In problem-based teaching it is incredibly important that teachers don’t simply “tell students how to get the answers”, or they lose the opportunity for students to develop their own capacity for logical reasoning and analytical thought.
Instead, teachers have a very different role.
Firstly, they start by asking students a problem that they don’t yet know how to solve (insightful problems)! This requires the students to think mathematically, and experiment to try and work together to figure out a solution.
Secondly, they look out for student misconceptions (where the student has a fundamental misunderstanding of a concept), and help the students to analyse these ideas to see if they really work. When students self-correct their misconceptions, their mathematical understanding deepens and they learn concepts far more quickly.
Thirdly, and possibly most importantly, the teacher’s primary job becomes to ask really great questions that encourage students to think deeply about a problem, access their prior knowledge about it, experiment with different ideas and then analyse how well these ideas work.
Teachers help students to focus on the fundamental principles and patterns in mathematics, therefore enabling deep-level understanding to develop without as much need for repetition and memorisation.
Finally, explanation of algorithms and formulae follow the inquiry process, and are cemented through routine, application and non-standard problems before progressing to a new topic.