Topic

Modelling Mathematics, surface and area

Learning Objectives

Methodology: Here we used the math-trail idea to join in and out in the two senses of bringing the in work outside (in-out) and the out work inside (out-in):
- in-out: recognizing in the city’s environment some objects or mathematical concepts
already sawn and explained in class (in this experimentation this was done in
meeting 2 and which will be the subject of study in this paper);
- out-in: starting from real city’s objects or experiences and then bringing them in
class, to introduce new objects or mathematical concepts (in this experimentation
this was done in meeting 4).

The whole experimentation aimed at helping students grasp mathematical concepts such as surface (the space included within a closed curve) and area (the number representing the measure of a surface), not through frontal lecture, but through discovery activities based on mathematical laboratory activities. 

Description of the learning process and activities

In-out-in methodology: (14 hours) Three of the five meetings were lessons in classroom, based on mathematical laboratory, with the involvement of the body, and two of them were outdoor activities, based on math trails designed by MCM around the city center of Catania.

A web portal and a mobile app for math-trail program were created and several math-trail tasks were designed around cities allover the world and uploaded into a system by the teachers. Then students walk the trail and its tasks by the help of mobile app or a .pdf file (generated by the web portal) to find and solve mathematical modelling tasks around the city. In this paper we present the results of an experimental teaching activity through MCM-Project for primary school students. 

Meeting 1 (21 February 2019): classroom activity carried out as a mathematics laboratory. It began with the Dido’s legend on ox hide, and the Dido’s problem on maximal surface figure among isoperimetric figures and it finished playing with a Tangram. The Tangram game had an important role to build a way to decompose polygons and, as we will see in the analysis section, students will apply this skill in the resolutions of some MCM-tasks of Meeting 2 (2 hours);
2) Meeting 2 (21 March 2019): outdoor activity based on a math trail on the recognition of polygons and calculation of perimeters and areas (4 hours);
3) Meeting 3 (22 March 2019): classroom activity. Some of the MCM-tasks that the children had encountered in Meeting 2 were commented on together, regarding solution strategies and data found on site. After that, the focus was on equidecompasability with a Tangram and other mathematical concepts that students would encounter in the second outdoor activity. Moreover, an appendix was devoted to errors and measurements. The lesson ended with a summary test allowing to verify
the acquired skills (2 hours);
4) Meeting 4 (11 April 2019): outdoor activity based on a math trail on circumference and circle and construction of a “human compass” (4 hours);
5) Meeting 5 (17 April 2019): classroom activity that included a focus on the use of compasses as drawing tools, an appendix on the acceptability range of a given measurement, the acceptability range of a given measurement. The lesson ended with a summary test allowing to verify the acquired skills (2 hours)."

Organisations involved

Carried out by the IDMI Goethe - University Frankfurt (Germany)

Resources required

A web portal and a mobile app for math-trail program were created and several math-trail tasks were designed around cities allover the world and uploaded into a system by the teachers (MCM app).

Other remarks

In the experimentation, children were the main characters of their learning. The experimenter was there for hints but he never said more than what was useful to help students in the modelling phase. If some child showed an idea that could be used as a strategy, he did not say “right” or “wrong”, but rather “if it is true, you are right”. In such a way, he led them to think by themselves, favouring a process of mathematization.
it is useful to plan preparation meetings (like the first in the experimentation proposed in this paper) or discussion meetings (like the third or the fifth) in class. In this way we can make sure that all pupils have the knowledge to afford the trail and, if some student miss something, the teacher has a chance to assess the class.