Thinking about Italy, I think that PBL is not used more widely because teachers are not particularly expert using this method.

Moreover, developing projects in a class requires time and, as each matter in Italy has not so many hours, we are only able to prepare pupils to superate the certification exam. In our country, educational system requires teachers 2.0 who use PBL but eventually they doesn't organize the school in the right way to do it: too many things to explain to pupils, too many matters and so on.

Morever, according to my experience, most of pupils of my school are lazy and when they have to solve a problem whose solution is unknown they prefer giving up instead thinking critically for finding a solution or they let the work do to only one student who is more motived (according to what I have seen in the video). These are the most important factors that stopping me from achieving what is outlined in the video.

Finally, I think that to envelope PBL pupils need to have previously learned necessary knowledge and it requires time and exercise. 
One last thing, we have a great challenge as educators to envelope critical thinking, collaboration and other social skills but, in Italy, we have not the right structure to envelope it.

 Sometimes, I, also, organize team works to develope students critical-communicative skills. Students, on their hand, according to my experience, need lots of exercises to assimilate knowledge and when they find something different in the structure of the exercise they prefer giving up instead of trying to find a solution, right or not. This reaction is very frequen, especially during team works.

 Using this course, I want to complement my approach with PBL, but, most of all,  I want to develope students critical thinking and I want to involve as many students as possible.

 Another negativte aspect, according to my experience, is that sometimes, after the test, students don't remember what they have studied: they consider arguements we have discussed completely unconnected and forgetable after the test; to solve this problem, I try to make practical examples, so pupils can fix their ideas and, as many times I can, I recall necessary previous knowledge.

 As said before, I have tried to use something similar to PBL but the laziness of my students in the use of critical thinking makes me understand that something doesn't work in my approach. Moreover students tend to solve problems on their own without communicating ideas.

 Finally, as I teach math in a secondary school, a subject topic I teach that I can use to experiment with PBL is solid geometry: for example in my a school, in a second class of a professional istitute, I suggested to project and decorate a more-plan cake.

 

This project is addressed to the second class of a professional istitute and I think to implement it in a classroom after discussing solid geometry and after recalling rational numbers and their applications such as proportions. 

The learning goals I would like to achive with the class are:

I agree with you that we haven't time to explain the whole programme required and to realize, also, interesting projects, even if they should be very useful.

I think, reading your text that you already suggest some interesting team works, according to the theory of PBL.

The only thing I want to point out is that I think that 14 y.o. students are too young to suggest them such a project. According to my experience they haven't the right competence to elaborate something like an app. Moreover I think that if you want to realize the project you have described you have to look for something which is eventually achievable using the instruments they own.

Second assessment
Probably you have already understand the limit of your project because you said that the project is not sudent driven and so this is in contrast with PBL but I understand perfectly your difficulty.

As regards your project the arguement is really interesting and real world connected.

Finally, the individual assessment of student is not so easy but you can follow them during the work if they do it at school so that you can assess different aspects.

The only problem is that students don't have to give up after the failure...as their teachers we have to avoid this, stimulating their minds with open questions which could sound interesting, building for them a suitable environment where they could develop critical thinking enjoying the situation and avoiding to give our students all the answers before they have tried and tried to find one.