As a pre-service teacher, I think that relational understanding and instrumental understanding are not two different types of understanding but two stages of understanding. The two stages of understanding not only apply to mathematical learning but also apply to every single subject area of teaching and learning.
I view instrumental understanding as the basic and fundamental stage of the understanding. This then leads to the ultimate stage of understanding, namely, relational understanding.
Ideally, as educators, we want every student to have in depth relational understanding in every topic that we teach. However, there are limitations, such as under-funded public school systems, limited resources, diversity of the classroom, backwash effect of examinations, and over-burdened syllabi. In reality, the goal of relational understanding of every student in a given classroom is the biggest challenge that all math teachers are facing.
Thus, teaching is an art, an art of juggling between what needs to be done and what actually can be done from the given resources, between rushing through over-burdened syllabi and thoroughly cover in-depth material, and between obtaining superficial instrumental understanding and the ultimate goal of relational understanding. Although public education is not always the top priority of the government; we, public educators, do need to have a life-long passion of teaching in order to fight through all the possible obstacles and not to be discouraged in reaching the ultimate goal of relational understanding.
I like your observation that instrumental and relational understanding are not opposing approaches, but just two phases of learning. This is the beginning of an important insight!
ReplyDeleteHowever, I worry when you blame "the usual suspects" for the lack of progress to the relational in so many math classes. Lack of funding, full syllabus, the effect of examinations -- sure, these do affect our teaching, but I see many teachers carrying on with exclusively instrumental ways of teaching even when provincial exams are abolished, items are dropped from the syllabus and funding becomes available. I don't really believe that all those reasons are the real reasons that many teachers resort to a purely instrumental way of teaching. What do you think? How to get to a more in-depth understanding of this issue, rather than simply blaming external forces?