Teaching Mathematics Using An Approach That Is Both Conceptual And Procedural -
Nine Keys To The Hybrid Conceptual-Procedural Approach
Key #6 Unpacked
Key #6 explains why the hybrid approach is student-centred and why the focus is on understanding first and procedures second.
The hybrid conceptual approach is a highly-structured, mostly student-centred, conceptually-based approach but one that also incorporates the explicit teaching of procedures.
There are two important principles required for the hybrid conceptual approach to be successful:
The hybrid conceptual approach must be (predominantly) student-centred. Yes, it is highly structured. Yes, the teacher still has control over the direction of the learning. But it cannot be driven by the teacher-centredness that drives the procedural approach.
The first aim of any teacher using the hybrid conceptual approach is to have students understand what it is they are doing within an activity. The second aim of the teacher - also important but nevertheless it must follow the first - is the teaching of procedures.
Key #6 unpacked
How can a student-centred, conceptual-based approach incorporate the explicit teaching of procedures? In order to answer this question, we need to unpack student-centred learning.
Student-centred learning receives a bad rap by many mathematics educationalists. The problem with discussing it is that student-centred learning assumes many different forms. Some of those forms lack structure. It is this (assumption of) lack of structure that I suspect is the reason for the bad rap. However, student-centred learning can also be highly structured, and in the hybrid conceptual approach it is.
The challenge of student-centredness and the need to deal with a wide spread of students within a unit of work.
One consequence of any student-centred unit of work is that it increases the degree to which students are spread throughout a unit. I suspect, by the way, this increase in student spread is a major reason for the reluctance in some teachers to adopt student-centred units of work.
The reality is that unless teachers know how to effectively deal with a wide student spread they often revert to the safety of teacher-centredness. This is why, through Learn Implement Share, we spend considerable time showing teachers how to embrace and manage student spread. Dealing with student spread is not difficult once you become comfortable with some strategies. However, it requires some guidance. Learning to master increased student spread, for example, is not something many teachers can easily adopt through a quick browse of a couple of articles.
Mini-Lessons as a vehicle for the explicit teaching of procedures
In a nutshell, because of the increased student spread associated with the hybrid conceptual approach, many of the explicit ‘teaching spots’ - integral to any unit - occur to small groups of students rather than as whole-class lectures. This is because students reach the various points within a unit requiring specific instruction at different times.
On the surface, this explicit teaching via mini-lessons appears to be inefficient - why would we want to teach something several times when we can teach it once to the whole class? Yet there are powerful advantages to delivering instructions multiple times via mini-lessons. These advantages are premised on the creating of ‘a need’ in students to learn what it was we want them to learn.
The main point is this - it is much more effective to teach students some content when they are ready and asking for it than it is to teach the content when we decide they need it.
One of the best ways to deliver information to students when they need it and when they are seeking it is to use mini-lessons. Don’t forget, that in the hybrid conceptual approach we are using mini-lessons BECAUSE the unit is student-centred and THEREFORE much more engaging for students. Remember - placing students at the centre of their learning raises their engagement.
Although there’s more to the use of mini-lessons, in essence, the above explains why we use them and how the process works. In the hybrid conceptual approach, most of the teaching of procedures occurs via mini-lessons but some, of course, occurs to the whole class.
What about flipped mastery learning?
Anyone who has embraced the flipped mastery approach to learning - delivering comprehensive online units to students - will know that flipped mastery is very student-centred and deals with student spread naturally. However, many teachers are yet to embrace the flipped-mastery approach and in some situations, flipped-mastery isn’t appropriate due to technology limitations. Hence it is not a focus of this article.