La validación matemática como evidencia de comprensión en la era de la inteligencia artificial generativa
DOI:
https://doi.org/10.69639/arandu.v13i3.2579Palabras clave:
educación matemática, comprensión matemática, validación matemática, argumentación matemática, inteligencia artificial generativaResumen
La producción de soluciones matemáticas correctas ha sido asumida tradicionalmente como evidencia suficiente de comprensión matemática. La irrupción de la inteligencia artificial generativa, capaz de producir demostraciones y procedimientos correctos sin intervención del estudiante, vuelve insostenible ese supuesto y obliga a la educación matemática a preguntarse qué tipo de evidencia permite inferir comprensión matemática cuando la producción de soluciones puede externalizarse. Este artículo, de naturaleza teórica y conceptual, articula la epistemología de las matemáticas, la investigación sobre comprensión y prueba, y la teoría de la evaluación para fundamentar el constructo de validación matemática, entendido como la capacidad de interpretar, examinar, justificar, verificar, corregir y argumentar resultados matemáticos asumiendo responsabilidad epistémica sobre su aceptación o rechazo. Se sostiene que la validación constituye una práctica matemática de alto nivel cognitivo, ligada a la demostración, la argumentación y la construcción de significado, y que ofrece una evidencia de comprensión matemática más robusta que la producción. Se discuten sus implicaciones para la evaluación matemática y se delimitan las contribuciones del constructo al campo.
Descargas
Citas
Ajjawi, R., Tai, J., Dollinger, M., Dawson, P., Boud, D., & Bearman, M. (2023). From authentic assessment to authenticity in assessment: Broadening perspectives. Assessment & Evaluation in Higher Education, 49(4), 499–510. https://doi.org/10.1080/02602938.2023.2271193
Baird, J.-A., Andrich, D., Hopfenbeck, T. N., & Stobart, G. (2017). Assessment and learning: Fields apart? Assessment in Education: Principles, Policy & Practice, 24(3), 317–350. https://doi.org/10.1080/0969594X.2017.1319337
Balacheff, N. (2010). Bridging knowing and proving in mathematics: A didactical perspective. En G. Hanna, H. N. Jahnke, & H. Pulte (Eds.), Explanation and proof in mathematics (pp. 115–135). Springer. https://doi.org/10.1007/978-1-4419-0576-5_9
Bearman, M., Tai, J., Dawson, P., Boud, D., & Ajjawi, R. (2024). Developing evaluative judgement for a time of generative artificial intelligence. Assessment & Evaluation in Higher Education, 49(6), 893–905. https://doi.org/10.1080/02602938.2024.2335321
Biesta, G. J. J. (2009). Good education in an age of measurement: On the need to reconnect with the question of purpose in education. Educational Assessment, Evaluation and Accountability, 21(1), 33–46. https://doi.org/10.1007/s11092-008-9064-9
Biesta, G. J. J. (2010). Good education in an age of measurement: Ethics, politics, democracy. Paradigm Publishers.
Biesta, G. (2015). What is education for? On good education, teacher judgement, and educational professionalism. European Journal of Education, 50(1), 75–87. https://doi.org/10.1111/ejed.12109
Black, P., & Wiliam, D. (1998). Assessment and classroom learning. Assessment in Education: Principles, Policy & Practice, 5(1), 7–74. https://doi.org/10.1080/0969595980050102
Black, P., & Wiliam, D. (2009). Developing the theory of formative assessment. Educational Assessment, Evaluation and Accountability, 21(1), 5–31. https://doi.org/10.1007/s11092-008-9068-5
Brookhart, S. M. (2003). Developing measurement theory for classroom assessment purposes and uses. Educational Measurement: Issues and Practice, 22(4), 5–12. https://doi.org/10.1111/j.1745-3992.2003.tb00139.x
Brousseau, G. (2002). Theory of didactical situations in mathematics: Didactique des mathématiques, 1970–1990 (N. Balacheff, M. Cooper, R. Sutherland y V. Warfield, Eds. y Trads.). Kluwer Academic.
Chinn, C. A., Buckland, L. A., & Samarapungavan, A. (2011). Expanding the dimensions of epistemic cognition: Arguments from philosophy and psychology. Educational Psychologist, 46(3), 141–167. https://doi.org/10.1080/00461520.2011.587722
Chirikov, I., Smirnov, I., & Kizilcec, R. F. (2026). Generative AI use and misuse call for assessment reform in higher education. Science, 392(6800), 818–820. https://doi.org/10.1126/science.aec5115
Colom Cañellas, A. J. (2004). La Teoría de la Educación en su doble dimensionalidad: Como teoría acerca de la realidad y como teoría acerca del saber educativo. Teoría de la Educación. Revista Interuniversitaria, 16, 19–48.
Crooks, N. M., & Alibali, M. W. (2014). Defining and measuring conceptual knowledge in mathematics. Developmental Review, 34(4), 344–377. https://doi.org/10.1016/j.dr.2014.10.001
Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1–2), 103–131. https://doi.org/10.1007/s10649-006-0400-z
Ernest, P. (2002). The philosophy of mathematics education. RoutledgeFalmer. https://doi.org/10.4324/9780203497012
Ernest, P. (2018). The philosophy of mathematics education: An overview. En P. Ernest (Ed.), The philosophy of mathematics education today (pp. 13–35). Springer. https://doi.org/10.1007/978-3-319-77760-3_2
Fullat, O. (1992). Filosofías de la educación: Paideia. CEAC.
García Carrasco, J. (1993). El concepto de educabilidad y el proceso educacional. Teoría de la Educación. Revista Interuniversitaria, 5, 11–32.
Gargallo López, B. (1995). Estrategias de aprendizaje: Estado de la cuestión. Propuestas para la intervención educativa. Teoría de la Educación. Revista Interuniversitaria, 7, 53–76.
Hamami, Y., & Morris, R. L. (2020). Philosophy of mathematical practice: A primer for mathematics educators. ZDM – Mathematics Education, 52(6), 1113–1126. https://doi.org/10.1007/s11858-020-01159-5
Hamami, Y., & Morris, R. L. (2024). Understanding in mathematics: The case of mathematical proofs. Noûs. https://doi.org/10.1111/nous.12489
Harel, G. (2008). DNR perspective on mathematics curriculum and instruction, Part I: Focus on proving. ZDM, 40(3), 487–500. https://doi.org/10.1007/s11858-008-0104-1
Harel, G. (2024). Epistemological justification. ZDM – Mathematics Education. https://doi.org/10.1007/s11858-024-01603-w
Hofer, B. K., & Pintrich, P. R. (1997). The development of epistemological theories: Beliefs about knowledge and knowing and their relation to learning. Review of Educational Research, 67(1), 88–140. https://doi.org/10.3102/00346543067001088
Jover, G., Gonzálvez, V., & Prieto, M. (2017). Una filosofía de la educación del siglo XXI. Síntesis.
Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001). Adding it up: Helping children learn mathematics. National Academy Press. https://doi.org/10.17226/9822
Kollosche, D. (2021). Styles of reasoning for mathematics education. Educational Studies in Mathematics, 107(3), 471–486. https://doi.org/10.1007/s10649-021-10046-z
Messick, S. (1989). Validity. En R. L. Linn (Ed.), Educational measurement (3.ª ed., pp. 13–103). Macmillan.
Messick, S. (1995). Validity of psychological assessment: Validation of inferences from persons’ responses and performances as scientific inquiry into score meaning. American Psychologist, 50(9), 741–749. https://doi.org/10.1037/0003-066X.50.9.741
Michener, E. R. (1978). Understanding understanding mathematics. Cognitive Science, 2(4), 361–383. https://doi.org/10.1207/s15516709cog0204_3
Newton, P. E. (2007). Clarifying the purposes of educational assessment. Assessment in Education: Principles, Policy & Practice, 14(2), 149–170. https://doi.org/10.1080/09695940701478321
Niss, M., & Højgaard, T. (2019). Mathematical competencies revisited. Educational Studies in Mathematics, 102(1), 9–28. https://doi.org/10.1007/s10649-019-09903-9
Panadero, E., Andrade, H., & Brookhart, S. (2018). Fusing self-regulated learning and formative assessment: A roadmap of where we are, how we got here, and where we are going. The Australian Educational Researcher, 45(1), 13–31. https://doi.org/10.1007/s13384-018-0258-y
Rittle-Johnson, B., & Schneider, M. (2015). Developing conceptual and procedural knowledge of mathematics. En R. Cohen Kadosh & A. Dowker (Eds.), The Oxford handbook of numerical cognition (pp. 1118–1134). Oxford University Press. https://doi.org/10.1093/oxfordhb/9780199642342.013.014
Rycroft-Smith, L. (2023). Conceptions and consequences of mathematical argumentation, justification, and proof. Research in Mathematics Education, 25(2), 119–124. https://doi.org/10.1080/14794802.2023.2165537
Sadler, D. R. (1989). Formative assessment and the design of instructional systems. Instructional Science, 18(2), 119–144. https://doi.org/10.1007/BF00117714
Schoenfeld, A. H. (2016). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics (Reprint). Journal of Education, 196(2), 1–38. https://doi.org/10.1177/002205741619600202
Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses, and mathematizing. Cambridge University Press.
Shepard, L. A., Penuel, W. R., & Pellegrino, J. W. (2018). Using learning and motivation theories to coherently link formative assessment, grading practices, and large-scale assessment. Educational Measurement: Issues and Practice, 37(1), 21–34. https://doi.org/10.1111/emip.12189
Skovsmose, O. (1994). Towards a philosophy of critical mathematics education. Kluwer. https://doi.org/10.1007/978-94-017-3556-8
Stiggins, R. (2005). From formative assessment to assessment for learning: A path to success in standards-based schools. Phi Delta Kappan, 87(4), 324–328. https://doi.org/10.1177/003172170508700414
Stylianides, A. J., & Ball, D. L. (2008). Understanding and describing mathematical knowledge for teaching: Knowledge about proof for engaging students in the activity of proving. Journal of Mathematics Teacher Education, 11(4), 307–332. https://doi.org/10.1007/s10857-008-9077-9
Tai, J., Ajjawi, R., Boud, D., Dawson, P., & Panadero, E. (2018). Developing evaluative judgement: Enabling students to make decisions about the quality of work. Higher Education, 76(3), 467–481. https://doi.org/10.1007/s10734-017-0220-3
Torrance, H. (2012). Formative assessment at the crossroads: Conformative, deformative and transformative assessment. Oxford Review of Education, 38(3), 323–342. https://doi.org/10.1080/03054985.2012.689693
Touriñán López, J. M. (1987). Teoría de la educación: La educación como objeto de conocimiento. Anaya.
Wiggins, G. (2020). The case for authentic assessment. Practical Assessment, Research, and Evaluation, 2, Artículo 2. https://doi.org/10.7275/ffb1-mm19
Wiliam, D. (2011). What is assessment for learning? Studies in Educational Evaluation, 37(1), 3–14. https://doi.org/10.1016/j.stueduc.2011.03.001
Yorke, M. (2003). Formative assessment in higher education: Moves towards theory and the enhancement of pedagogic practice. Higher Education, 45(4), 477–501. https://doi.org/10.1023/A:1023967026413
Publicado
Cómo citar
Número
Sección
Licencia
Derechos de autor 2026 Marco Vinicio Gutiérrez Montenegro
Esta obra está bajo una licencia internacional Creative Commons Atribución 4.0.
