The Effectiveness of Visualization of Proofs in Learning Mathematics by Using Discovery Learning Viewed from Conceptual Understanding

Angga Kristiyajati, Ariyadi Wijaya


The purpose of this study is to explore the effectiveness of the use of visualization of proofs upon discovery learning models in mathematics learning in terms of understanding concepts. This study is an experimental design which used quantitative methods to obtain data on student conceptual understanding. The sampling technique used was stratified random sampling. The population sample used in this study was 11th grade secondary students, chosen from 11 IPA 2 of SMAN 8 Yogyakarta, 11 IPA 3 of SMAN 8 Yogyakarta, 11 IPA 3 of SMAN 2 Yogyakarta, 11 IPA 4 of SMAN 2 Yogyakarta, 11 IPA 1 of SMAN 11 Yogyakarta, and 11 IPA 2 of SMAN 11 Yogyakarta. In each school, two classes were chosen with one class was given a discovery learning treatment with visualization of proofs (PWW) and the other class was given a discovery learning treatment without visualization of proofs. The measurement instrument used in this study was an essay test instrument with five questions. Discovery learning is said to be effective if the average value of conceptual understanding is at least 75. Based on the results of this study, although the data obtained did not meet the assumptions of normality, the number of members of the sample were more than 30 so that the data analysis could use parametric statistics. Using a hypothesis testing with a significance level of 0.05, it was found that the use of visualization of proofs was effective in mathematics discovery learning models if it was viewed from conceptual understanding.


conceptual understanding; visualization of proofs; discovery learning

Full Text:



Allen, M. J., & Yen, W. M. (1979). Introduction to measurement theory. California: Brooks/Cole Publishing Company.

Alsina, C., & Nelsen, R. B. (2010). An invitation to proofs without words. European Journal of Pure and Applied Mathematics, 3(1), 118–127. Retrieved from

Aso, K. (2001). Visual images as educational materials in mathematics. Community College Journal of Research and Practice, 25(5–6), 355–360. Retrieved from:

Aziz, R. A., Tarmedi, E., & Kusmarni, Y. (2018). Developing students’ information literature skill through the application of learning discovery learning model in social studies learning. International Journal Pedagogy of Social Studies, 3(1), 9–20.

Balım, A. G. (2009). The effects of discovery learning on students’ success and inquiry learning skills. Egitim Arastirmalari-Eurasian Journal of Educational Research, 200(35), 1–20.

Berger, P. D., Maurer, R. E., & Celli, G. B. (2018). Experimental Design with Applications in Management, Engineering, and the Sciences. Boston: Springer.

Hanna, G. (2000). Proof, Explanation and exploration: an overview. Educational Studies in Mathematics, 44(1–3), 5–23.

Hanna, G. (2014). Mathematical proof, argumentation, and reasoning. In S. Lerman (Ed.), Encyclopedia of Mathematics Education (pp. 404–408). London: Springer.

Hanna, G., & Villiers, M. de. (2008). ICMI Study 19: Proof and proving in mathematics education. ZDM Mathematics Education, 40(2), 329–336.

Hasnida, N., Ghazali, C., & Zakaria, E. (2011). Students’ procedural and conceptual understanding of mathematics. Australian Journal of Basic and Applied Sciences, 5(7), 684–691.

Kemdikbud. (2015). Rencana Strategis Kementerian Pendidikan dan Kebudayaan 2015-2019 [Strategic Plan of the Ministry of Education and Culture 2015-2019]. Jakarta: Kementerian Pendidikan dan Kebudayaan.

Kementerian Pendidikan dan Kebudayaan. (2016). Peraturan Menteri Pendidikan dan Kebudayaan Republik Indonesia Nomor 22.Tahun 2016 Tentang Standar Proses Pendidikan Dasar dan Menengah [Regulation of the Minister of Education and Culture of the Republic of Indonesia Number 22. 2016 concerning Basic and Secondary Education Process Standards]. Jakarta: Kementerian Pendidikan dan Kebudayaan.

Knipping, C. (2004). Challenges in teaching mathematical reasoning and proof – Introduction. ZDM Mathematics Education, 36(5), 127–128.

Knuth, E. J. (2002). Proof as a tool for learning mathematics. Mathematics Teacher, 95, 486–490.

National Research Council. (2010). Adding + it UP ! Helping Children Learn Mathematics. In J. Klipatrick, J. Swafford, & B. Findell (Eds.), October. Washington, DC: National Academy Press.

NCTM. (2000). Principles and Standard for Schools Mathematics. Reston: The National Council of Teachers of Mathematics.

Nitko, A. J., & Brookhart, S. M. (2011). Educational assessment of students (6th ed.). Boston: Pearson Education, Inc.

Philips, L. M., Norris, S. P., & Macnab, J. S. (2010). Visualization in Mathematics, Reading and Science Education.

Prince, M. J., & Felder, R. M. (2006). Inductive Teaching and Learning Methods: Definitions, Comparisons, and Research Bases. Journal of Engineering Education, 95(2), 123–138.

PUSPENDIK. (2014). Materi Pelatihan Guru Implementasi Kurikulum 2013 Tahun 2014 [Teacher Training Materials Implementation of 2013 Curriculum 2014]. Jakarta: BPSDMPKPMP KEMDIKBUD.

PUSPENDIK. (2016). Aplikasi Pamer 2016 [2016 Exhibition application]. Jakarta: Kementerian Pendidikan dan Kebudayaan.

PUSPENDIK. (2017). Aplikasi Pamer 2017. Jakarta: Kementerian Pendidikan dan Kebudayaan.

Rieber, L. P., Tzeng, S.-C., & Tribble, K. (2004). Discovery learning, representation, and explanation within a computer-based simulation: finding the right mix. Learning and Instruction, 14(3), 307–323.

Syah, M. (2014). Psikologi Pendidikan [Educational Psychology]. Bandung: Remaja Rosdakarya.

Walpole, R. E., Myers, R. H., & Ye, K. (2007). Probability & Statistics for Enginers & Scientist. London: Pearson Education International.



  • There are currently no refbacks.

Indexed by:


Southeast Asian Mathematics Education Journal
SEAMEO Regional Centre for QITEP in Mathematics
Jl. Kaliurang Km 6, Sambisari, Condongcatur, Depok, Sleman
Yogyakarta, Indonesia
Telp. +62 274 889955

p-ISSN: 2089-4716 | e-ISSN: 2721-8546

Southeast Asian Mathematics Education Journal is licensed under a Creative Commons Attribution 4.0 International License

View My Stats

 Supported by: