Abstract:

Many non-formal learning contexts are almost unlimited resources for school teaching and learning, facilitating the connection between contents and real world situations that provide meaningful learning for students. On the other hand, through the movement, they offer opportunities for students interact with the environment and peers, promoting health, also social and affective dimensions.
If these potentialities are taken advantage of, it is possible to reduce the excessive time children spend sitting in classrooms, most of the time performing routine activities that demotivate them and do not promote the engagement necessary for effective learning.
The role of affect in student learning is indisputable. In mathematics classes affective traits are fundamental because they can influence the interactions with cognition, the structuring of the social reality of the classroom, the self-concept of the mathematics learner, the way he/she learns and the limitations that can bring to effective learning, and can condition students' thoughts and actions.
Among the various affective traits which influence learning is engagement which, in turn, is influenced by other factors such as the motivation and interest of the student. Engagement in a task presupposes that the student does not simply perform or simulate the task mechanically, but connects to it cognitively, behaviourally and affectively.
Taking into account, the importance of engagement in learning, but also that nonformal contexts can serve to promote connections between mathematics and different areas of knowledge and real world and also to motivate students, mathematical tasks were constructed and implemented outdoors, namely in a pedagogical farm, a protected area, an urban space (town) and a public garden. The tasks were organized and implemented through a mathematical trail over four months with primary students (age 8-9).
The aim of the study was to understand the contribution of non-formal learning contexts to the students' engagement in solving mathematical tasks and, consequently, to mathematics learning. To this end, the following guiding questions were defined: 1. How can we characterize students' engagement, at the cognitive level, during the accomplishment of the mathematical trails? 2. How can we characterize students' engagement, at the behavioural and affective level, during the accomplishment of the mathematical trails?
We opted a qualitative and interpretative methodology and a case study design.
Data were collected through participant observation, audio and video recordings, interviews and written documents (students' tasks resolutions). With regard to data analysis, the categories and subcategories were based on the literature, although some were adjusted according to the data that emerged. In terms of cognitive engagement, most students invested in trying to solve the tasks, looking for appropriate strategies to facilitate the correct resolution of each task and the mobilisation of the necessary knowledge. In terms of behavioural engagement, several aspects related to attention, collaboration and commitment were recorded. In terms of affective engagement, there was evidence of interest and satisfaction, collaboration and also some anxiety and frustration.

Keywords:

Mathematical tasks, Engagement, Outdoor, Connections.