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All Journal Journal of Honai Math Studies in Learning and Teaching Edumatica: Jurnal Pendidikan Matematika Hipotenusa: Journal of Mathematical Society
Ogbonnaya, Ugorji
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Education Mathematics Social Sciences Other

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Visualization techniques for proofs: Implications for enhancing conceptualization and understanding in mathematical analysis Muzangwa, Jonatan; Ogbonnaya, Ugorji
Journal of Honai Math Vol. 7 No. 2 (2024): Journal of Honai Math
Publisher : Universitas Papua

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.30862/jhm.v7i2.603

Abstract

Visual images are frequently utilized to elucidate concepts in general mathematics and geometry; however, their application in mathematical analysis remains uncommon. This paper demonstrates how visual imagery can enhance the proof of certain theorems in mathematical analysis. It emphasizes the importance of visualization in the learning and understanding of mathematical concepts, particularly within mathematical analysis, where diagrams are seldom employed. The paper focuses on the reasoning processes used by mathematicians in proving selected fundamental theorems of mathematical analysis. It provides illustrative examples where visual images are instrumental in performing specific subtasks within proof development and in completing the proofs. The proofs discussed include the sum of the first n natural numbers, the sum rule of integration, the mean value theorem for derivatives, the mean value theorem for integrals, and Young’s Inequality. This paper underscores that visual images serve not only as persuasive tools but also as bridges between symbolic representations and real-world understanding.
Algebraic Question Types in Grade 10 Mathematics Textbook: A Cognitive Perspective Sibiya, Mandlenkosi Richard; Sekao, David; Ogbonnaya, Ugorji
Studies in Learning and Teaching Vol. 6 No. 1 (2025): April
Publisher : CV Sinergi Ilmu dan Publikasi

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.46627/silet.v6i1.531

Abstract

Mathematics education depends on textbooks as the primary source of information for both teachers and learners while providing essential tasks that develop cognitive skills. The types of tasks used in these resources directly affect learners’ cognitive growth and show why textbooks must contain various question types. Current research lacks examination of mathematics textbooks in South African schools, even though learners consistently perform poorly in mathematics. This study investigated whether a widely used Grade 10 Mathematics textbook challenged learners to various question types that help build problem-solving skills and high-level thinking. A deductive content analysis approach evaluated 1 360 questions from the textbook’s algebra content, including algebraic expressions, exponents, equations and inequalities, number patterns, and functions. The research results indicated that most questions (62%) required routine calculations while the content lacked questions that required higher-order cognitive skills such as problem-solving, argumentation and interpretation. The assessed textbook appears to fail learners by not adequately developing critical thinking skills that learners need for both future advanced math classes and practical applications. The study recommends more diverse question types in mathematics textbooks to develop learners’ cognitive skills and problem-solving abilities.
Looking Back dalam Menyelesaikan Masalah Representasi Aljabar pada Siswa Iilonga, Hesekiel K.; Ogbonnaya, Ugorji
Edumatica : Jurnal Pendidikan Matematika Vol 14 No 1 (2024): Edumatica: Jurnal Pendidikan Matematika (April 2024)
Publisher : Department of Mathematics Education, Faculty of Teacher Training and Education, Universitas Jambi

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.22437/edumatica.v14i01.29404

Abstract

Solving mathematics problems is a complex cognitive process that involves a series of steps. These steps typically include understanding the problem, devising a plan, carrying out the plan, and looking back (checking the solution). While each of these steps is crucial, the ‘looking back’ step has not received much attention in mathematics education research. This study investigated Grade 10 mathematics students’ looking back in solving algebraic word problems in Namibia. The study followed a qualitative approach. The sample of the study was 351 Grade 10 students from ten secondary schools in the Ohangwena Region in Namibia. Data was collected using the Algebra Word problem-solving Achievement Test and Interview. The results show that, in general, the students did not look back on their solutions to the algebraic word problem. While some students indicated that looking back necessitates the consolidation of their work, others said looking back on their solutions could be time-wasting and confusing. These findings point to the need for mathematics teachers in Namibia to incorporate and model the looking-back step of mathematical problem-solving in their mathematics teaching.
Mathematics Teachers' Understanding of the Algorithms for Solving Quadratic Function Problems Ibeawuchi, Emmanuel; Ogbonnaya, Ugorji
Hipotenusa: Journal of Mathematical Society Vol. 7 No. 2 (2025): Hipotenusa: Journal of Mathematical Society
Publisher : Program Studi Tadris Matematika Universitas Islam Negeri (UIN) Salatiga

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.18326/hipotenusa.v7i2.5536

Abstract

This study investigated secondary school mathematics teachers' understanding of the mathematical concepts underlying the algorithms used in teaching quadratic functions. It examined three constructs: i) why the graph of a parabola is a curve and not a straight line; ii) why the solution of a quadratic equation remains unchanged when the equation is multiplied by -1; and iii) the justification for the counter-intuitive shift of the parabola in horizontal transformations. A qualitative case study design was employed, involving five secondary school mathematics teachers selected through a combination of purposive and convenience sampling. Data were collected using a Subject Matter Knowledge Questionnaire (SMKQ) that focused on teachers' understanding of quadratic functions and their underlying algorithms. Responses were analysed through conventional qualitative content analysis. Findings revealed that while most teachers demonstrated adequate procedural skills, they exhibited limited conceptual understanding of the meaning underlying the algorithms in the constructs investigated. Their reasoning often reflected a focus on knowing how rather than understanding why, indicating a gap between procedural fluency and conceptual depth. The study concludes that strengthening teachers' conceptual understanding of mathematical algorithms is essential for improving instructional quality and learner outcomes. It recommends targeted professional development programmes that integrate both procedural and conceptual aspects of mathematical knowledge.
Co-Authors Ibeawuchi, Emmanuel Iilonga, Hesekiel K. Muzangwa, Jonatan Sekao, David Sibiya, Mandlenkosi Richard