Teaching Mathematics using an approach that is both conceptual and procedural -
Nine keys to the hybrid conceptual-procedural approach
Richard Andrew
Bullet #3 centres around how students best acquire conceptual understanding
The necessary prerequisite to learning mathematics which is best provided by a quality conceptual approach is conceptual understanding in students (students actually understanding what it is they are working on in class).
Key #3 unpacked
Proceduralists tend to hold the view that ‘If students gain sufficient practice answering questions through the correct use of procedures they will eventually understand what it is they are working on’. This viewpoint tends to lead proceduralists to assume that the procedural approach is at least equal to the conceptual approach in regard to bringing understanding to students.
However, I propose that the only teachers who make such a claim - that proceduralism is on par with conceptualism in regard to imparting understanding to students - are those who have never experienced delivering a quality conceptually-based unit.
This is because once we experience delivering a quality, well-structured, conceptually-based unit (and compare it to our experience of a procedurally-driven unit) it becomes obvious that the conceptual approach is superior at enabling students to understand what it is they are working on.
As an example - or perhaps as an analogy (because this example deals with one minor aspect of mathematics rather than an entire unit) let’s consider the teaching of the rounding of decimals. You’ll be familiar with the situation where you discover a number of students within a yr 7-10 class who should by now, be competent when rounding decimals yet clearly are not. We can either reteach rounding procedurally or we can present the concept conceptually. If we were to reteach rounding procedurally we would likely:
One of the fastest, most effective ways to conceptually approach the rounding of decimals is to workshop some open-ended questions.
To prepare the ground we’ll likely demonstrate, via Socratic questioning, diagrams and/or teacher-held equipment, what the first and second decimal places mean. Then we’ll move into the conceptual teaching of decimal rounding. It looks something like this:
Note this is actually a teacher-directed approach. However, it is also very conceptually-based and can occur at different stages within a student-centred, conceptually-based unit.
Note also that the initial focus is 100% on understanding and not at all on procedures or rules. Importantly, the above process is highly engaging for the students - as well as the teacher.
The above is just one example of why a (highly structured) conceptual approach is superior with respect to enabling students to acquire an understanding of concepts.
However, I propose that the only teachers who make such a claim - that proceduralism is on par with conceptualism in regard to imparting understanding to students - are those who have never experienced delivering a quality conceptually-based unit.
This is because once we experience delivering a quality, well-structured, conceptually-based unit (and compare it to our experience of a procedurally-driven unit) it becomes obvious that the conceptual approach is superior at enabling students to understand what it is they are working on.
As an example - or perhaps as an analogy (because this example deals with one minor aspect of mathematics rather than an entire unit) let’s consider the teaching of the rounding of decimals. You’ll be familiar with the situation where you discover a number of students within a yr 7-10 class who should by now, be competent when rounding decimals yet clearly are not. We can either reteach rounding procedurally or we can present the concept conceptually. If we were to reteach rounding procedurally we would likely:
- explain what a decimal is,
- explain the rules for rounding decimals
- demonstrate some examples
- have students work through some examples.
One of the fastest, most effective ways to conceptually approach the rounding of decimals is to workshop some open-ended questions.
To prepare the ground we’ll likely demonstrate, via Socratic questioning, diagrams and/or teacher-held equipment, what the first and second decimal places mean. Then we’ll move into the conceptual teaching of decimal rounding. It looks something like this:
- Write 3.7 (or similar) on the board.
- Say “OK students, there was a two-decimal-place number and we rounded it to one decimal place. The original number was 3 decimal 7 something ( 3 . 7 _ ) What might our original number have been?”
- We absolutely know that some students will have no idea what the answer is, and this is part of the plan.
- We also know that some students will offer some correct possibilities.
- By using the 9-second rule (not asking for an answer for 9 seconds) we wait for all students to ponder the question.
- A student then offers an answer (it doesn’t matter if it’s correct or incorrect)
- ‘Ben’ suggests 3.74
- We give no indication whether this is a correct solution or not.
- We say “What do you think folks … could the original number have been 3.74?” One student says “yes” and you ask why. You extract the correct reasons from the students.
- We then repeat the process several times ensuring at least one correct ‘3 . 6 _ ‘ answer is supplied.
- The vast majority of students who previously possessed a patchy understanding of rounding will be experiencing light bulb (aha) moments. This is powerful because they are seeing in front of their eyes how the rounding of decimals works. They see the process at work and it makes sense. In contrast, the procedural approach forces students to decipher the confusing and very long-winded rules used to explain when we round down and when we round up.
- Conversely, in the conceptual approach the rules are provided by the students AFTER the aha moments when we explicitly reinforce them.
- We then continue with the original lesson.
Note this is actually a teacher-directed approach. However, it is also very conceptually-based and can occur at different stages within a student-centred, conceptually-based unit.
Note also that the initial focus is 100% on understanding and not at all on procedures or rules. Importantly, the above process is highly engaging for the students - as well as the teacher.
The above is just one example of why a (highly structured) conceptual approach is superior with respect to enabling students to acquire an understanding of concepts.
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