## Procedural vs Conceptual Knowledge

in Mathematics Education

Working Towards A Conceptual Approach To Teaching Mathematics - Without Ignoring Procedures

In the article ‘A Distinction Between Conceptual Knowledge and Procedural Knowledge‘ J. E. Schwartz writes poignantly about the difference between conceptual knowledge and procedural knowledge, how math education has historically focused on the latter, and how having procedural knowledge without conceptual knowledge thwarts a person’s mathematical development. To quote Schwartz –
‘Chances are, when you learned elementary mathematics, you learned to perform mathematical procedures. Known to mathematicians as algorithms, these procedures enabled you to find answers to problems according to set rules. If, for example, you think of division in terms of “divide, multiply, subtract, bring down” then you learned a division procedure (or algorithm). For another example, if you think only in terms of cross multiplying as a way of approaching problems involving proportions, chances are you learned only a procedure for solving mathematical proportions. At this point you may be wondering, “What else is there? What else would a person learn in a mathematics class?” The answer is, there is a great deal more to mathematics! These mathematical procedures are much like recipes that efficiency experts have developed to enable people to go straight to specific kinds of answers when confronted with particular kinds of well-defined problems.’ |

Today much is written about the need to improve the teaching and learning process in schools. The inference is that we need to be employing quality maths teaching strategies. But what are quality maths teaching strategies?

To begin to answer this question, let’s look at the process of imparting procedural knowledge, a process which has been widely used since we began teaching mathematics in schools centuries ago.

## Terminology

Words are symbols, as are phrases. This means that different words hold different meanings for different people. Some words are more consistently understood than others. For example, the ideas that come to mind for mathematics teachers when they encounter the words ‘procedural approach’ tend to be somewhat similar. Presumably, this is because most of us were taught mathematics via a procedural approach.

Conversely, the words ‘conceptual approach’ conjures up different meanings for different teachers. This is because conceptual approaches to mathematics teaching vary enormously.

Conversely, the words ‘conceptual approach’ conjures up different meanings for different teachers. This is because conceptual approaches to mathematics teaching vary enormously.

So, let’s talk briefly about terminology. As is my style, I will state this as simply as possible.

At Learn Implement Share, we advocate the type of approach outlined in point 4., above. For this reason, we use the following terms interchangeably:

We also use the term ‘a procedural approach’ interchangeably with ‘a Procedural-first, Understanding-second approach’, which is explained below.

Again, the reason for using these terms interchangeably is to minimize the chance of being misunderstood. In my early days of working with teachers, I would use the term Conceptual Approach to infer one that prioritized understanding but NOT at the expense of procedures. I was dismayed to discover that many teachers assumed I was advocating the exclusion of procedures!

- Conceptual knowledge is the result of a student successfully acquiring conceptual understanding.
- Procedural knowledge is the result of a student successfully learning a procedure.
- A conceptual approach (conceptually-based approach) can be many things:
- A Montessori-type approach
- A very open, unstructured approach (“here’s a set of equipment, see what you can discover about …”) with minimal emphasis on the teaching of procedures
- A relatively structured approach using lots of hands-on equipment with minimal emphasis on the teaching of procedures
- A highly structured approach, requiring minimal hands-on equipment, with activities designed to have students use their own logical reasoning and be immersed in the mathematics upon which the procedures - that will follow - are based.

At Learn Implement Share, we advocate the type of approach outlined in point 4., above. For this reason, we use the following terms interchangeably:

- Conceptual approach.
- Hybrid conceptual-procedural approach.
- Hybrid conceptual approach.
- An Understanding-first, Procedures-second approach.

We also use the term ‘a procedural approach’ interchangeably with ‘a Procedural-first, Understanding-second approach’, which is explained below.

Again, the reason for using these terms interchangeably is to minimize the chance of being misunderstood. In my early days of working with teachers, I would use the term Conceptual Approach to infer one that prioritized understanding but NOT at the expense of procedures. I was dismayed to discover that many teachers assumed I was advocating the exclusion of procedures!

## A ‘Procedures-first, Understanding-second’ approach explained

It would be tempting, at this point, to describe, step-by-step the procedural (Procedures-first, Understanding-second) approach - i.e. demonstrate the first procedure, have students copy, show the procedure in context, have students practice, and so on. But this would be missing the point. In my view, the Procedures-first, Understanding-second approach is less about WHAT happens and more about the mindset, or intention, of the teacher using such an approach.

If I’m using a Procedures-first, Understanding-second approach - as I did for many years - then my intention and (almost) entire focus as I enter a lesson is two-fold:

As has already been stated, how I go about achieving this endeavour is less significant than the above mindset. Of course, a Procedures-first, Understanding-second approach will likely include some higher-order thinking and collaborative pedagogies. However, at its core, the main focus is on the teaching of procedures with the hope that understanding will follow with practice.

In the article ‘Why Students Need To Understand What They Are Working On For The Majority Of Lesson Time' I explain why a lack of student understanding is problematic. In essence, we want to avoid students spending significant chunks of time not understanding what they are working on in lessons. Unfortunately, by default, a Procedures-first, Understanding-second approach can only result in having many students spending significant time in most lessons not truly understanding what they are working on.

Note, this is not a criticism of teachers. Teachers, by and large, are doing their best with what they know. The criticism here is aimed squarely at the Procedures-first, Understanding-second approach which, by default, is designed to teach procedures; not to generate conceptual understanding.

Regardless, I have reason to believe that the ‘Procedures-first, Understanding-second’ is still the most widely used approach in mathematics classrooms today. And I believe there are some compelling reasons for its continued popularity.

If I’m using a Procedures-first, Understanding-second approach - as I did for many years - then my intention and (almost) entire focus as I enter a lesson is two-fold:

- To teach the next set of procedures for the unit as effectively as I can, and
- To have students gain sufficient practice with those procedures so that they (hopefully) gain an understanding of how and why those procedures work.

As has already been stated, how I go about achieving this endeavour is less significant than the above mindset. Of course, a Procedures-first, Understanding-second approach will likely include some higher-order thinking and collaborative pedagogies. However, at its core, the main focus is on the teaching of procedures with the hope that understanding will follow with practice.

In the article ‘Why Students Need To Understand What They Are Working On For The Majority Of Lesson Time' I explain why a lack of student understanding is problematic. In essence, we want to avoid students spending significant chunks of time not understanding what they are working on in lessons. Unfortunately, by default, a Procedures-first, Understanding-second approach can only result in having many students spending significant time in most lessons not truly understanding what they are working on.

Note, this is not a criticism of teachers. Teachers, by and large, are doing their best with what they know. The criticism here is aimed squarely at the Procedures-first, Understanding-second approach which, by default, is designed to teach procedures; not to generate conceptual understanding.

Regardless, I have reason to believe that the ‘Procedures-first, Understanding-second’ is still the most widely used approach in mathematics classrooms today. And I believe there are some compelling reasons for its continued popularity.

## Three reasons in favour of a procedural (Procedures-first-Understanding-second) approach:

- A Procedures-first-Understanding-second approach is easier to use year after year. It requires little modification over time compared to alternative approaches.
- The approach is teacher-centric, and teacher-centricity is what the majority of teachers are most familiar with. Moving away from a teacher-centred approach is difficult for many teachers simply because it is unfamiliar and because, rarely, is any concrete roadmap provided.
- The approach can be reasonably effective at enabling students – those gifted with a good memory – to gain high results in short-term mathematics tests. And this appears to be the primary outcome sought by most parents and school administrators.

Note that none of the above three reasons is educationally sound. Rather, they are simply reasons why teachers find a procedural approach natural to use and difficult to move away from.

## Two reasons why changing from a traditional procedural approach can be difficult.

- Teachers typically do not know what a successful alternative to a traditional procedural approach looks like. Perhaps they trialled some conceptually-based activities in the past and saw them fail or not provide any advantages over their more familiar procedural approach. What's more, much of the professional learning (that advocates a shift towards a conceptual approach), rarely gives participants a sufficient, easy-to-follow roadmap for implementing a conceptual approach.
- Teachers hold misconceptions about what they perceive a conceptual approach to be.
- One major misconception is that any alternative to a procedural approach will require more time, time that they do not have.
- Another is that alternatives to the procedural approach almost always involve the extensive use of hands-on materials - clearly a problematic scenario considering the nature of many high school mathematics classes.
- Perhaps the greatest misconception is that any alternative to a traditional procedural approach places insufficient emphasis on the teaching of procedures.

## An alternative to a procedural approach

When leading teachers towards an alternative to a procedural approach, any resistance they have to embracing a (hybrid) conceptual approach always centres around one or more of the above misconceptions. Of course, in each case the opposite is true - a quality hybrid conceptual-procedural approach, if implemented well, will

- Save time
- Not need to involve the extensive use of equipment, and
- Allow for the explicit teaching of procedures.

## Characteristics of an effective conceptual (Understanding-first, Procedures-second) approach :

- The approach will be somewhat student-centred. However, it will be student-centred in a way that is highly-structured, allowing the teacher to have a strong sense of directing the learning.
- Mathematical understanding in students is fostered at the beginning of units through the use of activities and strategies which allow students to 'get their hands dirty' before procedures are introduced. Allowing students to understand what it is that they are working on BEFORE they encounter rules and procedures does two things: 1) it maximizes the all-important ‘aha moments’, and 2) it gives students a greater sense of ownership over their learning.
- Collaboration between students is actively fostered.
- Metacognition and other higher-order thinking processes are encouraged.
- Student engagement is improved. This is a critical outcome given that students need to be authentically engaged in order to learn.
- Differentiation of instruction occurs easily - a natural by-product of the student-centred approach.

As J.E.Schwartz explains:

‘In a conceptually oriented mathematics class, the bulk of (the) time is spent helping the students develop insight. Activities and tasks are presented to provide learners with experiences that provide opportunities for new understandings. Once the students gain understanding, then there is a need for some time to be spent on practice.’

## See the Understanding-first, Procedures-second approach infused in one specific unit of work via this mini-course.

A procedural approach, on the other hand, tends to limit the teacher's potential in achieving the above outcomes.

To further illuminate the point, Schwartz says:

To further illuminate the point, Schwartz says:

‘The concept of division and the procedure of solving division problems are not the same thing. In today’s mathematics classrooms, we are teaching concepts first and foremost. Procedures are learned too, but not without a conceptual understanding. One of the benefits to emphasizing conceptual understanding is that a person is less likely to forget concepts than procedures. If conceptual understanding is gained, then a person can reconstruct a procedure that may have been forgotten. On the other hand, if procedural knowledge is the limit of a person’s learning, there is no way to reconstruct a forgotten procedure. Conceptual understanding in mathematics, along with procedural skill, is much more powerful than procedural skill alone.’

Note that in the above quote, Schwartz references ‘In today’s mathematics classrooms’. I assume this to be an out-of-context statement. I think it should read ‘In mathematics classrooms in which a successful conceptual approach is used’, because I strongly believe that a Procedures-first, Understanding-second approach remains the most commonly used today.

The drive to improve educational standards across the globe are (arguably) worthy initiatives. However, in order for real change to occur, teachers require much more than research-backed theories. What is required to transition to successful conceptually-based approaches are 'feet-on-the-ground', easy-to-implement, tried-and-proven strategies; preferably with an effective form of long-term guidance.

The drive to improve educational standards across the globe are (arguably) worthy initiatives. However, in order for real change to occur, teachers require much more than research-backed theories. What is required to transition to successful conceptually-based approaches are 'feet-on-the-ground', easy-to-implement, tried-and-proven strategies; preferably with an effective form of long-term guidance.

## Adopting a conceptual approach

Once teachers are presented with a quality, long-term road map, one that requires them to implement the strategies with their students and share their implementation experiences with their colleagues, the transition to a conceptual approach becomes inevitable.

J. E. Schwartz’ book ‘Elementary Mathematics Pedagogical Content Knowledge: Powerful Ideas for Teachers‘, written with Elementary-Middle school teachers in mind but also relevant to High School teachers, will also assist with this journey.

J. E. Schwartz’ book ‘Elementary Mathematics Pedagogical Content Knowledge: Powerful Ideas for Teachers‘, written with Elementary-Middle school teachers in mind but also relevant to High School teachers, will also assist with this journey.

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