American Journal of Applied Mathematics and Statistics
ISSN (Print): 2328-7306 ISSN (Online): 2328-7292 Website: Editor-in-chief: Mohamed Seddeek
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American Journal of Applied Mathematics and Statistics. 2024, 12(1), 10-14
DOI: 10.12691/ajams-12-1-2
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Assessing the Effectiveness of the APOS/ACE Instructional Treatment with the Help of Neutrosophic Triplets

Michael Gr. Voskoglou1,

1Department of Mathematical Sciences, Graduate T. E. I. of Western Greece, Patras, Greece

Pub. Date: February 01, 2024

Cite this paper:
Michael Gr. Voskoglou. Assessing the Effectiveness of the APOS/ACE Instructional Treatment with the Help of Neutrosophic Triplets. American Journal of Applied Mathematics and Statistics. 2024; 12(1):10-14. doi: 10.12691/ajams-12-1-2


The APOS/ACE instructional treatment for teaching mathematics was introduced in the USA by Prof. Ed Dubinsky and his research team during the 1990’s The central idea of the APOS/ACE treatment is that one can always find a suitable computer task for helping students to build the mental constructions that lead to the learning of the corresponding mathematical topic. In this work a method is presented for assessing the overall performance of a student group when the instructor is not sure about the accuracy of the individual grades assigned to the students. This method is developed using neutrosophic sets as tools and writing their elements in the form of neutrosophic triplets and it is used here for evaluating the effectiveness of the APOS/ACE instructional treatment for teaching mathematics. The outcomes of the classroom application performed for this purpose provide a strong indication that the APOS/ACE approach benefits the mediocre and the weak in mathematics students more than the good students, but this requires further experimental research.

APOS/ACE Neutrosophic Set (NS) Neutrosophic Assessment Neutrosophic Triplet (NT) GPA Index

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[1]  Voskoglou, M.Gr., Methods for Assessing Human-Machine Performance under Fuzzy Conditions, Mathematics, 7, 230, 2019.
[2]  Van Broekhoven, E., De Baets, B., Fast and accurate centre of gravity defuzzification of fuzzy systems outputs defined on trapezoidal fuzzy partitions, Fuzzy Sets Syst., 6, 157, 904–918, 2006.
[3]  Voskoglou, M.Gr., Broumi, S., Smarandache, F., A Combined Use of Soft and Neutrosophic Sets for Student Assessment with Qualitative Grades, Journal of Neutrosophic and Fuzzy Systems, 4(1), 15-20, 2022
[4]  Asiala, M., Brown, A., DeVries, D.J., Dubinsky, E., Mathews, D. & Thomas, K., A framework for research and curriculum development in undergraduate mathematics education, Research in Collegiate Mathematics Education II, CBMS Issues in Mathematics Education, 6, 1-32, 1996.
[5]  Dubinsky, E., McDonald, M.A., APOS: A constructivist theory of learning in undergraduate mathematics education research, in D. Holton et al. (Eds.), The Teaching and Learning of Mathematics at University Level: An ICMI Study, Kluwer Academic Publishers, Dordrecht, Netherlands, pp. 273-280, 2001.
[6]  Arnon, I., Cottrill, J., Dubinsky, E., Oktac¸ A., Roa, S., Trigueros, M. & Weller, K., APOS Theory: A Framework for Research and Curriculum Development in Mathematics Education, Springer, N. Y., Heidelberg, Dordrecht, London, 2014.
[7]  Voskoglou, M. Gr., An application of the APOS/ACE approach teaching the irrational numbers, Journal of Mathematical Sciences and Mathematics Education, 8(1), 30-47, 2013
[8]  Voskoglou, M.Gr., Fuzzy Logic in the APOS/ACE Instructional Treatment of Mathematics, American Journal of Educational Research, 3(3), 330-339, 2015.
[9]  Borji, V., Voskoglou, M.Gr., Applying the APOS Theory to Study the Student Understanding of the Polar Coordinates, American Journal of Educational Research, 4(16), 1149-1156, 2016.
[10]  Borji, V., Voskoglou, M.Gr., Designing an ACE Approach for Teaching the Polar Coordinates, American Journal of Educational Research, 5(3), 303-309, 2017.
[11]  Voskoglou, M. Gr., A Markov Chain Model for the APOS/ACE Instructional Treatment of Mathematics, International Journal of Education and Learning Systems, 4, 5(3), 1-6, 2019.
[12]  Piaget, J., Genetic Epistemology, Columbia University Press, New York and London, 1970.
[13]  Weller, K., Clark, J.M., Dubinsky, E., Loch, S., McDonald, M., Students performance and attitudes in courses based on APOS theory and the ACE teaching cycle, in A. Selden et al. (Eds.), Research in Collegiate Mathematics Education V, Providence, RI, American Mathematical Society, pp. 97-181, 2003.
[14]  Weller, K., Arnon, I.., Dubinsky, E., Pre - service Teachers’ Understanding of the Relation between a Fraction or Integer and Its Decimal Expansion: Strength and Stability of Belief, Canadian Journal of Science, Mathematics and Technology Education, 11(2), 129-159, 2011.
[15]  Zadeh, L.A., Fuzzy Sets. Inf. Control, 8, 338–353, 1965.
[16]  Atanassov, K.T., Intuitionistic Fuzzy Sets, Fuzzy Sets and Systems, 20(1), 87-96, 1986.
[17]  Smarandache, F., Neutrosophy / neutrosophic probability, set, and logic, Proquest, Michigan, USA, 1998.
[18]  Wang, H., Smarandanche, F., Zhang, Y. & Sunderraman, R., Single valued neutrosophic sets, Review of the Air Force Academy (Brasov), 1(16), 10-14, 2010.
[19]  Smarandache, F., Indeterminacy in neutrosophic theories and their applications, International Journal of Neutrosophic Science, 15(2), 89-97, 2021.
[20]  Cuong, B.C., Picture fuzzy sets, Journal of Computer Science and Cybernetics, 30(4), 409-420, 2014.
[21]  Smarandache, F., Algebra, retrieved from https://, 2023.
[22]  Smarandache, F., Geometries, retrieved from https:// , 2023.
[23]  Salama, A.A. & Alblowi, S.A., Neutrosophic sets and neutrosophic topological spaces, IOSR Journal of Mathematics, 3(4), 31-35, 2013.
[24]  Voskoglou, M. Gr., An Application of Neutrosophic Sets to Decision Making, Neutrosophic Sets and Systems, 53, 1-9, 2023.
[25]  Smarandache, F., Smarandache Notions, retrieved from, 2023.
[26]  Smarandache, F., Plithogeny, Plithogenic Set, Logic, Probability and Statistics, Pons, Brussels, Belgium, 2017.
[27]  Smarandache, F., Plithogenic Set, an Extension of Crisp, Fuzzy, Intuitionistic Fuzzy and Neutrosophic Set – Revisited, Neutrosophic Sets and Systems, 21, 153-166, 2018.
[28]  Ganathy, S., Nagarajan, D., Broumi, S., & Lathamaheswari, M., Plithogenic sets and their applications in decision making, Neutrosophic Sets and Systems, 38, 453-459, 2020.
[29]  Quek, S.G., Salvachandram, G., Smarandache, F., Vimala, J., Bui, Q.-T. & Gerogiannis, V., Entropy Measures for Plithogenic Sets and Applications in Multi – Attribute Decision Making, Mathematics, 8, 965, 2020.
[30]  Smarandache, F., Subtraction and Division of Neutrosophic Numbers, Critical Review, XIII, 103-110, 2016.
[31]  Voskoglou, M.Gr., Finite Markov Chain and Fuzzy Logic Assessment Models, Create Space Independent Platform (Amazon), Columbia, SC, 2017.