Pursuing a mathematical mindset ... 3 schools on a journey ( part 2)

At our most recent Maths Flexible grouping  meeting, one of the major topics of discussion coming from our PLG's was a question from teachers around how students are supposed to acquire knowledge which is new to them when engaging in flexible grouping, problem solving mathematics tasks. The in-school leaders shared how this issue can form a stumbling block for some teachers when exploring this new pedagogy, and that we needed an answer that recognised the concerns of our teachers, but still allowed for growth and change.

I decided to return to the work of Jo Boaler and found in her recent book 'Mathematical Mindsets' what I think are some useful thoughts in relation to this issue.    
I have paraphrased it slightly so this post is not too long.

 " when I share open, inquiry-based mathematics tasks with teachers, such as the growing shapes task, they often ask questions such as, " I get that these tasks are engaging and create good mathematical discussions, but how do students learn new knowledge, such as trig functions? Or how to factorise? They can't discover it." This is a reasonable question, and we do have important research knowledge about this issue. It is true that while ideal mathematics discussions are those in which students use mathematical methods and ideas to solve problems, there are times when teachers need to introduce students to new methods and ideas. In the vast majority of mathematics classrooms this happens in a routine of teachers showing methods to students, which students then practice through textbook questions. In better mathematics classrooms, students go beyonds practicing isolated methods and use them to solve applied problems, but the order remains - teachers show methods, then students use them.
In an important study, researchers compared 3 ways of teaching mathematics

(1) Teacher shows the methods - students then solved problems with them.
(2) Students were left to discover methods through exploration.
(3) The third was a reversal of the typical sequence: the students were given problems to solve, and they did not know methods to solve them, but they were given opportunity to solve the problems, they became curious, and their brains were primed to learn new methods, so that when the teachers taught the methods, students paid greater attention to them and were more motivated to learn them. 
The researchers published their results with the title " A time for telling", and they argued that the question is not " Should we tell or explain methods?" but "When is the best time to do this?" Their study showed clearly that the best time was after students had explored the problems. Jo Boaler (pg 66, 2016).

The key component of this seems to be that what is important is that teachers give their students opportunities to explore mathematical ideas, using equipment and sharing their thoughts with others ( mixed ability plug here) and THEN students can go on to participate in targeted teaching sessions- as needed from that point. 

Check out part 1.