How to Solve It: A New Aspect of Mathematical Method
Integrity signals
Registry
Cites
Statistics
Claims the paper makes
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
The TTRL approach-systematically generating and solving related, often simpler, problem variants to build towards a solution for a complex target problem-is inspired by a core tenet of human mathematical problem solving, as described by Polya.
“Indeed, this TTRL approach—systematically generating and solving related, often simpler, problem variants to build towards a solution for a complex target problem—is inspired by a core tenet of human mathematical problem solving, as notably described by Polya75.”
From Olympiad-level formal mathematical reasoning with reinforcement learning (2026) · cites this paper as [75]
Cites
Still being checked.
Cited by
Sorted by citations
Minimax Estimation of Functionals of Discrete Distributions
Jiao, J., Venkat, K., Han, Y., & Weissman, T. (2015) · IEEE Transactions on Information Theory · doi:10.1109/tit.2015.2412945 · Cited by 199 · see citing works
Sörensen, K., Sevaux, M., & Glover, F. (2018) · Springer eBooks · doi:10.1007/978-3-319-07124-4_4 · Cited by 161 · see citing works
Gamification through leaderboards: An empirical study in engineering education
Ortíz-Rojas, M., Chiluiza, K., & Valcke, M. (2019) · Computer Applications in Engineering Education · doi:10.1002/cae.12116 · Cited by 156 · see citing works
Son, A. L., Darhim, D., & Fatimah, S. (2020) · Journal on Mathematics Education · doi:10.22342/jme.11.2.10744.209-222 · Cited by 133 · see citing works
Özreçberoğlu, N., & Çağanağa, Ç. K. (2018) · Eurasia Journal of Mathematics Science and Technology Education · doi:10.29333/ejmste/82536 · Cited by 103 · see citing works
Improving mathematical problem solving skills through visual media
Widodo, S. A., Darhim, & Ikhwanudin, T. (2018) · Journal of Physics Conference Series · doi:10.1088/1742-6596/948/1/012004 · Cited by 83 · see citing works
(2011) · SpringerReference · doi:10.1007/springerreference_18578 · Cited by 72 · see citing works
Sörensen, K., Sevaux, M., & Glover, F. (2018) · Springer eBooks · doi:10.1007/978-3-319-07153-4_4-1 · Cited by 60 · see citing works
Conceptualizing Perseverance in Problem Solving as Collective Enterprise
Sengupta‐Irving, T., & Agarwal, P. (2017) · Mathematical Thinking and Learning · doi:10.1080/10986065.2017.1295417 · Cited by 52 · see citing works
Teaching materials of algebraic equation
Widodo, S. A., Prahmana, R. C. I., Purnami, A. S., & Turmudi, T. (2017) · Journal of Physics Conference Series · doi:10.1088/1742-6596/943/1/012017 · Cited by 46 · see citing works
Reducing techno-anxiety in high school teachers by improving their ICT problem-solving skills
Muñoz, O. R., Penalba, F. A., Sánchez, J. F., & Santos, O. (2017) · Behavior and Information Technology · doi:10.1080/0144929x.2016.1221462 · Cited by 29 · see citing works
Framework of Mathematization in Physics from a Teaching Perspective
Pospiech, G. (2019) · Springer eBooks · doi:10.1007/978-3-030-04627-9_1 · Cited by 27 · see citing works
The Art of Grid Fields: Geometry of Neuronal Time
SHILNIKOV, A. L., & Maurer, A. P. (2016) · Frontiers in Neural Circuits · doi:10.3389/fncir.2016.00012 · Cited by 24 · see citing works
Measurement of Creative Teaching in Mathematics Class
Kandemir, M. A., Tezci, E., Shelley, M., & Demırli, C. (2019) · Creativity Research Journal · doi:10.1080/10400419.2019.1641677 · Cited by 21 · see citing works
A Learning-Embedded Attributed Petri Net to Optimize Student Learning in a Serious Game
Liang, J., Tang, Y. G., Hare, R., Wu, B., & Wang, F. (2021) · IEEE Transactions on Computational Social Systems · doi:10.1109/tcss.2021.3132355 · Cited by 21 · see citing works
Learning from the parallel pathways of Makers to broaden pathways to engineering
Foster, C., Wigner, A., Lande, M., & Jordan, S. S. (2018) · International Journal of STEM Education · doi:10.1186/s40594-017-0098-8 · Cited by 20 · see citing works
Olympiad-level formal mathematical reasoning with reinforcement learning
Hubert, T., Mehta, R., Sartran, L., Horváth, M. Z., Žužić, G., Wieser, E., Huang, A., Schrittwieser, J., Schroecker, Y., Masoom, H., et al. (2026) · Nature · doi:10.1038/s41586-025-09833-y · Cited by 19 · see citing works
“These exemplar pairs illustrate how the LLM can generate variants by emulating problem-solving heuristics that build on classical strategies 75 and their formal counterparts in proof engineering, such as simplification 76 , generalization 77 , lemma proposal 78 , exploring analogies 79 and learning…”
Innovation Contests for High-Tech Procurement
Paik, J. H., Scholl, M., Sergeev, R. A., Randazzo, S., & Lakhani, K. R. (2020) · Research-Technology Management · doi:10.1080/08956308.2020.1707007 · Cited by 19 · see citing works
Trust Me, I’m a Doctor: A PhD Survival Guide
Deconinck, K. (2015) · The Journal of Economic Education · doi:10.1080/00220485.2015.1071223 · Cited by 18 · see citing works
The Missing Curriculum in Physics Problem-Solving Education
Williams, M. (2018) · Science & Education · doi:10.1007/s11191-018-9970-2 · Cited by 17 · see citing works
Boaler, J., Brown, K., LaMar, T., Leshin, M. S., & Selbach-Allen, M. E. (2022) · Education Sciences · doi:10.3390/educsci12100694 · Cited by 17 · see citing works
Hwang, S., Yeo, S., & Son, T. (2021) · lnternational Electronic Journal of Elementary Education · doi:10.26822/iejee.2021.208 · Cited by 14 · see citing works
What do mathematicians wish to teach teachers about the discipline of mathematics?
Hoffmann, A., & Even, R. (2023) · Journal of Mathematics Teacher Education · doi:10.1007/s10857-023-09577-4 · Cited by 13 · see citing works
Kuncoro, K. S., Junaedi, I., & Dwijanto, D. (2018) · Journal of Physics Conference Series · doi:10.1088/1742-6596/983/1/012150 · Cited by 13 · see citing works
Eye tracking technology on children's mathematical education: systematic review
Paskovske, A., & Klizienė, I. (2024) · Frontiers in Education · doi:10.3389/feduc.2024.1386487 · Cited by 9 · see citing works
Gender Differences in High School Students’ Beliefs about Mathematical Problem Solving
Sintema, E. J., & Jita, T. (2022) · International Journal of Learning Teaching and Educational Research · doi:10.26803/ijlter.21.10.22 · Cited by 9 · see citing works
Cognitive complexity measures for educational texts: Empirical validation of linguistic parameters
Kupriyanov, R. V., Владиславовна, Б. О., & Aleksandrova, O. I. (2023) · Russian Journal of Linguistics · doi:10.22363/2687-0088-35817 · Cited by 9 · see citing works
Velleman, D. J. (2019) · Cambridge University Press eBooks · doi:10.1017/9781108539890 · Cited by 8 · see citing works
Analysis-Design-Justification (ADJ): A Framework to Develop Problem-Solving Skills
Acuña, R., & Bansal, A. (2021) · IEEE Global Engineering Education Conference (EDUCON) · doi:10.1109/educon46332.2021.9454055 · Cited by 8 · see citing works
Real-time capture of student reasoning while writing
Franklin, S., & Hermsen, L. M. (2014) · Physical Review Special Topics - Physics Education Research · doi:10.1103/physrevstper.10.020121 · Cited by 8 · see citing works
Teaching Formation to Develop Computational Thinking
Villalba‐Condori, K. O. (2017) · Advances in IT standards and standardization research (AISSR) book series/Advances in IT standards and standardization research series · doi:10.4018/978-1-5225-4944-4.ch004 · Cited by 7 · see citing works
Healthcare Transformation: The Electronic Health Record
Edberg, D. T., & Wendel, J. (2018) · Springer eBooks · doi:10.1007/978-3-319-93003-9_7 · Cited by 7 · see citing works
Solving open problems with students as a first research experience
Flórez, R., & Mukherjee, A. (2017) · Teaching Mathematics and its Applications An International Journal of the IMA · doi:10.1093/teamat/hrx008 · Cited by 7 · see citing works
Flórez, R., & Mukherjee, A. (2018) · PRIMUS · doi:10.1080/10511970.2018.1506530 · Cited by 7 · see citing works
Teaching study skills in mathematics service courses—how to cope with students’ refusal?
Feudel, F., & Dietz, H. M. (2017) · Teaching Mathematics and its Applications An International Journal of the IMA · doi:10.1093/teamat/hrx016 · Cited by 7 · see citing works
Aldon, G., Front, M., & Gardes, M. (2017) · Éducation & didactique · doi:10.4000/educationdidactique.2810 · Cited by 6 · see citing works
Pulos, J. M., Peltier, C. J., Williams-Diehm, K. L., & Martin, J. E. (2019) · Beyond Behavior · doi:10.1177/1074295619871009 · Cited by 6 · see citing works
Jupri, A., Usdiyana, D., & Sispiyati, R. (2021) · JRAMathEdu (Journal of Research and Advances in Mathematics Education) · doi:10.23917/jramathedu.v6i2.13263 · Cited by 10 · see citing works
Methodological Classification of Innovative Engineering Projects
Zwart, S. D., & de Vries, M. J. (2016) · Philosophy of engineering and technology · doi:10.1007/978-3-319-33717-3_13 · Cited by 6 · see citing works
Inverse reformulations of a modelling problem proposed by prospective teachers in Spain
Martinez-Luaces, V. E., Fernández-Plaza, J. A., Rico, L., & Ruiz-Hidalgo, J. F. (2019) · International Journal of Mathematical Education in Science and Technology · doi:10.1080/0020739x.2019.1683773 · Cited by 6 · see citing works
Inganah, S., Vidyastuti, A. N., & Sah, R. W. A. (2023) · AL-ISHLAH Jurnal Pendidikan · doi:10.35445/alishlah.v15i1.2454 · Cited by 6 · see citing works
Characteristics of pre-service teachers’ performance in problem posing
Masriyah, M., Kurniasari, I., & Palupi, E. L. W. (2018) · Journal of Physics Conference Series · doi:10.1088/1742-6596/1088/1/012115 · Cited by 6 · see citing works
A study on the performance comparison of metaheuristic algorithms on the learning of neural networks
Lai, K. H., Zainuddin, Z., & Ong, P. (2017) · AIP conference proceedings · doi:10.1063/1.4995871 · Cited by 6 · see citing works
Binary Artificial Bee Colony Algorithms for {0-1} Advertisement Problem
Asuman, A., & Babayiğit, B. (2019) · National Conference on Electrical and Electronics Engineering (ELECO) · doi:10.1109/iceee2019.2019.00025 · Cited by 5 · see citing works
Boamah, S., Asemani, E., Koranteng, E. K., & Mensah, R. O. (2025) · Discover Education · doi:10.1007/s44217-025-00619-y · Cited by 4 · see citing works
Levels, stickers, and strategies: how gamified math tasks foster elementary student engagement
Pan, Z., Wang, K., Zhu, J., & Smith, M. K. (2026) · Education and Information Technologies · doi:10.1007/s10639-025-13866-1 · Cited by 13 · see citing works
Doing Mathematics with Purpose: Mathematical Text Types
Dostal, H. M., & Robinson, R. (2017) · The Clearing House A Journal of Educational Strategies Issues and Ideas · doi:10.1080/00098655.2017.1357409 · Cited by 4 · see citing works
The development of mathematics teacher professional competencies through social media
Jupri, A., Marwati, R., Sispiyati, R., & Rosjanuardi, R. (2022) · Jurnal Elemen · doi:10.29408/jel.v8i1.4782 · Cited by 4 · see citing works
Periodic Pólya urns, the density method and asymptotics of Young tableaux
Banderier, C., Marchal, P., & Wallner, M. (2020) · The Annals of Probability · doi:10.1214/19-aop1411 · Cited by 4 · see citing works
Magenheim, J., & Schulte, C. (2020) · Encyclopedia of Education and Information Technologies · doi:10.1007/978-3-030-10576-1_253 · Cited by 4 · see citing works
Wei, Y., Zhong, Y., & Pi, F. (2025) · Physical Review Physics Education Research · doi:10.1103/physrevphyseducres.21.010117 · Cited by 3 · see citing works
Mathematical and computational awareness before and after the pandemic
Bubnó, K., & Takács, V. L. (2022) · Frontiers in Education · doi:10.3389/feduc.2022.933339 · Cited by 3 · see citing works
Jupri, A., & Hidayat, A. S. (2022) · Jurnal Pendidikan Matematika · doi:10.22342/jpm.16.3.17820.257-268 · Cited by 3 · see citing works
Yamamoto, S., Tobe, Y., Tawatsuji, Y., & Hirashima, T. (2022) · Research and Practice in Technology Enhanced Learning · doi:10.58459/rptel.2023.18028 · Cited by 3 · see citing works
Computer programming as a tool to improve mathematic skills in basic education
Barroso, R., Castro, A., & Rocha, Á. (2018) · Iberian Conference on Information Systems and Technologies (CISTI) · doi:10.23919/cisti.2018.8399192 · Cited by 3 · see citing works
Gaming the Lecture Hall: Using Social Gamification to Enhance Student Motivation and Participation
Mader, S., & Bry, F. (2019) · Advances in intelligent systems and computing · doi:10.1007/978-3-030-11932-4_53 · Cited by 3 · see citing works
Son, A. L., Darhim, & Fatimah, S. (2021) · KALAMATIKA Jurnal Pendidikan Matematika · doi:10.22236/kalamatika.vol6no1.2021pp57-70 · Cited by 3 · see citing works
Avila-Santos, A. P., Rodrigues, L., da Silva, T. E. V., Vieira, T. M. D. A., Marinho, M., Macario, V., Dermeval, D., Bittencourt, I. I., & Isotani, S. (2024) · Anais do ... Simpósio Brasileiro de Informática na Educação/Anais do Simpósio Brasileiro de Informática na Educação · doi:10.5753/sbie.2024.242500 · Cited by 2 · see citing works
Keep asking: What do I want? What do I have? What do I do?
Ung, L., & Dahabreh, I. J. (2025) · European Journal of Epidemiology · doi:10.1007/s10654-025-01222-z · Cited by 2 · see citing works
Torff, B. (2018) · Advances in higher education and professional development book series · doi:10.4018/978-1-5225-6331-0.ch002 · Cited by 2 · see citing works
Sholehah, N. A., Juandi, D., & Kurniawan, S. (2022) · EDU-MAT Jurnal Pendidikan Matematika · doi:10.20527/edumat.v10i2.14763 · Cited by 2 · see citing works
Sevier, J., Fernandes, A., & Cifarelli, V. V. (2025) · School Science and Mathematics · doi:10.1111/ssm.18361 · Cited by 2 · see citing works
Feel-Phy: An Intelligent Web-Based Physics QA System
Fong, K. S., Bong, C. H., Ahmad, Z. B., & Idris, N. B. (2016) · Communications in computer and information science · doi:10.1007/978-981-10-2777-2_23 · Cited by 2 · see citing works
Cao, Y., Dong, L., & Li, X. (2017) · New frontiers of educational research · doi:10.1007/978-3-662-55781-5_10 · Cited by 2 · see citing works
Strategi Pemecahan Masalah Geometri Mahasiswa Calon Guru Matematika: Antara Prediksi dan Kenyataan
Jupri, A. (2022) · Jurnal Gantang · doi:10.31629/jg.v6i2.3539 · Cited by 2 · see citing works
Solihat, T. A. P., Roesdiana, L., & Haerudin, H. (2022) · Polinomial Jurnal Pendidikan Matematika · doi:10.56916/jp.v1i2.233 · Cited by 2 · see citing works
Programming and Mathematics in an Upper-Level University Problem-Solving Course
Lovrić, M. (2018) · PRIMUS · doi:10.1080/10511970.2017.1403524 · Cited by 2 · see citing works
Planning and Delivering a Cooperative Maths Lesson
Barczi-Veres, K. (2022) · European Journal of Mathematics and Science Education · doi:10.12973/ejmse.3.1.9 · Cited by 2 · see citing works
Synthetic Proof with the Support of Dynamic Geometry
Blažek, J., & Pech, P. (2019) · International Journal for Technology in Mathematics Education · doi:10.1564/tme_v26.3.01 · Cited by 2 · see citing works
Aziz, T. A., Hakim, L., & Makmuri, M. (2023) · AIP conference proceedings · doi:10.1063/5.0125755 · Cited by 1 · see citing works
Using Educational Robotics (ER) to Promote STEM Problem-Solving in Preservice Teachers
Jaipal-Jamani, K. (2024) · Education and human development · doi:10.5772/intechopen.1006520 · Cited by 2 · see citing works
Fostering the Development of Young Students’ Analytical Thinking by use of a Problem-solving Method
Lanting, L. S. (2025) · Journal of Research in Science Mathematics and Technology Education · doi:10.31756/jrsmte.4117si · Cited by 1 · see citing works
Mondal, P., & Vijaykumar, R. (2025) · Learning Research and Practice · doi:10.1080/23735082.2025.2517053 · Cited by 1 · see citing works
Problem-solving skills of prospective elementary teachers: A systematic literature review
Faizah, S. N., Yuliati, L., & Pristiani, R. (2025) · Multidisciplinary Reviews · doi:10.31893/multirev.2026253 · Cited by 1 · see citing works
HAREZMI EDUCATION MODEL IN PRIMARY MATHEMATICS EFFECTS ON PROBLEM-SOLVING AND CRITICAL THINKING
Filiz, T., & Erel, B. D. (2026) · International Online Journal of Primary Education · doi:10.55020/iojpe.1855886 · Cited by 1 · see citing works
Hodiyanto, H., Setyaningsih, E., Hasan, B., Palobo, M., & Koirala, B. N. (2026) · Riemann Research of Mathematics and Mathematics Education · doi:10.38114/riemann.v8i1.182 · Cited by 1 · see citing works
Foundation model-based generative AI for smart manufacturing: A paradigm shift
Zheng, H., Geng, Z., & Xu, X. W. (2026) · ENGINEERING Management · doi:10.1007/s42524-026-5359-0 · Cited by 2 · see citing works
Martínez-Macas, F. E., Galarza-Martínez, E. M., Bustamante-Cedillo, J. E., Galarza-Martinez, V. N., & Loachamin-Pachacama, E. A. (2026) · Revista Científica Zambos. · doi:10.69484/rcz/v5/n2/178 · Cited by 2 · see citing works
Kadunz, G. (2018) · ICME-13 monographs · doi:10.1007/978-3-319-70287-2_7 · Cited by 1 · see citing works
Süreli Çocuk Yayınlarındaki Matematik Problemlerinin İncelenmesi
Şirin, O. K. A. Y., & İncikabı, L. (2019) · Eğitimde Kuram ve Uygulama · doi:10.17244/eku.536105 · Cited by 1 · see citing works
On Mathematical Visualization in Education
Karpov, A. A., Klepov, V., & Nikitin, A. A. (2020) · Communications in computer and information science · doi:10.1007/978-3-030-37436-5_2 · Cited by 1 · see citing works
Interaction Between Subject and DGE by Solving Geometric Problems
Blažek, J., & Pech, P. (2022) · Mathematics education in the digital era · doi:10.1007/978-3-030-86909-0_9 · Cited by 1 · see citing works
Geometry, culture, and gender: An analysis of students’ problem solving ability
Wicaksono, A. B., Chasanah, A. N., Widyaningsih, E., & Gunawan, G. (2022) · AIP conference proceedings · doi:10.1063/5.0107940 · Cited by 1 · see citing works
Analisis Proses Pemecahan Masalah Siswa Dalam Mensketsa Luas Daerah Di Bawah Kurva
Azizah, N., Salwah, S., & Ikram, M. (2022) · Venn Journal of Sustainable Innovation on Education Mathematics and Natural Sciences · doi:10.53696/2964-867x.56 · Cited by 1 · see citing works
Wulan, E. R., Rahayu, D. S., Milla, Y. I. E., & Araiku, J. (2024) · Al-Jabar Jurnal Pendidikan Matematika · doi:10.24042/ajpm.v15i1.19317 · Cited by 1 · see citing works
A vueltas con la modelización del vaciado de depósitos
López, J.-C. C., López-Navarro, E., Pérez, C.-L., & Villanueva, R. J. (2024) · Modelling in Science Education and Learning · doi:10.4995/msel.2024.20904 · Cited by 1 · see citing works
Formal semantics of executable node and activity group of UML 2.5 activity diagram
Jamal, M., & Zafar, N. A. (2017) · International Conference on Communication Technologies (ComTech) · doi:10.1109/comtech.2017.8065769 · Cited by 1 · see citing works
Namgyal, T., & Kongmanus, K. (2018) · International Conference on Information and Education Technology (ICIET) · doi:10.1145/3178158.3178193
Magenheim, J., & Schulte, C. (2019) · Encyclopedia of Education and Information Technologies · doi:10.1007/978-3-319-60013-0_253-1 · Cited by 1 · see citing works
SUGGESTIONS FOR FURTHER READING
(2017) · McGill-Queen's University Press eBooks · doi:10.2307/j.ctt1w1vn61.8
Sinitsky, I., & Ilany, B.-S. (2016) · SensePublishers eBooks · doi:10.1007/978-94-6300-699-6_5
Löh, C., Stoiber, T., & Treude, J.-H. (2019) · Springer eBooks · doi:10.1007/978-3-662-58725-6_2
Decision-making odontoiatrico: un modello basato su alberi decisionali
D’Avanzo, E., Adinolfi, P., & Michelotti, A. (2019) · MECOSAN · doi:10.3280/mesa2018-105002
From Periods to Anabelian Geometry and Quantum Amplitudes
Ionescu, L. M. (2020) · Advances in Pure Mathematics · doi:10.4236/apm.2020.105014
Laily, T. R. N., Prastika, A. P. A., Marfuah, S., & Suharti, S. (2020) · AIP conference proceedings · doi:10.1063/5.0000690
Benson, S. R. (2021) · Mathematics Magazine · doi:10.1080/0025570x.2021.1843929
Wijaya, A. P., & Asnawati, R. (2018) · SSRN Electronic Journal · doi:10.2139/ssrn.3248150
Teaching Formation to Develop Computational Thinking
Villalba‐Condori, K. O. (2021) · IGI Global eBooks · doi:10.4018/978-1-6684-2411-7.ch019
(1975) · International Journal of Productivity and Performance Management · doi:10.1108/eb048315
Myths, Stereotypes, and Misconceptions on Mathematics Teaching and Learning
Martínez, E. P., Castro, J. L., & Campillo, R. M. L. (2022) · Advances in educational technologies and instructional design book series · doi:10.4018/978-1-7998-9128-4.ch007
- Authors
- POLYA, G.
- Venue
- Princeton University Press eBooks
- Publisher
- Princeton University Press
- Published
- 2014-10-26
- OpenAlex
- W4229814857
- Cited by
- 140
- References
- 0
- Citations by year
- 2026: 15 · 2025: 15 · 2024: 8 · 2023: 5 · 2022: 14 · 2021: 9 · 2020: 8 · 2019: 26 · 2018: 16 · 2017: 11 · 2016: 5 · 2015: 2 · 2014: 3
- FWCI
- 2.29
- Type
- book
- Language
- en
- Open access
- closed
- Locations
- Princeton University Press eBooks (publishedVersion)
- Topics
- Physical Sciences › Mathematics › Applied Mathematics › Mathematics Education and Pedagogy
- Physical Sciences › Mathematics › Theoretical Computer Science › History and Theory of Mathematics
- Health Sciences › Health Professions › Speech and Hearing › Problem Solving Skills Development
- Keywords
- problem solving, mathematical problem solving, mathematical thinking
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review