Developing Tangible Mathematics: Design and Validation of Rational Number Manipulatives for Grade 7

Abstract– This study reports the design and validation of tangible mathematics materials for enhancing grade 7 students' understanding of rational numbers, identified as the most challenging topic in a preceding needs assessment. Using design-based research methodology with iterative refinement, we developed acrylic manipulatives (fraction circles, bars, tiles, and operational pieces) featuring systematic color-coding and movable connections. Validation by five mathematics education experts yielded exceptional ratings for content quality (37.8/40), technical accuracy (16/16), and instructional design (22/24). Teacher feedback confirmed the materials' effectiveness for concept visualization while suggesting physical enhancements. Preliminary testing with 84 students demonstrated significant performance improvements in the experimental group using manipulatives compared to controls. Both quantitative data and classroom observations confirmed enhanced understanding, engagement, and problem-solving abilities with the tangible approach. These findings establish a strong foundation for implementing tangible mathematics in grade 7 classrooms, addressing the critical transition from concrete to abstract mathematical thinking.

manipulatives, design-based, understanding, problem-solving, engagement

1. Ainsworth, S. (1999). The functions of multiple representations. Computers & Education, 33(2-3), 131-152. [Google Scholar] [Crossref]

2. Akbaşlı, S., & Yeşilce, İ. (2018). Use of tangible materials and computer in mathematics teaching: Opinions of school principals. EURASIA Journal of Mathematics, Science and Technology Education, 14(6), 2523-2532. [Google Scholar] [Crossref]

3. Bartolini, M. G., & Martignone, F. (2020). Manipulatives in mathematics education. Encyclopedia of mathematics education, 487-494. [Google Scholar] [Crossref]

4. ] Bruner, J. S. (1966). Toward a theory of instruction. Harvard University Press. [Google Scholar] [Crossref]

5. Carraher, D. W., Schliemann, A. D., Brizuela, B. M., & Earnest, D. (2006). Arithmetic and algebra in early mathematics education. Journal for Research in Mathematics education, 37(2), 87-115. [Google Scholar] [Crossref]

6. Department of Education (2009). Guidelines and Processes for LRMDS Assessment and Evaluation [Google Scholar] [Crossref]

7. Fyfe, E. R., McNeil, N. M., Son, J. Y., & Goldstone, R. L. (2014). Concreteness fading in mathematics and science instruction: A systematic review. Educational Psychology Review, 26(1), 9-25. https://doi.org/10.1007/s10648-014-9260-8 [Google Scholar] [Crossref]

8. Kaput, J. J., Carraher, D. W., & Blanton, M. L. (Eds.). (2017). Algebra in the early grades. Routledge. [Google Scholar] [Crossref]

9. Lakoff, G., & Núñez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. Basic Books. [Google Scholar] [Crossref]

10. Laski, E. V., Jor'dan, J. R., Daoust, C., & Murray, A. K. (2015). What makes mathematics manipulatives effective? Lessons from cognitive science and Montessori education. Sage Open, 5(2), 2158244015589588. [Google Scholar] [Crossref]

11. Moyer-Packenham, P. S., & Westenskow, A. (2013). Effects of virtual manipulatives on student achievement and mathematics learning. International Journal of Virtual and Personal Learning Environments, 4(3), 35-50. [Google Scholar] [Crossref]

12. OECD. (2023). PISA 2022 Results (Volume I): Excellence and Equity in Education. PISA, OECD Publishing, Paris. https://doi.org/10.1787/48ebd440-en [Google Scholar] [Crossref]

13. Piaget, J. (1952). The origins of intelligence in children. International Universities Press. [Google Scholar] [Crossref]

14. Sweller, J. (1988). Cognitive load during problem solving: Effects on learning. Cognitive Science, 12(2), 257-285. https://doi.org/10.1207/s15516709cog1202_4 [Google Scholar] [Crossref]

15. Wilkie, K. J., & Sullivan, P. (2017). Exploring intrinsic and extrinsic motivational aspects of middle school students' aspirations for their mathematics learning. Educational Studies in Mathematics, 97(3), 235–254. https://doi.org/10.1007/s10649-017-9795-y [Google Scholar] [Crossref]

16. Witzel, B. S., Mercer, C. D., & Miller, M. D. (2003). Teaching algebra to students with learning difficulties: An investigation of an explicit instruction model. Learning Disabilities Research & Practice, 18(2), 121-131. https://doi.org/10.1111/1540-5826.00068 [Google Scholar] [Crossref]

• Patriarchy and Gender Roles in Shobhaa De's Second Thoughts and Easterine Kire’s Life on Hold
• Comparative Effects of Zeolite and Fly Ash on Soil Properties and Agricultural Productivity
• Assessing Interrelationships among Banking Satisfaction, Cooperative Perspectives and Rural Service Beliefs in Cooperative Banking in Northern Nigeria
• High-Performance Alloys and Composites’ Applications in Production Engineering
• Integrating Tidal Energy into Sub-Saharan Africa’s Power Mix: A Strategic Framework for Renewable Energy Expansion