The first phrase that I stopped to think was: "the beginnings which are different from its final form". This is very true to mathematics. We see the polished definitions, put together nicely in textbooks, yet the exercises don't resemble the process. Currently there is no process that directs students to create mathematics that is similar to how it's done in the real world. Some mathematical definitions take years to make, and big problems don't get solved like in Good Will Hunting, or like the assigned homework questions.
Another thing that I found important is the snippet from Feynman's lecture about just trying to find out more about the world. I know that lecture and I've watched it many times and Feynman's curiosity is contagious. If we can inspire our students to think the same way, to be curious about things, it will not only make them lifelong learners but push them towards discovery and contribution to our understanding of the world.
Asking "is there an observation that felt like this one?" is probably one of the most important questions to ask when it comes to guiding students for discovery. The reason is, underlying the actual words lies the fact that we are pushing the student towards looking for patterns. Looking for patterns is key in discovery. It's basically saying "this is like this" or "this is just a special case of this".
Beautiful, Marius! I love Feynman's attitude of curiosity and wonder (alongside scientific and mathematical rigour and modelling). How could we find ways to help our students see mathematics as more open and available to creative approaches?
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