EFFICACY OF THE ALGORITHMIC APPROACH IN INDEPENDENT PROBLEM-SOLVING IN DESCRIPTIVE GEOMETRY

ABSTRACT

This study experimentally evaluates the pedagogical efficacy of an algorithmic approach in solving descriptive geometry problems independently. Using a randomized controlled trial design, 88 first-year engineering students were randomly assigned to control and experimental groups. Baseline equivalence was confirmed by comparing participants' scores on the Purdue Spatial Visualization Test (PSVT:R) and a diagnostic test in descriptive geometry. Over a six-week intervention, problem-solving accuracy, solution speed, error rate, and spatial reasoning indicators were recorded. Results show that the experimental group, which used the algorithmic approach, significantly outperformed the control group: accuracy increased by 31% (p<0.001, Cohen's d=1.1), solution speed improved by 38% (p<0.001, d=1.4), error rate decreased by 45% (p<0.001), and spatial reasoning scores improved by 22% (p<0.001, d=0.9). The study demonstrates that the algorithmic approach reduces cognitive load by breaking down complex spatial tasks into systematic steps, thereby enhancing students' independent work skills and spatial reasoning. This methodology is a valuable pedagogical tool for improving the quality of education in descriptive geometry and engineering graphics, promoting individualized and effective learning.

АННОТАЦИЯ

В данном исследовании экспериментально оценивается педагогическая эффективность алгоритмического подхода при самостоятельном решении задач начертательной геометрии. Используя дизайн рандомизированного контролируемого исследования, 88 студентов первого курса инженерных специальностей были случайным образом распределены в контрольную и экспериментальную группы. Эквивалентность на старте подтверждена сравнением баллов участников по тесту Purdue Spatial Visualization Test (PSVT:R) и диагностическому тесту по начертательной геометрии. В течение шестинедельного вмешательства фиксировались точность решения задач, скорость решения, частота ошибок и показатели пространственного мышления. Результаты показывают, что экспериментальная группа, использовавшая алгоритмический подход, значительно превзошла контрольную группу: точность повысилась на 31% (p<0,001, d=1,1), скорость решения улучшилась на 38% (p<0,001, d=1,4), частота ошибок снизилась на 45% (p<0,001), а показатели пространственного мышления улучшились на 22% (p<0,001, d=0,9). Исследование демонстрирует, что алгоритмический подход снижает когнитивную нагрузку за счёт разложения сложных пространственных задач на систематические шаги, тем самым улучшая навыки самостоятельной работы и пространственного мышления студентов. Данная методика является ценным педагогическим инструментом для повышения качества обучения начертательной геометрии и инженерной графике, способствуя индивидуализированному и эффективному обучению.

Keywords: descriptive geometry, algorithmic approach, spatial reasoning, cognitive load, independent learning, experimental study, engineering graphics, randomized controlled trial.

Ключевые слова: начертательная геометрия, алгоритмический подход, пространственное мышление, когнитивная нагрузка, самостоятельное обучение, экспериментальное исследование, инженерная графика, рандомизированное контролируемое исследование.

1. Introduction

Descriptive geometry is a cornerstone of engineering education, essential for developing spatial visualization, technical drafting, and analytical problem-solving skills. Despite its importance, students often struggle with the subject due to the high cognitive demands of mentally manipulating and projecting complex three-dimensional objects onto two-dimensional planes.

Traditional instruction in descriptive geometry frequently relies on teacher-centered demonstrations and static examples, which may not adequately scaffold the problem-solving process. This can lead to cognitive overload, surface-level learning, and a reliance on memorized procedures rather than deep conceptual understanding.

To address these pedagogical challenges, this study investigates the efficacy of a structured algorithmic approach in facilitating independent problem-solving in descriptive geometry. An algorithmic approach decomposes complex spatial tasks into sequential, logical steps, providing learners with a clear, reusable framework. This method is theorized to reduce extraneous cognitive load, enhance self-regulation, and promote the development of robust spatial reasoning.

While algorithmic methods have been explored in mathematics and programming education, there is a notable lack of rigorous, controlled studies examining their application in descriptive geometry—a domain characterized by intense visuospatial processing. This research fills that gap by employing a randomized controlled trial (RCT) design to evaluate the impact of the algorithmic approach on problem-solving accuracy, speed, error reduction, and spatial ability among first-year engineering students.

The findings aim to provide evidence-based insights for curriculum design and instructional practice, contributing to both pedagogical theory (e.g., Cognitive Load Theory) and the improvement of STEM education outcomes in technical graphics and engineering fundamentals.

2. Materials and Methods

2.1 Participants and Randomization

A randomized controlled trial included 88 first-year engineering students from Termiz State University of Engineering and Agro-Technologies. Participants were randomly assigned using a computer-based generator (Random.org) into two groups:

• Experimental Group (EG, n=44): Received algorithmic-based instruction.
• Control Group (CG, n=44): Taught via traditional teacher-centered methods.

Baseline equivalence was confirmed using the Purdue Spatial Visualization Test (PSVT:R) and a 20-point diagnostic test in descriptive geometry. Independent samples t-tests showed no significant differences in PSVT scores (t(86)=0.42, p=0.67) or prior knowledge (t(86)=0.38, p=0.71).

Note: The study was later extended to 138 participants (including students from other universities) to enhance generalizability.

2.2 Experimental Design and Intervention

A 6-week parallel-group design was implemented. Both groups attended 90-minute sessions twice weekly in the same classroom with the same instructor.

EG Intervention: Algorithms were presented via paper handouts, whiteboard demonstrations, and short animated videos (2–4 min) created with GeoGebra 3D and Adobe After Effects. Videos were accessible anytime via QR codes and Moodle.

Example: For the "Methods of Reconstructing an Epure" algorithm, a YouTube link (https://youtu.be/4lB-Kx42iRE?si=TW5R8pLUVzlyIxgQ) was provided, demonstrating substitution, rotation, and parallel displacement methods.

CG Intervention: Traditional instruction involved static board drawings without algorithmic steps or multimedia aids.

Table 1.

Instructional Methods Comparison

2.3 Measurement Tools and Variables

Primary outcomes included:

• Spatial reasoning: PSVT:R (pre/post scores).
• Accuracy: 10-problem test (0–20 points).
• Speed: Average solution time per problem.
• Errors: Error rate and classification (design, measurement, analytical).

Secondary measures:

• Motivation: Intrinsic Motivation Inventory (IMI).
• Creativity: Torrance Tests of Creative Thinking (TTCT).

All instruments showed good reliability (Cronbach's α > 0.78).

2.4 Statistical Analysis

Data were analyzed in SPSS 26 using:

• Shapiro–Wilk normality test.
• Independent samples t-tests (EG vs. CG).
• Effect sizes (Cohen's *d*).
• Repeated measures ANOVA for longitudinal stability.
• Pearson correlation for relationships between variables.
• 95% confidence intervals for key indicators.

Significance was set at p < 0.05 with Bonferroni correction.

3. Results

The experimental group demonstrated statistically significant and substantial improvements across all measured outcomes compared to the control group (all p < 0.001).

Table 2.

Key Performance Indicators for the Experimental (EG) and Control Groups (CG)

The algorithmic approach led to a 47.3% increase in accuracy, a 38% reduction in solution time, and a 45% decrease in error rate. The most notable error reduction was in analytical (logical) errors (-62%). Spatial reasoning (PSVT) improved by 22% in the EG, significantly outperforming the CG (+7%).

Long-Term and Transfer Effects:

• Retention (6 months): EG retained 72% of knowledge vs. 48% for CG.
• Transfer to solid mechanics: EG showed a 25% higher problem-solving efficacy (p<0.01).

Additional Analyses:

• Stability: EG's accuracy gains remained stable over 6 weeks (ANOVA, F=4.15, p<0.05).
• Correlation: A strong positive correlation was found between PSVT improvement and test scores (r=0.68, p<0.01).

4. Discussion

The results confirm the high efficacy of the algorithmic approach in descriptive geometry education. Large effect sizes (d = 0.9–1.4) indicate not only statistical significance but also substantial practical impact.

4.1 Theoretical Implications

1. Cognitive Load Theory: By decomposing problems into sequential steps, the approach effectively reduced intrinsic and extraneous cognitive load, optimizing working memory use. This is evidenced by the 38% faster solution speed and 45% lower error rate.
2. Spatial Reasoning Development: The step-by-step process required continuous mental manipulation (rotation, projection), transforming passive learning into active spatial analysis, leading to a 22% PSVT gain.
3. Metacognitive Skills: Algorithms acted as an external scaffold, fostering self-regulation, planning, and monitoring. This was reflected in increased intrinsic motivation (EG: +24%, CG: +8%) and creativity (EG: +18% in graphic creativity).

4.2 Role of Multimedia

The integration of short instructional animations likely enhanced outcomes via Dual-Coding Theory, facilitating better understanding through combined visual and verbal channels, particularly for independent review.

4.3 Limitations and Future Research

• Internal Validity: Short intervention duration (6 weeks) and potential instructor bias.
• External Validity: Sample limited to engineering students from one university; generalizability requires testing in diverse contexts.
• Creativity Concern: Efficacy for open-ended, multi-solution problems remains unclear and warrants investigation.
• Future Directions: Longitudinal studies, cross-cultural comparisons, qualitative analysis of cognitive processes, and integration with AR/VR technologies are recommended.

5. Conclusion

This RCT provides robust evidence that the algorithmic approach significantly enhances independent problem-solving in descriptive geometry. It improves accuracy (47%), speed (38%), and precision (45% fewer errors) while also developing spatial reasoning (22% gain) and metacognitive skills. The large, statistically significant effects (p<0.001, d>0.8) underscore its pedagogical value. For optimal integration, the approach should be combined with other methods to ensure a balance between structured problem-solving and creative, adaptive thinking development.

References:

1. Abramovich, S., & Connell, M. L. (2021). Developing deep knowledge in middle school mathematics. Springer International Publishing.
2. Akhmedov, B. A. (2022). Algorithmic thinking and self-regulation in technical education. Journal of Educational Innovation, 15(2), 45–60.
3. Bhagat, K. K., Chang, C. Y., & Huang, R. (2017). Integrating GeoGebra with TPACK for improving mathematics teachers' professional development. In Proceedings of the 17th IEEE International Conference on Advanced Learning Technologies (pp. 313–314). IEEE.
4. Gungor, A., et al. (2022). The use of virtual reality in a chemistry lab and its effects on students' self-efficacy. Eurasia Journal of Mathematics, Science and Technology Education, 18(3), em2090.
5. Hansen, E. K. S., & Naalsund, M. (2022). The role of teacher actions in students' productive interaction. International Electronic Journal of Mathematics Education, 17(3), em0685.
6. Ivanov, V. P. (2021). Structured problem-solving methods in engineering education. Journal of Technical Pedagogy, 8(1), 22–35.
7. Karimov, R. S. (2021). Algorithmic models for developing spatial thinking in technical universities. Science and Education, 3(4), 112–125.
8. Mayer, R. E. (2020). Multimedia learning (3rd ed.). Cambridge University Press.
9. Paivio, A. (1986). Mental representations: A dual coding approach. Oxford University Press.
10. Piaget, J. (1950). The psychology of intelligence. Routledge.
11. Polya, G. (1957). How to solve it: A new aspect of mathematical method (2nd ed.). Princeton University Press.
12. Sweller, J. (1988). Cognitive load during problem solving: Effects on learning. Cognitive Science, 12(2), 257–285.