Key #5 refers to the authentic engagement of students
The authentic engagement of students is a fundamental pre-requisite for the learning of mathematics but it rarely features as part of the procedural vs conceptual debate. Unless students are authentically engaged in what they are working on - i.e. immersed in activities, understanding the mathematics and owning their learning - then their ability to learn is severely handicapped. A well-structured conceptual approach is superior when it comes to authentically engaging students i.e. immersing them in activities, having them understand what it is they are working on and owning their learning.
Key #5 unpacked
When student engagement is mentioned as part of any discussion on maths education it is rarely given the status it deserves. Perhaps this is because student engagement is difficult to measure and because it’s often confused with ‘the entertainment of students via games and jovial dispositions of the teacher’. This is why at Learn Implement Share we use the term 'Authentic Student Engagement'.
Students are authentically engaged when:
They want to be in the room.
They don’t need to be one of the ‘better’ students in order to enjoy learning.
They find the tasks interesting.
The activities (including well-constructed worksheets) enable them to be immersed in the work and understand what they re doing.
They have a sense of control over their learning because they are at the centre of it.
The student-centred nature of the activities enables them to have a sense of ownership over the learning.
The learning is social because collaboration and peer learning are key features of the approach.
Can’t students be equally engaged via a procedural approach? Some proceduralists will argue that the procedural approach is just as effective at engaging students and having them take ownership of their learning. No doubt there are some proceduralists who are better at engaging their students compared to some ‘not-so-engaging’ conceptualists. But that is an entirely different argument altogether! What I’m proposing here is that any quality procedural teacher will find that his/her ability to authentically engage students will significantly improve through embracing a highly-structured, (somewhat) student-centred, (hybrid) conceptual approach.