The Procedural vs Conceptual approaches to
mathematics teaching debate is flawed
10 steps which explain and map out a path forward
Richard Andrew 6th Sept 2018
The Procedural vs Conceptual approaches to
Bullet #3 - Conceptual Understanding
The necessary key which is best provided by a quality conceptual approach is conceptual understanding in students (students understanding what it is they are doing in class).
Unpacking bullet #3
Proceduralists tend to hold the view that ‘If students gain sufficient practice answering questions through the correct use of routines they will eventually understand what it is they are doing’. This viewpoint allows them to assume that the procedural approach is at least equal to the conceptual approach in regard to bringing understanding to students. However, I suggest teachers making this claim have not experienced delivering a quality conceptually-based unit because when one has delivered such a unit it becomes very obvious that a quality conceptual approach is superior at enabling students to understand what it is they are doing.
As an example - or perhaps as an analogy (because this example deals with one minor aspect of maths rather than an entire unit) let’s consider the teaching of the rounding of decimals. You’ll be familiar with the situation where you discover a number of students within a yr 7-10 class who should by now, be competent when rounding decimals yet clearly are not. We can either reteach rounding procedurally or we can do it conceptually. If we were to reteach rounding procedurally we would likely:
One of the fastest, most effective ways to conceptually approach the rounding of decimals is to workshop some open-ended questions.
To prepare the ground you’ll likely demonstrate, via Socratic questioning, diagrams and/or teacher-held equipment, what the first and second decimal places mean. And then you’ll move into the conceptual teaching of decimal rounding. And it looks something like this:
Note also that the initial focus is 100% on understanding and not at all on procedures or rules. Importantly, the above process is highly engaging for the students - as well as the teacher.
As an example - or perhaps as an analogy (because this example deals with one minor aspect of maths rather than an entire unit) let’s consider the teaching of the rounding of decimals. You’ll be familiar with the situation where you discover a number of students within a yr 7-10 class who should by now, be competent when rounding decimals yet clearly are not. We can either reteach rounding procedurally or we can do it conceptually. If we were to reteach rounding procedurally we would likely:
- explain what a decimal is,
- then explain the rules for rounding decimals
- then demonstrate some examples
- and finally, have students work through some examples.
One of the fastest, most effective ways to conceptually approach the rounding of decimals is to workshop some open-ended questions.
To prepare the ground you’ll likely demonstrate, via Socratic questioning, diagrams and/or teacher-held equipment, what the first and second decimal places mean. And then you’ll move into the conceptual teaching of decimal rounding. And it looks something like this:
- You write 3.7 (or similar) on the board.
- You say “OK students, there was a two-decimal-place number and we rounded it to one decimal place. The original number was 3 decimal 7 something ( 3 . 7 _ ) What might our original number have been?”
- You absolutely know that some students will have no idea what the answer is, and this is part of the plan.
- You also know that some students will offer some correct possibilities.
- By using the 9-second rule (not asking for an answer for 9 seconds) you wait for all students to ponder the question.
- A student then offers an answer (it doesn’t matter if it’s correct or incorrect)
- Ben suggests 3.74
- You give no indication whether this is a correct solution or not.
- You say “What do you think folks … could have the original number been 3.74?” One student says “yes” and you ask why. You extract the correct reasons from the students.
- You then repeat the process several times ensuring at least one correct ‘3 . 6 _ ‘ answer is supplied.
- The vast majority of students who previously possessed a patchy understanding of rounding will be experiencing light bulb (Aha) moments. This is powerful because they are seeing in front of their eyes how the rounding of decimals works. They see the process at work and it makes sense. In contrast, the Procedural approach forces students to decipher the confusing and very long-winded rules used to explain when we round down and when we round up.
- The rules are provided by the students AFTER the aha moments and you explicitly reinforce them.
- You then continue with the original lesson.
Note also that the initial focus is 100% on understanding and not at all on procedures or rules. Importantly, the above process is highly engaging for the students - as well as the teacher.