The following resources serve resources to inspire and prompt discussion around problem-solving and inquiry based approaches to teaching mathematics.
In this video, Conrad Wolfram argues that the emphasis in mathematics should not be on the calculations (or computation) but on problem-solving: posing the right questions, determining math formulation to contextualize questions, calculating, and then formulating conclusions.
TED's Description of this video of Dan Meyer's talk: "Today's math curriculum is teaching students to expect — and excel at — paint-by-numbers classwork, robbing kids of a skill more important than solving problems: formulating them. Dan Meyer shows classroom-tested math exercises that prompt students to stop and think."
In this video, Dan Finkel talks about how math education needs to change, to include questioning and struggle for longer lasting learning and engagement:
Start with a question;
Students need time to struggle;
You [the teacher] are not the answer key;
Say "yes" to your students' ideas;
Play!
In this video, Jo Boaler and some of her students outline four messages they feel are important to consider when teaching math:
Everyone Can Learn Math to High Levels;
Believe in Yourself;
Struggle and Mistakes are Really Important;
Speed is Not Important.