## 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 practise of those procedures. Following on, it would be tempting to describe what a traditional Procedures-first, Understanding-second approach looks like in action. However, the procedures-first, understanding-second is more of a mindset - an attitude - than an approach. When we use the traditional procedures-first, understanding-second approach, we hold the teaching of procedures as our #1 priority with understanding following as a second priority.

The problem with presenting maths in a way that prioritises the teaching of procedures over understanding is that, by default, it results in students spending significant amounts of lesson time not understanding what they are working on. To say this is problematic is an understatement because when students don’t understand the activities they are working through the experience is unpleasant and demoralising. It inhibits learning. Furthermore, 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 teaching approach needs to firstly bring conceptual understanding to students, before prioritising the teaching of 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 to adopt an Understanding-first, Procedures-second mindset.

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 teaching approach needs to firstly bring conceptual understanding to students, before prioritising the teaching of 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 to adopt an Understanding-first, Procedures-second mindset.

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

- Mathematical understanding in students is fostered through the use of activities and strategies which maximise the occurrence of aha moments for students.
- The activities used are inherently engaging because they require students to use their own logic and reasoning early on, rather than having to follow a procedure provided by the teacher.
- The initial activities lay the foundation for the procedures that follow; the procedures are presented traditionally, however students are more able to make sense of them because they are familiar with the underlying mathematics.
- The approach is somewhat student-centred, yet at the same time highly-structured and well-scaffolded, affording the teacher a sense of control, over the learning.
- Collaboration between students is integral to the approach.
- There is a strong focus on higher-order thinking, including metacognition.

## Be careful of speaking in binary

For a long time, I’ve been advocating the adoption of a conceptual approach by mathematics teachers. However, the message was muddied by the following misinterpretation: ‘A conceptual approach obviously means a lack of emphasis on the teaching of procedures’. I had never advocated NOT teaching procedures, but I had not included the word ‘procedures’ in the name. This misconception remains true today because I commonly receive comments to articles stating “I didn’t realise there was an approach that dealt with both concepts and procedures.”

More recently, I’ve made it clear to teachers that we are referring to a hybrid conceptual-procedural approach, or, as I've mentioned here, an Understanding-first, Procedures-second approach. In my view, however, it is still very much a conceptual approach because conceptual understanding is the initial focus.

More recently, I’ve made it clear to teachers that we are referring to a hybrid conceptual-procedural approach, or, as I've mentioned here, an Understanding-first, Procedures-second approach. In my view, however, it is still very much a conceptual approach because conceptual understanding is the initial focus.

## The challenge of transitioning to a conceptual approach

For many teachers of mathematics, the task of transitioning to a (hybrid) conceptual approach is a challenging one, not because a (hybrid) conceptual approach is especially difficult to implement. Rather, simply because it is unfamiliar to teachers who have always presented mathematics via a traditional procedural approach (Understanding-first, Procedures-second).

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

**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. Grant Wiggins

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, student interactions with the materials 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 3 Conceptually-Based Maths Activities that Don’t Require Hands-on Materials.

While it is true that many teachers have lost important lesson-time experimenting with conceptual-based teaching, it is often not the approach that is time-consuming, but rather the way the approach is being implemented in the classroom. When conceptual-based teaching is implemented poorly, students are likely to obtain little insight from the activities presented. However, when the approach is implemented well, conceptually-based teaching has actually been seen to save time in the classroom.

This misconception also ties into the previous one, for hands-on activities that are not well-scaffolded and that lack teacher direction can definitely be more time-consuming in the classroom.

This misconception, I suspect, stems from a fear of the unknown. 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, which is a mode of operating foreign to many teachers.

The uninformed view of the student-centred approach is that the teacher loses his or her sense of control. However, in the highly-structured student-centred approach advocated here, the opposite is true - teachers can feel like they have more control over the learning.

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

**Misconception 2:****Conceptually-based teaching is too time-consuming to be realistically implemented.**While it is true that many teachers have lost important lesson-time experimenting with conceptual-based teaching, it is often not the approach that is time-consuming, but rather the way the approach is being implemented in the classroom. When conceptual-based teaching is implemented poorly, students are likely to obtain little insight from the activities presented. However, when the approach is implemented well, conceptually-based teaching has actually been seen to save time in the classroom.

This misconception also ties into the previous one, for hands-on activities that are not well-scaffolded and that lack teacher direction can definitely be more time-consuming in the classroom.

**Misconception 3:****Conceptually-based teaching is too difficult to manage.**This misconception, I suspect, stems from a fear of the unknown. 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, which is a mode of operating foreign to many teachers.

The uninformed view of the student-centred approach is that the teacher loses his or her sense of control. However, in the highly-structured student-centred approach advocated here, the opposite is true - teachers can feel like they have more control over the learning.

In my experience, once high school mathematics teachers have been guided through the transition to (an Understanding-first, Procedures-second) conceptual approach they are impressed 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.

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## Calls to action

Below are links to two free tutorials which fit nicely into an Understanding-first, Procedures-second mindset. Note that the approaches covered in the tutorials are about much more than the specific topic in focus - i.e. avoid thinking “I know how to teach that topic, therefore I don’t need the tutorial’ (!!)

Click the links to gain detailed info. Then sign up if you are keen.

We'd love your thoughts at the bottom of the page. (Your email address will not be required)

Click the links to gain detailed info. Then sign up if you are keen.

**Tutorial #1**: Need-Choice-Levels Approach**Tutorial #2**: A conceptual approach showcased through one topicWe'd love your thoughts at the bottom of the page. (Your email address will not be required)