📄 social science

From learning obstacles to reasoning: Didactical design of a problem-based electronic module on trigonometric ratios

This study presents the development and validation of a problem-based electronic module for trigonometric ratios, designed to address specific student learning obstacles, which significantly improved reasoning scores in a trial despite revealing persistent challenges in side identification and contextual problem-solving that guide future revisions.

Original authors: Runisah Runisah, Nova Mulyana, Nandang Nandang, Rosyadi Rosyadi, Farid Gunadi, Naif Mastoor Alsulami

Published 2026-10-15

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Runisah Runisah, Nova Mulyana, Nandang Nandang, Rosyadi Rosyadi, Farid Gunadi, Naif Mastoor Alsulami

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Mathematics in the classroom is often a place where students memorize rules to pass a test, only to forget them the moment the exam is over. This is especially true for trigonometry, a branch of math that deals with the relationships between the angles and sides of triangles. While this subject is essential for later studies in physics and engineering, it is frequently taught as a set of rigid formulas to be recited rather than a way of thinking about space and shape. When students encounter real-world problems, they often struggle to see how these abstract rules apply, leading to a gap between what they can calculate and what they can actually understand. Researchers have long known that students get stuck on specific hurdles, such as mixing up which side of a triangle is which or failing to explain why their answer makes sense. The challenge for educators is not just to teach the formulas better, but to design learning experiences that directly address these specific mental roadblocks.

A team of researchers in Indonesia set out to tackle this problem by creating a new kind of digital learning tool for tenth-grade students. Instead of starting with a generic lesson plan, they began by listening to the students. They interviewed three students and two teachers to uncover exactly where the confusion lay. They found four main obstacles: students could not reliably identify the sides of a triangle relative to a specific angle, they relied on memorized tricks without understanding the meaning behind them, they rarely explained their reasoning, and they struggled to turn real-world stories into mathematical shapes. Using these findings, the researchers built a problem-based electronic module. This was a digital book accessible on phones that guided students through a series of real-life scenarios, such as calculating the height of a building using shadows, rather than just solving abstract equations. The design was carefully mapped out so that every activity was intended to fix one of the specific hurdles the students had identified.

The module was then tested in a single classroom of twenty-seven students. Before the lesson, the students took a test to measure their ability to reason through mathematical problems. After a single ninety-minute session using the new digital module, they took the same test again. The results showed a dramatic shift. The average score on the reasoning test jumped from less than fifteen points to over sixty points. Every single student improved their score, and the statistical analysis confirmed that this change was highly significant. The experts who reviewed the module beforehand rated it as very valid, and an observer watching the class in action found it very practical to use. The students themselves reported that the material was clear and engaging, with the vast majority agreeing that the digital format helped them learn.

However, the story does not end with a perfect success. When the researchers looked closely at how the students actually worked through the problems during the lesson, they found that the old habits had not completely vanished. While the scores improved, many students still struggled to correctly identify the sides of a triangle after working in groups, and several groups provided correct answers without being able to explain why they were right. In fact, only two out of seven groups managed to solve the very first problem of the lesson, which involved comparing the shadows of a child, a father, and a flagpole. Some groups got stuck on the details, while others simply wrote down their names or the given numbers without attempting to solve the puzzle. This revealed a crucial insight: while the digital tool was excellent at raising scores and engaging students, a single lesson was not enough time for everyone to fully overcome the deep-seated confusion about how triangles work.

The researchers concluded that the approach of designing from the students' specific mistakes was highly effective, but it also highlighted the limits of time. The module successfully guided students to higher reasoning scores, yet the persistent errors showed that some concepts, like knowing which side is which, require more than one explanation to truly stick. The study suggests that for these tools to work best, teachers need to spread the activities over more time, allowing students to revisit the difficult parts and practice explaining their thinking until it becomes second nature. The work demonstrates that when learning materials are built specifically to clear the path where students actually stumble, the results can be powerful, but true mastery still requires patience and repeated practice.

Technical Summary: From Learning Obstacles to Reasoning

Problem Statement
Trigonometric ratios remain a significant barrier for secondary students, particularly in Indonesia, where PISA results indicate low proficiency in mathematical problem-solving. Existing literature suggests that difficulties in trigonometry stem less from computational errors and more from conceptual misunderstandings, such as failing to interpret ratios as properties of angles, reliance on memorized formulas, and an inability to model contextual problems. Furthermore, while problem-based learning (PBL) and digital resources are widely studied, few interventions specifically address trigonometric ratios, and many digital designs rely on generic instructional models (e.g., ADDIE) rather than being grounded in empirically identified learning obstacles. This study addresses the gap in designing a problem-based electronic module that directly targets the specific epistemological and didactical obstacles students encounter.

Methodology
The study employed Didactical Design Research (DDR), an interpretive methodology proceeding through three distinct stages:

Prospective Analysis: Researchers identified learning obstacles through initial reasoning tests and semi-structured interviews with three Grade 10 students (representing high, medium, and low achievement) and two mathematics teachers. These obstacles were used to construct a Hypothetical Learning Trajectory (HLT) and design a problem-based electronic module (flipbook format).
Metapedadidactic Analysis: The module was implemented in a single Grade 10 class (27 students) over one lesson (2 × 45 minutes). The module served as the primary learning resource. Data collection included a pretest/posttest on mathematical reasoning, an observer's field notes, student questionnaires, and group worksheets.
Retrospective Analysis: The actual learning trajectory observed in the classroom was compared against the anticipated HLT to identify discrepancies between design intent and student performance.

Key Contributions

Obstacle-Driven Design: The study demonstrates a methodology where specific learning obstacles (identified as epistemological and didactical) directly inform the features of a digital learning resource, rather than relying on generic needs analysis.
Integration of HLT and Digital Media: It bridges the gap between theoretical learning trajectories and practical electronic module design, showing how specific module features (e.g., visual diagrams for side identification, structured worksheet prompts) can be mapped to anticipated student responses.
Diagnostic Retrospective Analysis: By comparing the HLT with the observed trajectory, the study reveals not just if a module works, but where and why specific design features fail to overcome persistent obstacles, offering a more nuanced evaluation than standard validity/practicality surveys.

Results

Identified Obstacles: Four primary obstacles were confirmed: (1) confusing sides relative to the reference angle (epistemological), (2) reliance on memorized formulas without conceptual meaning (didactical), (3) absence of mathematical justification (didactical), and (4) difficulty translating contextual problems into right-triangle models (epistemological).
Validity and Practicality: Three experts rated the module as "very valid" (means ranging from 4.33 to 4.82 on a 5-point scale). An observer rated the implementation as "very practical" (mean 4.23), though noted that the core problem-based activities and time management were areas for improvement.
Reasoning Gains: Students' mathematical reasoning scores increased significantly from a pretest mean of 14.78 to a posttest mean of 62.37 ( $t(26) = 14.59, p < .001$ ). The normalized gain was 0.56 (medium category). Every student scored higher on the posttest.
Retrospective Discrepancies: Despite the score gains, the retrospective analysis revealed that the design did not fully overcome all obstacles. Side confusion persisted after group discussion, and many students provided correct answers without complete justification. Crucially, only two of seven groups successfully solved the opening contextual problem, and the evaluation phase (checking steps and drawing conclusions) was largely unachieved, likely due to time constraints.

Significance and Claims
The paper claims that its primary significance lies in the traceability of the design process. By grounding the module in documented obstacles, the study allows for a precise diagnosis of where a design succeeds or fails. The authors argue that while the module was valid and practical, the persistence of certain obstacles (specifically side identification and justification) indicates that these are deep-seated epistemological issues requiring more than a single lesson or a single explanation.

The study concludes that problem-based materials must explicitly protect time for the justification and evaluation phases of reasoning, which are often lost when lessons are too short. It suggests that for designers, starting with documented obstacles rather than generic models provides a clearer path for revision. For teachers, it implies that naming triangle sides should be treated as a reasoning task to be revisited, rather than a fact to be memorized via mnemonics. The authors modestly note that the lack of a control group and the single-lesson duration limit the ability to claim the module's long-term effectiveness, positioning the results as evidence of directional change in reasoning rather than a definitive measure of the module's isolated impact.

Technical Summary: From Learning Obstacles to Reasoning

More like this