Conceptual understanding versus procedural knowledge has lead to a rich an insightful debate in the field of mathematics education over last 50 years. On this page I have posted the salient literature which has informed my view of conceptual understanding, and I invite your comments below.
As you peruse this literature, I invite you to reflect on your own experience (whether you are a teacher, student or parent):
1. Do most of your lessons start with procedural or conceptual knowledge?
2. Is the maths curriculum too focused towards one style of learning? Procedural or conceptual?
3. What are the benefits in your eyes of conceptual over procedural? Does procedural have benefits over conceptual understanding?
More fascinating research papers can be found here.
Crooks and Alibi review the literature on Conceptual Understanding.
A bit of a long read, but this literature review summarises the debates and issues around conceptual understanding in maths education.
More recently, Foster (2014) discussed this debate further.
I will leave you to read these amazing articles yourself, and I’d welcome your comments at the bottom of this page.
Relational Understanding vs Instrumental Understanding
Richard Skemp (1976) discussed the difference between relational and instrumental understanding. This is a debate I’ve had with myself and other teachers many times.
Starr on the other hand argues for a different view. He suggests that the common view of procedural knowledge is not sufficient to account for deep procedural knowledge. He points to mathematical procedures that are intuitive, heuristic and/or sophisticated, which require more than a simplistic understanding to be able to execute.
This line-of-thought has influenced my approach to teaching tedious procedures.
Rittle-Johnson and Siegler demonstrate evidence to support their view that conceptual and procedural knowledge develop iteratively.
Iteratively means here that incremental progress in one type of knowledge leads to progress in the other.
Pesek & Kirshner conducted an experiment in which two groups of students learned about area and perimeter. One group was taught exclusively through conceptual (relational) instruction. The other recieved procedural (instrumental) instruction before conceptual. The first group faired better in the post-test, even with less lesson time.
The researchers explain this by discussing how the instrumental instruction can lead to cognitive interference while learning conceptual ideas.
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