Why We Need An
Understanding-first, Procedures-second Mindset
When Teaching Mathematics
Most teachers agree that procedural knowledge is important if students are to be successful in mathematics. However, in this article, I’ll explain why there’s a problem with an over-emphasis on procedural teaching.
The conventional Procedures-first mindset explained.
In my early years of teaching, I was a Procedures-first teacher. Allow me to explain …
As a Procedures-first teacher, I would walk into a classroom having, as my top priority, to teach the next series of procedures of the current unit of work and to have students successfully gain practice with those procedures. Of course, I occasionally threw in some student-centred activities and a couple of open investigations. However, my number one priority was to teach procedures. In fact, I didn’t even consider this to be a priority because teaching procedures was what I assumed mathematics teaching was. I have reason to believe that, although nobody teaches ONLY via procedures, the ‘Procedures-first’ mindset remains the dominant approach employed in maths classrooms today. 
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What about understanding?
As a Procedures-first teacher, my view was that understanding in students followed the sufficient practice of procedures. This was the view preached to me when I was a student, and it was certainly the mantra I preached to my students when I began teaching.
But as a teacher using Procedures-first, I knew that for many students, understanding never came and that for many, their understanding was little more than a vague sense of how the procedures worked.
Procedures-first has, by default, a major flaw.
Note that I am not criticising teachers. We all do our best with what we know. But the fact is, we only know what we know and we don’t know what we don’t know. And one thing that the many exceptional proceduralists apparently don't understand is that there’s a major flaw with the Procedures-first approach.
The major flaw: Students spending too much time not understanding maths
If we enter maths lessons holding as our #1 aim to teach procedures, then, by default, we cannot help but have students NOT understanding - especially initially - as they tackle their work.
Logic and experience inform me that, by default - and through no fault of the teacher - a Procedures-first approach results in many students NOT understanding what they are working on for significant amounts of lesson time.
Furthermore, the understanding that follows the practice of a procedure is mostly an understanding of how the procedure works. Having an understanding of how a procedure works is clearly important. However, let's not confuse an understanding of a procedure with an understanding of the concept upon which the procedure is based. The two are entirely different types of understanding. I’m proposing that we need students to understand the related concept BEFORE we teach them the next procedure.
Logic and experience inform me that, by default - and through no fault of the teacher - a Procedures-first approach results in many students NOT understanding what they are working on for significant amounts of lesson time.
Furthermore, the understanding that follows the practice of a procedure is mostly an understanding of how the procedure works. Having an understanding of how a procedure works is clearly important. However, let's not confuse an understanding of a procedure with an understanding of the concept upon which the procedure is based. The two are entirely different types of understanding. I’m proposing that we need students to understand the related concept BEFORE we teach them the next procedure.
Why students need to understand related maths concepts before learning procedures
The Committee on Developments in the Science of Learning, (2000, p. 8) cited 'learning with understanding (as) one of the hallmarks of the new science of learning'. In other words, understanding needs to be present for learning to occur.
That understanding needs to be present for learning to occur is so obvious I have no idea why it needs to be researched. (When was the last time you successfully learned something complicated that you could make zero sense of?) But, there you have it - the claim has been backed up by research - understanding needs to be present in order for learning to occur!
If we flip the statement around, we can safely claim that when we have students working with procedures before they understand the related concepts, their learning is thwarted.
That understanding needs to be present for learning to occur is so obvious I have no idea why it needs to be researched. (When was the last time you successfully learned something complicated that you could make zero sense of?) But, there you have it - the claim has been backed up by research - understanding needs to be present in order for learning to occur!
If we flip the statement around, we can safely claim that when we have students working with procedures before they understand the related concepts, their learning is thwarted.
Resilience
I can hear the objections … but Richard, students NEED to grapple with concepts before they can truly understand them. To believe that students are going to understand a concept first time every time is ridiculous! Grappling with mathematics builds resilience. And I fully agree. However, we all know that when students don’t understand the maths they are working on for extended periods of time, they begin to dislike - even hate - mathematics; they switch off. And the door to meaningful learning shuts.
Also - and this is my key point on this one - there is a massive difference between students grappling with some maths with a level of confidence that they will eventually understand it and students who have no clue at all about the maths that is in front of them.
In conclusion, I’m proposing that the Procedures-first mindset results in WAY too many students spending WAY too much time in WAY too many lessons not understanding the activities they have been given. This leads to disengagement, which, in turn, blocks them from learning and forms the mindset ‘I suck at maths’.
The follow-up question to ask, then, is this: How can we have more students having more understanding of what they are working on for more time in more lessons?
Also - and this is my key point on this one - there is a massive difference between students grappling with some maths with a level of confidence that they will eventually understand it and students who have no clue at all about the maths that is in front of them.
In conclusion, I’m proposing that the Procedures-first mindset results in WAY too many students spending WAY too much time in WAY too many lessons not understanding the activities they have been given. This leads to disengagement, which, in turn, blocks them from learning and forms the mindset ‘I suck at maths’.
The follow-up question to ask, then, is this: How can we have more students having more understanding of what they are working on for more time in more lessons?
The Understanding-first, Procedures-second mindset
Rather than having, as our over-riding priority, to teach our students procedures, why not prepare students for the coming procedures by presenting them with well-structured activities that foster understanding of the very concepts upon which the coming procedures are based?
Teachers who walk into their mathematics classrooms with an Understanding-first, Procedures-second mindset realise the importance of procedures and procedural knowledge. However, they realise that having students understand what they are working on for the majority of lesson time is a key aim.
Teachers who walk into their mathematics classrooms with an Understanding-first, Procedures-second mindset realise the importance of procedures and procedural knowledge. However, they realise that having students understand what they are working on for the majority of lesson time is a key aim.
Some principles of an Understanding-first, Procedures-second mindset/approach:
The following bullets are common to any Understanding-first approach. More could be added.
If you are thinking ‘Well, Richard, this sounds fine in theory BUT in practice I can't see it working … “ then check out the article ‘The Understanding-first, Procedures-second Approach in Action - Three Examples’.
- Use activities that enable students to use their own thinking, in ways that have them working deeply with the concepts associated with the soon-to-be-taught procedures.
- (Therefore) use well-structured activities that have a degree of open-endedness and student-centeredness.
- Where possible and when appropriate, have students determine the associated procedures.
- Generally, strive to give students the opportunity to understand the maths they are working on from the get-go and for the majority of lesson time.
If you are thinking ‘Well, Richard, this sounds fine in theory BUT in practice I can't see it working … “ then check out the article ‘The Understanding-first, Procedures-second Approach in Action - Three Examples’.
Related Articles
Why Students Need To Understand The Concept Before We Teach Them The Procedure - here
The Understanding-first, Procedures-second Approach In Action - Three Examples - here
Why Compartmentalising Is A Bad Idea When Teaching Mathematics - here
70% Of Capable Students Are Failing Mathematics. What Can We do? - here
The Case For NOT Teaching Procedures - here
Call to Action
I'd love to hear your thoughts. What points do you agree with? What do you disagree with? Do share!
