The aim of this research is to share how didactic suitability criteria improve teachers’ didactic analysis competency organizing their reflection among a task.  The development and evaluation of the competence in didactic analysis entail contemplating a training (of future primary and secondary school teachers), based on reflection, which allows to analyze practices and identify those that are the quality ones (owing them or not). Along these lines, in the area of Mathematics Education, research has been carried out to know how the knowledge of teacher’s mathematical content is evident in their classes in the form of good practices. Beginning with the teacher’s collaboration, the core idea of the “Lesson Study" methodology (Fernández and Yoshida, 2004), a didactic task is planned, observed, and analyzed by a group of teachers. When this group of teachers has contact with a set of researchers, there is an opportunity to also practice the “Concept Study” methodology proposed by Brent and Renert (2013). In this methodology, researchers commit themselves to the practice of teachers in the examination and the development of models on mathematical understanding. The “Concept Study” combines elements of two relevant approaches in mathematics education research: "concept analysis" and "lesson study." The studies framed in the "concept analysis" are focused on the explanation of the logical structures and associations that are inherent to the mathematical concepts. 

As high-quality mathematics is the main objective of any scholar mathematics lesson, it is needed to consider the teacher's mathematical knowledge (MKT). The group of Deborah Ball et al. (Hill et al., 2008) considered the characteristics this knowledge must have to achieve quality teaching. They have found that, although there is a significant, reliable, and positive association between certain levels of MKT and the mathematical quality of instruction, there are also a considerable number of factors that mediate this relationship, facilitating or hindering the use of teacher knowledge in their practice. Rowland et al. (2005), interested in knowing how the knowledge of the teacher's mathematical content is evident in their classes, analyzed the sessions and recorded them on video, in order to characterize the activated knowledge of the teacher during the instruction. As a result, they proposed “The Knowledge Quartet”: foundation, transformation, connection, and contingency.

Some research on the teacher’s professional development underlines the importance of the competence called "looking meaningfully" at the student’s mathematical thinking (Mason, 2002). This competence allows the math teacher to see the mathematics teaching-learning situations in a professional way that differentiates him from the direction of looking at someone who is not a math teacher. This competence can be characterized as a set of three interrelated skills: identify the strategies used by the students, interpret the understanding revealed by the students and decide how to respond, taking into account the students' understanding. On the other hand, the importance of interpreting the math class from the double perspective of mathematical content and social interaction is argued by Planas and Iranzo (2009) in an analysis model presented for the description and interpretation of interaction processes in the mathematics classroom. They presented, as an example, how the operational use and integration of notions associated with different socio-cultural and semiotic theoretical traditions are proposed. 

Font et al. (2012) presented a list of generic and specific competencies proposed for the initial training of high-school teachers of mathematics. According to these authors, the competence in the didactic analysis (understood as design, apply and evaluate learning sequences, through didactic analysis techniques and quality criteria, to establish planning, implementation, evaluation cycles, and propose improvement proposals) is one of the specific key competencies that future high school math teachers should develop.

A geometric drawing task addressed to high-school students was analyzed, improved, implemented, and re-analyzed by a set of researchers and former teachers. To establish a hypothetical reflective cycle of a didactic task for teacher practice, the authors applied the criteria of didactic suitability (Godino, 2013; Breda, Font, and Pino-Fan, 2018). Didactic suitability is seen as the equilibrium between six main suitability facets: epistemic, cognitive, interactional, mediational, affective, and ecologic. Each suitability facet is defined by a set of components described by their own indicators. Then, a facet is reached when all the indicators of all its components are achieved. Conversely, the absence of some indicators points out the actions that can be added when redesigning the didactic sequence on the study. According to this, the reflective cycle proposed was: (1) identifying the didactic suitability of a mathematics task performed in the past, (2) using the missed suitability indicators to make a new task, (3) testing the new task with students, and to identify the didactic suitability of the new proposal. (1) Teacher’s instructions and the geometric drawing task performed by high-school students were analyzed using a checklist created according to Godino (2013) and Breda et al. (2018) didactic suitability indicators. At this first stage, the lowest facet of this task was interactional, and the highest corresponds to the ecological aspect. (2) The improvement of interactional and affective facets was the redesign objective, but it was also a challenge to preserve the high ecologic aspect previously reached. The start point was the curriculum caring in order to maintain high ecological suitability. Then the task was contextualized proposing it as an artisan balloon contest. Also, the original individual assignment was changed into a collaborative work of geometric drawing whose objective was to build an octahedron using the single faces drawn by eight different teams. In this stage, the authors also proposed that students must be in charge of identifying some of the didactic suitability indicators. Student and teacher questionnaires were prepared to collect the information. (3) This last stage was in the classroom, where students were organized into groups to draw one of the eight octahedra faces. Having the octahedra face done, a new team was formed with one member of each team. The new team was in charge of the final collaborative task: to assemble the geometric body. Finally, in a new session, the questionnaires were answered.

The reflective cycle proposed and the didactic suitability. Having the didactic suitability as the principal reflective tool was significant because it provides clear, organized criteria. Using them as a checklist systematized and simplified the analysis at the time that gives punctual orientations related to how to improve the sequence. Some graphic resources were used: tables, and bar, and hexagonal graphics. It was found that the global appreciation of the didactic suitability was easily interpreted from the hexagonal graphic, but on the other hand, the bar graphic was useful to locate the specific aspects that would be changed to raise some didactic suitability facet. As a complement, but not less important, the tables oriented the actions and procedures that could be applied in a new and improved task. The student opinions agreed on the assignment of a higher value to the interactive facet, which means the proposal achieved the objective: improving the interactional aspect by working on a collaborative task. In general, as students said, there was a rise in all the didactic suitability facets. The didactic sequence results. According to the proposal, the new task was designed to favor student interaction. It was observed that during a collaborative drawing task, the use of geometric language is promoted, and also drawing procedures were orally exposed and argued. Stressful episodes were not found, and the task was fully accomplished.

Breda, A., Font, V., & Pino-Fan, L.R. (2018). Criterios valorativos y normativos en la Didàctica de las Matemáticas: el caso del constructo idoneidad didáctica. Bolema 32(60) 255-278. Doi:10.1590/1980-4415v32n60a13 Brent, D. & Renert, M. (2013). Profound Understanding of Emergent Mathematics: broadening the construct of teachers' disciplinary knowledge. Educational Studies in Mathematics, 82(2), 245-265. Fernández, C. & Yoshida, M. (2004) Lesson study: a Japanese approach to improving mathematics teaching and learning. Mahwah, NJ: Erlbaum. Font, V., Giménez, J., Zorrilla, J., Larios, V., Dehesa, N., Aubanell, A., & Benseny, A. (2012). Competencias del profesor y competencias del profesor de matemáticas: Una propuesta. En V. Font, J. Giménez, V. Larios & J. Zorrilla (Eds.), Competencias del profesor de matemáticas de secundaria y bachillerato (pp. 61-70). Barcelona: Publicaciones de la Universitat de Barcelona. García-Mora, E. & Díez-Palomar, J. (2019). Discusión del uso del constructo “idoneidad didáctica” para el análisis de tareas de construcción de objetos geométricos.” En M.P. Bermúdez (Ed.), Avances en Ciencias de la Educación y del Desarrollo (pp.). Granada: Universidad de Granada-Asociación Española de Psicología Conductual. Godino, J.D. (2013). Indicadores de la idoneidad didáctica en procesos de enseñanza y aprendizaje de las matemáticas. Cuadernos de Investigación y Formación en Educación Matemática 8(119 111-132. Hill, H., Blunk, M., Charambous, Y., Lewis, J., Phelps, G., Sleep, L., & Ball, D. (2008). Mathematical Knowledge for Teaching and the Mathematical Quality of Instruction. An Exploratory Study. Cognition and Instruction, 26(4), 430-511. Mason, J. (2002). Researching your own practice. The discipline of noticing. London: Routledge-Falmer. Planas, N. & Iranzo, N. (2009). Consideraciones metodológicas para la interpretación de procesos de interacción en el aula de matemáticas. Revista Latinoamericana de Investigación en Matemática Educativa-RELIME, 12 (2), 179-213. Rowland, T., Huckstep, P. & Thwaites, A. (2005). Elementary teachers' mathematics subject knowledge: the knowledge quartet and the case of Naomi. Journal of Mathematics Teacher Education, 8(3), 255-281.