## 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

mathematics teaching debate is flawed

10 steps which explain and map out a path forward

Richard Andrew 6th Sept 2018

Disclaimer: Contains some course promotional statements.

The debate continues to rage - Is it the procedural approach to teaching mathematics that is the more effective or is it the conceptual approach that is superior for students learning mathematics?

This article proposes that the long-running ‘Procedural vs Conceptual’ debate is flawed.

I have unpacked the reasons why the debate is flawed and included some steps forwards in ten bullet points below. However, many of the bullets require expanding. Links to the expansions are provided throughout.

**One**

Both Procedural and Conceptual approaches to the teaching of mathematics contain necessary keys in order for students to be successful with high school mathematics.

The debate continues to rage - Is it the procedural approach to teaching mathematics that is the more effective or is it the conceptual approach that is superior for students learning mathematics?

This article proposes that the long-running ‘Procedural vs Conceptual’ debate is flawed.

I have unpacked the reasons why the debate is flawed and included some steps forwards in ten bullet points below. However, many of the bullets require expanding. Links to the expansions are provided throughout.

Both Procedural and Conceptual approaches to the teaching of mathematics contain necessary keys in order for students to be successful with high school mathematics.

The necessary key to learning mathematics which is best provided by a quality procedural approach is the explicit teaching of routines and procedures.

The necessary key to learning mathematics which is best provided by a quality conceptual approach is conceptual understanding in students (students understanding what it is they are doing in class).

The Procedural vs Conceptual debate assumes that the two approaches are mutually exclusive, that we use one approach to focus on the teaching of procedures and the other to focus on ‘teaching for understanding’.

However, the procedural vs conceptual debate is flawed because the common belief (that the two approaches are mutually exclusive) is a misconception. A hybrid approach exists that includes the essential keys from both approaches - the explicit teaching of procedures as well as ‘enabling students to understand what it is they are doing’.

The authentic engagement of students is a fundamentally important foundation for the learning of mathematics but it rarely features as part of the procedural vs conceptual debate. Unless students are authentically engaged in what it is they are doing - i.e. immersed in activities, understanding the mathematics and owning their learning - then their ability to learn is severely handicapped. There is little doubt that a quality conceptual approach is the superior of the two when it comes to authentically engaging students - immersing them in activities, having them understand what it is they are doing and owning their learning.

The hybrid conceptual approach needs to be highly-structured, mostly student-centred, conceptually-based and one that also incorporates the explicit teaching of procedures. It is important to note that the (successful) hybrid approach cannot be one that is teacher-directed and procedural for reasons that have been covered in the bullet expansions.

Adopting the hybrid approach requires a very different pedagogy to a traditional procedural approach. Although the ‘hybrid’ conceptual approach is inspiring and easy to administer once sufficient experience has been gained, the transition from a traditional procedural approach to the hybrid requires significant time and guidance.

The transition requires a paradigm shift in the teachers’ thinking and classroom management. Ideally, teachers making the shift will receive some quality, ongoing professional guidance which requires them to implement a range of well-structured, student-centred, conceptually-based activities.

Learn Implement Share provides long-term guidance to mathematics teachers regarding the transition via several comprehensive online courses. Each course caters for individuals as well as department-wide TEAMs. (Links to information are provided below.)

To see what a course looks like that addresses the hybrid conceptual approach check out the 2½ minute video below.

Engagement: Winning over your mathematics class: **Info** *Testimonials*

Conceptual Coordinate Geometry:**Info** **Testimonials**

Conceptual Coordinate Geometry:

Some final points need mentioning with regard to adopting the ‘hybrid’ conceptual approach.

One common misconception which needs to be dispelled is that conceptual approaches to teaching maths all require hands-on activities with lots of equipment. This misconception is held by many conceptualists as well as, I believe, most proceduralists. If this were true then it would be difficult to be an advocate for the conceptual approach because the free use of hands-on equipment in most high school maths classes is typically problematic. The example given when unpacking Bullet #3 (the rounding of decimals) likely involved some equipment but equipment that was handled only by the teacher in conjunction with quality, targeted, Socratic questioning.

In summary, the (hybrid) conceptual approach rarely requires students to be handling lots of equipment, certainly not in an unstructured way.

In summary, the (hybrid) conceptual approach rarely requires students to be handling lots of equipment, certainly not in an unstructured way.

A good way to express the difference between a quality conceptual approach and a quality procedural approach is as follows:

Teachers using a quality procedural approach have as their number one initial aim to teach routines and procedures well. Understanding follows as a secondary focus. (E.g. ‘This lesson I’m aiming for the students to successfully use the next five procedures and to gain practice with questions requiring those procedures. And with sufficient practice they will - hopefully - understand.)

Teachers using a quality conceptual approach have as their number one initial aim to have students understand what it is they are doing from (almost) the beginning of each activity. Then, when appropriate, students will be taught the related procedures. Conceptually-based activities typically have students ‘getting their hands dirty’ with the mathematics related to the up-coming formulas/procedures so that when students are explicitly taught those procedures they recognise the procedures and the underpinning mathematics.

Note that while the decimal rounding example was relatively easy to articulate it would be almost impossible to explain the ‘hybrid’ conceptual approach for an entire unit. It would require multiple examples each with several sequenced pages containing explanations, stories and videos as well as sufficient time to assimilate the information and implement the approach. The name for such an explanation is ‘comprehensive online course’.

Teachers using a quality procedural approach have as their number one initial aim to teach routines and procedures well. Understanding follows as a secondary focus. (E.g. ‘This lesson I’m aiming for the students to successfully use the next five procedures and to gain practice with questions requiring those procedures. And with sufficient practice they will - hopefully - understand.)

Teachers using a quality conceptual approach have as their number one initial aim to have students understand what it is they are doing from (almost) the beginning of each activity. Then, when appropriate, students will be taught the related procedures. Conceptually-based activities typically have students ‘getting their hands dirty’ with the mathematics related to the up-coming formulas/procedures so that when students are explicitly taught those procedures they recognise the procedures and the underpinning mathematics.

Note that while the decimal rounding example was relatively easy to articulate it would be almost impossible to explain the ‘hybrid’ conceptual approach for an entire unit. It would require multiple examples each with several sequenced pages containing explanations, stories and videos as well as sufficient time to assimilate the information and implement the approach. The name for such an explanation is ‘comprehensive online course’.

The term ‘hybrid’ was woven into this article as a way of explaining that the strengths of the procedural approach can be incorporated into a quality conceptual approach. However, in courses, we don’t refer to a ‘hybrid’ conceptual approach - we simply call it a quality conceptual approach which, naturally - in our view - incorporates the explicit teaching of procedures.

Your input is welcome. We’d love to know what you agree with, what surprised you, what you disagree with (and why), any questions you have, etc.

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