iosr-Jm   Volume-21 ~ Issue-5 ~ Sep. – Oct. 2025

Abstract : This paper presents a theoretical framework for understanding geometric objects through multiple representations in mathematics education. We establish mathematical foundations for transforming between different representational forms while maintaining the same geometric object and propose the concept of "representational fluency" as a theoretical construct for mathematics education. The primary contribution is a Translation Principle, which demonstrates that geometric constraints can be reformulated as the domain of a function, along with........

Keywords: geometric representation, mathematical equivalence, representational fluency, coordinate
transformations, mathematics education, theoretical framework.

[1]. Ainsworth, S. (2006). Deft: A Conceptual Framework For Considering Learning With Multiple Representations. Learning And Instruction, 16(3), 183-198.
[2]. Battista, M. T. (2007). The Development Of Geometric And Spatial Thinking. In F. K. Lester (Ed.), Second Handbook Of Research On Mathematics Teaching And Learning (Pp. 843-908). Information Age Publishing.
[3]. Duval, R. (2006). A Cognitive Analysis Of Problems Of Comprehension In Learning Of Mathematics. Educational Studies In Mathematics, 61(1-2), 103-131.
[4]. Edwards, C. H., & Penney, D. E. (2008). Calculus: Early Transcendentals (7th Ed.). Pearson Prentice Hall.
[5]. Goldin, G., & Shteingold, N. (2001). Systems Of Representations And The Development Of Mathematical Concepts. In A. A. Cuoco & F. R. Curcio (Eds.), The Roles Of Representation In School Mathematics (Pp. 1-23). National Council Of Teachers Of Mathematics.

Abstract : The chief interest of this article is to discuss non-archimedean pseudo- differential operator connected to coupled fractional Fourier transform. In this article, we some classes of p-adic complete inner product spaces, Bφ, k(Qp Qp), 0 k < ∞, connected to negative......

Keywords: Non-archimedean analysis, Pseudo-differential operators, Frac- tional Fourier transform, M-dissipative operators.

[1] Alexandra V Antoniouk, Andrei Yu Khrennikov, And Anatoly N Kochubei. Multidimensional Nonlinear Pseudo-Differential Evolution Equation With P- Adic Spatial Variables. Journal Of Pseudo-Differential Operators And Appli- Cations, 11:311–343, 2020.
[2] Andrei Khrennikov, Klaudia Oleschko, And Maria De Jesus Correa Lopez. Modeling Fluid’s Dynamics With Master Equations In Ultrametric Spaces Rep- Resenting The Treelike Structure Of Capillary Networks. Entropy, 18(7):249, 2016.
[3] Klaudia Oleschko And A Yu Khrennikov. Applications Of P-Adics To Geo- Physics: Linear And Quasilinear Diffusion Of Water-In-Oil And Oil-In-Water Emulsions. Theoretical And Mathematical Physics, 190(1):154–163, 2017.
[4] Ehsan Pourhadi, Andrei Khrennikov, Reza Saadati, Klaudia Oleschko, And María De Jesús Correa Lopez. Solvability Of The P-Adic Analogue Of Navier– Stokes Equation Via The Wavelet Theory. Entropy, 21(11):1129, 2019.
[5] Wilson A Zúñiga-Galindo. Pseudodifferential Equations Over Non- Archimedean Spaces, Volume 2174. Springer, 2016.

Abstract : This study investigates the relationship between Pell numbers and Pell-Lucas numbers, which follow the same recurrence relation but differ in initial conditions. The goal of this study is to establish and prove general identities connecting the two sequences through the Principle of Mathematical Induction. Several key identities involving sums, products, squares, and linear combinations were derived and validated.

Keywords: Pell numbers, Pell-Lucas numbers, relationship identities, principle of mathematical induction

[1]. M. Narayan Murty And Binayak Padhy, A Study On Pell And Pell-Lucas Numbers, Iosr Journal Of Mathematics, Volume 19, Issue 2 Series 1, Pp. 28-36, 2023. Doi: 10.9790/5728-1902012836
[2]. S.F.Santana And J.L. Diaz-Barrero, Some Properties Of Sums Involving Pell Numbers, Missouri Journal Of Mathematical Sciences, Doi.10.35834/2006/1801033, Vol.18, No.1, 2006.
[3]. O’regan, Gerard. Mathematical Induction And Recursion. In Guide To Discrete Mathematics: An Accessible Introduction To The History, Theory, Logic And Applications, Pp. 79-88. Cham: Springer International Publishing, 2021..

Abstract : This study explores the impact of solving the Rubik’s Cube on spatial reasoning abilities using inferential statistical methods. Within a sample of 30 participants there existed two cohorts, cubers (n = 15) and non-cubers (n = 15). A two-sample t-test was employed to analyse the outcomes after both groups completed a standardized spatial reasoning test. The findings suggested that cubers scored far higher in spatial reasoning (M = 72.74%) compared to non-cubers (M = 56.63%) because the t-statistic equaled 2.962 as p = 0.006, also this indicates a statistically important.....

[1]. Https://Www.Centraltest.Com/Blog/Spatial-Reasoning-Often-Overlooked-Key-Asset
[2]. Https://Www.Graphpad.Com/Quickcalcs/Pvalue1/
[3]. Https://Www.Assessmentday.Co.Uk/Free/Spatial/1/Index.Php?_Gl=1*1a8i5at*_Gcl_Au*Mtmxotkwntmxmc4xnzq3mza0mzcy*_Ga*Ndu2nti2mdu0lje3ndczmdqznzi.*_Ga_S04nddmhwq*Cze3ndg3otu5mjgkbzikzzekdde3ndg3otu5mzykajuyjgwwjggw
[4]. Https://Psycnet.Apa.Org/Record/2010-16524-002
[5]. Https://Www.Scirp.Org/Reference/Referencespapers?Referenceid=1359229.