## Conceptual approaches to teaching and learning mathematics need to be student-centered

... why student-centred approaches pose a major problem to high school teachers!

Part 1 in a 3-part series

Part 1 in a 3-part series

... why student-centred approaches pose a major problem to high school teachers!

Part 1 in a 3-part series

Part 1 in a 3-part series

This article, the first in a 3-part series, proposes that a conceptual approach to teaching mathematics should also be student-centered. It explains that the byproduct of a student-centered approach is an increase in the spread of students throughout a unit of work and that this increase in ‘student spread’ proves to be problematic for many high school teachers seeking to embrace a student-centered approach.

The second article Catering for student spread within mathematics classes offers some strategies which go a long way to solving the problem posed by increased student spread.

The third and final article Let’s encourage student spread in mathematics units argues the case in favour of encouraging student spread within a mathematics unit of work.

The second article Catering for student spread within mathematics classes offers some strategies which go a long way to solving the problem posed by increased student spread.

The third and final article Let’s encourage student spread in mathematics units argues the case in favour of encouraging student spread within a mathematics unit of work.

In the article Procedural knowledge vs conceptual knowledge in math education, I offer the following to give some idea of what a conceptual approach to teaching and learning mathematics can look like.

- Mathematical understanding in students is fostered through the use of activities and strategies specifically designed to engineer ‘aha moments’ in students.
- Student engagement is high, largely due to the use of quality, conceptually-based activities.
- Collaboration between students is actively fostered.
- Metacognition and other higher-order thinking processes are encouraged.
- Differentiation of instruction is commonplace, a natural by-product of the student-centred approach used.

Throughout history, the majority of math teaching has been teacher-directed. In other words, it has been the teacher who decides what students learn and when. It has been the teacher who sets the pace of learning. Traditionally, the teacher imparts mathematical instruction and demonstrates the proofs using a whole-class lecture format. Consequently, there has been little room for student collaboration and discovery. The traditional approach has typically focused on the direct teaching of mathematical rules and procedures. It has been a top-down, ‘teaching by telling’ approach.

A student-centered approach, in many ways, is the opposite of the above. In a student-centered approach, the student is at the centre of his or her learning. It is predicated on collaboration, inquiry, and discovery. A significant aspect in favour of a student-centered approach is the fact that it empowers students to begin to take ownership over their learning. In contrast, a teacher-centered approach has the teacher in charge of the students’ learning. This is never intentional; it is simply that 'the teacher being in charge of the students’ learning' is, by nature, a byproduct of a teacher-centered approach.

"Students need direct instruction" is a common cry from opponents of student-centered learning. In my view this cry is not entirely wrong; in fact, it is probably 'half right'. Students do require direct instruction. They need expert guidance. However, they also - absolutely - need to take ownership of their learning. And the reality is the only approach which engineers genuine student ownership of learning is one that is student-centered. Ironically, this is not obvious for a teacher until he or she has successfully embraced a student-centred approach.

I'm always intrigued when I see 'the student-centered approach' getting a bad rap when I read claims that student-centered approaches score poorly on a statistical scale in relationship to 'getting results'. First up, I'm not interested in 'results' until we have our students engaged, and drilling students through more teacher direction has never been an effective way to engage students. Secondly, what does "Student centred approaches are not successful" mean? How on earth do you control for that? Observe 50 different teachers using a student-centred approach and you'll see 50 very different approaches, some successful, some not. Teacher direction is way more 'tidy' and consistent as an approach to measure. Not so the student-centred approach.

What this boils down to is teacher skill. Give my child a highly skilled teacher-directed practitioner over her poorly-skilled student-centred counterpart any day. But give me the highly skilled, student-centred practitioner every day of the week because then I'll know my child is alive in the world of learning, making choices, taking ownership, loving learning.

I'm always intrigued when I see 'the student-centered approach' getting a bad rap when I read claims that student-centered approaches score poorly on a statistical scale in relationship to 'getting results'. First up, I'm not interested in 'results' until we have our students engaged, and drilling students through more teacher direction has never been an effective way to engage students. Secondly, what does "Student centred approaches are not successful" mean? How on earth do you control for that? Observe 50 different teachers using a student-centred approach and you'll see 50 very different approaches, some successful, some not. Teacher direction is way more 'tidy' and consistent as an approach to measure. Not so the student-centred approach.

What this boils down to is teacher skill. Give my child a highly skilled teacher-directed practitioner over her poorly-skilled student-centred counterpart any day. But give me the highly skilled, student-centred practitioner every day of the week because then I'll know my child is alive in the world of learning, making choices, taking ownership, loving learning.

When a sceptic hears the term 'student-centred approach to mathematics' the assumption often is - I believe - that students are involved in a lesson that looks like this: "Here's a bunch of equipment ... here are some reference books ... here's the internet ... see what you can learn about trigonometry. You've got two weeks". This is not at all what I'm advocating as a student centred approach to mathematics teaching. Rather, I'm advocating that a cleverly designed, well executed, student-centered approach MUST be highly scaffolded. It MUST contain quality instruction and guidance from the teacher. Such an approach will afford the teacher many opportunities for direct instruction.

We began this article talking about a conceptually-based approach to teaching and learning mathematics, and then segued into student-centered learning. I argue the two are closely related. I propose that a conceptually-based unit ideally needs to be student-centered and that a well constructed, well delivered, student-centered unit, by default, can be an ideal foundation to enable students to gain a conceptual understanding of the mathematics at hand.

As I mentioned above, I suspect when most teachers think of 'a conceptual approach' or 'student-centered learning' they imagine students surrounded by loads of hands-on materials yet learning very little of mathematical worth. Or they imagine students being left to their own devices to learn from un-scaffolded information. They imagine a teacher not teaching, not guiding, yet a teacher who is happy because "Hey, I’m doing student-centered learning – I’m swimming with the tide!" They imagine students having fun yet who are not particularly engaged mathematically. And they deduce - I would assume correctly - that the resulting test scores will be less than optimal.

The imaginings described above, however, are misconceptions which teachers who are not using a well-scaffolded, student-centred, conceptually-based approach tend to hold. (The article Misconceptions about conceptually-based math/s teaching contains more on this.)

The imaginings described above, however, are misconceptions which teachers who are not using a well-scaffolded, student-centred, conceptually-based approach tend to hold. (The article Misconceptions about conceptually-based math/s teaching contains more on this.)

That a well-scaffolded, student-centred conceptually-based approach is more engaging is obvious to any teacher experienced in delivering maths units in this way. It is not obvious, however, to teachers who haven't.

Time and again I have teachers who I have coaxed into implementing this approach tell me they are surprised by the increased depth of engagement of their students - deeper thinking, more passionate collaboration, a 'buzzier' classroom.

‘We can lead a horse to water but we can’t make him drink' is a commonly quoted analogy for life yet not typically applied to the classroom. Yet it is the for the classroom that the analogy is appropriate. Let’s find ways to have our ‘students drink at the trough of mathematics’. In my view, the only way to get our ‘students to drink at the trough of mathematics’ is through student engagement. And therein lies the case for using a well-scaffolded, student-centred, conceptually-based approach.

Time and again I have teachers who I have coaxed into implementing this approach tell me they are surprised by the increased depth of engagement of their students - deeper thinking, more passionate collaboration, a 'buzzier' classroom.

‘We can lead a horse to water but we can’t make him drink' is a commonly quoted analogy for life yet not typically applied to the classroom. Yet it is the for the classroom that the analogy is appropriate. Let’s find ways to have our ‘students drink at the trough of mathematics’. In my view, the only way to get our ‘students to drink at the trough of mathematics’ is through student engagement. And therein lies the case for using a well-scaffolded, student-centred, conceptually-based approach.

The main reason a teacher-directed approach is popular, apart from the "It’s the way I've always taught" reasoning, is that a teacher-directed approach is neat and tidy. We herd students together so they progress through a unit as uniformly as possible. We deliver the teaching segments to the whole class. We avoid, as much as possible, repeating ourselves. We resist students falling behind and we resist students getting too far ahead. On the surface, a teacher-directed approach appears sound and logical.

The problem with a teacher-directed approach, as described above, is that any given room-full of students never learn uniformly, not even when the students are closely streamed. They each learn at their own pace and they each learn slightly differently. 'Herding students' does not have any real effect on the rate students genuinely learn.

For many high-school mathematics teachers, when they embrace a student-centred approach they encounter 'a very big problem'. The very big problem is the increased student spread created by the student-centred approach. As soon as we cease our efforts to 'keep them all together', and students are free to progress at their own pace, the spread of students within a unit becomes significant. Many teachers are understandably threatened by this because they find themselves in unfamiliar territory - "My entire career I've taught in a way that minimises student spread. Now that I've released the brakes students are spread widely through the unit. How do I deal with this?"

The problem with a teacher-directed approach, as described above, is that any given room-full of students never learn uniformly, not even when the students are closely streamed. They each learn at their own pace and they each learn slightly differently. 'Herding students' does not have any real effect on the rate students genuinely learn.

For many high-school mathematics teachers, when they embrace a student-centred approach they encounter 'a very big problem'. The very big problem is the increased student spread created by the student-centred approach. As soon as we cease our efforts to 'keep them all together', and students are free to progress at their own pace, the spread of students within a unit becomes significant. Many teachers are understandably threatened by this because they find themselves in unfamiliar territory - "My entire career I've taught in a way that minimises student spread. Now that I've released the brakes students are spread widely through the unit. How do I deal with this?"

Three of the major challenges posed by increased student spread, from my experience, are:

Of course, not all high school mathematics teachers are bothered by student spread. A portion of those I work with are already comfortable dealing with student spread. But still, a high percentage are not.

I strongly suspect this apprehension of dealing with increased student spread is one of the reasons student-centered learning is not as widely accepted as it could be.

- Instruction (the biggest challenge) – how do I run my teaching sessions if some students are well ahead, and others are well behind 'the required point in the unit'?
- Early finishers – what do I do with them?
- Motivation – now that I’m not teaching my students as a whole class as much, I have fewer opportunities to 'do my performance'. So how do I motivate them?

Of course, not all high school mathematics teachers are bothered by student spread. A portion of those I work with are already comfortable dealing with student spread. But still, a high percentage are not.

I strongly suspect this apprehension of dealing with increased student spread is one of the reasons student-centered learning is not as widely accepted as it could be.

It is a challenge to write articles like this and not have it read as rhetorical ‘waffle’. And it is impossible to show teachers a path to follow only via articles. The only way I know how to successfully guide teachers into implementing a well-scaffolded, conceptually-based, student-centered approach is via a long-term, experiential guided learning journey. The articles will hopefully get people thinking and help lay some sort of foundation. However, the transition to the sort of approach being advocated here is a paradigm shift. To embrace a paradigm shift requires time, self-reflection, quality guidance, encouragement and practice.

The next article in this 3-part series Catering for student spread within mathematics classes explores the issue of increased student spread in greater depth.

The next article in this 3-part series Catering for student spread within mathematics classes explores the issue of increased student spread in greater depth.

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