Inductive vs. Deductive Learning in Calculus: Unraveling the Impact on Integral Understanding

Research Paper Recommended citation: Kulcsár, N. (2025). Inductive vs. Deductive Learning in Calculus: Unraveling the Impact on Integral Understanding. In Kangaslampi, R., Langie, G., Järvinen, H.-M., & Nagy, B. (Eds.), SEFI 53rd Annual Conference. European Society for Engineering Education (SEFI), Tampere, Finland. DOI: 10.5281/zenodo.17632066. This Conference Paper is brought to you for open access by the 53rd Annual Conference of the European Society for Engineering Education (SEFI) at Tampere University in Tampere, Finland. This work is licensed under a Creative Commons Attribution-NonCommercial-Share Alike 4.0 International License. Inductive vs. Deductive Learning in Calculus: Unraveling the Impact on Integral Understanding N Kulcsár Széchenyi István University, Győr, Hungary, 0000-0001-8525-5851 Conference Key Areas: Teaching mathematics and physics in engineering education Keywords: inductive learning, deductive learning, brain-based learning theory, project-based program, calculus ABSTRACT Our experience of learning mathematics in schools is often based on learning and applying formulas. This suggests that teaching follows a predominantly deductive approach in which students use predetermined rules and theorems to solve problems. This is the dominant method worldwide, but the question arises: is it possible to teach mathematics in other ways, for example by an inductive approach, which first introduces students to general relationships by starting with concrete examples? The aim of this study was to evaluate the effectiveness of both inductive and deductive learning methods in engineering mathematics education. The experimental group learned the basics of integration through group work using either an inductive or a deductive approach. Their acquired knowledge was assessed through a test, which was repeated one week later. Following the learning process, they also completed a questionnaire in which they described their learning experience, strengths, and challenges. The results indicate that both methods have advantages and challenges with no significant difference in overall effectiveness based on test scores. Students generally reported positive experiences, but some suggested improvements such as restructuring group sizes, providing a mix of both methods, and increasing opportunities for cross-group discussions. Overall, the findings suggest that integrating elements of both learning approaches could offer a more balanced approach, catering to different learning styles while maintaining engagement and conceptual clarity. Nárcisz Kulcsár [email protected] 1 INTRODUCTION “Mathematics is a deductive science: starting from certain premises, it arrives, by a strict process of deduction, at the various theorems which constitute it. […] No appeal to common sense, or ‘intuition,’ or anything except strict deductive logic, ought to be needed in mathematics after the premises have been laid down.”(Bertrand Russell,1919; Rott & Rott, 2021) “Mathematics is an experimental science. The formulation and testing of hypothesis play in mathematics a part not other than in chemistry, physics, astronomy, or botany. […] It matters little that the mathematician experiments with pencil and paper while the chemist uses testtube and retort, or the biologist stains and the microscope.” (Norbert Wiener, 1923; Rott & Rott, 2021) Do we prefer to think of mathematics as a system of rigorously derived statements, or as a discipline based on discovery and experimentation? Is it better to learn by building concepts and theorems deductively, or by first discovering the underlying relationships through examples? Our research explores this question by comparing inductive and deductive approaches to learning. The origin of deductive and inductive learning can be traced back to Aristotle, who established a unified methodology combining both reasoning types in his works, particularly in the "Posterior Analytics." He emphasized inducing principles from observations and deducing conclusions from these principles. In contrast, Francis Bacon later advocated for a more empirical approach, focusing on inductive reasoning to derive generalizations from specific observations. Both philosophers significantly influenced the development of scientific inquiry and the philosophy of science. (Stadler, 2004) 1.1 Inductive and Deductive Learning Approaches Inductive method proceeds from particular examples to general rules of formulae, concrete illustration to abstract rules, known to unknown and simple to complex. During the „bottom-up” inductive method students construct a formula with the help of a sufficient number of concrete, actual and real examples. The following learning steps are used in this approach (Fig 1): 1. Appearance/presentation of Examples 2. Observation/Reflection 3. Generalization (Simplification) 4. Testing and verification (authentication) (Atta et al., 2015) Prince and Felder (2007) pointed out six different teaching methods which are integrally related to the following teaching methods: discovery learning, inquiry-based learning, problem-based learning, project-based learning, case-based teaching, and just-in-time teaching. Studies show that students taught through inductive methods demonstrate significant improvements in reasoning and achievement, particularly in mathematics (Elsayed & Almahri, 2023). In deductive method we proceed from general to particular and from abstract to concrete. In this method, teachers provide the needed rules, the new concepts and then students apply them to solve the problems. This „top-down” logic is more teachercentered than the inductive approach (Atta et al., 2015). The following learning steps are used in this approach: 1. Clear recognition of the problem 2. Search for a tentative hypothesis 3. Formulating a tentative hypothesis 4. Verification (Atta et al., 2015). The Moore method, a long-standing approach in university-level calculus instruction, is a highly structured, student-centered deductive strategy. When implemented in proof-oriented calculus courses, it has been shown to promote deep conceptual understanding by requiring students to independently construct definitions, theorems, and proofs with minimal direct instruction (Mahavier, 1999). Prince and Felder (2006) highlight that traditional engineering instruction is deductive but inductive methods are especially effective in STEM education due to active engagement and deeper cognitive processing. However, purely inductive approaches can impose cognitive overload (Kirschner et al., 2006), suggesting a need for guided discovery. The selection of an appropriate teaching approach depends not only on the characteristics of the subject matter but also on the size of the group. According to Shoaib (2010), the deductive method can be used in large classrooms, while the inductive method is effective in small groups. Hutton (2020) emphasize that flipped-classroom models, which blend direct instruction with in-class active problem-solving, offer a promising framework for scaling hybrid induction–deduction models in large calculus courses. Fig. 1. Comparison of the inductive and deductive learning approach 1.2 Brain-based Learning Theory From the end of the 20th century onwards, neurologists, biologists, psychologists, educators, and physicians have used research findings related to the brain's information-processing work to enhance learning and teaching, forming the basic principles for Brain-based Learning Theory (Connell, 2009). This theory suggests that Deduction Theory Hypothesis Observation Confirmation Induction Theory Hypothesis Pattern Observation General rules, Principles, Laws Specific cases, examples Induction Deduction learning is more effective when it aligns with how the brain naturally processes information—by recognizing patterns, making connections, engaging emotions, and constructing meaningful experiences. The Brain-based Learning Theory aligns with both inductive and deductive learning, but it generally favors inductive approaches while recognizing the value of deductive methods in certain situations. Inductive learning, where students first explore examples and then derive general principles, supports this by: • Encouraging active engagement and curiosity. • Allowing learners to build their own cognitive structures, which enhances retention. • Providing contextual and meaningful learning experiences, which improves motivation. While Brain-based Learning emphasizes constructivist approaches (Feketéné Szakos, 2014; Connell, 2009) like inductive learning, deductive learning can still be beneficial, especially when: • Learners need efficiency, such as when solving routine engineering problems. • There is a strong foundational knowledge base that allows for logical extensions. • Time constraints require direct instruction before deeper exploration. 1.3 Objective The present study aimed to answer the following questions: 1. Is deductive or inductive learning more effective for immediate understanding? 2. Is deductive or inductive learning more effective for short and long-term recall? 3. How do learners evaluate the deductive and inductive learning methods regarding clarity, effectiveness, and engagement? 4. What are the common challenges and benefits associated with inductive and deductive approaches? 2 METHODOLOGY 2.1 Research instruments and procedure Within the subject Calculus 2, the topic of integration has been chosen for the investigation. 2 groups were formed, one for the inductive and one for the deductive learning approach. The groups were seated in two interconnecting rooms. During the lesson, each group received printed instructions and worked collaboratively. The instructor was available for support, but minimal guidance was provided to encourage peer problem solving and group autonomy. 2 worksheets were prepared, one for the deductive and one for the inductive learning group. Furthermore, 2 tests were used to test the acquired knowledge, tests were the same for the two groups. A questionnaire was also developed and filled in by the students after the experiment. The worksheet developed for the deductive group used the following structure: 1. Definition of the indefinite integral 2. Primitive functions of the elementary functions in a table form 3. Dialog cards (Flip cards) - H5P format - were created in moodle system. Students had 10 minutes to practice with them to memorize the integrals of the elementary functions. 4. ∫𝑓(𝑥)𝑑𝑥 = 𝐹(𝑥) + 𝑐. Explain, why is +c at the end? 5. ∫1 𝑥𝑑𝑥 = ln|𝑥|+ 𝑐. Explain, why is x in an absolute value? 6. 3 rules were listed (multiplication by constant, sum rule, difference rule) and students had to use these rules for 5 integrals. 7. One rule was introduced with formula (composite function with linear inner function): ∫𝑓(𝑎𝑥 + 𝑏)𝑑𝑥 = 𝐹(𝑎𝑥+𝑏) 𝑎+ 𝑐 and students had to use these rules for 6 integrals. The worksheet developed for the inductive group used the following structure: 1. Definition of the indefinite integral 2. Primitive functions of the elementary functions in a table form. The antiderivatives of the elementary functions were missing. Students had to figure out the antiderivatives based on derivation. 3. ∫𝑓(𝑥)𝑑𝑥 = 𝐹(𝑥)+ 𝑐. Explain, why is +c at the end? 4. ∫1 𝑥𝑑𝑥 = ln|𝑥|+ 𝑐. Explain, why is x in an absolute value? 5. 3 examples were listed and students had to formulate the rules (multiplication by constant, sum rule, difference rule) and other 2 tasks were listed to test their acquired knowledge. 6. 4 examples were listed and students had to formulate a rule (composite function with linear inner function) and other 2 tasks were listed to test their acquired knowledge. After completing the worksheets together, everybody individually wrote a test with 6 tasks where he had to use the learnt rules (multiplication with constant, sum rule, difference rule, and composite function with linear inner function). One week later, students completed a similar test with 6 tasks to investigate what they remember. A questionnaire was also filled out by them about the learning method they had tried the previous week. The questionnaire contained 9 open-ended and 5 closed-ended (on a 5 point Likert scale) questions in the following structure: 1. Learning Experience and Understanding 2. Example-Driven vs. Problem-Solving Approach 3. Motivation and Emotional Factors 4. Summary and Suggestions This study was designed as a single-session intervention within the semester, rather than a long-term instructional approach. The aim was to explore how students respond to different learning strategies (inductive vs. deductive) during one guided lesson, not to assess the cumulative effects of an entire semester of instruction. 2.2 Participants The group size and the classroom facilities in the project-based vehicle engineering bachelor's program made it possible to carry out the research. The English-language program attracts many foreign students, so students come from different countries with different knowledge. The experimental group consisted of 21 participants, 10 in the inductive learning group and 11 in the deductive learning group. All students registered for the subject for the first time. All students had already learned the basics of differentiation, but the topic of integration was new to most of them, although some had prior exposure. 5 students have studied the basics of integration in secondary school, that is why I asked them to participate in the deductive learning group. 3 RESULTS 3.1 Results of the tests 21 students completed the first test and 17 students the second. The tests consisted of 6 tasks and were worth 6 points. Only perfect solutions were worth points. Table 1 shows some results of the tests. Table 1. Test results in the different groups Mean of the first test results Missing “+C” Missing absolute value in the case of lnx Mean of the second test results Missing “+C” Missing absolute value in the case of lnx Inductive Learning Group 4 3 2 3.13 1 4 Deductive Learning Group 3.36 3 1 3.78 0 4 A paired sample t-test was used to measure the difference between the test scores to see if there was a significant change over time between the first and second tests. Although the results show a deterioration in the inductive learning group and an improvement in the deductive learning group, the change is not significant. In the inductive learning group, neither the paired sample correlation (p=0.055) nor the paired t-test (p=0.086) was significant. However, in the deductive learning group, the paired sample correlation (p=0.004) was significant, but the sample paired test (p=1) was not. This means that in this group the individual differences remained relatively stable across the tests. Namely, students who scored higher in the first test also tended to score higher in the second, but the change was not significant here also. An independentsamples t-test was used to see if there was a significant difference between the scores of the two groups on the first and second tests. Although it seems that the inductive learning group did better on the first test and the deductive learning group on the second test, in neither the first (p=0.439) nor the second test (p=0.475) was there a significant difference between the test scores. Table 1 also shows the frequency (how many students made the mistake) of two common mistakes. One is forgetting +c, and the other one is that the integral of 1/x is not lnx but ln|x|. In the second test compared to the first test, the first error was made by fewer students, while the second error was made by more students. 3.2 Results of the questionnaire Eighteen students completed the post-lesson questionnaire (10 from the inductive group, 8 from the deductive group). Overall, both groups reported positive experiences with their respective learning approaches, especially highlighting the value of group work. Students frequently mentioned that collaboration helped them clarify concepts, solve problems, and stay motivated. In terms of perceived effectiveness, students rated their group's learning method highly (mean = 4.11), with no significant difference between the groups (inductive = 3.89, deductive = 4.33; t=-1.437, p = 0.180). The material was also considered understandable (mean = 4.27), and no significant difference was found between the groups in this respect. Students found the method interesting (mean = 4.00), and motivation peaked during problem solving, successful collaboration, and moments of understanding. Most students did not report frustration. Those who did attributed it to not immediately finding the right approach or formula, but noted that peer or teacher support helped them overcome it. Students described the environment as safe and supportive, which enabled risk-taking and experimentation. When asked about long-term understanding, a slight majority (10 out of 18) preferred inductive learning, emphasizing deeper comprehension and recall. Others valued the structure and clarity of the deductive approach. Many expressed interest in trying the alternative method, recognizing that both have value depending on the learner and the topic. Suggestions for improvement included using smaller or more balanced groups, combining inductive and deductive elements, and providing more initial guidance for difficult concepts. Several students reflected on their own learning habits and reported gaining insight into new ways of understanding and practicing mathematics. 4 DISCUSSION AND CONCLUSIONS The results of the study suggest that inductive and deductive learning each have their weaknesses and strengths, with no overall noteworthy difference regarding effectiveness on tests. While the inductive group scored slightly higher on the first test, their score decreased slightly by the second test, whereas marginal improvement was experienced by the deductive group. Neither of these changes, however, was statistically significant. From the questionnaire results, both learning methods were found effective and comprehensible to students, with an overall preference for inductive learning since it was more engaging and created deeper conceptual insight. The majority of students enjoyed the potential of deriving formulas from examples, which they referred to as being helpful to memorization, intuition, and problem-solving capability. Some students considered the approach challenging, particularly when faced with abstract mathematical ideas or when they failed to identify patterns. On the other hand, the deductive method was structured and efficient, minimizing early confusion but sometimes leading to rote memorization without deep comprehension. Collaborative work was a key aspect of the learning experience, and students valued the opportunity to share their thoughts and refine their results through peer discussion. This social aspect was seen as clarifying confusion and enhancing motivation. Although students reported generally positive experiences, proposals for alteration included restructuring group size, offering a combination of both approach of learning, and enhancing discussion between groups. Concerning personal learning processes, students embraced the need to address topics from various viewpoints, practice extensively with examples, and blend solo problem-solving with group collaboration. Some noted that this experience has shown that they value persistence and exploring problems before relying on external sources. These observations indicate that a combination of both inductive and deductive aspects of learning might be the optimal solution, targeting various learning styles while preserving interest and conceptual comprehension. Overall, the results indicate that while both methods are effective, a hybrid model that uses example-based discovery alongside structured formula presentation is likely to be optimum in the teaching of engineering mathematics. Although the present study involved a relatively small number of participants, it served as a methodological pilot to explore how different instructional strategies affect students’ experiences and learning processes in a controlled setting. The aim was not to produce generalizable conclusions, but to identify key insights that could inform future research. For wider applicability, similar designs could be implemented in larger university cohorts using scalable formats such as flipped classrooms, peer instruction, or digital collaboration platforms to preserve active engagement while adapting to large-group contexts. REFERENCES Atta, M. A., Ayaz, M., & Nawaz, Q. (2015). Comparative study of inductive & deductive methods of teaching mathematics at elementary level. Gomal University Journal of Research, 31(1), 20-28. Connell, J. D. (2009). The global aspects of brain-based learning. Educational horizons, 88(1), 28-39. Elsayed, A. M., & Almahri, A. M. (2023). 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