A methodology for developing skills in linear functions based on the IDEAL method

The practical orientation of educational content and strengthening of inter-disciplinary connections are one of the main tasks of the modern school. In mathematics, linear functions are a simple and effective tool for modeling everyday, economic and natural processes. However, for many students, mastering linear equations and graphs is limited only as a formulaic skill, and the ability to apply the obtained models in real situations remains insufficient. In order to eliminate this gap, it is necessary to systematically introduce problem-oriented teaching and research-oriented methods into the classroom.

The IDEAL method (Identify — Define — Explore — Act — Look back) structures the learning process in stages and develops logical thinking from problem identification to conclusion. This method encourages students to actively research, formulate hypotheses and evaluate the results obtained. Using the IDEAL method in the context of linear functions allows students to systematically develop the skills of building a model, understanding the meaning of parameters, finding solutions using graphs and checking the conclusions obtained.

Research problem: Do 8th grade students have sufficient skills in formulating and proving practical problems using linear functions, and how can the IDEAL method improve these skills? This issue directly affects the results of teaching and learning in school mathematics, since it is important to consider that linear models are widely used in everything from economic problems to speed, cost, and revenue problems in physics.

The purpose of the study is to improve the skills of 8th grade students in solving applied problems using linear functions and justifying the obtained solutions by using the IDEAL method in the classroom. The tasks include: 1) diagnosing the initial level of students; 2) developing lesson plans using problem and interdisciplinary contexts (economics, physics, everyday life); 3) developing tasks and assessment tools used at each stage of the IDEAL method; 4) testing the effectiveness of the implementation and analyzing the results.

The significance of this study is that it contributes to the development of students' analytical thinking, modeling, and reflection skills. In addition, the use of linear functions in a practical context shows students the relevance of mathematics in everyday life and increases interest in the subject. Formative and summative assessment elements (e.g., research journals, self- and peer-assessment descriptors) are used to assess student learning, which allows for more accurate monitoring of the learning process and the introduction of result-oriented adjustments.

The following sections describe the specific structure of the lessons according to the stages of the IDEAL method, and present tasks, resources, and assessment criteria for each stage. In addition, pilot results and practical recommendations are presented, and problems and solutions for the wider use of the IDEAL method are discussed.

The "IDEAL" method was used to develop students’ skills in solving complex and real-world problems collaboratively. The “Thinkers” group analyzed the flight distance of a golf ball using a quadratic function graph and concluded its relationship with free-fall acceleration. The “Theorists” group aimed to determine when a rabbit population would reach its maximum using a parabolic model. However, due to insufficient attention to data collection and analysis, they initially miscalculated the time of population decline. Guided questions helped students identify their mistakes and revise their predictions, ultimately leading to correct conclusions. For assessment, the self-designed “Research Journal” was employed, allowing students to critically reflect on their learning and create their own descriptors. This method effectively evaluated group work against established criteria. In the consolidation phase, students modeled trucks passing through parabolic tunnels, representing the shapes using quadratic functions and correctly solving the problem. The “Achievement Measure” tool was used to precisely assess student progress, and the “Plus-Minus-Interesting” method supported planning for the next lesson by identifying strengths and areas for improvement in teaching and learning. During the lesson introduction, the “Quick Questions” method engaged students, reviewed prior knowledge, and highlighted key terms such as “quadratic function” and “parabola.” When two students struggled with vertex coordinates, recalling formulas helped them arrive at the correct answers. Constructive feedback was given for determining the axis of symmetry, and applied problem-solving reinforced the use of quadratic functions. Socrative.com helped gauge students’ prior knowledge. The “Thinkers” applied quadratic functions to calculate the golf ball’s flight distance in relation to free-fall acceleration. They computed different scenarios, concluding that lower acceleration results in longer flight distance. Independent research online allowed them to relate findings to athletes’ performances in different countries. This process fostered critical thinking and collaborative problem-solving. Meanwhile, the “Theorists” explored rabbit population growth and extinction risks using the IDEAL framework. They plotted the parabolic graph, identified the vertex for maximum population at 16 months, and determined the extinction point by finding the zeros of the function—predicting it would occur at 40 months. Group discussions integrated biological and geographical factors, encouraging multiple perspectives and reasoned conclusions. Overall, lesson planning emphasized effective use of instructional resources, assessment strategies, and consideration of students’ cognitive styles. The approach successfully promoted analytical thinking, research skills, and collaborative learning, demonstrating the value of structured, inquiry-based methods in mathematics education.

Students’ collaborative group work supported joint decision-making, as reflected in their well-structured descriptors. During the study’s planning phase, we decided that students would actively create descriptors within their groups. Both groups successfully analyzed quadratic function graphs and drew conclusions. Group 1 calculated distances and noted their dependence on 𝑔 g, while Group 2 determined that population decline corresponded to the intersection of the quadratic function with the x-axis. The IDEAL method helped students develop skills in drawing conclusions from applied problems. A significant outcome was the growth of students’ self-regulation and awareness of strategies to improve learning. A Level B student’s precise self-assessment demonstrated this development. Guiding questions such as, “Why did you rate yourself as ‘good’?” helped students reflect critically. Careful planning and implementation of teaching and assessment strategies ensured the lesson achieved its intended outcomes. However, some students still need support in systematically analyzing information based on function properties. This highlighted the importance of modeling, leading to a new focus: “Developing students’ skills in retrieving and organizing information through modeling based on given characteristics.” Based on the study results, the following professional development actions are planned for teachers in intellectual schools: Conduct a webinar on effective IDEAL method implementation via Teams; Organize a seminar-training focused on differentiated instruction using resources; Publish a methodological guide on “Developing Students’ Reasoning Skills”; Disseminate study results through media outlets. This study demonstrates that structured collaborative learning, the IDEAL method, and modeling effectively foster critical thinking, reasoning, and self-regulated learning. Students’ active participation in planning, analysis, and reflection contributed to the lesson’s success and highlighted areas for further development.

Setyadi, Mardiyana, and Triyanto (2019) report that the IDEAL model is an effective tool for studying, categorizing, and analyzing students’ problem-solving skills, and can be used as a key instrument for improving learning. Similarly, Smelova (2023) notes that through the IDEAL method, students can independently formulate problems, identify solution strategies, and create their own plans for solving them. Problem-solving using IDEAL indicators involves identifying and describing a problem and constructing effective plans to resolve it. This process engages other executive functions, including attention control, planning, and task initiation. Through the IDEAL model, students can manage their time, regulate emotions, and apply organizational skills to solve problems. Over time, students develop self-monitoring abilities, can use working memory effectively, and exercise self-control to influence future problem-solving processes. The strategies provided by IDEAL indicators are especially important because not all students can solve the same problem correctly. Students with a history of behavioral or learning difficulties often struggle to select effective strategies in stressful situations. Without a step-by-step model for problem-solving, including problem identification and solution planning, many students rely on ineffective behaviors when facing challenging tasks. The IDEAL model serves as a practical framework to teach students with diverse behavioral profiles how to achieve positive outcomes in difficult situations. It provides a structured approach that guides learners through the steps of recognizing the problem, planning strategies, executing actions, and evaluating results. According to Bransford and Stein (1993), IDEAL is a widely recognized method for promoting effective problem-solving and equipping students to make constructive decisions, even under stress or uncertainty. Overall, the IDEAL framework supports the development of cognitive, emotional, and organizational skills while fostering independent thinking and resilience in problem-solving, making it a valuable tool for educators seeking to enhance students’ learning and decision-making abilities.