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.”

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 procedurallybased approach which hopes that understanding will naturally occur as a result of students learning those procedures. We need to adopt an Understandingfirst, Proceduressecond mindset.
An Understandingfirst, Proceduressecond 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 studentcentred, yet at the same time highlystructured and wellscaffolded, affording the teacher a sense of control, over the learning.
 Collaboration between students is integral to the approach.
 There is a strong focus on higherorder 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 conceptualprocedural approach, or, as I've mentioned here, an Understandingfirst, Proceduressecond 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 conceptualprocedural approach, or, as I've mentioned here, an Understandingfirst, Proceduressecond 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 (Understandingfirst, Proceduressecond).
Conceptuallybased approaches to teaching: Four common misconceptions
Misconception 1: Conceptuallybased teaching is synonymous with handson activities (which are extremely difficult to manage with my students.)
Few would disagree that handson activities are difficult to manage in many high school classrooms. In an ideal world, all students would explore handson materials within a wellstructured activity, in a highlyengaged, inquiring and selfdirected manner.
Clearly, many high school students have become removed from that ideal world! Therefore, releasing handson materials enmass 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 handson doesn’t mean it’s mindson':
Few would disagree that handson activities are difficult to manage in many high school classrooms. In an ideal world, all students would explore handson materials within a wellstructured activity, in a highlyengaged, inquiring and selfdirected manner.
Clearly, many high school students have become removed from that ideal world! Therefore, releasing handson materials enmass 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 handson doesn’t mean it’s mindson':
The belief that conceptuallybased teaching is synonymous with handson activities is, in my view, a misconception. Grant Wiggins
A quality, conceptual approach can incorporate some use of handson materials as part of wellscaffolded, teacherled 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 studentcentred yet with a level of teacher direction.
Misconception 2: Conceptuallybased teaching is too timeconsuming to be realistically implemented.
While it is true that many teachers have lost important lessontime experimenting with conceptualbased teaching, it is often not the approach that is timeconsuming, but rather the way the approach is being implemented in the classroom. When conceptualbased teaching is implemented poorly, students are likely to obtain little insight from the activities presented. However, when the approach is implemented well, conceptuallybased teaching has actually been seen to save time in the classroom.
This misconception also ties into the previous one, for handson activities that are not wellscaffolded and that lack teacher direction can definitely be more timeconsuming in the classroom.
Misconception 3: Conceptuallybased teaching is too difficult to manage.
This misconception, I suspect, stems from a fear of the unknown. A conceptuallybased 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 studentcentredness, which is a mode of operating foreign to many teachers.
The uninformed view of the studentcentred approach is that the teacher loses his or her sense of control. However, in the highlystructured studentcentred 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 Understandingfirst, Proceduressecond) 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: Conceptuallybased teaching is too timeconsuming to be realistically implemented.
While it is true that many teachers have lost important lessontime experimenting with conceptualbased teaching, it is often not the approach that is timeconsuming, but rather the way the approach is being implemented in the classroom. When conceptualbased teaching is implemented poorly, students are likely to obtain little insight from the activities presented. However, when the approach is implemented well, conceptuallybased teaching has actually been seen to save time in the classroom.
This misconception also ties into the previous one, for handson activities that are not wellscaffolded and that lack teacher direction can definitely be more timeconsuming in the classroom.
Misconception 3: Conceptuallybased teaching is too difficult to manage.
This misconception, I suspect, stems from a fear of the unknown. A conceptuallybased 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 studentcentredness, which is a mode of operating foreign to many teachers.
The uninformed view of the studentcentred approach is that the teacher loses his or her sense of control. However, in the highlystructured studentcentred 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 Understandingfirst, Proceduressecond) 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 studentcentred, conceptual approach offers more avenues for a teacher’s charisma to shine  there are more chances to connect with students oneonone and in small groups  although the charisma will be expressed somewhat differently.
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 studentcentred, conceptual approach offers more avenues for a teacher’s charisma to shine  there are more chances to connect with students oneonone and in small groups  although the charisma will be expressed somewhat differently.
Summary
From the outset, a conceptuallybased approach to teaching mathematics appears inefficient and counterintuitive. However, with measured guidance and numerous feetontheground examples of conceptuallybased 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 Understandingfirst, Proceduressecond 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.
Mini Course #1: NeedChoiceLevels Approach
Mini Course #2: A conceptual approach showcased through one topic
We'd love your thoughts below.
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If you don't create an account, please state your name at the start of your comment. Thanks.
Click the links to gain detailed info. Then sign up if you are keen.
Mini Course #1: NeedChoiceLevels Approach
Mini Course #2: A conceptual approach showcased through one topic
We'd love your thoughts below.
NOTE: Create a Hyvor account before commenting (click LOGIN) that way you'll be notified of replies and you won't be anonymous.
If you don't create an account, please state your name at the start of your comment. Thanks.