## Conceptual Mathematics Teaching:

4 Common Misconceptions Held By Teachers

In the article Procedural vs conceptual knowledge in mathematics education, I expand on James E Schwartz’s piece based on procedural and conceptual knowledge. To paraphrase Schwartz -

“Mathematical procedures, known to mathematicians as algorithms (are) procedures (which) enable (us) to find answers to problems according to set rules.”

## The traditional procedural approach

I argue that mathematical education has historically focused on imparting procedural knowledge through a ‘Procedures-first, Understanding-second' approach’ where teachers have as their first priority to teach procedures to students and where understanding hopefully follows with sufficient practice of those procedures. It would be tempting to describe below what the traditional Procedures-first, Understanding-second approach looks like in action. However, the important thing to note is the intent of the teacher. When we use the traditional procedures-first, understanding-second approach we hold the teaching of procedures as our #1 priority with understanding as our second priority.

The problem with presenting maths in a way that has the teaching of procedures as the #1 priority is that, by default, it results in students spending significant amounts of lesson time not understanding what they are working on, and not understanding what one is working on is unpleasant and demotivating. It inhibits learning. It makes the role of the teacher unnecessarily difficult.

Mathematical understanding is the realm of conceptual knowledge. In Procedural vs conceptual knowledge in mathematics education I propose that in order for students to acquire conceptual knowledge, the approach needs to be one that firstly aims to bring conceptual understanding to students and as a second priority, teaches procedures. In other words, we need a conceptual approach that also teaches procedures rather than a procedurally-based approach which 'hopes that understanding will naturally occur as a result of students learning those procedures'. We need an Understanding-first, Procedures-second approach.

Mathematical understanding is the realm of conceptual knowledge. In Procedural vs conceptual knowledge in mathematics education I propose that in order for students to acquire conceptual knowledge, the approach needs to be one that firstly aims to bring conceptual understanding to students and as a second priority, teaches procedures. In other words, we need a conceptual approach that also teaches procedures rather than a procedurally-based approach which 'hopes that understanding will naturally occur as a result of students learning those procedures'. We need an Understanding-first, Procedures-second approach.

## An Understanding-first, Procedures-second approach - what it looks like

- Mathematical understanding in students is fostered through the use of activities and strategies which enable ‘aha’ moments to occur within students.
- The activities used are inherently engaging.
- The approach is somewhat student-centred, yet at the same time highly-structured and well-scaffolded.
- Activities are structured in such a way that students are able to use their own logic and reasoning early on in the activities, rather than having to follow a teacher-provided routine.
- The initial activities lay the foundation for the procedures that follow; the procedures are presented traditionally but the students are better-able to make sense of them because they are familiar with the underlying mathematics.
- Collaboration between students is heavily promoted.
- There is a strong focus on meta cognition and other higher-order thinking.

There exists a great deal of confusion about conceptual approaches to teaching mathematics. A decade ago when presenting conceptual approaches it was common for teachers to assume I was suggesting that the teaching of procedures was unimportant. Now I try to make it clear to teachers that we are referring to a hybrid conceptual-procedural approach, or, as I've already mentioned, an Understanding-first, Procedures-second approach. In my view though, it is still very much a conceptual approach because conceptual understanding is the #1 focus.

## The challenge of transitioning to a conceptual approach

For many teachers of mathematics, the task of transitioning to a

Adding to the challenge of making this transition are the misconceptions commonly possessed by teachers who have not yet embraced this conceptual approach.

*(hybrid) conceptual approach*is a challenging one, not because a (hybrid) conceptual approach is especially difficult but simply because it is unfamiliar to teachers who have only taught 'traditionally'.Adding to the challenge of making this transition are the misconceptions commonly possessed by teachers who have not yet embraced this conceptual approach.

## Conceptually-based approaches to teaching: Four common misconceptions

Below are four common misconceptions held by teachers about conceptual approaches to teaching mathematics.

Few would disagree that hands-on activities are difficult to manage in many high school classrooms. In an ideal world, all students would explore hands-on materials within a well-structured activity, in a highly-engaged, inquiring and self-directed manner.

Clearly, many high school students have become removed from that ideal world! Therefore, releasing hands-on materials en-mass to your average high school mathematics class is likely to be ineffective in bringing about conceptual understanding. The late Grant Wiggins excellently explored this point in his Experiential Learning article by expanding on the statement “Just because it’s hands-on doesn’t mean it’s minds-on”.

**Misconception 1:****Conceptually-based teaching is synonymous with hands-on activities and hands-on activities are extremely difficult to manage with my students.**Few would disagree that hands-on activities are difficult to manage in many high school classrooms. In an ideal world, all students would explore hands-on materials within a well-structured activity, in a highly-engaged, inquiring and self-directed manner.

Clearly, many high school students have become removed from that ideal world! Therefore, releasing hands-on materials en-mass to your average high school mathematics class is likely to be ineffective in bringing about conceptual understanding. The late Grant Wiggins excellently explored this point in his Experiential Learning article by expanding on the statement “Just because it’s hands-on doesn’t mean it’s minds-on”.

The belief that conceptually-based teaching is synonymous with hands-on activities is, in my view, a misconception.

A quality, conceptual approach can incorporate some use of hands-on materials as part of well-scaffolded, teacher-led activities infused with multiple leading questions. However, students' interaction with the material may be best governed by the teacher in accordance with the capacity for the particular group of students for learning. In other words, the activities are essentially student-centred yet with a level of teacher direction. For example here are

This misconception usually stems from a couple of sources. Firstly, it can be linked to misconception 1, i.e. that a conceptual approach is one where lots of materials are distributed to students who are expected to spend considerable time exploring the materials to gain the required insights. Commonly, teachers with misconception 2 have experienced ‘losing’ significant time because such activities provided very little insight for students due to student disengagement, poor implementation or both. Such an experience causes teachers to retreat to the comfort of the traditional approach.

Secondly, teachers can conclude that conceptually-based teaching is more time consuming than procedural teaching because the approach – even when not involving hands-on materials – simply take longer than the procedural approach they are used to.

The reason this is a misconception, however, is that there are

This misconception comes about because a conceptually-based approach to teaching mathematics requires a very different pedagogy to a traditional procedural approach. One of the main differences is that a conceptual approach requires a degree of student-centred-ness and teachers who are not used to allowing students to be 'at the centre of their learning' such a change is often accompanied by a feeling of being out of control.

However, there are highly successful, highly skilled primary teachers (and high school teachers) all over the world who are able to successfully engage their students in ways that allow for individual progression; where multiple activities occur simultaneously and whose students are learning MUCH more than a set routines for answering text questions. Ask these teachers if what they are doing is difficult for them to manage and of course they will say no. To these teachers, a conceptually-based approach is an obvious one to use.

In my experience, once high school mathematics teachers have been guided through the transition to (an Understanding-first, Procedures-second) conceptual approach they become delighted by the improved engagement and understanding of their students and by how easy the conceptual approach is to manage.

This is absolutely a misconception and stems from the idea that teaching conceptually and assuming an 'active facilitator role' means the teacher becomes passive and has less opportunity to ‘work the group with charisma'. Arguably, a student-centred, conceptual approach offers more avenues for a teacher’s charisma to shine - there are more chances to connect with students one-on-one and in small groups - although the charisma will be expressed somewhat differently.

*three**Conceptually-Based Maths Activities that Don’t Require Hands-on Materials.***Misconception 2:****Conceptually-based teaching is too time-consuming to be realistically implemented.**This misconception usually stems from a couple of sources. Firstly, it can be linked to misconception 1, i.e. that a conceptual approach is one where lots of materials are distributed to students who are expected to spend considerable time exploring the materials to gain the required insights. Commonly, teachers with misconception 2 have experienced ‘losing’ significant time because such activities provided very little insight for students due to student disengagement, poor implementation or both. Such an experience causes teachers to retreat to the comfort of the traditional approach.

Secondly, teachers can conclude that conceptually-based teaching is more time consuming than procedural teaching because the approach – even when not involving hands-on materials – simply take longer than the procedural approach they are used to.

The reason this is a misconception, however, is that there are

*conceptually-based approaches which have proven to be more efficient than the traditional*. However, as has already been mentioned, an effective, conceptually-based approach requires a very different pedagogy to a traditional, procedural, 'let’s-keep-them-all-together' approach.**Misconception 3:****Conceptually-based teaching is too difficult to manage.**This misconception comes about because a conceptually-based approach to teaching mathematics requires a very different pedagogy to a traditional procedural approach. One of the main differences is that a conceptual approach requires a degree of student-centred-ness and teachers who are not used to allowing students to be 'at the centre of their learning' such a change is often accompanied by a feeling of being out of control.

However, there are highly successful, highly skilled primary teachers (and high school teachers) all over the world who are able to successfully engage their students in ways that allow for individual progression; where multiple activities occur simultaneously and whose students are learning MUCH more than a set routines for answering text questions. Ask these teachers if what they are doing is difficult for them to manage and of course they will say no. To these teachers, a conceptually-based approach is an obvious one to use.

In my experience, once high school mathematics teachers have been guided through the transition to (an Understanding-first, Procedures-second) conceptual approach they become delighted by the improved engagement and understanding of their students and by how easy the conceptual approach is to manage.

**Misconception 4:****I’ll lose my role as teacher, and therefore also my enjoyment that comes with that role.**This is absolutely a misconception and stems from the idea that teaching conceptually and assuming an 'active facilitator role' means the teacher becomes passive and has less opportunity to ‘work the group with charisma'. Arguably, a student-centred, conceptual approach offers more avenues for a teacher’s charisma to shine - there are more chances to connect with students one-on-one and in small groups - although the charisma will be expressed somewhat differently.

## Summary

From the outset, a conceptually-based approach to teaching mathematics appears inefficient and counter-intuitive. However, with measured guidance and numerous ‘feet on the ground’ examples of conceptually-based strategies and resources, the challenge can be mastered with relative ease.

**A Teacher PD course - Engaging students through a quality conceptual approach to math**

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