Main Article Content
Abstract
This study investigates the impact of integrating both concrete and virtual manipulatives on the mathematics achievement of fifth-grade students across different achievement levels (low, average, and high). Utilizing a quasi-experimental design with pre- and post-tests, a convenience sample of fifth-grade students was randomly assigned to either a control or experimental group. The data collection instruments, consisting of separate pre- and post-tests on the same mathematical concepts, underwent thorough validity and reliability testing. Initial assessments demonstrated that the achievement levels between the control and experimental groups were comparable prior to the intervention. The experimental group received instruction that incorporated both concrete and virtual manipulatives, whereas the control group followed traditional teaching methods. Following a 12-week intervention period, a post-test was administered. The data were analyzed using parametric paired-sample t-tests and one-way ANCOVA, ensuring that all underlying assumptions were satisfied. The findings revealed significant improvements in post-test scores among students in the experimental group, regardless of their initial achievement levels. Although low-achieving students in the control group also showed progress, their gains were less substantial compared to those in the experimental group. This study highlights the potential benefits of incorporating both concrete and virtual manipulatives in fifth-grade mathematics instruction to enhance academic achievement.
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References
- Abdul-Karim, H., Kasimu, O., Adul-Rahaman, A., Kanimam, Y. S. Imoro, M., & Dokurugu, M. E. (2023). Assessing the impact of algebra tiles as visual and manipulative aids on students’ algebraic understanding. American Journal of Educational Research, 11(10), 705-711. https://pubs.sciepub.com/education/11/10/9/index.html
- Akpan, E. T., Charles-Ogan, G. I., Eze, F. B., Okafor-Agbala, U. C., & Onyeka, E. C. (2023). Technology enhanced learning: utilization of symboLAb manipulative instruction and performance of students in quadratic graphs. Asian Journal of Advanced Research and Reports, 17(11), 32-42. https://sdfswk3623.s3.ap-northeast-2.amazonaws.com/Akpan17112023AJARR99909.pdf
- Andrabi, T., Jishnu, D., Asim, I., Khwaja, Vishwanathan, T., & Zajonc, T. (2008). Learning and educational achievements in Punjab schools (LEAPS): Insights to inform the education policy debate. Harvard University. https://static1.squarespace.com/static/618f98a44478fc2e7b844cb4/t/6203c90517f63c6853836acc/1644415240287/LEAPS_report_report.pdf
- Arrogante, O. (2022). Sampling techniques and sample size calculation: How and how many participants should I select for my research? Intensive Care Nursing, 33(1), 44-47. https://doi.org/10.1016/j.enfie.2021.03.004
- ASER Pakistan. (2024). Annual status of education report (ASER) Pakistan, National (Rural). https://aserpakistan.org/document/2024/aser_national_2023.pdf
- Asian Development Bank (ADB) (2023). ADB brief no. 283: Strengthening school teaching in Pakistan. Manila. https://www.adb.org/sites/default/files/publication/935721/adb-brief-283-strengthening-school-teaching-pakistan.pdf
- Bajpai, N., & Pandey, J. (2024). The effectiveness of experiential learning on mathematics achievement among fifth grade school students: A quasi-experimental study. International Journal of Indian Psychology, 12(1), 525-535. https://ijip.in/pdf-viewer/?id=42881
- Back, J. (2019). Manipulative in the primary classroom. NRICH, 3, 1-8. https://nrich.maths.org/content/id/10461/Manipulatives%20in%20the%20Primary%20Classroom.pdf
- Bhutta, S. M., & Rizvi, N. F. (2022). Assessing teachers’ pedagogical practices and students’ learning outcomes in science and mathematics across primary and secondary school level: A nationwide study (2018-21). Aga Khan University, Institute for Educational Development, Karachi, Pakistan, 1-4. https://ecommons.aku.edu/pakistan_ied_pdck/317
- Bone, E. K., Bouck, E. C., & Satsangi, R. (2023). Comparing concrete and virtual manipulatives to teach algebra to middle school students with disabilities. Exceptionality, 31(1), 1-17. https://doi.org/10.1080/09362835.2021.1938057
- Bornaa, C. S., Okwan, B., & Iddrisu, A. B. et al. (2023). Cooperative learning with manipulatives and students’ performance in mathematics problem solving. Journal of Education and Practice, 14(19), 1-12. https://www.researchgate.net/publication/372807317_Cooperative_Learning_with_Manipulatives_and_Students%27_Performance_in_Mathematics_Problem_Solving
- Bouck, E. C., Christopher, W., & Bone, E. (2018). Manipulative apps to support students with disabilities in mathematics. Intervention in School and Clinic, 53(3). https://doi.org/10.1177/1053451217702115
- Bouck, E. C., Park, J., Satsangi, R., Cwiakala, K., & Levy, K. (2019). Using the virtual-abstract instructional sequence to support acquisition of algebra. Journal of Special Education Technology, 34(4), 253–268. https://doi.org/10.1177/0162643419833022
- Bouck, E. C., Park, J., & Stenzel, K. (2020). Virtual manipulatives as assistive technology to support students with disabilities with mathematics. Preventing School Failure: Alternative Education for Children and Youth, 64(4), 281-289. https://doi.org/10.1080/1045988X.2020.1762157
- Bouck, E., Long, H., & Bae, Y. (2023). Exploring the virtual-representational-abstract instructional sequence across the learning stages for struggling students. Behavior Modification, 47(3), 590-614. https://doi.org/10.1177/01454455221129998
- Byrne, E. M., Jensen, H., Thomsen, Bo S., & Ramchandani, P. G. (2023). Educational interventions involving physical manipulatives for improving children’s learning and development: a scoping review. Review of Education, 11(2), 1-42. https://doi.org/10.1002/rev3.3400
- Coles, A., & Sinclair, N. (2019). Re-thinking ‘concrete to abstract’ in mathematics education: Towards the use of symbolically structured environments. Canadian Journal of Science, Mathematics and Technology Education, 19, 465-480. https://doi.org/10.1007/s42330-019-00068-4
- Crowe, A. (2022, retrieved 21 May 2024). Math manipulatives: How can they be used to enhance the classroom experience? https://www.prodigygame.com/main-en/blog/math-manipulatives/
- Dhindsa, N. (2020). Teach-to-one blended mathematics’ impact on middle school students’ mathematics achievement. [Doctoral Dissertations, Walden University]. https://scholarworks.waldenu.edu/cgi/viewcontent.cgi?article=10753&context=dissertations
- Dinsmoor, K. (2022). Math Manipulatives. In S. L. Mason (Ed.), Student-centered approaches in K–12 and higher education. EdTech Books. https://edtechbooks.org/student_centered/math_manipulatives
- Ernest, P., Skovsmose, O., Bendegem, J. P. van., Bicudo, M., Miarka, R., Kvasz, L., & Moeller, R. (2016). The Philosophy of Mathematics Education. In Editor Gabriele, K. (2016) ICME-13 tropical surveys. Springer International Publishing: AG Switzerland. https://library.oapen.org/bitstream/id/d99d2551-2ef4-4bc9-9ad3-cde7d561dee7/1002255.pdf
- Foulkes, M., Sella, F., Wege, T. E., & Gilmore, C. (2023). The effect of concreteness on mathematical manipulatives. Mind, Brain, and Education, 17(3), 185-196. https://doi.org/10.1111/mbe.12374
- Gilligan-Lee, K.A., Hawes, Z.C., Williams, A.Y., Farran, E.K., & Mix, K. S. (2023). Hands-on: Investigating the role of physical manipulatives in spatial training: Registered report. Child Development, 94(5), 1205-1221. https://doi.org/10.1111/cdev.1396
- Ghodbane, T., & Achachi, H. H. E. (2019). Narrowing the achievement gap between EFL students in oral expression through cooperative learning. Arab World English Journal (AWEJ), 10(2), 365-378. https://dx.doi.org/10.24093/awej/vol10no2.28
- Gravito. L. C., Sagayno, R. C., Alcantara, R. T., & Tesorio, G. T. (2023). Homemade manipulative materials for teaching grade iv mathematics. European Journal of Humanities and Educational Advancements (EJHEA), 4(2), 11-15. https://www.researchgate.net/publication/368364062_European_Journal_of_Humanities_and_Educational_Advancements_EJHEA_HOMEMADE_MANIPULATIVE_MATERIALS_FOR_TEACHING_GRADE_IV_MATHEMATICS
- Guan, H., Qin, X. L., Rao, Y. S., & Cao, S. (2020). Domain model of web-based dynamic geometry software and its applications. Journal of Computer Applications, 40(4). https://doi.org/10.11772/j.issn.1001-9081.2019091672
- Guan, H., Li, J., Rao, Y., Chen, R., & Xu, Z. (2024). Comparative effects of dynamic geometry system and physical manipulatives on Inquiry-based Math Learning for students in Junior High School. Education and Information Technologies, 1-23. https://doi.org/10.1007/s10639-024-12663-6
- Gülen, S. (2020). A study to determine the ability of fifth-grade students in reflecting their knowledge about sun, earth, and moon by different measurement tools. Science Education International, 31(1), 41-51. https://files.eric.ed.gov/fulltext/EJ1247785.pdf
- Haji Ismail, N. F., Shahrill, M., & Asamoah, D. (2023). Learning through virtual manipulatives: Investigating the impact of Gizmos-based lessons on students’ performance in integers. Contemporary Mathematics and Science Education, 4(1), 2-12. https://doi.org/10.30935/conmaths/12857
- Hawes, Z. C. K., Gilligan-Lee, K. A., & Mix, K. S. (2022). Effects of spatial training on mathematics performance: A meta-analysis. Developmental Psychology, 58(1), 112–137. https://doi.org/10.1037/dev0001281
- Iqbal, M. Z., Shams, J. A., & Nazir, M. (2020). Effect of using mathematics manipulatives on the student’s academic achievement. Journal of Science Education, 11(1), 1-15. https://ojs.aiou.edu.pk/index.php/jse/article/view/1713
- Jablonski, S., & Matthias, L. (2023). Teaching and learning of geometry: A literature review on current developments in theory and practice. Education Sciences, 13(7), 682-692. https://doi.org/10.3390/educsci13070682
- Johnson, J. R. R. (2022). Educators’ perceptions of implementing concrete and virtual math manipulatives in title I schools. [Doctoral dissertation, Trevecca Nazarene University]. Trevecca Nazarene University ProQuest Dissertations Publishing. https://www.proquest.com/docview/2688493214/fulltextPDF/6ED214C144F4401CPQ/1?accountid=15156&sourcetype=Dissertations%20&%20Theses
- Jones, J. P., & Tiller, M. (2017). Using concrete manipulatives in mathematical instruction. Dimensions of Early Childhood, 45(1), 18–23. https://files.eric.ed.gov/fulltext/EJ1150546.pdf
- Julie, (2021, retrieved May 21, 2024). Math manipulatives: How do they aid student learning? https://wordpress.oise.utoronto.ca/robertson/2021/10/18/math-manipulatives/
- Kabel, M., Hwang, J., & Hwang, J. (2021). Lessons learned from a rural classroom study: Transitioning from concrete to virtual manipulatives to teach math fact fluency to students with learning disabilities. Journal of Curriculum Studies Research, 3(1), 42-68. https://doi.org/10.46303/jcsr.2021.7
- Kemp, K. (2021). Test corrections appear to benefit lower-achieving students in an introduction to biology major course: Results of a single-site, one-semester study. Journal of Microbiology & Biology Education, 22(2), 122-21. https://doi.org/10.1128/jmbe.00122-21
- Lange J. (2021). The importance of using manipulatives in math class. [Master thesis & Capstone projects, Northwestern College]. Northwestern College, Iowa Research Repository. https://nwcommons.nwciowa.edu/cgi/viewcontent.cgi?article=1291&context=education_masters
- Large Scale Assessment (LSA) (2021). Grade-5 main findings report – 2021. Cambridge Education Mott MacDonald: Lahore, Pakistan. https://pec.edu.pk/system/files/LSA%20Main%20Findings%20Report%202021.pdf#overlay-context=publications
- Lee, C. Y., & Chen, M. J. (2015). Effects of worked examples using manipulatives on fifth graders' learning performance and attitude toward mathematics. Journal of Educational Technology & Society, 18(1), 264-275. http://www.jstor.org/stable/jeductechsoci.18.1.264
- Liu, D., Lory, C., Lei, Q., Cai, W., Mao, Y., & Yang, X. (2024). Using explicit instruction and virtual manipulatives to teach measurement concepts for students with autism spectrum disorder. The Journal of Special Education, 58(1), 3-13. https://doi.org/10.1177/00224669231171558
- Long, H. M., Bouck, E. C., & Kelly, H. (2023). An evidence-based practice synthesis of virtual manipulatives for students with ASD and IDD. Focus on Autism and Other Developmental Disabilities, 38(3), 147-157. https://doi.org/10.1177/10883576221121654
- Mainali, B. (2021). Representation in teaching and learning mathematics. International Journal of Education in Mathematics, Science, and Technology (IJEMST), 9(1), 1-21. https://doi.org/10.46328/ijemst.1111
- Mazo, J. C. (2024). Virtual manipulative: Its effects as an instructional tool in teaching mathematics. Multidisciplinary International Journal of Research and Development, 3(4), 15-35. https://www.mijrd.com/papers/v3/i4/MIJRDV3I40002.pdf
- Mills, G., & Gay, L. (2018). Educational research: Competencies for analysis and applications. Pearson Education.
- Milton, J. H., Flores, M. M., Hinton, V. M., Dunn, C., & Darch, C. B. (2023). Using the concrete-representational-abstract sequence to teach conceptual understanding of place value, rounding, and expanded notation. Learning Disabilities Research & Practice, 38(1), 15-25. https://doi.org/10.1111/ldrp.12299
- Mirza, M., & Iqbal, M. Z. (2014). Impact of collaborative teaching (CT) on mathematics students’ achievement in Pakistan. Journal of Research & Reflections in Education, 8(1), 13-21. https://ue.edu.pk/jrre/articles/81002.pdf
- Moyer-Packenham, P. S., & Bolyard, J. J. (2016). Revisiting the Definition of a Virtual Manipulative. In: Moyer-Packenham, P. (eds) International Perspectives on Teaching and Learning Mathematics with Virtual Manipulatives. Mathematics education in the digital era, vol 7. Springer, Cham. https://doi.org/10.1007/978-3-319-32718-1_1
- Muhammad, S. M., Sani, M. A., Abdullahi, F., & Bayaro, A. (2023). The effect using manipulatives on the performance of pupils in primary school mathematics. Journal of Mathematical Sciences & Computational Mathematics, 4(2), 223-232. https://doi.org/10.15864/jmscm.4206
- National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author.
- National Curriculum Council – Pakistan (2020). Single national curriculum mathematics grade I-V 2020: One nation, one curriculum. Ministry of Federal Education and Professional Training, Islamabad. Government of Pakistan. https://mofept.gov.pk/SiteImage/Misc/files/SNC%20Mathematics%201-5.pdf
- Nash, E. (2023, retrieved May 21, 2024). Math manipulatives: A student-centered approach to teaching mathematics. https://greatminds.org/math/blog/eureka/math-manipulatives-a-student-centered-approach-to-teaching-mathematics
- Ng, OL., & Ye, H. (2022). Mathematics learning as embodied making: primary students’ investigation of 3D geometry with handheld 3D printing technology. Asia Pacific Education Review, 23, 311–323. https://doi.org/10.1007/s12564-022-09755-8
- Nguyen, T., Watts, T. W., Duncan, G. J., Clements, D. H., Sarama, J. S., Wolfe, C., & Spitler, M. E. (2016). Which preschool mathematics competencies are most predictive of fifth grade achievement? Early Childhood Research Quarterly, 36(3), 550-560. https://doi.org/10.1016/j.ecresq.2016.02.003
- Nurjannah, S., & Kusnandi, (2021). Literature study: The role of abstraction ability to strengthen students’ early knowledge in mathematics learning. International Conference on Mathematics and Science Education (ICMScE), 1806, 1-6. https://iopscience.iop.org/article/10.1088/1742-6596/1806/1/012064/pdf
- Nurhasanah, F., Kusumah, Y. S., & Sabandar, J. (2017). Concept of triangle: examples of mathematical abstraction in two different contexts. International Journal on Emerging Mathematics Education, 1(1), 53-70. http://dx.doi.org/10.12928/ijeme.v1i1.5782
- O’Meara, N., Johnson, P., & Leavy, A. (2019). A comparative study investigating the use of manipulatives at the transition from primary to post-primary education. International Journal of Mathematical Education in Science and Technology, 51(6), 835-857. https://doi.org/10.1080/0020739X.2019.1634842
- Pallant, J. (2020). SPSS survival manual: A step by step guide to data analysis using IBM SPSS. Routledge. https://doi.org/10.4324/9781003117452
- Park, J., & Bouck, E. (2022). Teacher-delivered virtual manipulative mathematics intervention to individuals with extensive support needs. Research in Developmental Disabilities, 131, https://doi.org/10.1016/j.ridd.2022.104339
- Pavlou, Y., Zacharias, C. Z., & Papaevripidou, M. (2024). Comparing the impact of physical and virtual manipulatives in different science domains among preschoolers. Science Education, 1-29. https://doi.org/10.1002/sce.21869
- Peltier, C., Morin, K. L., Bouck, E. C., Lingo, M. E., Pulos, J., Scheffler, F. A., & Deardorff, M. E. (2020). A meta-analysis of single case research using mathematics manipulatives with students at risk or identified with a disability. Journal of Special Education, 54(1), 3-15. https://doi.org/10.1177/0022466919844516
- Provincial Assessment of Students Learning (PASL) (2020). PASL report 2018-19/PEC. UNICEF publisher. https://pec.edu.pk/system/files/PASL%20Report%20PEC_compressed.pdf#overlay-context=publications
- Prosser, S. K., & Bismarck, S. F. (2023). Concrete–representational–abstract (CRA) instructional approach in an algebra I inclusion class: Knowledge retention versus students’ perception. Education Science, 13, 1061. https://doi.org/10.3390/educsci13101061
- Root, J. R., Cox, S. K., Gilley, D., & Wade, T. (2021). Using a virtual-representational-abstract integrated framework to teach multiplicative problem solving to middle school students with developmental disabilities. Journal of Autism and Developmental Disorders, 51(7), 2284-2296. https://doi.org/10.1007/s10803-020-04674-2
- Ruel, E., Wagner, W. E., & Gillespie; B. J. (2015). The practice of survey research: Theory and applications. Sage Publications, Inc.
- Salifu, A. S. (2022). The effects of balance model and algebra tiles manipulative in solving linear equations in one variable. Contemporary Mathematics and Science Education, 3(2), 1-10. https://doi.org/10.30935/conmaths/12028
- Sari, M. H., & Aydoğdu, S. (2020). The effect of concrete and technology-assisted learning tools on place value concept, achievement in mathematics and arithmetic performance of primary school students. International Journal of Curriculum and Instruction, 12(1), 197-224. https://ijci.globets.org/index.php/IJCI/article/view/312/159
- Satsangi, R., & Bouck, E. C. (2015). Using virtual manipulative instruction to teach the concepts of area and perimeter to secondary students with learning disabilities. Learning Disability Quarterly, 38, 174–186. https://doi.org/10.1177/0731948714550101
- Satsangi, R., & Miller, B. (2017). The case for adopting virtual manipulatives in mathematics education for students with disabilities. Preventing school failure: Alternative Education for Children and Youth, 61, 303–310. https://doi.org/10.1080/1045988X.2016.1275505
- Satsangi, R., Hammer, R., & Evmenova, A. S. (2018). Teaching multistep equations with virtual manipulatives to secondary students with learning disabilities. Learning Disabilities Research & Practice, 33(2), 99–111. https://doi.org/10.1111/ldrp.12166
- Satsangi, R., Raines, A. R., & Fraze, K. (2021). Virtual manipulatives for teaching algebra: A research to practice guide for secondary students with a learning disability. Learning Disabilities: A Multidisciplinary Journal, 26(1), 46-57. https://doi.org/10.18666/LDMJ-2021-V26-I1-10349
- Satsangi, R., & Raines, A. R. (2023). Examining virtual manipulatives for teaching computations with fractions to children with mathematics difficulty. Journal of Learning Disabilities, 56(4), 295–309. https://doi.org/10.1177/00222194221097710
- Serin, H. (2023). The role of technology in mathematics education: Promoting student achievement. International Journal of Social Sciences & Educational Studies, 10(2), 390-395. https://www.semanticscholar.org/reader/44bf2d839400dcf034639e0cd02a4bbb93414981
- Shafiq, M., Hashmi, M. A., & Zafar, S. (2023). Impact of teacher’s teaching strategies on the academic achievements of middle school students in mathematics. Global Educational Studies Review, 8(2), 245-256. https://doi.org/10.31703/gesr.2023(VIII-II).23
- Shah, S. S. H., Munir, T. A., Sabir, M., & Tipu, S. A. (2013). Psychometric analysis of MCQs used in assessing the students at entrance to a medical college. Annals of King Edward Medical University, 18(3), 296. https://doi.org/10.21649/akemu.v18i3.419
- Shen, Z. (2023). Effect of manipulative-based interventions on improving mathematics computation outcomes for students with mathematics difficulties. [Master Dissertation, University of Taxes at Austin]. https://repositories.lib.utexas.edu/server/api/core/bitstreams/4bed7143-7d75-44cb-881d-6a557ab30fa4/content
- Shin, M., Park, J., Grimes, R., & Bryant, D. P. (2021). Effects of using virtual manipulatives for students with disabilities: three-level multilevel modeling for single-case data. Exceptional Children, 87(4), 418–437. https://doi.org/10.1177/00144029211007150
- Shin, M., Simmons, M., Meador, A., Goode, F. J., Deal, A., & Jackson, T. (2023). Mathematics instruction for students with learning disabilities: applied examples using virtual manipulatives. Intervention in School and Clinic, 58(3), 198–204. https://doi.org/10.1177/10534512221081268
- Shurr, J., Bouck, E. C., Bassette, L., & Park, J. (2021). Virtual versus concrete: A comparison of mathematics manipulatives for three elementary students with autism. Focus on Autism and Other Developmental Disabilities, 36(2), 71-82. https://doi.org/10.1177/1088357620986944
- Shuxratovna, R. N. (2024). Pedagogical possibilities of implementing the CPA (concrete-pictorial-abstract) approach. International Journal of Pedagogics, 4(2), 68-76. https://doi.org/10.37547/ijp/Volume04Issue02-13
- Simon, M. A. (2022). Contributions of the learning through activity theoretical framework to understanding and using manipulatives in the learning and teaching of mathematical concepts. The Journal of Mathematical Behavior, 66, 100945. https://doi.org/10.1016/j.jmathb.2022.100945
- Stigberg, S., Stigberg, H., & Maugesten, M. (2022). Making manipulatives for mathematics education. FabLearn Europe, 1, 1-9. https://doi.org/10.1145/3535227.3535228
- Stratton, S. J. (2019). Quasi-experimental design (pre-test and post-test studies) in prehospital and disaster research. Prehospital and Disaster Medicine, 34(6), 573-574. https://doi.org/10.1017/S1049023X19005053
- Sugiarti, L., & Retnawati, H. (2019). Analysis of student difficulties on algebra problem solving in junior high school. Journal of Physics: Conference Series, 1320(1), 012103. https://iopscience.iop.org/article/10.1088/1742-6596/1320/1/012103/pdf
- Sulistiowati, D. L., Herman, T., & Jupri, A. (2019). Student difficulties in solving geometry problem based on Van Hiele thinking level. Journal of Physics: Conference Series, 1157(1), 042118. https://iopscience.iop.org/article/10.1088/1742-6596/1157/4/042118/pdf
- Surynková, P. (2023). Concrete 3D printed manipulatives in geometry learning and problem solving: Usefulness of instructional 3D printed tools in the topic of polygons and tilings. Thirteenth Congress of the European Society for Research in Mathematics Education (CERME13), Alfréd Rényi Institute of Mathematics; Eötvös Loránd University of Budapest: Hungary. https://hal.science/hal-04419375/document
- Toth, E. E., Morrow, B. L. & Ludvico, L. R. (2009). Designing blended inquiry learning in a laboratory context: a study of incorporating hands-on and virtual laboratories. Innovative Higher Education, 33(5), 333–344. https://doi.org/10.1007/s10755-008-9087-7
- Tjandra, C. (2023). Effectiveness of using manipulatives in mathematics teaching in inclusive education programs in an elementary school. Dharmas Education Journal, 4(1), 168-178. https://doi.org/10.56667/dejournal.v4i1.944
- Ukdem, Ş., & Çetin, H. (2022). Investigating the impact of interventions using concrete and virtual manipulatives on 3rd grade students' fraction concept and motivation. International Online Journal of Education and Teaching (IOJET), 9(3), 1113-1131. https://files.eric.ed.gov/fulltext/EJ1352288.pdf
- Walter, M., & Max, J. (2012). Social research methods. University of Oxford Press. https://www.academia.edu/3564150/Social_Research_Methods_2nd_Edition_
- Wang, T. L., Tseng, Y. K. (2018). The comparative effectiveness of physical, virtual, and virtual-physical manipulatives on third-grade students’ science achievement and conceptual understanding of evaporation and condensation. International Journal of Science and Mathematics Education, 16, 203–219. https://doi.org/10.1007/s10763-016-9774-2
- Wilkie, K. J., & Hopkins, S. (2024). Primary students’ relational thinking and computation strategies with concrete-to-symbolic representations of subtraction as difference. The Journal of Mathematical Behavior, 73, 1-18. https://doi.org/10.1016/j.jmathb.2023.101121
- Yaduvanshi, S., & Singh, S. (2019). Fostering achievement of low-, average-, and high-achievers’ students in biology through structured cooperative learning (STAD Method). Education Research International, 1-10. https://doi.org/10.1155/2019/1462179
- Yakubova, G., Chen, B. B., Al-Dubayan, M. N., & Gupta, S. (2024). Virtual instruction in teaching mathematics to autistic students: effects of video modeling, virtual manipulatives, and mathematical games. Journal of Special Education Technology, 39(1), 51-66. https://doi.org/10.1177/01626434231177875
- Ye, B. B., & Feng, M. M. (2019). From materialization, Electronicization to informatization: The evolution of primary school mathematics teaching aids. Curriculum Teaching Material and Method, 7, 68–75. https://doi.org/10.19877/j.cnki.kcjcjf.2019.07.011
- Zacharias, Z. C. & Olympiou, G. (2011). Physical versus virtual manipulatives: rethinking physics experimentation. Learning and Instruction, 21, 317–331. https://doi.org/10.1016/j.learninstruc.2010.03.001
- Zacharia, Z. C., & de Jong, T. (2014). The effects on students’ conceptual understanding of electric circuits of introducing virtual manipulatives within a physical manipulatives-oriented curriculum. Cognition and Instruction, 32, 101-158. https://www.jstor.org/stable/43941139
- Zengin, Y. (2023). Effectiveness of a professional development course based on information and communication technologies on mathematics teachers’ skills in designing technology-enhanced task. Education and Information Technologies. https://doi.org/10.1007/s10639-023-11728-2
References
Abdul-Karim, H., Kasimu, O., Adul-Rahaman, A., Kanimam, Y. S. Imoro, M., & Dokurugu, M. E. (2023). Assessing the impact of algebra tiles as visual and manipulative aids on students’ algebraic understanding. American Journal of Educational Research, 11(10), 705-711. https://pubs.sciepub.com/education/11/10/9/index.html
Akpan, E. T., Charles-Ogan, G. I., Eze, F. B., Okafor-Agbala, U. C., & Onyeka, E. C. (2023). Technology enhanced learning: utilization of symboLAb manipulative instruction and performance of students in quadratic graphs. Asian Journal of Advanced Research and Reports, 17(11), 32-42. https://sdfswk3623.s3.ap-northeast-2.amazonaws.com/Akpan17112023AJARR99909.pdf
Andrabi, T., Jishnu, D., Asim, I., Khwaja, Vishwanathan, T., & Zajonc, T. (2008). Learning and educational achievements in Punjab schools (LEAPS): Insights to inform the education policy debate. Harvard University. https://static1.squarespace.com/static/618f98a44478fc2e7b844cb4/t/6203c90517f63c6853836acc/1644415240287/LEAPS_report_report.pdf
Arrogante, O. (2022). Sampling techniques and sample size calculation: How and how many participants should I select for my research? Intensive Care Nursing, 33(1), 44-47. https://doi.org/10.1016/j.enfie.2021.03.004
ASER Pakistan. (2024). Annual status of education report (ASER) Pakistan, National (Rural). https://aserpakistan.org/document/2024/aser_national_2023.pdf
Asian Development Bank (ADB) (2023). ADB brief no. 283: Strengthening school teaching in Pakistan. Manila. https://www.adb.org/sites/default/files/publication/935721/adb-brief-283-strengthening-school-teaching-pakistan.pdf
Bajpai, N., & Pandey, J. (2024). The effectiveness of experiential learning on mathematics achievement among fifth grade school students: A quasi-experimental study. International Journal of Indian Psychology, 12(1), 525-535. https://ijip.in/pdf-viewer/?id=42881
Back, J. (2019). Manipulative in the primary classroom. NRICH, 3, 1-8. https://nrich.maths.org/content/id/10461/Manipulatives%20in%20the%20Primary%20Classroom.pdf
Bhutta, S. M., & Rizvi, N. F. (2022). Assessing teachers’ pedagogical practices and students’ learning outcomes in science and mathematics across primary and secondary school level: A nationwide study (2018-21). Aga Khan University, Institute for Educational Development, Karachi, Pakistan, 1-4. https://ecommons.aku.edu/pakistan_ied_pdck/317
Bone, E. K., Bouck, E. C., & Satsangi, R. (2023). Comparing concrete and virtual manipulatives to teach algebra to middle school students with disabilities. Exceptionality, 31(1), 1-17. https://doi.org/10.1080/09362835.2021.1938057
Bornaa, C. S., Okwan, B., & Iddrisu, A. B. et al. (2023). Cooperative learning with manipulatives and students’ performance in mathematics problem solving. Journal of Education and Practice, 14(19), 1-12. https://www.researchgate.net/publication/372807317_Cooperative_Learning_with_Manipulatives_and_Students%27_Performance_in_Mathematics_Problem_Solving
Bouck, E. C., Christopher, W., & Bone, E. (2018). Manipulative apps to support students with disabilities in mathematics. Intervention in School and Clinic, 53(3). https://doi.org/10.1177/1053451217702115
Bouck, E. C., Park, J., Satsangi, R., Cwiakala, K., & Levy, K. (2019). Using the virtual-abstract instructional sequence to support acquisition of algebra. Journal of Special Education Technology, 34(4), 253–268. https://doi.org/10.1177/0162643419833022
Bouck, E. C., Park, J., & Stenzel, K. (2020). Virtual manipulatives as assistive technology to support students with disabilities with mathematics. Preventing School Failure: Alternative Education for Children and Youth, 64(4), 281-289. https://doi.org/10.1080/1045988X.2020.1762157
Bouck, E., Long, H., & Bae, Y. (2023). Exploring the virtual-representational-abstract instructional sequence across the learning stages for struggling students. Behavior Modification, 47(3), 590-614. https://doi.org/10.1177/01454455221129998
Byrne, E. M., Jensen, H., Thomsen, Bo S., & Ramchandani, P. G. (2023). Educational interventions involving physical manipulatives for improving children’s learning and development: a scoping review. Review of Education, 11(2), 1-42. https://doi.org/10.1002/rev3.3400
Coles, A., & Sinclair, N. (2019). Re-thinking ‘concrete to abstract’ in mathematics education: Towards the use of symbolically structured environments. Canadian Journal of Science, Mathematics and Technology Education, 19, 465-480. https://doi.org/10.1007/s42330-019-00068-4
Crowe, A. (2022, retrieved 21 May 2024). Math manipulatives: How can they be used to enhance the classroom experience? https://www.prodigygame.com/main-en/blog/math-manipulatives/
Dhindsa, N. (2020). Teach-to-one blended mathematics’ impact on middle school students’ mathematics achievement. [Doctoral Dissertations, Walden University]. https://scholarworks.waldenu.edu/cgi/viewcontent.cgi?article=10753&context=dissertations
Dinsmoor, K. (2022). Math Manipulatives. In S. L. Mason (Ed.), Student-centered approaches in K–12 and higher education. EdTech Books. https://edtechbooks.org/student_centered/math_manipulatives
Ernest, P., Skovsmose, O., Bendegem, J. P. van., Bicudo, M., Miarka, R., Kvasz, L., & Moeller, R. (2016). The Philosophy of Mathematics Education. In Editor Gabriele, K. (2016) ICME-13 tropical surveys. Springer International Publishing: AG Switzerland. https://library.oapen.org/bitstream/id/d99d2551-2ef4-4bc9-9ad3-cde7d561dee7/1002255.pdf
Foulkes, M., Sella, F., Wege, T. E., & Gilmore, C. (2023). The effect of concreteness on mathematical manipulatives. Mind, Brain, and Education, 17(3), 185-196. https://doi.org/10.1111/mbe.12374
Gilligan-Lee, K.A., Hawes, Z.C., Williams, A.Y., Farran, E.K., & Mix, K. S. (2023). Hands-on: Investigating the role of physical manipulatives in spatial training: Registered report. Child Development, 94(5), 1205-1221. https://doi.org/10.1111/cdev.1396
Ghodbane, T., & Achachi, H. H. E. (2019). Narrowing the achievement gap between EFL students in oral expression through cooperative learning. Arab World English Journal (AWEJ), 10(2), 365-378. https://dx.doi.org/10.24093/awej/vol10no2.28
Gravito. L. C., Sagayno, R. C., Alcantara, R. T., & Tesorio, G. T. (2023). Homemade manipulative materials for teaching grade iv mathematics. European Journal of Humanities and Educational Advancements (EJHEA), 4(2), 11-15. https://www.researchgate.net/publication/368364062_European_Journal_of_Humanities_and_Educational_Advancements_EJHEA_HOMEMADE_MANIPULATIVE_MATERIALS_FOR_TEACHING_GRADE_IV_MATHEMATICS
Guan, H., Qin, X. L., Rao, Y. S., & Cao, S. (2020). Domain model of web-based dynamic geometry software and its applications. Journal of Computer Applications, 40(4). https://doi.org/10.11772/j.issn.1001-9081.2019091672
Guan, H., Li, J., Rao, Y., Chen, R., & Xu, Z. (2024). Comparative effects of dynamic geometry system and physical manipulatives on Inquiry-based Math Learning for students in Junior High School. Education and Information Technologies, 1-23. https://doi.org/10.1007/s10639-024-12663-6
Gülen, S. (2020). A study to determine the ability of fifth-grade students in reflecting their knowledge about sun, earth, and moon by different measurement tools. Science Education International, 31(1), 41-51. https://files.eric.ed.gov/fulltext/EJ1247785.pdf
Haji Ismail, N. F., Shahrill, M., & Asamoah, D. (2023). Learning through virtual manipulatives: Investigating the impact of Gizmos-based lessons on students’ performance in integers. Contemporary Mathematics and Science Education, 4(1), 2-12. https://doi.org/10.30935/conmaths/12857
Hawes, Z. C. K., Gilligan-Lee, K. A., & Mix, K. S. (2022). Effects of spatial training on mathematics performance: A meta-analysis. Developmental Psychology, 58(1), 112–137. https://doi.org/10.1037/dev0001281
Iqbal, M. Z., Shams, J. A., & Nazir, M. (2020). Effect of using mathematics manipulatives on the student’s academic achievement. Journal of Science Education, 11(1), 1-15. https://ojs.aiou.edu.pk/index.php/jse/article/view/1713
Jablonski, S., & Matthias, L. (2023). Teaching and learning of geometry: A literature review on current developments in theory and practice. Education Sciences, 13(7), 682-692. https://doi.org/10.3390/educsci13070682
Johnson, J. R. R. (2022). Educators’ perceptions of implementing concrete and virtual math manipulatives in title I schools. [Doctoral dissertation, Trevecca Nazarene University]. Trevecca Nazarene University ProQuest Dissertations Publishing. https://www.proquest.com/docview/2688493214/fulltextPDF/6ED214C144F4401CPQ/1?accountid=15156&sourcetype=Dissertations%20&%20Theses
Jones, J. P., & Tiller, M. (2017). Using concrete manipulatives in mathematical instruction. Dimensions of Early Childhood, 45(1), 18–23. https://files.eric.ed.gov/fulltext/EJ1150546.pdf
Julie, (2021, retrieved May 21, 2024). Math manipulatives: How do they aid student learning? https://wordpress.oise.utoronto.ca/robertson/2021/10/18/math-manipulatives/
Kabel, M., Hwang, J., & Hwang, J. (2021). Lessons learned from a rural classroom study: Transitioning from concrete to virtual manipulatives to teach math fact fluency to students with learning disabilities. Journal of Curriculum Studies Research, 3(1), 42-68. https://doi.org/10.46303/jcsr.2021.7
Kemp, K. (2021). Test corrections appear to benefit lower-achieving students in an introduction to biology major course: Results of a single-site, one-semester study. Journal of Microbiology & Biology Education, 22(2), 122-21. https://doi.org/10.1128/jmbe.00122-21
Lange J. (2021). The importance of using manipulatives in math class. [Master thesis & Capstone projects, Northwestern College]. Northwestern College, Iowa Research Repository. https://nwcommons.nwciowa.edu/cgi/viewcontent.cgi?article=1291&context=education_masters
Large Scale Assessment (LSA) (2021). Grade-5 main findings report – 2021. Cambridge Education Mott MacDonald: Lahore, Pakistan. https://pec.edu.pk/system/files/LSA%20Main%20Findings%20Report%202021.pdf#overlay-context=publications
Lee, C. Y., & Chen, M. J. (2015). Effects of worked examples using manipulatives on fifth graders' learning performance and attitude toward mathematics. Journal of Educational Technology & Society, 18(1), 264-275. http://www.jstor.org/stable/jeductechsoci.18.1.264
Liu, D., Lory, C., Lei, Q., Cai, W., Mao, Y., & Yang, X. (2024). Using explicit instruction and virtual manipulatives to teach measurement concepts for students with autism spectrum disorder. The Journal of Special Education, 58(1), 3-13. https://doi.org/10.1177/00224669231171558
Long, H. M., Bouck, E. C., & Kelly, H. (2023). An evidence-based practice synthesis of virtual manipulatives for students with ASD and IDD. Focus on Autism and Other Developmental Disabilities, 38(3), 147-157. https://doi.org/10.1177/10883576221121654
Mainali, B. (2021). Representation in teaching and learning mathematics. International Journal of Education in Mathematics, Science, and Technology (IJEMST), 9(1), 1-21. https://doi.org/10.46328/ijemst.1111
Mazo, J. C. (2024). Virtual manipulative: Its effects as an instructional tool in teaching mathematics. Multidisciplinary International Journal of Research and Development, 3(4), 15-35. https://www.mijrd.com/papers/v3/i4/MIJRDV3I40002.pdf
Mills, G., & Gay, L. (2018). Educational research: Competencies for analysis and applications. Pearson Education.
Milton, J. H., Flores, M. M., Hinton, V. M., Dunn, C., & Darch, C. B. (2023). Using the concrete-representational-abstract sequence to teach conceptual understanding of place value, rounding, and expanded notation. Learning Disabilities Research & Practice, 38(1), 15-25. https://doi.org/10.1111/ldrp.12299
Mirza, M., & Iqbal, M. Z. (2014). Impact of collaborative teaching (CT) on mathematics students’ achievement in Pakistan. Journal of Research & Reflections in Education, 8(1), 13-21. https://ue.edu.pk/jrre/articles/81002.pdf
Moyer-Packenham, P. S., & Bolyard, J. J. (2016). Revisiting the Definition of a Virtual Manipulative. In: Moyer-Packenham, P. (eds) International Perspectives on Teaching and Learning Mathematics with Virtual Manipulatives. Mathematics education in the digital era, vol 7. Springer, Cham. https://doi.org/10.1007/978-3-319-32718-1_1
Muhammad, S. M., Sani, M. A., Abdullahi, F., & Bayaro, A. (2023). The effect using manipulatives on the performance of pupils in primary school mathematics. Journal of Mathematical Sciences & Computational Mathematics, 4(2), 223-232. https://doi.org/10.15864/jmscm.4206
National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author.
National Curriculum Council – Pakistan (2020). Single national curriculum mathematics grade I-V 2020: One nation, one curriculum. Ministry of Federal Education and Professional Training, Islamabad. Government of Pakistan. https://mofept.gov.pk/SiteImage/Misc/files/SNC%20Mathematics%201-5.pdf
Nash, E. (2023, retrieved May 21, 2024). Math manipulatives: A student-centered approach to teaching mathematics. https://greatminds.org/math/blog/eureka/math-manipulatives-a-student-centered-approach-to-teaching-mathematics
Ng, OL., & Ye, H. (2022). Mathematics learning as embodied making: primary students’ investigation of 3D geometry with handheld 3D printing technology. Asia Pacific Education Review, 23, 311–323. https://doi.org/10.1007/s12564-022-09755-8
Nguyen, T., Watts, T. W., Duncan, G. J., Clements, D. H., Sarama, J. S., Wolfe, C., & Spitler, M. E. (2016). Which preschool mathematics competencies are most predictive of fifth grade achievement? Early Childhood Research Quarterly, 36(3), 550-560. https://doi.org/10.1016/j.ecresq.2016.02.003
Nurjannah, S., & Kusnandi, (2021). Literature study: The role of abstraction ability to strengthen students’ early knowledge in mathematics learning. International Conference on Mathematics and Science Education (ICMScE), 1806, 1-6. https://iopscience.iop.org/article/10.1088/1742-6596/1806/1/012064/pdf
Nurhasanah, F., Kusumah, Y. S., & Sabandar, J. (2017). Concept of triangle: examples of mathematical abstraction in two different contexts. International Journal on Emerging Mathematics Education, 1(1), 53-70. http://dx.doi.org/10.12928/ijeme.v1i1.5782
O’Meara, N., Johnson, P., & Leavy, A. (2019). A comparative study investigating the use of manipulatives at the transition from primary to post-primary education. International Journal of Mathematical Education in Science and Technology, 51(6), 835-857. https://doi.org/10.1080/0020739X.2019.1634842
Pallant, J. (2020). SPSS survival manual: A step by step guide to data analysis using IBM SPSS. Routledge. https://doi.org/10.4324/9781003117452
Park, J., & Bouck, E. (2022). Teacher-delivered virtual manipulative mathematics intervention to individuals with extensive support needs. Research in Developmental Disabilities, 131, https://doi.org/10.1016/j.ridd.2022.104339
Pavlou, Y., Zacharias, C. Z., & Papaevripidou, M. (2024). Comparing the impact of physical and virtual manipulatives in different science domains among preschoolers. Science Education, 1-29. https://doi.org/10.1002/sce.21869
Peltier, C., Morin, K. L., Bouck, E. C., Lingo, M. E., Pulos, J., Scheffler, F. A., & Deardorff, M. E. (2020). A meta-analysis of single case research using mathematics manipulatives with students at risk or identified with a disability. Journal of Special Education, 54(1), 3-15. https://doi.org/10.1177/0022466919844516
Provincial Assessment of Students Learning (PASL) (2020). PASL report 2018-19/PEC. UNICEF publisher. https://pec.edu.pk/system/files/PASL%20Report%20PEC_compressed.pdf#overlay-context=publications
Prosser, S. K., & Bismarck, S. F. (2023). Concrete–representational–abstract (CRA) instructional approach in an algebra I inclusion class: Knowledge retention versus students’ perception. Education Science, 13, 1061. https://doi.org/10.3390/educsci13101061
Root, J. R., Cox, S. K., Gilley, D., & Wade, T. (2021). Using a virtual-representational-abstract integrated framework to teach multiplicative problem solving to middle school students with developmental disabilities. Journal of Autism and Developmental Disorders, 51(7), 2284-2296. https://doi.org/10.1007/s10803-020-04674-2
Ruel, E., Wagner, W. E., & Gillespie; B. J. (2015). The practice of survey research: Theory and applications. Sage Publications, Inc.
Salifu, A. S. (2022). The effects of balance model and algebra tiles manipulative in solving linear equations in one variable. Contemporary Mathematics and Science Education, 3(2), 1-10. https://doi.org/10.30935/conmaths/12028
Sari, M. H., & Aydoğdu, S. (2020). The effect of concrete and technology-assisted learning tools on place value concept, achievement in mathematics and arithmetic performance of primary school students. International Journal of Curriculum and Instruction, 12(1), 197-224. https://ijci.globets.org/index.php/IJCI/article/view/312/159
Satsangi, R., & Bouck, E. C. (2015). Using virtual manipulative instruction to teach the concepts of area and perimeter to secondary students with learning disabilities. Learning Disability Quarterly, 38, 174–186. https://doi.org/10.1177/0731948714550101
Satsangi, R., & Miller, B. (2017). The case for adopting virtual manipulatives in mathematics education for students with disabilities. Preventing school failure: Alternative Education for Children and Youth, 61, 303–310. https://doi.org/10.1080/1045988X.2016.1275505
Satsangi, R., Hammer, R., & Evmenova, A. S. (2018). Teaching multistep equations with virtual manipulatives to secondary students with learning disabilities. Learning Disabilities Research & Practice, 33(2), 99–111. https://doi.org/10.1111/ldrp.12166
Satsangi, R., Raines, A. R., & Fraze, K. (2021). Virtual manipulatives for teaching algebra: A research to practice guide for secondary students with a learning disability. Learning Disabilities: A Multidisciplinary Journal, 26(1), 46-57. https://doi.org/10.18666/LDMJ-2021-V26-I1-10349
Satsangi, R., & Raines, A. R. (2023). Examining virtual manipulatives for teaching computations with fractions to children with mathematics difficulty. Journal of Learning Disabilities, 56(4), 295–309. https://doi.org/10.1177/00222194221097710
Serin, H. (2023). The role of technology in mathematics education: Promoting student achievement. International Journal of Social Sciences & Educational Studies, 10(2), 390-395. https://www.semanticscholar.org/reader/44bf2d839400dcf034639e0cd02a4bbb93414981
Shafiq, M., Hashmi, M. A., & Zafar, S. (2023). Impact of teacher’s teaching strategies on the academic achievements of middle school students in mathematics. Global Educational Studies Review, 8(2), 245-256. https://doi.org/10.31703/gesr.2023(VIII-II).23
Shah, S. S. H., Munir, T. A., Sabir, M., & Tipu, S. A. (2013). Psychometric analysis of MCQs used in assessing the students at entrance to a medical college. Annals of King Edward Medical University, 18(3), 296. https://doi.org/10.21649/akemu.v18i3.419
Shen, Z. (2023). Effect of manipulative-based interventions on improving mathematics computation outcomes for students with mathematics difficulties. [Master Dissertation, University of Taxes at Austin]. https://repositories.lib.utexas.edu/server/api/core/bitstreams/4bed7143-7d75-44cb-881d-6a557ab30fa4/content
Shin, M., Park, J., Grimes, R., & Bryant, D. P. (2021). Effects of using virtual manipulatives for students with disabilities: three-level multilevel modeling for single-case data. Exceptional Children, 87(4), 418–437. https://doi.org/10.1177/00144029211007150
Shin, M., Simmons, M., Meador, A., Goode, F. J., Deal, A., & Jackson, T. (2023). Mathematics instruction for students with learning disabilities: applied examples using virtual manipulatives. Intervention in School and Clinic, 58(3), 198–204. https://doi.org/10.1177/10534512221081268
Shurr, J., Bouck, E. C., Bassette, L., & Park, J. (2021). Virtual versus concrete: A comparison of mathematics manipulatives for three elementary students with autism. Focus on Autism and Other Developmental Disabilities, 36(2), 71-82. https://doi.org/10.1177/1088357620986944
Shuxratovna, R. N. (2024). Pedagogical possibilities of implementing the CPA (concrete-pictorial-abstract) approach. International Journal of Pedagogics, 4(2), 68-76. https://doi.org/10.37547/ijp/Volume04Issue02-13
Simon, M. A. (2022). Contributions of the learning through activity theoretical framework to understanding and using manipulatives in the learning and teaching of mathematical concepts. The Journal of Mathematical Behavior, 66, 100945. https://doi.org/10.1016/j.jmathb.2022.100945
Stigberg, S., Stigberg, H., & Maugesten, M. (2022). Making manipulatives for mathematics education. FabLearn Europe, 1, 1-9. https://doi.org/10.1145/3535227.3535228
Stratton, S. J. (2019). Quasi-experimental design (pre-test and post-test studies) in prehospital and disaster research. Prehospital and Disaster Medicine, 34(6), 573-574. https://doi.org/10.1017/S1049023X19005053
Sugiarti, L., & Retnawati, H. (2019). Analysis of student difficulties on algebra problem solving in junior high school. Journal of Physics: Conference Series, 1320(1), 012103. https://iopscience.iop.org/article/10.1088/1742-6596/1320/1/012103/pdf
Sulistiowati, D. L., Herman, T., & Jupri, A. (2019). Student difficulties in solving geometry problem based on Van Hiele thinking level. Journal of Physics: Conference Series, 1157(1), 042118. https://iopscience.iop.org/article/10.1088/1742-6596/1157/4/042118/pdf
Surynková, P. (2023). Concrete 3D printed manipulatives in geometry learning and problem solving: Usefulness of instructional 3D printed tools in the topic of polygons and tilings. Thirteenth Congress of the European Society for Research in Mathematics Education (CERME13), Alfréd Rényi Institute of Mathematics; Eötvös Loránd University of Budapest: Hungary. https://hal.science/hal-04419375/document
Toth, E. E., Morrow, B. L. & Ludvico, L. R. (2009). Designing blended inquiry learning in a laboratory context: a study of incorporating hands-on and virtual laboratories. Innovative Higher Education, 33(5), 333–344. https://doi.org/10.1007/s10755-008-9087-7
Tjandra, C. (2023). Effectiveness of using manipulatives in mathematics teaching in inclusive education programs in an elementary school. Dharmas Education Journal, 4(1), 168-178. https://doi.org/10.56667/dejournal.v4i1.944
Ukdem, Ş., & Çetin, H. (2022). Investigating the impact of interventions using concrete and virtual manipulatives on 3rd grade students' fraction concept and motivation. International Online Journal of Education and Teaching (IOJET), 9(3), 1113-1131. https://files.eric.ed.gov/fulltext/EJ1352288.pdf
Walter, M., & Max, J. (2012). Social research methods. University of Oxford Press. https://www.academia.edu/3564150/Social_Research_Methods_2nd_Edition_
Wang, T. L., Tseng, Y. K. (2018). The comparative effectiveness of physical, virtual, and virtual-physical manipulatives on third-grade students’ science achievement and conceptual understanding of evaporation and condensation. International Journal of Science and Mathematics Education, 16, 203–219. https://doi.org/10.1007/s10763-016-9774-2
Wilkie, K. J., & Hopkins, S. (2024). Primary students’ relational thinking and computation strategies with concrete-to-symbolic representations of subtraction as difference. The Journal of Mathematical Behavior, 73, 1-18. https://doi.org/10.1016/j.jmathb.2023.101121
Yaduvanshi, S., & Singh, S. (2019). Fostering achievement of low-, average-, and high-achievers’ students in biology through structured cooperative learning (STAD Method). Education Research International, 1-10. https://doi.org/10.1155/2019/1462179
Yakubova, G., Chen, B. B., Al-Dubayan, M. N., & Gupta, S. (2024). Virtual instruction in teaching mathematics to autistic students: effects of video modeling, virtual manipulatives, and mathematical games. Journal of Special Education Technology, 39(1), 51-66. https://doi.org/10.1177/01626434231177875
Ye, B. B., & Feng, M. M. (2019). From materialization, Electronicization to informatization: The evolution of primary school mathematics teaching aids. Curriculum Teaching Material and Method, 7, 68–75. https://doi.org/10.19877/j.cnki.kcjcjf.2019.07.011
Zacharias, Z. C. & Olympiou, G. (2011). Physical versus virtual manipulatives: rethinking physics experimentation. Learning and Instruction, 21, 317–331. https://doi.org/10.1016/j.learninstruc.2010.03.001
Zacharia, Z. C., & de Jong, T. (2014). The effects on students’ conceptual understanding of electric circuits of introducing virtual manipulatives within a physical manipulatives-oriented curriculum. Cognition and Instruction, 32, 101-158. https://www.jstor.org/stable/43941139
Zengin, Y. (2023). Effectiveness of a professional development course based on information and communication technologies on mathematics teachers’ skills in designing technology-enhanced task. Education and Information Technologies. https://doi.org/10.1007/s10639-023-11728-2