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The objective of this study was to investigate how prospective secondary mathematics teachers apply inquiry-based teaching principles to modify tasks that support students' engagement in specific mathematical practices. The research employed the theory of goal-oriented decision-making to describe and explain the use of inquiry-based teaching principles as a conceptual tool by these prospective teachers. The study involved two cohorts, comprising 43 prospective teachers (20 in one cohort and 23 in the other) enrolled in a Secondary Education Teaching program. Data were collected from written reports documenting the implementation of two professional tasks, where participants modified textbook assignments to promote exploratory teaching. An inductive analysis was conducted in two phases. The findings revealed that prospective teachers consistently applied inquiry-based teaching principles when they set specific mathematical practices as student learning objectives, such as analyzing particular cases, identifying patterns and relationships, and formulating conjectures and generalizations. However, when these mathematical practices were not established as learning objectives, teachers struggled to apply inquiry-based teaching principles consistently during task modification. These results suggest that inquiry-based teaching principles are an effective conceptual tool for prospective teachers' instructional reasoning. Nonetheless, for consistent application, it is crucial to establish a coherent network of logical connections between the conceptual tool and the intended learning objectives.
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References
- Artigue, M., & Blomhøj, M. (2013). Conceptualizing inquiry-based education in mathematics. ZDM Mathematics Education, 45(6), 797-810. https://doi-org/10.1007/s11858-013-0506-6
- Ayalon, M., Naftalief, E., Levenson, E., & Levy, S. (2021). Prospective and in-service mathematics teachers’ attention to a rich mathematics task while planning its implementation in the classroom. International Journal of Science and Mathematics Education, 19, 1695-1716. https://doi.org/10.1007/s10763-020-10134-1
- Aytekin-Kazanç, E., & Işıksal-Bostan, M. (2024). A prospective teacher’s conceptions about the nature of mathematical tasks and professional knowledge within task unfolding. Journal of Mathematics Teacher Education, 27(3), 521-546, https://doi.org/10.1007/s10857-023-09584-5
- Barnhart, T., & van Es, E. (2015). Studying teacher noticing: Examining the relationship among pre-service science teachers’ ability to attend, analyze and respond to student thinking. Teaching and Teacher Education, 45, 83-93. https://doi.org/10.1016/j.tate.2014.09.005
- Barquero, B., & Jenssen, B.E. (2020). Impact of theoretical perspectives on the design of mathematical modelling tasks. AIEM – Avances de Investigación en Educación Matemática, nº 17, 98-113. https://doi.org/10.35763/aiem.v0i17.317
- Chin, S.L., Choy, B.H., & Leong, Y.H. (2022). Overlaps and shifts of instructional goals in the design of a set of mathematics tasks. Mathematics Education Research Journal, 34, 523–549. https://doi.org/10.1007/s13394-022-00425-x
- Choy, B.H. (2016). Snapshots of mathematics teacher noticing during task design. Mathematics Education Research Journal, 28, 421-440. https://doi.org/10.1007/s13394-016-0173-3
- Coles, A., & Brown, L. (2016). Task design for ways of working: making distinctions in teaching and learning mathematics. Journal of Mathematics Teacher Education, 19, 149-168. https://doi.org/10.1007/s10857-015-9337-4
- Corbin, J., & Strauss, A. (2008). Basics of qualitative research: Techniques and procedures for developing grounded theory (3rd ed.). Sage.
- D'Amore, B., & Fandiño Pinilla, M. I. (2007). Relaciones entre área y perímetro: convicciones de maestros y de estudiantes1. Revista latinoamericana de investigación en matemática educativa, 10(1), 39-68. https://www.scielo.org.mx/pdf/relime/v10n1/v10n1a3.pdf
- Dietiker, L., Males, L., Amador, J., & Earnest, D. (2018). Curricular Noticing: A framework to describe teachers' interactions with curricular materials. Journal for Research in Mathematics Education, 49(5), 521-532. https://doi.org/10.5951/jresematheduc.49.5.0521
- Dorier, JL., & Maass, K. (2020). Inquiry-Based Mathematics Education. En S. Lerman (eds.). Encyclopedia of Mathematics Education (pp.384-388). Springer. https://doi.org/10.1007/978-3-030-15789-0_176
- Fernández, C., Moreno, M., & Sánchez-Matamoros, G. (2024). Prospective secondary teachers’ noticing of students’ thinking about the limit concept: pathways of development. ZDM–Mathematics Education, 1-15. https://doi.org/10.1007/s11858-024-01573-z
- Greefrath, G. (2020). Mathematical modelling competence. Selected current research developments. AIEM – Avances de Investigación en Educación Matemática, nº 17, 38-51.
- Grossman, P., Hammerness, K., & McDonald, M. (2009). Redefining teaching, re-imagining teacher education. Teachers and Teaching: Theory and Practice, 15(2), 273-289. https://doi.org/10.1080/13540600902875340
- Grossman, P. (2018). Teaching Core Practices in Teacher Education. Cambridge, MA: Harvard Education Press.
- Hiebert, J., Morris, A., Berk, D., & Jansen, A. (2007). Preparing teacher to learn from teaching. Journal of Teacher Education, 58(1), 47-61. https://doi.org/10.1177/0022487106295726
- Ivars, P., Fernández, C., & Llinares, S. (2020). Using a hypothetical learning trajectory to propose instructional activities. Enseñanza de las Ciencias, 38(3), 105-124. https://doi.org/10.5565/rev/ensciencias.2947
- Jaworski, B. (1992). Mathematics teaching: What is it? For the Learning of Mathematics, 12(1), 8–14. https://flm-journal.org/Articles/457BFC43568382BB10454681823D3.pdf
- Jacobs, V. R., & Spangler, D. A. (2017). Research on core practices in K–12 mathematics teaching. In J. Cai (Ed.), Compendium for Research in Mathematics Education. NCTM.
- Kazemi, E., & Waege, K. (2015). Learning to teach within practice-based methods courses. Mathematics Teacher Education and Development, 17(2), 125-145. https://mted.merga.net.au/index.php/mted/article/view/229
- Kim, H., Metzger, M., & Heaton, R. (2020). Teacher planning sessions as professional opportunities to learn: An elementary mathematics teachers’ re-conceptualization of instructional triangles. International Journal of Science and Mathematics Education, 18, 1207-1227. https://doi.org/10.1007/s10763-019-10019-y
- Lampert, M. (2012). Improving teaching and teachers: A “Generative Dance?”. Journal of Teacher Education, 63(5), 361-367. https://doi.org/10.1177/0022487112447111
- Lee, E-J., Lee, K-H., & Park, M. (2019). Developing preservice teachers’ abilities to modify mathematical tasks: using noticing-oriented activities. International Journal of Science and Mathematics Education, 17, 965-985. https://doi.org/10.1007/s10763-018-9891-1
- Lee, E-J., Hwang, S., & Yeo, S. (2023). Preservice teachers’ tasks identification and modification related to cognitive demand. International Journal of Science and Mathematics Education, 22, 911-935, https://doi.org/10.1007/s10763-023-10410-w
- Leong, Y.H., Yeo, B.W.J., & Choy, B.H. (2022). Instructional materials as a site to study teachers’ planning and learning. Mathematics Education Research Journal, 34, 575-598. https://doi.org/10.1007/s13394-022-00430-0
- Matsumoto-Royo, K., & Ramirez-Montoya, M.S. (2021). Core practices in practice-based teacher education: A systematic literature review of its teaching and assessment process. Studies in Educational Evaluation, 70, 1 01047. https://doi.org/10.1016/j.stueduc.2021.101047
- McDonald, M., Kazemi, E., & Kavanagh, S. (2013). Core practices and pedagogies of teacher education: A call for a common language and collective activity. Journal of Teacher Education, 64(5), 378-386. https://doi.org/10.1177/0022487113493807
- Mitchell, R., & Marin, K. A. (2015). Examining the use of structured analysis framework to support prospective teacher noticing. Journal of Mathematics Teacher Education, 18, 551-575. https://doi.org/10.1007/s10857-014-9294-3
- Moreno, M., Sánchez-Matamoros, G., Callejo, M. L., Pérez-Tyteca, P., & Llinares, S. (2021). How prospective kindergarten teachers develop their noticing skills: The instrumentation of a learning trajectory. ZDM–Mathematics Education, 53, 57-72. https://doi.org/10.1007/s11858-021-01234-5
- Paredes, S., Cáceres, M.J., Diego-Mantecón, J.M., Blanco, T.F., & Chamoso, J.M. (2020). Creating realistic mathematics tasks involving suthenticity, cognitive domains, and openness characteristics: A study with pre-service teachers. Sustainability, 12(22), 9656. https://doi.org/10.3390/su12229656
- Parrish, C. W., Snider, R. B., & Creager, M. A. (2022). Investigating how secondary prospective teachers plan to launch cognitively demanding tasks. Journal of Mathematics Teacher Education, 26 (3), 395-423. https://doi.org/10.1007/s10857-022-09534-7
- Riard, A., & Kaur, B. A (2022). A‘stimulus-based interview’ approach to illuminate teachers’ orientations and resources related to task selection and modification. Mathematics Education Research Journal, 34, 599–630. https://doi.org/10.1007/s13394-022-00417-x
- Schoenfeld, A. H. (2010). How We Think. A Theory of Goal-Oriented Decision Making and its Educational Applications. Routledge.
- Schoenfeld, A. H. (2011). Toward professional development for teachers grounded in a theory of decision making. ZDM – Mathematics education, 43, 457–469. https://doi.org/10.1007/s11858-011-0307-8
- Schoenfeld, A. H. (2015). How we think: A theory of human decision-making, with a focus on teaching. In the Proceedings of the 12th International Congress on Mathematical Education: Intellectual and Attitudinal Challenges (pp. 229-243). Springer International Publishing.
- Smith, M. (2014). Tools as a Catalyst for Practitioners’ Thinking. Mathematics Teacher Educator, 3(1), 3-7. https://doi.org/10.5951/mathteaceduc.3.1.0003
- Smith, M. S., & Stein, M. K. (2011). Five practices for orchestrating productive mathematics discussions. National Council of Teachers of Mathematics
- Stone, M. E. (1994). Teaching relationships between area and perimeter with the Geometer's Sketchpad. The Mathematics Teacher, 87(8), 590-594. https://doi.org/10.5951/MT.87.8.0590
- Sullivan, P., Clarke, D., & Clarke, B. (2013). Teaching with Tasks for Effective Mathematics. Science+Business Media.
- Thomas, M., & Yoon, C. (2014). The impact of conflicting goals on mathematical teaching decisions. Journal of Mathematics Teacher Education, 17(3), 227-243. https://doi.org/10.1007/s10857-013-9241-8
- Towers, J. (2010). Learning to teach mathematics through inquiry: A focus on the relationship between describing and enacting inquiry-oriented teaching. Journal of Mathematics Teacher Education, 13(3), 243-263. https://doi.org/10.1007/s10857-009-9137-9
- Trouche, L., Gitirana, V., Miyakawa, T., Pepin, B., & Wang, Ch. (2019). Studying mathematics teachers interactions with curriculum materials through different lenses: Towards a deeper understanding of the processes at stake. International Journal of Educational Research, 93, 53-67. https://doi.org/10.1016/j.ijer.2018.09.002
- Trouche, L., Gueudet, G., & Pepin, B. (2019). The “Resource” Approach to Mathematics Education. Springer Nature. https://doi.org/10.1007/978-3-030-20393-1
- Watson, A., & Ohtani, M. (2021). Task Desing in Mathematics Education, an ICMI Study 22. Springer nature. https://doi.org/10.1007/978-3-319-09629-2
- Zeichner, K. (2012). The turn once again toward practice-based teacher education. Journal of Teacher Education, 63(5), 376-382. https://doi.org/10.1177/0022487112445789
References
Artigue, M., & Blomhøj, M. (2013). Conceptualizing inquiry-based education in mathematics. ZDM Mathematics Education, 45(6), 797-810. https://doi-org/10.1007/s11858-013-0506-6
Ayalon, M., Naftalief, E., Levenson, E., & Levy, S. (2021). Prospective and in-service mathematics teachers’ attention to a rich mathematics task while planning its implementation in the classroom. International Journal of Science and Mathematics Education, 19, 1695-1716. https://doi.org/10.1007/s10763-020-10134-1
Aytekin-Kazanç, E., & Işıksal-Bostan, M. (2024). A prospective teacher’s conceptions about the nature of mathematical tasks and professional knowledge within task unfolding. Journal of Mathematics Teacher Education, 27(3), 521-546, https://doi.org/10.1007/s10857-023-09584-5
Barnhart, T., & van Es, E. (2015). Studying teacher noticing: Examining the relationship among pre-service science teachers’ ability to attend, analyze and respond to student thinking. Teaching and Teacher Education, 45, 83-93. https://doi.org/10.1016/j.tate.2014.09.005
Barquero, B., & Jenssen, B.E. (2020). Impact of theoretical perspectives on the design of mathematical modelling tasks. AIEM – Avances de Investigación en Educación Matemática, nº 17, 98-113. https://doi.org/10.35763/aiem.v0i17.317
Chin, S.L., Choy, B.H., & Leong, Y.H. (2022). Overlaps and shifts of instructional goals in the design of a set of mathematics tasks. Mathematics Education Research Journal, 34, 523–549. https://doi.org/10.1007/s13394-022-00425-x
Choy, B.H. (2016). Snapshots of mathematics teacher noticing during task design. Mathematics Education Research Journal, 28, 421-440. https://doi.org/10.1007/s13394-016-0173-3
Coles, A., & Brown, L. (2016). Task design for ways of working: making distinctions in teaching and learning mathematics. Journal of Mathematics Teacher Education, 19, 149-168. https://doi.org/10.1007/s10857-015-9337-4
Corbin, J., & Strauss, A. (2008). Basics of qualitative research: Techniques and procedures for developing grounded theory (3rd ed.). Sage.
D'Amore, B., & Fandiño Pinilla, M. I. (2007). Relaciones entre área y perímetro: convicciones de maestros y de estudiantes1. Revista latinoamericana de investigación en matemática educativa, 10(1), 39-68. https://www.scielo.org.mx/pdf/relime/v10n1/v10n1a3.pdf
Dietiker, L., Males, L., Amador, J., & Earnest, D. (2018). Curricular Noticing: A framework to describe teachers' interactions with curricular materials. Journal for Research in Mathematics Education, 49(5), 521-532. https://doi.org/10.5951/jresematheduc.49.5.0521
Dorier, JL., & Maass, K. (2020). Inquiry-Based Mathematics Education. En S. Lerman (eds.). Encyclopedia of Mathematics Education (pp.384-388). Springer. https://doi.org/10.1007/978-3-030-15789-0_176
Fernández, C., Moreno, M., & Sánchez-Matamoros, G. (2024). Prospective secondary teachers’ noticing of students’ thinking about the limit concept: pathways of development. ZDM–Mathematics Education, 1-15. https://doi.org/10.1007/s11858-024-01573-z
Greefrath, G. (2020). Mathematical modelling competence. Selected current research developments. AIEM – Avances de Investigación en Educación Matemática, nº 17, 38-51.
Grossman, P., Hammerness, K., & McDonald, M. (2009). Redefining teaching, re-imagining teacher education. Teachers and Teaching: Theory and Practice, 15(2), 273-289. https://doi.org/10.1080/13540600902875340
Grossman, P. (2018). Teaching Core Practices in Teacher Education. Cambridge, MA: Harvard Education Press.
Hiebert, J., Morris, A., Berk, D., & Jansen, A. (2007). Preparing teacher to learn from teaching. Journal of Teacher Education, 58(1), 47-61. https://doi.org/10.1177/0022487106295726
Ivars, P., Fernández, C., & Llinares, S. (2020). Using a hypothetical learning trajectory to propose instructional activities. Enseñanza de las Ciencias, 38(3), 105-124. https://doi.org/10.5565/rev/ensciencias.2947
Jaworski, B. (1992). Mathematics teaching: What is it? For the Learning of Mathematics, 12(1), 8–14. https://flm-journal.org/Articles/457BFC43568382BB10454681823D3.pdf
Jacobs, V. R., & Spangler, D. A. (2017). Research on core practices in K–12 mathematics teaching. In J. Cai (Ed.), Compendium for Research in Mathematics Education. NCTM.
Kazemi, E., & Waege, K. (2015). Learning to teach within practice-based methods courses. Mathematics Teacher Education and Development, 17(2), 125-145. https://mted.merga.net.au/index.php/mted/article/view/229
Kim, H., Metzger, M., & Heaton, R. (2020). Teacher planning sessions as professional opportunities to learn: An elementary mathematics teachers’ re-conceptualization of instructional triangles. International Journal of Science and Mathematics Education, 18, 1207-1227. https://doi.org/10.1007/s10763-019-10019-y
Lampert, M. (2012). Improving teaching and teachers: A “Generative Dance?”. Journal of Teacher Education, 63(5), 361-367. https://doi.org/10.1177/0022487112447111
Lee, E-J., Lee, K-H., & Park, M. (2019). Developing preservice teachers’ abilities to modify mathematical tasks: using noticing-oriented activities. International Journal of Science and Mathematics Education, 17, 965-985. https://doi.org/10.1007/s10763-018-9891-1
Lee, E-J., Hwang, S., & Yeo, S. (2023). Preservice teachers’ tasks identification and modification related to cognitive demand. International Journal of Science and Mathematics Education, 22, 911-935, https://doi.org/10.1007/s10763-023-10410-w
Leong, Y.H., Yeo, B.W.J., & Choy, B.H. (2022). Instructional materials as a site to study teachers’ planning and learning. Mathematics Education Research Journal, 34, 575-598. https://doi.org/10.1007/s13394-022-00430-0
Matsumoto-Royo, K., & Ramirez-Montoya, M.S. (2021). Core practices in practice-based teacher education: A systematic literature review of its teaching and assessment process. Studies in Educational Evaluation, 70, 1 01047. https://doi.org/10.1016/j.stueduc.2021.101047
McDonald, M., Kazemi, E., & Kavanagh, S. (2013). Core practices and pedagogies of teacher education: A call for a common language and collective activity. Journal of Teacher Education, 64(5), 378-386. https://doi.org/10.1177/0022487113493807
Mitchell, R., & Marin, K. A. (2015). Examining the use of structured analysis framework to support prospective teacher noticing. Journal of Mathematics Teacher Education, 18, 551-575. https://doi.org/10.1007/s10857-014-9294-3
Moreno, M., Sánchez-Matamoros, G., Callejo, M. L., Pérez-Tyteca, P., & Llinares, S. (2021). How prospective kindergarten teachers develop their noticing skills: The instrumentation of a learning trajectory. ZDM–Mathematics Education, 53, 57-72. https://doi.org/10.1007/s11858-021-01234-5
Paredes, S., Cáceres, M.J., Diego-Mantecón, J.M., Blanco, T.F., & Chamoso, J.M. (2020). Creating realistic mathematics tasks involving suthenticity, cognitive domains, and openness characteristics: A study with pre-service teachers. Sustainability, 12(22), 9656. https://doi.org/10.3390/su12229656
Parrish, C. W., Snider, R. B., & Creager, M. A. (2022). Investigating how secondary prospective teachers plan to launch cognitively demanding tasks. Journal of Mathematics Teacher Education, 26 (3), 395-423. https://doi.org/10.1007/s10857-022-09534-7
Riard, A., & Kaur, B. A (2022). A‘stimulus-based interview’ approach to illuminate teachers’ orientations and resources related to task selection and modification. Mathematics Education Research Journal, 34, 599–630. https://doi.org/10.1007/s13394-022-00417-x
Schoenfeld, A. H. (2010). How We Think. A Theory of Goal-Oriented Decision Making and its Educational Applications. Routledge.
Schoenfeld, A. H. (2011). Toward professional development for teachers grounded in a theory of decision making. ZDM – Mathematics education, 43, 457–469. https://doi.org/10.1007/s11858-011-0307-8
Schoenfeld, A. H. (2015). How we think: A theory of human decision-making, with a focus on teaching. In the Proceedings of the 12th International Congress on Mathematical Education: Intellectual and Attitudinal Challenges (pp. 229-243). Springer International Publishing.
Smith, M. (2014). Tools as a Catalyst for Practitioners’ Thinking. Mathematics Teacher Educator, 3(1), 3-7. https://doi.org/10.5951/mathteaceduc.3.1.0003
Smith, M. S., & Stein, M. K. (2011). Five practices for orchestrating productive mathematics discussions. National Council of Teachers of Mathematics
Stone, M. E. (1994). Teaching relationships between area and perimeter with the Geometer's Sketchpad. The Mathematics Teacher, 87(8), 590-594. https://doi.org/10.5951/MT.87.8.0590
Sullivan, P., Clarke, D., & Clarke, B. (2013). Teaching with Tasks for Effective Mathematics. Science+Business Media.
Thomas, M., & Yoon, C. (2014). The impact of conflicting goals on mathematical teaching decisions. Journal of Mathematics Teacher Education, 17(3), 227-243. https://doi.org/10.1007/s10857-013-9241-8
Towers, J. (2010). Learning to teach mathematics through inquiry: A focus on the relationship between describing and enacting inquiry-oriented teaching. Journal of Mathematics Teacher Education, 13(3), 243-263. https://doi.org/10.1007/s10857-009-9137-9
Trouche, L., Gitirana, V., Miyakawa, T., Pepin, B., & Wang, Ch. (2019). Studying mathematics teachers interactions with curriculum materials through different lenses: Towards a deeper understanding of the processes at stake. International Journal of Educational Research, 93, 53-67. https://doi.org/10.1016/j.ijer.2018.09.002
Trouche, L., Gueudet, G., & Pepin, B. (2019). The “Resource” Approach to Mathematics Education. Springer Nature. https://doi.org/10.1007/978-3-030-20393-1
Watson, A., & Ohtani, M. (2021). Task Desing in Mathematics Education, an ICMI Study 22. Springer nature. https://doi.org/10.1007/978-3-319-09629-2
Zeichner, K. (2012). The turn once again toward practice-based teacher education. Journal of Teacher Education, 63(5), 376-382. https://doi.org/10.1177/0022487112445789