Abstract:

Professional qualification is essential in all areas of activity. Learning new methods and techniques, keeping up with news in the sector, and learning from successful experiences are ways to improve processes, deliveries, and results.
Continuing education aims to engage education professionals in improvement processes, such as participation in programmes, courses, symposiums, workshops, research and study groups, and institutional projects, among others. The objective is to provide continuous training that leads to the improvement of teaching practices and points outs ways to overcome difficulties based on theoretical and practical foundations. Learning new teaching and learning methods, understanding behavioural changes, and conversing with specialised professionals all contribute to the development of lesson plans that are more appealing and consistent with the skills and demands of the next generation. Keep this in mind, we conducted a workshop to teach in-service teachers about "open middle problems."

"Open Middle Problems" are challenges or problems that have clear and well-defined starting and ending points but are open in the "middle", that is, they allow different strategies and ways of approaching them and sometimes even different solutions. This type of challenge, which we can think of as "puzzles", will help students develop perseverance and creativity skills. To arrive at a solution, students may have to complete several iterations as they develop mathematical understanding, identify misconceptions, and increase their mathematical knowledge. Math problems that encourage debate and critical thinking.
During the workshop, we discussed with the participants the theoretical framework that outlined the selection of mathematical tasks for 7th grade to 12th grade.

This work aims to present a proposal for a workshop to training teachers on open-middle problems in order to work on critical thinking with their students. The approach is a qualitative and quantitative case study.
The analysis of the discussions and behaviour of the teachers during the workshop revealed that the activities prompted reflection and assisted the teachers in discovering new approaches to mathematics teaching.

Acknowledgements:
Vanda Santos: This work is funded by national funds through the FCT - Foundation for Science and Technology, I.P., within the scope of the project UIDB/00194/2020 and in the scope of the framework contract foreseen in the numbers 4, 5 and 6 of the article 23, of the Decree-Law 57/2016, of August 29, changed by Law 57/2017, of July 19.
Nuno Bastos was supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (FCT-Fundação para a Ciência e a Tecnologia), within project UIDB/04106/2020.

Keywords:

In-service teachers, math problems, critical thinking, technology, reasoning.