## Misconceptions about conceptual math/s teaching

11th April, 2017

11th April, 2017

In the article Procedural knowledge 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.”

I argue that mathematical education has historically focused on imparting procedural knowledge through a ‘rules-based approach’. I use the term ‘rules-based approach to refer to the traditional approach most of us were taught by, which looks something like the following:

“Here’s a rule …

here’s a theory behind the rule …

here’s an example involving the rule …

(write it down) …

here’re some questions for you to practice requiring the rule …

now here’s the next rule … ”

(and repeat).

The problem with a teaching and learning process that focuses on mathematical procedures is that the learning tends to occur at the expense of mathematical understanding.

Mathematical understanding is the realm of conceptual knowledge. In the Procedural knowledge vs conceptual knowledge in mathematics education article I propose that in order for students to acquire conceptual knowledge, the approach needs to have as it's main focus conceptual understanding. It needs to be 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 procedures'.

Mathematical understanding is the realm of conceptual knowledge. In the Procedural knowledge vs conceptual knowledge in mathematics education article I propose that in order for students to acquire conceptual knowledge, the approach needs to have as it's main focus conceptual understanding. It needs to be 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 procedures'.

- 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 typically student-centred, yet at the same time well-structured and well-scaffolded.
- The student-centred-ness of the approach is inherently engaging.
- Collaboration between students is heavily promoted.
- There is a strong focus on meta cognition and other higher-order thinking.
- Instruction is differentiated.

Shifting from a traditional approach to teaching mathematics using a conceptual approach is a challenging transition for teachers to make, requiring quality guidance and time. Adding to the challenge of making such a transition are the misconceptions commonly possessed by teachers who have not yet made this shift.

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

**Misconception 1: ****Conceptually-based teaching is synonymous with hands-on activities. 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, questioning and self-directed manner.

Clearly, many high school students have become removed from that ideal world! Therefore, releasing students into an abundant 'hands-on-materials situation' is likely to be ineffective in bringing about conceptual understanding. Grant Wiggins excellently explores 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 point I'm steering towards is this -

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, questioning and self-directed manner.

Clearly, many high school students have become removed from that ideal world! Therefore, releasing students into an abundant 'hands-on-materials situation' is likely to be ineffective in bringing about conceptual understanding. Grant Wiggins excellently explores 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 point I'm steering towards is this -

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 is best governed by the teacher in accordance with the capacity for the particular group of students for learning. In other words, the activities are student-centred but with a significant amount of teacher direction.

**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 the activities failing, sometimes dismally, through 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. I experienced this time-saving effect over many years. So, too, have many of the teachers I have worked with.

**Misconception 3: ****Conceptually-based teaching is too difficult to manage.**

Is a successful conceptually-based approach too difficult for a teacher to manage? It can be - if the teacher is not equipped with the necessary pedagogies then he/she is unlikely to cope. 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. Of course, it isn’t. To these teachers, a conceptually-based approach is an obvious one to use. The real issue is that the required skill set is very different to a traditional approach and therefore needs to be acquired.

**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 a facilitator role means the teacher becomes passive and has less opportunity to ‘work the group’ and to exude charisma into the classroom. 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.

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 the activities failing, sometimes dismally, through 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. I experienced this time-saving effect over many years. So, too, have many of the teachers I have worked with.

Is a successful conceptually-based approach too difficult for a teacher to manage? It can be - if the teacher is not equipped with the necessary pedagogies then he/she is unlikely to cope. 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. Of course, it isn’t. To these teachers, a conceptually-based approach is an obvious one to use. The real issue is that the required skill set is very different to a traditional approach and therefore needs to be acquired.

This is absolutely a misconception and stems from the idea that teaching conceptually and assuming a facilitator role means the teacher becomes passive and has less opportunity to ‘work the group’ and to exude charisma into the classroom. 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.

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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