Hermeneutic Foundations for Solving Equations and Their Systems
This article addresses issues related to schoolchildren's ability to solve different types of equations. It emphasises the importance of understanding the origins of algorithms and rules for solving equations, rather than merely memorising them. With this in mind, the authors identify three hermeneutic principles. Firstly, performing the same operation on both sides of an equation maintains the equality. Secondly, when dividing or multiplying both parts of an inequality by a negative number, the inequality sign must be reversed. Thirdly, when solving non-rational equations and their systems, the original equations must be reduced to an elementary equation or a form with only one function, which can be reduced to a rational equation by replacing the variable. After solving this equation by inverse replacement, an elementary equation is obtained that can be solved in the simplest way. By knowing these rules, students will be able to solve any equation, from the most basic to the most complex, throughout their mathematics studies, from the first to the eleventh grade.
While nobody left any comments to this publication.
You can be first.
Ermak, E. A. (2016) “Search for New Opportunities for Implementing the Hermeneutic Approach in Teaching Mathematics”, Chelovek kak subyekt sotsialno-pedagogicheskogo vzaimodeystviya: Materialy Mezhdunarodnoy nauchno-metodicheskoy konferentsii, posvyashchennoy pamyati professora L.M. Luzinoy, Pskov, 18-19 dekabrya 2015 goda [Man as a Subject of Social and Pedagogical Interaction: Proceedings of the International Scientific and Methodological Conference Dedicated to the Memory of Professor L. M. Luzina, Pskov, December 18-19, 2015], Pskov State University Publishing House, Pskov, Russia, 217-224 (in Russ).
Filippova, M. A. (2025), “Rules for memorizing the names of components when solving equations”, Educational social network nsportal.ru [Online], available at: https://nsportal.ru/nachalnaya-shkola/matematika/2023/09/06/pravila-na-zauchivan ie-nazvaniy-komponentov-pri-reshenii (Accessed 02 December 2025) (in Russ.).
Lapitsky, M. K. and Duginov, E. V. (2025), “Methods of solving problems on conservation laws”, Pedagogicheskaya innovatika i nepreryvnoye obrazovaniye v XXI veke: Sb. nauch. trudov III Mezhdunarodnoy nauchno-prakticheskoy konferentsii, Kirov, 14 maya 2025 goda [Pedagogical innovation and continuous education in the 21st century: collection of scientific papers of the III International scientific and practical conference, Kirov, May 14, 2025], Vyatka State University of Agrotechnology Publishing House, Kirov, 222-226 (in Russ.).
Lysenko, F. F. and Kulabukhov, S. V. (eds) (2024), Matematika. Podgotovka k EGE-2025. Profilnyy uroven. 40 trenirovochnykh variantov po demoversii 2025 goda: Uchebno-metodicheskoye posobiye [Mathematics. Preparation for the Unified State Exam – 2025. Profile Level. 40 Training Options Based on the 2025 Demo Version: Educational and Methodological Guide], Legion, Rostov-on-Don, Russia (in Russ.).
Mokienko, O. P. (2011), “Hermeneutic approach in teaching”, Science Vector of Togliatti State University. Series: Pedagogy, Psychology, 3(6), 204-206 (in Russ.). EDN: OCQCNH
Mushenok, Yu. V. (2025), “Errors in solving equations and their overcoming”, Bulletin of Scientific Conferences, 8-2(120), 62-64 (in Russ.). EDN: JUEFWV
Razbirayemsya v reshenii lineynykh uravneniy (2025) [Understanding the Solution of Linear Equations], Available at: https://externat.foxford.ru/polezno-znat/wiki-algebra-metody-resheniya-sistem-linejnyh-uravnenij?ysclid=mev9ie05ki599025509 (Accessed 02 December 2025) (in Russ.).
Skanavi, M. I. (2013), Sbornik zadach po matematike dlya postupayushchikh vo vtuzy [Collection of Mathematics Problems for Applicants to Technical Universities], 6th ed., ONIKS-LIT; Mir i Obrazovaniye, Moscow, Russia (in Russ.).
Sotnikova, O. A., Fefilova, E. F. and Goza, N. I. (2008), Germenevticheskiy podkhod k obucheniyu matematike (teoreticheskiy aspekt): Monografiya [Hermeneutic Approach to Teaching Mathematics (Theoretical Aspect): Monograph], Publishing House of the Komi Republic Academy of Public Administration and Management, Syktyvkar, Russia (in Russ).