly differential equations, remains underexplored. While case studies suggest positive outcomes, there is a need for additional research on the effectiveness of JIT approaches in enhancing conceptual understanding, bridging prerequisite knowledge gaps, and improving long-term retention in this specific area. Furthermore, more work is required to develop structured frameworks and digital tools that facilitate JIT delivery in math-intensive subjects. Problem Statement Differential equations is a fundamental part of mathematics and engineering curricula, yet they remain one of the most challenging subjects for undergraduate students due to their abstract nature and reliance on prior mathematical knowledge (Camacho et al., 2008; Ganter & Barker, 2004 ). The subject requires a strong understanding of prerequisite topics such as intermediate algebra, calculus, and linear algebra. However, many students enter differential equations courses with noticeable knowledge gaps that impede their ability to progress. Traditional lecture-based instruction often fails to address prior knowledge gaps effectively, leading to disengagement and poor performance (Freeman et al., 2014 ). JIT learning, an instructional strategy that delivers content in response to students’ immediate needs, has demonstrated potential in enhancing engagement and conceptual understanding across various STEM disciplines. The integration of JIT learning principles into the instruction of differential equations, however, remains underexplored and lacks a systematic framework. There is limited empirical evidence regarding how JIT approaches can be utilized to reinforce prerequisite knowledge, support real-time conceptual development, and enhance student performance, specifically in differential equations courses. There is a clear need for research exploring the integration of JIT learning strategies into differential equations curricula. Such an investigation could lead to more adaptive, responsive, and student-centered learning environments that promote conceptual understanding, improve academic outcomes, and foster long-term retention. Moreover, this gap highlights a critical need to investigate and develop effective JIT learning strategies tailored for differential equations instruction. Addressing this need could result in more adaptive and student-centered learning environments, enhancing comprehension, retention, and the ability to apply differential equations in real-world contexts. Research questions (1) How do students perceive JIT pedagogy implemented as bonus points for differential equations? (2) Was JIT pedagogy beneficial to student learning of the course material? (3) Was there any difference in student learning outcomes in students who used JIT compared to students who did not use JIT? Methods Participants This study examines data collected from a private, not-for-profit university in Daytona Beach, Florida. University ( 2024 ) has approximately 7900 undergraduate students attending the Daytona Beach Campus in fall 2024, with around 3800 students pursuing majors such as engineering, meteorology, space physics, computational math, engineering physics, and astronomy and astrophysics. Within this population of students, thirteen sections of Differential Equations were taught in spring 2025, with 38 to 40 students in each class. Although approximately 500 students on average take Differential Equations, only three sections participated. Students enrolled in these three sections have an equal chance of participating in the study. A convenience sampling method was used to collect data. According to Webster (n.d.), convenience sampling is chosen for research based on the availability and accessibility of the subjects. In these three sections, 119 students were enrolled and able to participate. This number is expected to decrease due to students dropping out and those opting not to participate. However, Fraenkel, Wallen, and Hyun ( 2019 ) noted that 30 to 50 participants are needed to demonstrate some degree of relationship. Students are selected on a voluntary basis and encouraged to sign a consent form that outlines their rights as volunteers and the procedures of the study. The initial contact with the subjects occurred in person to openly discuss the study and its expectations. An IRB-certified representative met with each class to discuss the survey and provide the link to complete it. Instruments The instruments used for this study included a Microsoft Forms survey and a questionnaire in the End of the Course Evaluations on the Canvas Learning Management System (LMS). The Microsoft Forms survey contained informed consent and 21 questions. The questions included Likert-scale questions, “choose all that apply” questions, and open-ended questions (see Appendix A) and took approximately 15 minutes to complete. The LMS survey contained branching questions (EOC; see Appendix B) and took approximately 10 minutes to complete. The EOC is a semesterly evaluation available to students at the end of each course, administered by the Office of Institutional Research, University. Professors can add questions to enhance the instruction and structure of their courses beyond the universal standard questions. This EOC survey was used to: (a) expand outreach to include more students who may not have attended class on the day the Microsoft Forms survey was provided, (b) allow students to change their minds about participating in the survey, and (c) provide a more in-depth description of their experience with the JIT handouts. Survey Validity and Reliability For this research, no survey or questionnaire exists. To show the validity and reliability of the surveys, two questions from the original Microsoft Forms survey were asked again in the EOC survey. According to Ranganathan, Caduff, and Frampton ( 2024 ), reliability and validity are defined as the “accuracy and consistency of the research tool” (p.1). To determine reliability, three classes of differential equations were given the same survey to show consistency among the results. Additionally, the measured stability among the results is similar among the classes. To determine validity, the survey was relevant to what was being studied (i.e., JIT handouts and student perception). Moreover, there were Likert-like questions, multiple-check questions, and open-ended questions that addressed the construction of the handouts and students’ perception of the handouts, thereby improving accuracy and relevance to move beyond traditional student satisfaction or favoritism. Lastly, clarity was evident in the responses to the survey questions due to the consistency among the results. P rocedure Students were provided 10 JIT handouts in Canvas with suggestions for when they would be most beneficial during the semester. Due dates were suggested, with the final due date being the second-to-last week of class. Students could turn in no more than two JIT handouts per week to maximize review, retention, and reasonable feedback return. Students were rewarded with up to two bonus points on their overall grade. Data were collected in two online formats. Participants were asked to complete an online Microsoft Forms survey about their experiences with JIT learning and then to answer two questions in the LMS at the end of the semester. The Microsoft Forms survey included an informed consent. If a student chose not to participate, the survey would close. If the student opted to participate, a second page would open to the survey questions. An IRB-certified representative from the proctored the data collection for the Microsoft Forms survey to eliminate professor bias. The representative was given a QR code with the link to distribute to students. Students could use their phones or computers to participate in this anonymous online survey. The LMS Evaluation questions opened at the end of the semester, and students completed the semesterly course evaluation; the additional questions for this study were included at the end of this semesterly evaluation, allowing students to participate or decline further participation. The question branched based on the student’s selected answer. First, the student could participate in the question or decline. Second, students were asked one question that contained one additional part. If the student chose “yes,” they submitted JIT handouts throughout the semester and were guided to the next question that addressed how JIT improved their study habits. If the student chose “no,” they did not submit any JIT handouts throughout the semester but still wanted to participate in the study were guided to the next question, which asked why they chose not to submit JITs throughout the semester. Results 119 students were eligible to participate in the study. Of this population, 51 participated in the online Microsoft Forms survey and 74 participated in the EOC survey. EOC Survey Results Of the 74 participants, 48 students submitted at least one JIT handout, 25 did not submit any JIT handouts, and 1 student declined to participate. Students who submitted JITs were asked, “How did JIT resources influence your study habits?” Student responses indicated positive influences such as being a good source for review, improving their understanding of the material, better managing deadlines, reinforcing uncomfortable topics, enhancing problem-solving mastery, enabling faster recall, and providing extra practice to close knowledge gaps. While most comments were positive, some negative and neutral feedback included that the JITs didn’t help, that they only completed them for bonus points, that the JITs didn’t change their study habits, and that they had no major impact on their studying or exams. Figure 1 shows students who did not submit any JITs being asked, “Why did you not participate in one or more JIT bonuses?” Figure 1 Response Results Note Student results to the question, “Why did you not participate in one or more JIT bonuses?” “Other” responses ranged from doing the JITs but not submitting them because they did not need them for a grade boost, were not a main priority, already knew the material, and lacked enough time due to external circumstances. Microsoft Forms Survey Results Out of the 51 participants, 36 students reported that they submitted at least one JIT during the course. Student responses are categorized into three main themes: (a) Learning materials and delivery, (b) Learning experience and effectiveness, and (c) Suggestions and improvements. Learning Materials and Delivery Accessibility and Usability Students found the JITs easily accessible in Canvas, with 49 students indicating “extremely easy" or “somewhat easy.” One student remained neutral. Overall, students rated the relevance of the JITs as very relevant (71%), somewhat relevant (22%), and not applicable (6%) to the course material. In total, 98% of the students felt that the JIT handouts aligned well with the differential equations course. Barriers In response to the question, “What barriers did you experience when accessing the JIT learning materials?” students identified Canvas, an LMS learning platform, as the most significant barrier due to loading issues. Other obstacles included personal procrastination, basic algebra skills that were worse than originally thought, the inability to turn in more than two JITs per week, and the need to print them out. JIT Handout Student Feedback Figures 2 and 3 address completion and recommendations of the JIT handouts. Figure 2 illustrates which handouts students completed, and Fig. 3 displays the recommendations for which handouts students would have liked more practice. Figure 2 JIT Specifics Note Responses to the question, “Which JIT handouts did you complete? Select all that apply.” Figure 3 Responses to Handouts Note Responses to the question, “Of the ones graded, which JIT handout did you feel you needed more problems to practice? Select all that apply.” Learning Experience and Effectiveness Relevance Results revealed that 71% of students in the differential equations course found the JIT handouts “extremely relevant,” while 22% of the students surveyed indicated “somewhat relevant.” The remaining students stated “N/A.” Expectations and Impact Depth and Detail to Learning Needs. In general, students commented that the recall of prior knowledge on specific topics was sufficiently detailed. One student noted, “Each JIT has a variety of problems ranging from simple to intricate which provides enough depth.” Confidence Levels. Students expressed increased confidence in differential equations and in the JITs. The JITs provided memory recall, reference, and better preparation for the course and the exams. Students felt more confident learning differential equations due to (a) basic math skill improvement, (b) easier transitions between concepts, and (c) applicability to what they were learning in class. Study Habits. While most of the differential equations students felt that the JITs did not affect their study habits, a few students noted that the JITs reduced their review time when preparing for exams and made them more prepared for what was happening in class. Task Completion Efficiency. Student noted improved speed in their algebra and less time second-guessing their prior knowledge. One student said: It has helped a lot with the nitty gritty of math. Although the new math is hard I did not need to spend time thinking about how to do a trig derivative or factor it made me be able to focus on diff eq. Overall Grade and Class Performance . Students focused on the JITs as bonus points, believing that they did not significantly impact their grade or overall performance. However, one student commented, “It has impacted for the better. It allows me to focus more on the class material with little practice for the older topics, rather than trying to relearn older topics before the test.” Student Self-reflection . Students were asked, “What are some positive outcomes you predict for yourself? Please choose all that apply.” Fig. 4 displays these student responses. Figure 4 Student Responses to Predicting Positive Learning Outcomes Note Student responses to positive learning outcomes Students were asked the same question from the EOC survey, “How did JIT resources influence your study habits?” Students indicated that the JITs helped with specific exams, to understand what was happening in class, gain confidence, and feel motivated to finish assignments. Responses also noted that they were not mandatory, and students appreciated earning bonus points toward their final grade. Other students remarked that it reduced the time they needed to study for exams. One student summarized the JITs: I think that are an amazing idea to review content. It’s great when the professor puts time and effort in even subjects they aren’t required to teach. This prompts the student to help fill in the gaps that they could have had. For example, in high school I remember being sick for a week while “completing the square” was taught. I thought it was a small math application but now taking DE, I see the importance of it. Now instead of going crazy all over google and asking AI to teach me, the JIT exists to help me fill in that gap. It’s like a knowledge bandage, that helps students fill in the gaps in their knowledge. I plan to download the JITs just to have, even the ones I haven’t completed yet for a grade. I think every class should have this incorporate within the courses. Fewer positive responses aligned with the EOC, as students did not see the value, complete the tasks, or have enough time. Some students also mentioned that it did not decrease their procrastination or that their stress increased due to the additional work in the course. Suggestions and Improvements The differential equations students who participated in the survey provided several suggestions. Figure 5 summarizes some of those suggestions. Figure 5 JIT Format Suggestions Note Responses to “What format would you prefer for JIT learning? Select all that apply.” Additional suggestions included embedding the video into Canvas and creating short modules, adding it to the homework packets, and adding more questions and topics. Students suggested adding more problems and handouts on trigonometry, the unit circle, implicit differentiation, fractions, and basic algebra. They recommended breaking the integration handout into individual JITs (e.g., u-substitution and integration by parts), including Taylor series, providing more applications relevant to their major, and placing a differential equations problem at the end to link the JIT handout to the classwork. Discussion Positive common themes among the differential equations students: better focus, more prepared, and more confident. For example, students experienced improved speed and reduced second-guessing of prior knowledge. This allowed them to focus on differential equations. As one student said: It has helped a lot with the nitty gritty of math. Although the new math is hard I did not need to spend time thinking about how to do a trig derivative or factor it made me be able to focus on diff eq. Negative and neutral common themes among the differential equations students: no value, already knew, and not enough time to do them. Students who did not do the JITs varied in reasons. For example, one student noted: I would say it primarily results from not having enought time to do the JITs as well as also having a pretty thourough knowledge of background concepts in the class. I could have made more time to do them but that required an extra level of dedication/motivation that I felt was better put into other classes/extracurriculars. While another student commented, “I already knew the material, but I did look at them to double check that I knew everything I needed to. I used them to keep myself on course, I just didn't complete any of them.” Improvement suggestions for the JITs included videos to go with the handouts, keeping it all on Canvas, more topics, and more handouts. Some students felt that there were not enough problems in the handouts and suggested individual handouts rather than condensing the topics into one handout (e.g., individual derivative handouts). Conclusions Overall, students who participated in the JITs felt that these activities enhanced their understanding of differential equations. They dedicated less time to researching foundational concepts and more time to mastering differential equations. While some students chose not to engage with the JITs, it’s crucial to understand their reasons and how the professor can inspire a different kind of motivation beyond bonus points. One student recommended allowing online submissions, while another suggested integrating them into the homework packets. The students offered valuable suggestions for improvement and expressed encouragement to retain the JITs as part of the curriculum. The literature suggests that Just-in-Time learning is a promising pedagogical approach for enhancing the teaching of complex STEM subjects. In the case of differential equations, where success is highly dependent on prior knowledge and timely intervention, JIT strategies can provide valuable preparation for personalized, responsive instruction. Moreover, JIT learning presents a potential solution by delivering content exactly when students need it, allowing instructors to customize lessons based on real-time feedback (Novak et al., 1999 ; Simkins & Maier, 2010 ). Although JIT learning has shown success in various STEM fields, its specific application to teaching differential equations remains underexplored and not systematically implemented. However, further research is required to validate and refine the use of JIT in this context. Recommendations Further research is recommended on several ideas. First, what kind of motivation do students need to improve their prior knowledge skills? Second, additional research on learning incentives to increase student participation for differential equations prior knowledge, and last, additional research on how students perceive JIT pedagogy are recommended. Declarations Participants were asked to fill out a Microsoft form about their experiences with Just in Time (JIT) learning, and they were also asked to answer two more questions in the End of the Course Evaluations on the Canvas Learning Management System (EOC). Specifically, data were collected from two forms, both online. Participants were asked to anonymously fill out a Microsoft Form about their experiences with JIT learning. Additionally, they were asked to anonymously answer two more questions in the EOC End of the Course at the end of the spring 2025 semester. A representative from the Center of Teaching and Learning Excellence, CTLE, Dr. Teha Cooks, proctored the data collection for the Microsoft Forms survey to remove professor bias. The LMS questions were added to the end of the course evaluation for the class, and students were asked to complete the course evaluation at their leisure before the university course evaluation deadline. Acknowledgments This author thanks the Center for Teaching and Learning Excellence at Embry-Riddle Aeronautical University in Daytona Beach for providing support and the opportunity to conduct this research. Moreover, this author thanks Dr. Teha Cooks for her support, encouragement, and guidance in implementing this research. Sources of Funding The Center for Teaching and Learning Excellence at Embry-Riddle Aeronautical University in Daytona Beach, Florida, offers semesterly competitive grants for faculty. This author competed for the Classroom Impact Grant and was awarded a small stipend. Financial or Non-financial Interests Stakeholders in this study are faculty and students in differential equations. The benefits of this research can help faculty: (a) improve students’ prior learning experiences in differential equations, (b) reduce class time spent on teaching prior knowledge, and (c) show students how to improve their prior knowledge study habits. Students benefit from this study by: (a) learning how to manage prior knowledge gaps, (b) learning how to improve study habits, and (c) developing confidence in their math skills. This author declares no competing interests. Ethical approval On April 3, 2025, this project received Approval # 25-150 for the Exempt category as per 45 CFR 46.105 from the Embry-Riddle Aeronautical University IRB Committee Chair. Author Contributions This paper is the sole work of this author. This author designed the study, set up unbiased data collection, and wrote this manuscript. Data Availability The data sets generated for this study are available on request to the corresponding author at [email protected] . Conflicts of Interest On behalf of all authors, the corresponding author states that there is no conflict of interest. References Arslan S (2010) Traditional instruction of differential equations and conceptual learning. Teach Math its applications: Int J IMA 29(2):94–107 Bibi A, Zamri SNS, Abedalaziz NAM, Ahmad M (2017) Teaching and learning of differential equations: A critical review to explore potential area for reform movement. Int J Innovative Res Multidisciplinary Field 3(6):225–235 Camacho M, Perdomo J, Santos-Trigo M (2009) Revisiting university students’ knowledge that involves basic differential equation questions. PNA 3(3):123–133 University (2024) Fact and figures . Retrieved from https://news.edu/media-resources/facts-and-figures Foster C (2021) Problem solving and prior knowledge. The Mathematical Association’s Mathematics in School , 50(4), pp. 6–8. Retrieved from https://www.foster77.co.uk/Foster,%20Mathematics%20in%20School,%20Problem%20solving%20and%20prior%20knowledge.pdf Fraenkel JR, Wallen NE, Hyun H (2019) How to design and evaluate research in education (7th ed.). McGraw-Hill Higher Education Freeman S, Eddy SL, McDonough M, Smith MK, Okoroafor N, Jordt H, Wenderoth MP (2014) Active learning increases student performance in science, engineering, and mathematics. Proceedings of the National Academy of Sciences , 111(23), 8410–8415. https://doi.org/10.1073/pnas.1319030111 Ganter SL, Barker W (eds) (2004) The curriculum foundations project: Voices of the partner disciplines. Mathematical Association of America, Washington, DC Juter K, Olsen J-F (2014) Just-in-time teaching in undergraduate mathematics. NOMAD Nordic Stud Math Educ 19(2):77–96. https://doi.org/10.7146/nomad.v19i2.148633 Killi S, Morrison A (2015) Just-in-time teaching, Just-in-need learning: Designing towards optimized pedagogical outcomes. Univers J Educational Res 3:742–750. https://doi.org/10.13189/ujer.2015.031013 Mayer RE (2005) The Cambridge handbook of multimedia learning. Cambridge University Press Novak GM, Patterson ET, Gavrin AD, Christian W (1999) Just-in-time teaching: Blending active learning with web technology. Prentice Hall Ranganathan P, Caduff C, Frampton CM (2024) Designing and validating a research questionnaire-Part 2. Perspect Clin Res 15(1):42–45 Simkins SP, Maier MH (2010) Just-in-Time Teaching: Across the disciplines, across the academy. Stylus Publishing Webster W (n.d.). What is sampling? Retrieved from https://www.qualtrics.com/experience-management/research/sampling-methods/ Additional Declarations The authors declare no competing interests. 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Select all that apply.”\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-9020528/v1/3dd328681ff75d482cdc3994.png"},{"id":103879354,"identity":"af0b3c43-e8fc-444d-9a46-07991b96e471","added_by":"auto","created_at":"2026-03-04 04:49:24","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":102428,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003eResponses to Handouts\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eNote.\u003c/em\u003eResponses to the question, “Of the ones graded, which JIT handout did you feel you needed more problems to practice? Select all that apply.”\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-9020528/v1/0f2178b2c196447247a39c03.png"},{"id":103879357,"identity":"8247b87e-32db-4ce9-b90e-e764cc5a6588","added_by":"auto","created_at":"2026-03-04 04:49:24","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":106456,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003eStudent Responses to Predicting Positive Learning Outcomes\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eNote.\u003c/em\u003e Student responses to positive learning outcomes\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-9020528/v1/6444ebdb1a6efe08805f43b7.png"},{"id":103879358,"identity":"81fb02fe-37a1-42c7-a7f8-d35f5ad41e47","added_by":"auto","created_at":"2026-03-04 04:49:24","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":53576,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003eJIT Format Suggestions\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eNote.\u003c/em\u003eResponses to “What format would you prefer for JIT learning? Select all that apply.”\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-9020528/v1/c3e7409b84070c018e3741e5.png"},{"id":104408087,"identity":"392c6451-ec79-4c92-924f-817d82b83c8b","added_by":"auto","created_at":"2026-03-11 12:41:30","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1398495,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9020528/v1/9c3071eb-d65f-41c1-9e65-58d9875d5d72.pdf"},{"id":104401528,"identity":"cfa7f631-5f0a-4508-94be-9a5ee32560f0","added_by":"auto","created_at":"2026-03-11 12:12:56","extension":"docx","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":414140,"visible":true,"origin":"","legend":"","description":"","filename":"Appendix.docx","url":"https://assets-eu.researchsquare.com/files/rs-9020528/v1/a726e304885369c81ca1e880.docx"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eAssessing Just-in-Time Implementation in Differential Equations\u003c/p\u003e","fulltext":[{"header":"Introduction","content":"\u003cp\u003eDifferential equations require extensive knowledge of Intermediate Algebra, Calculus II, and problem-solving strategies. When students reach differential equations, it is expected that they may have prior knowledge gaps because several classes have passed since Intermediate Algebra. Topics in Intermediate Algebra include solving systems of equations, completing the square, factoring, the quadratic formula, and exponent rules. These topics serve as a foundation for differential equations. Foster (\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2021\u003c/span\u003e) noted that prior knowledge gaps are not detrimental as long as the student can contribute something to the table, like relevant learned techniques. Differential equations employ various algorithmic solving techniques based on the classification of the equations (Camacho, Perdomo, \u0026amp; Santos-Trigo, \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2009\u003c/span\u003e; Foster, \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). How can students improve their recall of prior knowledge when these algorithmic equations require extensive knowledge of Intermediate Algebra or Calculus? Just-in-Time learning (JIT) is one tool designed to help students strengthen their foundational knowledge. The JIT handouts aim to motivate students both intrinsically and extrinsically, foster active learning, encourage classroom engagement, and provide remediation and support for students at all levels (Killi \u0026amp; Morrison, \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2015\u003c/span\u003e; Novak, Patterson, Gavrin, \u0026amp; Christian, \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e1999\u003c/span\u003e). The goal is to alleviate the frustration associated with learning new and old material simultaneously in differential equations and to create a more meaningful learning experience.\u003c/p\u003e"},{"header":"Literature Review: Just-in-Time Learning and Differential Equations Instruction","content":"\u003cp\u003eDifferential equations is essential for modeling dynamic systems in engineering, physics, and applied mathematics. However, differential equations is often regarded as one of the most challenging subjects in undergraduate mathematics due to their abstract nature, various problem-solving strategies, and reliance on prerequisite knowledge such as calculus, algebra, and linear algebra (Foster, \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2021\u003c/span\u003e; Ganter \u0026amp; Barker, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2004\u003c/span\u003e). Students frequently enter differential equations courses without sufficient mastery of these foundational topics, resulting in frustration, disengagement, and poor academic performance.\u003c/p\u003e \u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003eChallenges in Traditional Instruction\u003c/h2\u003e \u003cp\u003eAlthough widely used, traditional lecture-based instruction tends to be passive and does not adequately meet students' individual learning needs. Freeman et al. (\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2014\u003c/span\u003e) conducted a meta-analysis of STEM education and found that traditional lectures lead to significantly lower student performance compared to active learning methods. Arslan (\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e2010\u003c/span\u003e) discovered that procedural learning (i.e., how to do something) supports conceptual learning (i.e., why something happens), but in a traditional teaching setting, procedural learning did not result in students applying that knowledge to new situations. Additionally, Bibi et al. (\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2017\u003c/span\u003e) noted that professors rely on algebraic, graphical, and numerical techniques to teach differential equations. In the context of differential equations, where cognitive load is already high due to complex concepts and calculations, lecture-based delivery often fails to reinforce prerequisite knowledge when it is most needed.\u003c/p\u003e \u003c/div\u003e\n\u003ch3\u003eThe Just-in-Time Learning Approach\u003c/h3\u003e\n\u003cp\u003eJIT learning is an approach that delivers educational content based on students\u0026rsquo; immediate needs, often identified through formative assessments conducted before class. Novak et al. (\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e1999\u003c/span\u003e), pioneers of Just-in-Time Teaching, demonstrated that providing instructors with pre-class feedback enables them to modify lectures in real time, making instruction more responsive and learner-centered. JIT learning emphasizes timeliness, adaptability, and addressing misconceptions before they are reinforced.\u003c/p\u003e \u003cp\u003eSimkins and Maier (\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2010\u003c/span\u003e) provided Just-in-Time Teaching suggestions across academic disciplines and found that it improved engagement, accountability, and performance by encouraging students to come prepared and actively participate in class. This is especially relevant for differential equations, where a lack of foundational understanding of calculus or algebra can significantly impede new learning. JIT strategies can assist instructors in identifying and addressing these deficiencies early.\u003c/p\u003e\n\u003ch3\u003eApplication to Differential Equations\u003c/h3\u003e\n\u003cp\u003eAlthough JIT learning has been successfully applied in physics, biology, and economics, its implementation in teaching differential equations remains limited. Juter and Olsen (\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2014\u003c/span\u003e) investigated Just-in-Time Teaching implementation in undergraduate calculus courses and reported improvements in student engagement, problem-solving ability, and work habits. Their findings support the hypothesis that JIT can provide the necessary scaffolding to help students connect new concepts with prior knowledge.\u003c/p\u003e \u003cp\u003eMayer\u0026rsquo;s (\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2005\u003c/span\u003e) Cognitive Theory of Multimedia Learning supports the pedagogical rationale behind JIT strategies. According to Mayer, students learn more effectively when instructional content is aligned with their cognitive readiness and delivered in manageable chunks that reduce cognitive overload. In a subject like differential equations, where concepts are hierarchical and complex, JIT learning helps minimize extraneous load and focus attention on essential material.\u003c/p\u003e\n\u003ch3\u003eLimitations and Gaps\u003c/h3\u003e\n\u003cp\u003eDespite promising findings, the integration of JIT into mathematics instruction, particularly differential equations, remains underexplored. While case studies suggest positive outcomes, there is a need for additional research on the effectiveness of JIT approaches in enhancing conceptual understanding, bridging prerequisite knowledge gaps, and improving long-term retention in this specific area. Furthermore, more work is required to develop structured frameworks and digital tools that facilitate JIT delivery in math-intensive subjects.\u003c/p\u003e\n\u003ch3\u003eProblem Statement\u003c/h3\u003e\n\u003cdiv id=\"Sec8\" class=\"Section2\"\u003e\u003cp\u003eDifferential equations is a fundamental part of mathematics and engineering curricula, yet they remain one of the most challenging subjects for undergraduate students due to their abstract nature and reliance on prior mathematical knowledge (Camacho et al., 2008; Ganter \u0026amp; Barker, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2004\u003c/span\u003e). The subject requires a strong understanding of prerequisite topics such as intermediate algebra, calculus, and linear algebra. However, many students enter differential equations courses with noticeable knowledge gaps that impede their ability to progress.\u003c/p\u003e \u003cp\u003eTraditional lecture-based instruction often fails to address prior knowledge gaps effectively, leading to disengagement and poor performance (Freeman et al., \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2014\u003c/span\u003e). JIT learning, an instructional strategy that delivers content in response to students\u0026rsquo; immediate needs, has demonstrated potential in enhancing engagement and conceptual understanding across various STEM disciplines. The integration of JIT learning principles into the instruction of differential equations, however, remains underexplored and lacks a systematic framework. There is limited empirical evidence regarding how JIT approaches can be utilized to reinforce prerequisite knowledge, support real-time conceptual development, and enhance student performance, specifically in differential equations courses.\u003c/p\u003e \u003cp\u003eThere is a clear need for research exploring the integration of JIT learning strategies into differential equations curricula. Such an investigation could lead to more adaptive, responsive, and student-centered learning environments that promote conceptual understanding, improve academic outcomes, and foster long-term retention. Moreover, this gap highlights a critical need to investigate and develop effective JIT learning strategies tailored for differential equations instruction. Addressing this need could result in more adaptive and student-centered learning environments, enhancing comprehension, retention, and the ability to apply differential equations in real-world contexts.\u003c/p\u003e \u003c/div\u003e\n\u003ch3\u003eResearch questions\u003c/h3\u003e\n\u003cp\u003e(1) How do students perceive JIT pedagogy implemented as bonus points for differential equations?\u003c/p\u003e \u003cp\u003e(2) Was JIT pedagogy beneficial to student learning of the course material?\u003c/p\u003e \u003cp\u003e(3) Was there any difference in student learning outcomes in students who used JIT compared to students who did not use JIT?\u003c/p\u003e"},{"header":"Methods","content":"\u003cdiv id=\"Sec11\" class=\"Section2\"\u003e \u003ch2\u003eParticipants\u003c/h2\u003e \u003cp\u003eThis study examines data collected from a private, not-for-profit university in Daytona Beach, Florida. University (\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e2024\u003c/span\u003e) has approximately 7900 undergraduate students attending the Daytona Beach Campus in fall 2024, with around 3800 students pursuing majors such as engineering, meteorology, space physics, computational math, engineering physics, and astronomy and astrophysics. Within this population of students, thirteen sections of Differential Equations were taught in spring 2025, with 38 to 40 students in each class. Although approximately 500 students on average take Differential Equations, only three sections participated. Students enrolled in these three sections have an equal chance of participating in the study. A convenience sampling method was used to collect data. According to Webster (n.d.), convenience sampling is chosen for research based on the availability and accessibility of the subjects. In these three sections, 119 students were enrolled and able to participate. This number is expected to decrease due to students dropping out and those opting not to participate. However, Fraenkel, Wallen, and Hyun (\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2019\u003c/span\u003e) noted that 30 to 50 participants are needed to demonstrate some degree of relationship. Students are selected on a voluntary basis and encouraged to sign a consent form that outlines their rights as volunteers and the procedures of the study. The initial contact with the subjects occurred in person to openly discuss the study and its expectations. An IRB-certified representative met with each class to discuss the survey and provide the link to complete it.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec12\" class=\"Section2\"\u003e \u003ch2\u003eInstruments\u003c/h2\u003e \u003cp\u003eThe instruments used for this study included a Microsoft Forms survey and a questionnaire in the End of the Course Evaluations on the Canvas Learning Management System (LMS). The Microsoft Forms survey contained informed consent and 21 questions. The questions included Likert-scale questions, \u0026ldquo;choose all that apply\u0026rdquo; questions, and open-ended questions (see Appendix A) and took approximately 15 minutes to complete. The LMS survey contained branching questions (EOC; see Appendix B) and took approximately 10 minutes to complete. The EOC is a semesterly evaluation available to students at the end of each course, administered by the Office of Institutional Research, University. Professors can add questions to enhance the instruction and structure of their courses beyond the universal standard questions. This EOC survey was used to: (a) expand outreach to include more students who may not have attended class on the day the Microsoft Forms survey was provided, (b) allow students to change their minds about participating in the survey, and (c) provide a more in-depth description of their experience with the JIT handouts.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec13\" class=\"Section2\"\u003e \u003ch2\u003eSurvey Validity and Reliability\u003c/h2\u003e \u003cp\u003eFor this research, no survey or questionnaire exists. To show the validity and reliability of the surveys, two questions from the original Microsoft Forms survey were asked again in the EOC survey. According to Ranganathan, Caduff, and Frampton (\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2024\u003c/span\u003e), reliability and validity are defined as the \u0026ldquo;accuracy and consistency of the research tool\u0026rdquo; (p.1). To determine reliability, three classes of differential equations were given the same survey to show consistency among the results. Additionally, the measured stability among the results is similar among the classes. To determine validity, the survey was relevant to what was being studied (i.e., JIT handouts and student perception). Moreover, there were Likert-like questions, multiple-check questions, and open-ended questions that addressed the construction of the handouts and students\u0026rsquo; perception of the handouts, thereby improving accuracy and relevance to move beyond traditional student satisfaction or favoritism. Lastly, clarity was evident in the responses to the survey questions due to the consistency among the results.\u003c/p\u003e \u003cp\u003eP\u003cb\u003erocedure\u003c/b\u003e\u003c/p\u003e \u003cp\u003eStudents were provided 10 JIT handouts in Canvas with suggestions for when they would be most beneficial during the semester. Due dates were suggested, with the final due date being the second-to-last week of class. Students could turn in no more than two JIT handouts per week to maximize review, retention, and reasonable feedback return. Students were rewarded with up to two bonus points on their overall grade.\u003c/p\u003e \u003cp\u003eData were collected in two online formats. Participants were asked to complete an online Microsoft Forms survey about their experiences with JIT learning and then to answer two questions in the LMS at the end of the semester.\u003c/p\u003e \u003cp\u003eThe Microsoft Forms survey included an informed consent. If a student chose not to participate, the survey would close. If the student opted to participate, a second page would open to the survey questions. An IRB-certified representative from the proctored the data collection for the Microsoft Forms survey to eliminate professor bias. The representative was given a QR code with the link to distribute to students. Students could use their phones or computers to participate in this anonymous online survey.\u003c/p\u003e \u003cp\u003eThe LMS Evaluation questions opened at the end of the semester, and students completed the semesterly course evaluation; the additional questions for this study were included at the end of this semesterly evaluation, allowing students to participate or decline further participation. The question branched based on the student\u0026rsquo;s selected answer. First, the student could participate in the question or decline. Second, students were asked one question that contained one additional part. If the student chose \u0026ldquo;yes,\u0026rdquo; they submitted JIT handouts throughout the semester and were guided to the next question that addressed how JIT improved their study habits. If the student chose \u0026ldquo;no,\u0026rdquo; they did not submit any JIT handouts throughout the semester but still wanted to participate in the study were guided to the next question, which asked why they chose not to submit JITs throughout the semester.\u003c/p\u003e \u003c/div\u003e"},{"header":"Results","content":"\u003cp\u003e119 students were eligible to participate in the study. Of this population, 51 participated in the online Microsoft Forms survey and 74 participated in the EOC survey.\u003c/p\u003e \u003cdiv id=\"Sec15\" class=\"Section2\"\u003e \u003ch2\u003eEOC Survey Results\u003c/h2\u003e \u003cp\u003eOf the 74 participants, 48 students submitted at least one JIT handout, 25 did not submit any JIT handouts, and 1 student declined to participate. Students who submitted JITs were asked, \u0026ldquo;How did JIT resources influence your study habits?\u0026rdquo; Student responses indicated positive influences such as being a good source for review, improving their understanding of the material, better managing deadlines, reinforcing uncomfortable topics, enhancing problem-solving mastery, enabling faster recall, and providing extra practice to close knowledge gaps. While most comments were positive, some negative and neutral feedback included that the JITs didn\u0026rsquo;t help, that they only completed them for bonus points, that the JITs didn\u0026rsquo;t change their study habits, and that they had no major impact on their studying or exams.\u003c/p\u003e \u003cp\u003eFigure 1 shows students who did not submit any JITs being asked, \u0026ldquo;Why did you not participate in one or more JIT bonuses?\u0026rdquo;\u003c/p\u003e \u003cp\u003e \u003cb\u003eFigure 1\u003c/b\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec16\" class=\"Section2\"\u003e \u003ch2\u003eResponse Results\u003c/h2\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003cstrong\u003eNote\u003c/strong\u003e \u003cp\u003eStudent results to the question, \u0026ldquo;Why did you not participate in one or more JIT bonuses?\u0026rdquo;\u003c/p\u003e \u003c/p\u003e \u003cp\u003e\u0026ldquo;Other\u0026rdquo; responses ranged from doing the JITs but not submitting them because they did not need them for a grade boost, were not a main priority, already knew the material, and lacked enough time due to external circumstances.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec17\" class=\"Section2\"\u003e \u003ch2\u003eMicrosoft Forms Survey Results\u003c/h2\u003e \u003cp\u003eOut of the 51 participants, 36 students reported that they submitted at least one JIT during the course. Student responses are categorized into three main themes: (a) Learning materials and delivery, (b) Learning experience and effectiveness, and (c) Suggestions and improvements.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec18\" class=\"Section2\"\u003e \u003ch2\u003eLearning Materials and Delivery\u003c/h2\u003e \u003cdiv id=\"Sec19\" class=\"Section3\"\u003e \u003ch2\u003eAccessibility and Usability\u003c/h2\u003e \u003cp\u003eStudents found the JITs easily accessible in Canvas, with 49 students indicating \u0026ldquo;extremely easy\" or \u0026ldquo;somewhat easy.\u0026rdquo; One student remained neutral. Overall, students rated the relevance of the JITs as very relevant (71%), somewhat relevant (22%), and not applicable (6%) to the course material. In total, 98% of the students felt that the JIT handouts aligned well with the differential equations course.\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv id=\"Sec20\" class=\"Section2\"\u003e \u003ch2\u003eBarriers\u003c/h2\u003e \u003cp\u003eIn response to the question, \u0026ldquo;What barriers did you experience when accessing the JIT learning materials?\u0026rdquo; students identified Canvas, an LMS learning platform, as the most significant barrier due to loading issues. Other obstacles included personal procrastination, basic algebra skills that were worse than originally thought, the inability to turn in more than two JITs per week, and the need to print them out.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec21\" class=\"Section2\"\u003e \u003ch2\u003eJIT Handout Student Feedback\u003c/h2\u003e \u003cp\u003eFigures 2 and 3 address completion and recommendations of the JIT handouts. Figure\u0026nbsp;2 illustrates which handouts students completed, and Fig.\u0026nbsp;3 displays the recommendations for which handouts students would have liked more practice.\u003c/p\u003e \u003cp\u003e \u003cb\u003eFigure 2\u003c/b\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec22\" class=\"Section2\"\u003e \u003ch2\u003eJIT Specifics\u003c/h2\u003e \u003cp\u003e \u003cdiv description=\"A graph with different colored barsAI-generated content may be incorrect.\" class=\"Drawing\" id=\"759700144\" name=\"Picture 1\"\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003cstrong\u003eNote\u003c/strong\u003e \u003cp\u003eResponses to the question, \u0026ldquo;Which JIT handouts did you complete? Select all that apply.\u0026rdquo;\u003c/p\u003e \u003c/p\u003e \u003cp\u003e \u003cb\u003eFigure 3\u003c/b\u003e \u003c/p\u003e \u003cdiv id=\"Sec23\" class=\"Section3\"\u003e \u003ch2\u003eResponses to Handouts\u003c/h2\u003e \u003cp\u003e \u003cdiv description=\"A graph with colorful barsAI-generated content may be incorrect.\" class=\"Drawing\" id=\"2103415937\" name=\"Picture 1\"\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003cstrong\u003eNote\u003c/strong\u003e \u003cp\u003eResponses to the question, \u0026ldquo;Of the ones graded, which JIT handout did you feel you needed more problems to practice? Select all that apply.\u0026rdquo;\u003c/p\u003e \u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv id=\"Sec24\" class=\"Section2\"\u003e \u003ch2\u003eLearning Experience and Effectiveness\u003c/h2\u003e \u003cdiv id=\"Sec25\" class=\"Section3\"\u003e \u003ch2\u003eRelevance\u003c/h2\u003e \u003cp\u003eResults revealed that 71% of students in the differential equations course found the JIT handouts \u0026ldquo;extremely relevant,\u0026rdquo; while 22% of the students surveyed indicated \u0026ldquo;somewhat relevant.\u0026rdquo; The remaining students stated \u0026ldquo;N/A.\u0026rdquo;\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec26\" class=\"Section3\"\u003e \u003ch2\u003eExpectations and Impact\u003c/h2\u003e \u003cp\u003e \u003cb\u003eDepth and Detail to Learning Needs.\u003c/b\u003e In general, students commented that the recall of prior knowledge on specific topics was sufficiently detailed. One student noted, \u0026ldquo;Each JIT has a variety of problems ranging from simple to intricate which provides enough depth.\u0026rdquo;\u003c/p\u003e \u003cp\u003e \u003cb\u003eConfidence Levels.\u003c/b\u003e Students expressed increased confidence in differential equations and in the JITs. The JITs provided memory recall, reference, and better preparation for the course and the exams. Students felt more confident learning differential equations due to (a) basic math skill improvement, (b) easier transitions between concepts, and (c) applicability to what they were learning in class.\u003c/p\u003e \u003cp\u003e \u003cb\u003eStudy Habits.\u003c/b\u003e While most of the differential equations students felt that the JITs did not affect their study habits, a few students noted that the JITs reduced their review time when preparing for exams and made them more prepared for what was happening in class.\u003c/p\u003e \u003cp\u003e \u003cb\u003eTask Completion Efficiency.\u003c/b\u003e Student noted improved speed in their algebra and less time second-guessing their prior knowledge. One student said:\u003cdiv class=\"BlockQuote\"\u003e\u003cp\u003eIt has helped a lot with the nitty gritty of math. Although the new math is hard I did not need to spend time thinking about how to do a trig derivative or factor it made me be able to focus on diff eq.\u003c/p\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003e \u003cb\u003eOverall Grade and Class Performance\u003c/b\u003e. Students focused on the JITs as bonus points, believing that they did not significantly impact their grade or overall performance. However, one student commented, \u0026ldquo;It has impacted for the better. It allows me to focus more on the class material with little practice for the older topics, rather than trying to relearn older topics before the test.\u0026rdquo;\u003c/p\u003e \u003cp\u003e \u003cb\u003eStudent Self-reflection\u003c/b\u003e. Students were asked, \u0026ldquo;What are some positive outcomes you predict for yourself? Please choose all that apply.\u0026rdquo; Fig.\u0026nbsp;4 displays these student responses.\u003c/p\u003e \u003cp\u003e \u003cb\u003eFigure 4\u003c/b\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec27\" class=\"Section3\"\u003e \u003ch2\u003eStudent Responses to Predicting Positive Learning Outcomes\u003c/h2\u003e \u003cp\u003e \u003cdiv description=\"A graph with different colored barsAI-generated content may be incorrect.\" class=\"Drawing\" id=\"1858985117\" name=\"Picture 1\"\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003cstrong\u003eNote\u003c/strong\u003e \u003cp\u003eStudent responses to positive learning outcomes\u003c/p\u003e \u003c/p\u003e \u003cp\u003eStudents were asked the same question from the EOC survey, \u0026ldquo;How did JIT resources influence your study habits?\u0026rdquo; Students indicated that the JITs helped with specific exams, to understand what was happening in class, gain confidence, and feel motivated to finish assignments. Responses also noted that they were not mandatory, and students appreciated earning bonus points toward their final grade. Other students remarked that it reduced the time they needed to study for exams.\u003c/p\u003e \u003cp\u003eOne student summarized the JITs:\u003cdiv class=\"BlockQuote\"\u003e\u003cp\u003eI think that are an amazing idea to review content. It\u0026rsquo;s great when the professor puts time and effort in even subjects they aren\u0026rsquo;t required to teach. This prompts the student to help fill in the gaps that they could have had. For example, in high school I remember being sick for a week while \u0026ldquo;completing the square\u0026rdquo; was taught. I thought it was a small math application but now taking DE, I see the importance of it. Now instead of going crazy all over google and asking AI to teach me, the JIT exists to help me fill in that gap. It\u0026rsquo;s like a knowledge bandage, that helps students fill in the gaps in their knowledge. I plan to download the JITs just to have, even the ones I haven\u0026rsquo;t completed yet for a grade. I think every class should have this incorporate within the courses.\u003c/p\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eFewer positive responses aligned with the EOC, as students did not see the value, complete the tasks, or have enough time. Some students also mentioned that it did not decrease their procrastination or that their stress increased due to the additional work in the course.\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv id=\"Sec28\" class=\"Section2\"\u003e \u003ch2\u003eSuggestions and Improvements\u003c/h2\u003e \u003cp\u003eThe differential equations students who participated in the survey provided several suggestions. Figure\u0026nbsp;5 summarizes some of those suggestions.\u003c/p\u003e \u003cp\u003e \u003cb\u003eFigure 5\u003c/b\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec29\" class=\"Section2\"\u003e \u003ch2\u003eJIT Format Suggestions\u003c/h2\u003e \u003cp\u003e \u003cdiv description=\"A white background with black dotsAI-generated content may be incorrect.\" class=\"Drawing\" id=\"705795147\" name=\"Picture 1\"\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003cstrong\u003eNote\u003c/strong\u003e \u003cp\u003eResponses to \u0026ldquo;What format would you prefer for JIT learning? Select all that apply.\u0026rdquo;\u003c/p\u003e \u003c/p\u003e \u003cp\u003eAdditional suggestions included embedding the video into Canvas and creating short modules, adding it to the homework packets, and adding more questions and topics.\u003c/p\u003e \u003cp\u003eStudents suggested adding more problems and handouts on trigonometry, the unit circle, implicit differentiation, fractions, and basic algebra. They recommended breaking the integration handout into individual JITs (e.g., u-substitution and integration by parts), including Taylor series, providing more applications relevant to their major, and placing a differential equations problem at the end to link the JIT handout to the classwork.\u003c/p\u003e \u003c/div\u003e"},{"header":"Discussion","content":"\u003cp\u003ePositive common themes among the differential equations students: better focus, more prepared, and more confident. For example, students experienced improved speed and reduced second-guessing of prior knowledge. This allowed them to focus on differential equations. As one student said:\u003cdiv class=\"BlockQuote\"\u003e\u003cp\u003eIt has helped a lot with the nitty gritty of math. Although the new math is hard I did not need to spend time thinking about how to do a trig derivative or factor it made me be able to focus on diff eq.\u003c/p\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eNegative and neutral common themes among the differential equations students: no value, already knew, and not enough time to do them. Students who did not do the JITs varied in reasons. For example, one student noted:\u003cdiv class=\"BlockQuote\"\u003e\u003cp\u003eI would say it primarily results from not having enought time to do the JITs as well as also having a pretty thourough knowledge of background concepts in the class. I could have made more time to do them but that required an extra level of dedication/motivation that I felt was better put into other classes/extracurriculars.\u003c/p\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhile another student commented, \u0026ldquo;I already knew the material, but I did look at them to double check that I knew everything I needed to. I used them to keep myself on course, I just didn't complete any of them.\u0026rdquo;\u003c/p\u003e \u003cp\u003eImprovement suggestions for the JITs included videos to go with the handouts, keeping it all on Canvas, more topics, and more handouts. Some students felt that there were not enough problems in the handouts and suggested individual handouts rather than condensing the topics into one handout (e.g., individual derivative handouts).\u003c/p\u003e"},{"header":"Conclusions","content":"\u003cp\u003eOverall, students who participated in the JITs felt that these activities enhanced their understanding of differential equations. They dedicated less time to researching foundational concepts and more time to mastering differential equations. While some students chose not to engage with the JITs, it\u0026rsquo;s crucial to understand their reasons and how the professor can inspire a different kind of motivation beyond bonus points. One student recommended allowing online submissions, while another suggested integrating them into the homework packets. The students offered valuable suggestions for improvement and expressed encouragement to retain the JITs as part of the curriculum.\u003c/p\u003e\n\u003cp\u003eThe literature suggests that Just-in-Time learning is a promising pedagogical approach for enhancing the teaching of complex STEM subjects. In the case of differential equations, where success is highly dependent on prior knowledge and timely intervention, JIT strategies can provide valuable preparation for personalized, responsive instruction. Moreover, JIT learning presents a potential solution by delivering content exactly when students need it, allowing instructors to customize lessons based on real-time feedback (Novak et al., \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e1999\u003c/span\u003e; Simkins \u0026amp; Maier, \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2010\u003c/span\u003e). Although JIT learning has shown success in various STEM fields, its specific application to teaching differential equations remains underexplored and not systematically implemented. However, further research is required to validate and refine the use of JIT in this context.\u003c/p\u003e \n \u003cdiv id=\"Sec32\" class=\"Section2\"\u003e \u003ch2\u003eRecommendations\u003c/h2\u003e \u003cp\u003eFurther research is recommended on several ideas. First, what kind of motivation do students need to improve their prior knowledge skills? Second, additional research on learning incentives to increase student participation for differential equations prior knowledge, and last, additional research on how students perceive JIT pedagogy are recommended.\u003c/p\u003e \u003c/div\u003e"},{"header":"Declarations","content":"\u003cp\u003eParticipants were asked to fill out a Microsoft form about their experiences with Just in Time (JIT) learning, and they were also asked to answer two more questions in the End of the Course Evaluations on the Canvas Learning Management System (EOC). \u0026nbsp;Specifically, data were collected from two forms, both online. Participants were asked to anonymously fill out a Microsoft Form about their experiences with JIT learning. \u0026nbsp;Additionally, they were asked to anonymously answer two more questions in the EOC End of the Course at the end of the spring 2025 semester. A representative from the Center of Teaching and Learning Excellence, CTLE, Dr. Teha Cooks, proctored the data collection for the Microsoft Forms survey to remove professor bias. The LMS questions were added to the end of the course evaluation for the class, and students were asked to complete the course evaluation at their leisure before the university\u003cstrong\u003e\u0026nbsp;\u003c/strong\u003ecourse evaluation deadline.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAcknowledgments\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis author thanks the Center for Teaching and Learning Excellence at Embry-Riddle Aeronautical University in Daytona Beach for providing support and the opportunity to conduct this research. Moreover, this author thanks Dr. Teha Cooks for her support, encouragement, and guidance in implementing this research.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eSources of Funding\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe Center for Teaching and Learning Excellence at Embry-Riddle Aeronautical University in Daytona Beach, Florida, offers semesterly competitive grants for faculty. This author competed for the Classroom Impact Grant and was awarded a small stipend.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFinancial or Non-financial Interests\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eStakeholders in this study are faculty and students in differential equations. The benefits of this research can help faculty: (a) improve students\u0026rsquo; prior learning experiences in differential equations, (b) reduce class time spent on teaching prior knowledge, and (c) show students how to improve their prior knowledge study habits. \u0026nbsp;Students benefit from this study by: (a) learning how to manage prior knowledge gaps, (b) learning how to improve study habits, and (c) developing confidence in their math skills. This author declares no competing interests.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eEthical approval\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eOn April 3, 2025, this project received Approval # 25-150 for the Exempt category as per 45 CFR 46.105 from the Embry-Riddle Aeronautical University IRB Committee Chair.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor Contributions\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis paper is the sole work of this author. This author designed the study, set up unbiased data collection, and wrote this manuscript.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eData Availability\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe data sets generated for this study are available on request to the corresponding author at [email protected].\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConflicts of Interest\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eOn behalf of all authors, the corresponding author states that there is no conflict of interest.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eArslan S (2010) Traditional instruction of differential equations and conceptual learning. 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Perspect Clin Res 15(1):42\u0026ndash;45\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eSimkins SP, Maier MH (2010) Just-in-Time Teaching: Across the disciplines, across the academy. Stylus Publishing\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eWebster W (n.d.). \u003cem\u003eWhat is sampling?\u003c/em\u003e Retrieved from \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://www.qualtrics.com/experience-management/research/sampling-methods/\u003c/span\u003e\u003cspan address=\"https://www.qualtrics.com/experience-management/research/sampling-methods/\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"Embry–Riddle Aeronautical University","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Differential equations, Just-in-Time learning, Prior knowledge gaps in math","lastPublishedDoi":"10.21203/rs.3.rs-9020528/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9020528/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis study examined how differential equations students: (a) perceived Just-in-Time (JIT) learning to lessen prior knowledge gaps, (b) benefited from the JIT learning handouts, and (c) improved student learning outcomes. Results indicated that students positively benefited from the JITs, gained confidence in learning differential equations, and spent less time researching or studying prior knowledge. Students who did not complete the JITs results indicated that they already knew the material, did not have time to complete the JITs, or suggested that the handouts could have been better integrated into the course. The differential equations students provided suggestions for improvement and, overall, recommended keeping the JITs as a part of the course.\u003c/p\u003e","manuscriptTitle":"Assessing Just-in-Time Implementation in Differential Equations","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-03-04 04:49:19","doi":"10.21203/rs.3.rs-9020528/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"318d0463-2c6e-48f3-bdeb-e6fb518d266f","owner":[],"postedDate":"March 4th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":63847599,"name":"Algebra"}],"tags":[],"updatedAt":"2026-03-04T04:49:19+00:00","versionOfRecord":[],"versionCreatedAt":"2026-03-04 04:49:19","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9020528","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9020528","identity":"rs-9020528","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}