The following papers present an overview of the OSA theoretical notions, its motivation, and anthropological and semiotic assumptions. The OSA is an emerging theoretical system that attempts to articulate different points of view and theoretical notions about mathematical knowledge, its teaching and learning. It is supported and nurtured by contributions from the various disciplines and technologies interested in human cognition and its development: epistemology, psychology, sociology, semiotics, educational sciences.

- Godino, J. D., Batanero, C. y Font, V. (2019). The onto-semiotic approach: implications for the prescriptive character of didactics. [Versión en español].
*For the Learning of Mathematics, 39*(1), 37- 42 - Godino, J. D. Batanero, C. y Font, V. (2007). The onto-semiotic approach to research in mathematics education.
*ZDM. The International Journal on Mathematics Education*, 39 (1-2), 127-135. - Font, V., Godino, J. D. y Gallardo, J. (2013). The emergence of objects from mathematical practices.
*Educational Studies in Mathematics*, 82, 97–124.

In the following papers the notions of institutional and personal meaning of mathematical objects are developed and exemplified, laying the foundations for an anthropological - pragmatic and semiotic conception of mathematical knowledge, both from an institutional (social and cultural) and personal (cognitive) point of view. Likewise, the notions of configuration of practices, objects, processes and relationships (semiotic functions), as analytical tools for institutional and personal mathematical practices are developed and exemplified.

- Godino, J. D. & Batanero, C. (1998). Clarifying the meaning of mathematical objects as a priority area of research in Mathematics Education. In: A. Sierpinska & J. Kilpatrick (Ed.),
*Mathematics education as a research domain: A search for identity*(pp. 177-195). Dordrecht: Kluwer, A. P. - Godino, J. D., Burgos, M. & Gea, M. (2021). Analysing theories of meaning in mathematics education from the onto-semiotic approach.
*International Journal of Mathematical Education in Science and Technology.*The Version of Record of this manuscript has been published and is available in https://www.tandfonline.com/doi/full/10.1080/0020739X.2021.1896042

- Godino, J. D., Font, V., Wilhelmi, M. R. y Lurduy, O. (2011). Why is the learning of elementary arithmetic concepts difficult? Semiotic tools for understanding the nature of mathematical objects.
*Educational Studies in Mathematics,*77 (2), 247-265. . - Beltrán-Pellicer, P. y Godino, J. D. (2020). An onto-semiotic approach to the analysis of the affective domain in mathematics education.
*Cambridge Journal of Education, 50(1)*, 1-20. DOI: 10.1080/0305764X.2019.1623175 - Pino-Fan, L., Font, V., Gordillo, W., Larios, V. y Breda, A. (2018). Analysis of the meanings of the antiderivative used by students of the first engineering courses.
*International Journal of Science and Mathematics Education*,*16*(6), 1091-1113. - Claudia, L. F., Kusmayadi, T. A. & Fitriana, L. (2021). Semiotic analysis of mathematics problems-solving: configure mathematical objects viewed from high mathematical disposition.
*Journal of Physics: Conference Series, Vol. 1808*. Annual Engineering and Vocational Education Conference (AEVEC) 2020 18-19 September 2020, city, Indonesia. - Godino, J. D.,Giacomone, B., Blanco, T. F., Wilhelmi, M. R. y Contreras, A. (2016). Onto-semiotic configurations underlying diagramatic reasoning.
*Proceedings of the 40th Annual Meeting of the International Group for the Psychology of Mathematics Education (PME 40)*, Szegel, Hungary, 3-7 August, 2016. - Wilhelmi, M. R., Godino, J. D. & Lacasta, E. (2011). Epistemic configurations associated to the notion of equility in real numbers.
*Quaderni di Ricerca in Didattica (Mathematics),21*, 53-82. - Santi, G. (2011). Objectification and semiotic function.
*Educational Studies Mathematics*, 77, 285-311. - Castro, W. F. and Godino, J. D. (2009). Cognitive configurations of pre-service teachers when solving an arithmetic-algebraic problem.
*CERME 6, Group 4: Algebraic Thinking*. Lyon, France, 2009 - Font, V. & Contreras, A. (2008). The problem of the particular and its relation to the general in mathematics education.
*Educational Studies in Mathematics*, 69, 33-52. - Font, F., Godino, J. D. & D'Amore, B. (2007). An onto-semiotic approach to representations in mthematics education.
*For the Learning of Mathematics, 27*(2), 2-14. - Godino, J. D. and Batanero, C. (2003). Semiotic functions in teaching and learning mathematics.In, Anderson, A. Sáenz-Ludlow, S. Zellweger and V. V. Cifarelli (Eds),
*Educational Perspectives on Mathematics as Semiosis: From Thinking to Interpreting to Knowing*(pp. 149-167). Ottawa: LEGAS. - Recio, A. M. and Godino, J. D. (2001). Institutional and personal meanings of mathematical proof.
*Educational Studies in Mathematics*, 48, 83-99. - Godino, J. D. (1996). Mathematical concepts, their meaning and understanding. In L. Puig & A. Gutierrez (Eds.),
*Proceedings of the 20th PME Conference.*Valencia. (Vol 2, pp. 417-424) - Godino, J. D. (1994). Ecology of mathematical knowledge: an alternative vision of the popularization of mathematics. In A. Joseph, F. Mignot, F. Murat, B. Prum & R. Rentschler (Eds),
*First European Congress of Mathematics*(Vol. 3, pp. 150-156), Paris, July 6-10, 1992. Basel: Birkhauser Verlag.

The following papers introduce and exemplify the notions developed in the OSA to analyze and design mathematical instruction processes (teaching - learning of specific contents), taking into account the epistemic, ecological, cognitive, affective, interactional (teaching and teaching roles) and mediational facets.

- Godino, J. D. & Burgos, M. (2020). Interweaving transmission and inquiry in mathematics and sciences instruction. In K. O. Villalba-Condori et al. (Eds.), CISETC 2019, CCIS 1191 (pp. 6–21). Springer Nature Switzerland AG. The final publication is available at https://doi.org/10.1007/978-3-030-45344-2_2
- Godino, J. D., Rivas, H., Burgos, M. & Wilhelmi, M. D. (2018). Analysis of didactical trajectories in teaching and learning mathematics: overcoming extreme objectivist and constructivist positions.
*International Electronic Journal of Mathematics Education*, 14(1), 147-161. - Godino, J. D., Batanero, C., Contreras, A., Estepa,A. Lacasta,E. & Wilhelmi, M. R. (2013). Didactic engineering as design-based research in mathematics education.
*Proceedings CERME8*, Turkey. Available from, http://cerme8.metu.edu.tr/wgpapers/WG16/WG16_Godino.pdf - Godino, J. D. ( 2002). Studying the median: A framework to analyse instructional processes in Statistics Education. In, B. Phillips (Ed.), Proceedings of the ICOTS-6. CDROM. IASE.

The didactic suitability of an instructional process is defined as the degree to which the process (or a part of it) meets certain characteristics allowing to be classified as optimal or adequate. This required achieving the adaptation between the students’ personal meanings (learning) and the intended or implemented institutional meanings (teaching), taking into account the circumstances and resources available (environment). The learning optimization can take place locally through a mixed model of instruction that articulates the inquiry, collaboration and transmission of knowledge, a model managed by didactic suitability criteria interpreted and adapted to the context by the teacher. The following articles describe and exemplify the use of this tool.

- Godino, J.D., Batanero, C. Font, V., Contreras, A. & Wilhelmi, M. R. (2016). The theory of didactical suitability: Networking a system of didactics principles for mathematics education form different theoretical perspectives.
*TSG51.13th International Congress on Mathematical Education*. Hamburg. - Sánchez, A., Font, V. & Breda, A. (2021). Significance of creativity and its development in mathematics classes for preservice teachers who are not trained to develop students’ creativity.
*Mathematics Education Research Journal*. Available online: https://rdcu.be/ceRcU - Beltrán-Pellicer, P. & Godino, J. D. (2020). An onto-semiotic approach to the analysis of the affective domain in mathematics education.
*Cambridge Journal of Education, 50*(1), 1-20. - Hummes, V., Font, V. Breda, A. (2019). Combined use of the Lesson Study and the criteria of Didactical Suitability for the development of the reflection on the own practice in the training of mathematics teachers.
*Acta Scientiae, 21*(1), 64-82. - Breda, A., Pino-Fan, L. y Font, V. (2017). Meta Didactic-Mathematical Knowledge of Teachers: Criteria for the reflection and assessment on teaching practice.
*EURASIA Journal of Mathematics Science and Technology Education*, 13 (6), 1893-1918. DOI 10.12973/eurasia.2017.01207a. - Godino, J. D., Burgos, M., Beltrán-Pellicer, P., Gea, M. & Giacomone, ,B. (2019). Structuring the system of didactical suitability criteria for mathematics instruction processes. Poster presentation. CIEAEM71, Braga, Portugal.
- Ninow, V. & Kaiber, C. T.(2019). Affine function: An analysis from the perspective of the epistemic and cognitive suitability of the Onto-semiotic Approach.
*Acta Scientiae, 21*(6), 130-149. - Beltrán-Pellicer, P., Medina, A. & Quero, M. (2018). Movies and TV series fragments in mathematics: Epistemic suitability of instructional designs.
*International Journal of Innovation in Science and Mathematics Education,*26(1), 16–26, 2018. - Borji, V., Font, V., Alamolhodaei, H., & Sánchez, A. (2018). Application of the complementarities of two theories, APOS and OSA, for the analysis of the university students’ understanding on the graph of the function and its derivative.
*EURASIA Journal of Mathematics, Science and Technology Education, 14*(6), 2301-2315. - Pino-Fan, L., Guzmán, I., Font, V., & Duval, R. (2016). Analysis of the underlying cognitive activity in the resolution of a task on derivability of the function f(x)=|x|. An approach from two theoretical perspectives.
*PNA, 11*(2), 97-124. - Drijvers, P. Godino, J. D., Font, V. & Trouche, L. (2013). One episode, two lenses. A reflective analysis of student learning with computer algebra from instrumental and onto-semiotic perspectives.
*Educational Studies in Mathematics*, 82, 23–49. - Font, V., Trigueros, M., Badillo, E., Rubio, N. (2016). Mathematical objects through the lens of two different theoretical perspectives: APOS and OSA.
*Educational Studies in Mathematics. 91*(1), 107-122. - Pino-Fan, L., Guzmán, I., Font, V., & Duval, R. (2015). The theory of registers of semiotic representation and the onto-semiotic approach to mathematical cognition and instruction: linking looks for the study of mathematical understanding. In Beswick, K., Muir, T., & Wells, J. (Ed.),
*Proceedings of the 39th Conference of the International Group for the Psychology of Mathematics Education*(Vol. 4, pp. 33-40). Hobart, Australia: PME. - Font, V., Godino, J. D. and Contreras, A. (2008). From representation to onto-semiotic configurations in analysing mathematics teaching and learning processes. En, L. Radford, G. Schubring, y F. Seeger (eds.),
*Semiotics in Mathematics Education: Epistemology, History, Classroom, and Culture*(pp. 157–173). Rotterdam: Sense Publishers. - Font, V., Godino, J. D. & D’Amore, B. (2007). An onto-semiotic approach to representations in mathematics education.
*For the Learning of Mathematics*, 27(2), 1–7. - Godino, J. D., Font, V. (2010). The theory of representations as viewed from the onto-semiotic approach to mathematics education.
*Mediterranean Journal for Research in Mathematics Education, 9*(1), 189-210. - Molina, O., Font, V. & Pino-Fan, L. (2021). Norms That Regulate the Theorem Construction Process in an Inquiry Classroom of 3D Geometry: Teacher’s Management to Promote Them.
*Mathematics 2021, 9*, 2296. https://doi.org/10.3390/math9182296 - Burgos, M. & Godino, J. D. (2021). Assessing the epistemic analysis competence of prospective primary school teachers on proportionality tasks.
*International Journal of Science and Mathematics Education*. https://doi.org/10.1007/s10763-020-10143-0 Disponible en https://rdcu.be/cdwWj - Breda, A., Seckel, M. J., Farsani, D., Silva, J. F., & Calle, E. (2021). Teaching and learning of mathematics and criteria for its improvement from the perspective of future teachers: a view from the Ontosemiotic Approach.
*Mathematics Teaching Research Journal, 13*(1), 31-51. - Burgos, M., Beltrán-Pellicer, P. y Godino, J. D. (2020). The issue of didactical suitability in mathematics educational videos: experience of analysis with prospective primary school teachers.
*Revista Española de Pedagogía, 78*(275), 27-49. doi: https://doi.org/10.22550 - Erbilgin, E. & Arikan, S. (2021). Using lesson study to support preservice elementary teachers’ learning to teach mathematics.
*Mathematics Teacher Education and Development (MTED), 23*(1), 113-134. - Giacomone, B., Beltrán-Pellicer, P. & Godino, J. D. (2019). Cognitive analysis on prospective mathematics teachers’ reasoning using area and tree diagrams.
*International Journal of Innovation in Science and Mathematics Education, 27*(2), 18–32. - Carvalho, J. I. F de, Pietropaolo, R. C. & Campos, T. M. M. (2019). Developing secondary school teachers’ didactic–mathematical knowledge about probability.
(JIEEM),12,2,134-144.*Jornal Internacional de Estudos em Educação Matemática* - Neto, T., Kamuele, L. & Natividade, M. de (2019). Assessing the didactic and mathematical knowledge of prospective mathematics teachers in Namibe, Angola.
*Afrika Matematika*. http://link.springer.com/article/10.1007/s13370-019-00747-3 - Blanco, T. F., Godino, J. D. Sequeiros, P. G. & Diego-Mantecón, J. M. (2019). Skill Levels on Visualization and Spatial Reasoning in Pre-service Primary Teachers.
*Universal Journal of Educational Research, 7*(12), 2647-2661. - Breda, A., Pino-Fan, L. y Font, V. (2017). Meta didactic-mathematical knowledge of teachers: criteria for the reflections and assessment on theaching practice.
*Eurasia Journal of Mathematics, Science & Technology Education, 13*(6), 1893-1918. - Pino-Fan, L., Godino, J. D., & Font, V. (2018). Assessing key epistemic features of didactic-mathematical knowledge of prospective teachers: the case of the derivative.
*Journal of Mathematics Teacher Education, 21*(1), 63-94. doi: http://dx.doi.org/10.1007/s10857-016-9349-8. - Mallart, A.; Font, V. & Díez-Palomar, J. (2018). Case study on mathematics pre-service teachers’ difficulties in problem posing.
*EURASIA Journal of Mathematics, Science and Technology Education, 14*(4), 1465–1481. - Arteaga, P., Batanero, C., Contreras, J. M., & Cañadas, G. R. (2015) Statistical graphs complexity and reading levels: a study with prospective teachers.
*Statistique et Enseignement*, 6(1), 3-23. - Batanero, C., Contreras, J. M. y Díaz, C. y Sánchez, E. (2015). Prospective teachers’ semiotic conflicts in computing probabilities from a two-way table.
*Mathematics Education*, 10 (1), 3-16. - Pino-Fan, L., Assis, A., & Castro, W. F. (2015). Towards a methodology for the characterization of teachers' didactic-mathematical knowledge.
*Eurasia Journal of Mathematics, Science & Technology Education*, 11(6), 1429-1456. - Batanero, C. Arteaga, P., Serrano, L. & Ruiz. B. (2014). Prospective Primary School Teachers' Perception of Randomness. In E. J Chernoff, y B. Sriraman, (Eds.),
*Probabilistic thinking: presenting plural perspectives*. (pp. 345-366) .Advances in Mathematics Education Series. Springer. - Arteaga, P., & Batanero, C. (2011). Relating graph semiotic complexity to graph comprehension in statistical graphs produced by prospective teachers.
*Paper presented at CERME 8*, Rzeszow, Polonia, 2011 - Godino, J. D., Batanero, C., Roa, R. and Wilhelmi, M. R. (2008). Assessing and developing pedagogical content and statistical knowledge of primary school teachers throuch project work. ICMI/IASE 2008. In C. Batanero, G. Burrill, C. Reading & A. Rossman (Eds.),
*Joint ICMI/IASE Study: Teaching Statistics in School Mathematics. Challenges for Teaching and Teacher Education. Proceedings of the ICMI Study 18 and 2008 IASE Round Table Conference.* - Batanero C., Biehler, R., Maxara, C., Engel, J. & Vogel, M. (2005). Using simulation to bridge teachers´ content and pedagogical knowledge in probability.
*Paper presented at the ICMI Study 15*. Aguas de Lindoia, Brazil. - Aké, L., Godino, J. D., Gonzato, M. y Wilhelmi, M. R. (2013). Proto-algebraic levels of mathematical thinking. In Lindmeier, A. M. & Heinze, A. (Eds.). Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education, Vol. 2, pp. 1-8. Kiel, Germany: PME.
- Godino, J. D., Neto, T., Wilhelmi, M. R., Aké, L., Etchegaray, S. & Lasa, A. (2015). Algebraic reasoning levels in primary and secondary education. In Konrad Krainer and Naďa Vondrová. CERME 9 -Ninth Congress of the European Society for Research in Mathematics Education, Feb 2015, Prague, Czech Republic. (pp. 426-432), Proceedings of the Ninth Congress of the European Society for Research in Mathematics Education.
- Godino, J. D., Wilhelmi, M. R., Neto, T., Blanco, T. F., Contreras, A., Díaz-Batanero, C., Estepa, A. y Lasa, A. (2015). Assessing prospective primary school teachers’ didacticmathematical knowledge on elementary algebraic reasoning.
*Revista de Educación*, 370, 188-215. - Montiel, M., Wilhelmi, M. R., Vidakovic, D. y Elstak, I. (2012). Vectors, change of basis and matrix representation: onto-semiotic approach in the analysis of creating meaning.
*International Journal of Mathematical Education in Science and Technology, 43*(1), 11–32. - Godino, J. D., Font, V., Wilhelmi, M. R. y Lurduy, O. (2011). Why is the learning of elementary arithmetic concepts difficult? Semiotic tools for understanding the nature of mathematical objects. (Versión en español)
*Educational Studies in Mathematics*, 77 (2), 247-265 - Badillo, E., Font, V. & Edo, M. (2015). Analyzing the responses of 7 - 8 year olds when solving partitioning problems.
*International Journal of Science and Mathematics Education*, 13(4), 811-836. DOI: 10.1007/s10763-013-9495-8 - Burgos, M., Bueno, S., Godino, J.D., & Pérez, O. (2021). Onto-semiotic complexity of the Definite Integral. Implications for teaching and learning Calculus.
*REDIMAT – Journal of Research in Mathematics Education, 10*(1), 4-40. - Pino-Fan, L., Font, V., Gordillo, W., Larios, V. y Breda, A. (2018). Analysis of the meanings of the antiderivative used by students of the first engineering courses.
*International Journal of Science and Mathematics Education*,*16*(6), 1091-1113. - Montiel, M., Wilhelmi, M., Vidakovic, D. & Elstak, I. (2009). Using the onto-semiotic approach to identify and analyze mathematical meaning when transiting between different coordinate systems in a multivariate context.
*Educational Studies in Mathematics*, 72(2), 139–160. - Wilhelmi, M. R., Godino, J. D. y Lacasta, E. (2007). Didactic effectiveness of mathematical definitions: The case of the absolute value.
*International Electronic Journal of Mathematics Education,*2 (2): 72-90. - Lugo-Armenta, J.G. & Pino-Fan, L.R. (2021). Inferential Reasoning of Secondary School Mathematics Teachers on the Chi-square Statistic.
*Mathematics 2021, 9*, 2416. https://doi.org/10.3390/math9192416 - Lugo-Armenta, J. G., Pino-Fan. L. R. & Ruiz-Hernández, B. R. (2021). Chi-square Reference Meanings: a Historical-epistemological Overview.
*Revemop, 3,*e202108, 1-33. - Carvalho, J. I. F de, Pietropaolo, R. C. & Campos, T. M. M. (2019). Developing secondary school teachers’ didactic–mathematical knowledge about probability. Jornal Internacional de Estudos em Educação Matemática (JIEEM),12,2,134-144.
- Arteaga, P., Batanero, C., Contreras, J. M., & Cañadas, G. R. (2015) Statistical graphs complexity and reading levels: a study with prospective teachers.
*Statistique et Enseignement*, 6(1), 3-23. - Batanero, C., Cañadas, G. R. Díaz, C. y Gea, M. M. (2015). Judgment of Association between Potential Factors and Associated Risk in 2x2 Tables: A Study with Psychology Students.
*The Mathematics Enthusiast*, 12 (1), 347-363. - Batanero, C., Contreras, J. M. y Díaz, C. y Sánchez, E. (2015). Prospective teachers’ semiotic conflicts in computing probabilities from a two-way table.
*Mathematics Education*, 10 (1), 3-16. - Batanero, C. Arteaga, P., Serrano, L. & Ruiz. B. (2014). Prospective Primary School Teachers' Perception of Randomness. In E. J Chernoff, y B. Sriraman, (Eds.),
*Probabilistic thinking: presenting plural perspectives*. (pp. 345-366) .Advances in Mathematics Education Series. Springer. - Batanero, C. y Díaz, C. (2007). Meaning and understanding of mathematics. The case of probability. En J. P. van Bendegen y K. François (Eds.),
*Philosophical dimensions in mathematics education*(pp. 107-128). New York: Springer. - Batanero, C., Tauber, L. y Sánchez, V. (2004).Student's reasoning about the normal distribution. En D. Ben-Zvi y J.B. Garfield (Eds),
*The Challenge of developing statistical literacy, reasoning, and thinking*(pp. 257-276). Dordrecht: Kluwer. - Godino, J. D., Batanero, C. y Roa, R. (2005). An onto-semiotic analysis of combinatorial problems and the solving processes by university students.
*Educational Studies in Mathematics*,60 (1), 3-36. - Batanero, C., Godino, J. D., & Estepa, A. (1998). Building the meaning of statistical association through data analysis activities. In a. Olivier & k. Newstead:
*Proceedings of the 22 Conference of theIinternational Group for the Psychology of Mathematics Education*(v.1, pp. 221-236 (research forum). University of Stellembosch, South Africa

List of papers in which the concordance and complementarities of the OSA with other theoretical frameworks used in Mathematics Education are studied.

OSA papers with applications to the field of teachers’ education.

OSA papers with applications to the field of algebra.

OSA papers with applications to the field of arithmetics.

OSA papers with applications to the field of calculus.

OSA papers with applications to the field of statistical education.