## Procedural vs Conceptual Knowledge in Mathematics Education

Preparing to embrace a conceptual approach to teaching mathematics

Preparing to embrace a conceptual approach to teaching mathematics

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.

Words are symbols, as are phrases. This means that different words mean different things to different people. Some words are more consistently understood than others. The ideas that come to mind for mathematics teachers when they encounter the words ‘procedural approach’, for example, 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. Therefore, let’s talk briefly about terminology. As is my style, I will state this a simply as possible.

- 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 coming procedures are based.

Learn Implement Share advocates the type of approach outlines in point d., above. And 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’, explained below.

Again, the reason for using interchangeable terms is because terms are only labels, and I want to minimize the chance of being misunderstood.

- Conceptual approach.
- Hybrid conceptual-procedural approach.
- Hybrid conceptual approach.
- An Understanding-first, Procedures-second approach.
- (Researchers will have devised other terms to refer to the same sort of approach - I’ll add more as I become aware of them.)

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

Again, the reason for using interchangeable terms is because terms are only labels, and I want to minimize the chance of being misunderstood.

It would be tempting, at this point, to describe, step-by-step the procedural (Procedures-first, Understanding-second) approach - 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.

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

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, but at its core, the main focus is on the teaching of procedures with the hope that understanding will follow with practice. In the article ‘There’s an Elephant in the (Maths Class)Room! - Why students understanding … I explain why this 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, (i.e. through no fault of the teacher) a Procedures-first, Understanding-second approach can not help but have many students spending significant time on most lessons not understanding what they are working on.

Nevertheless, I’m confident in suggesting that ‘Procedures-first, Understanding-second’ is still the most widely employed approach by most teachers of mathematics today. I believe there are some compelling reasons for its continued popularity.

If I’m using a Procedures-first, Understanding-second approach - as I most certainly 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.

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, but at its core, the main focus is on the teaching of procedures with the hope that understanding will follow with practice. In the article ‘There’s an Elephant in the (Maths Class)Room! - Why students understanding … I explain why this 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, (i.e. through no fault of the teacher) a Procedures-first, Understanding-second approach can not help but have many students spending significant time on most lessons not understanding what they are working on.

Nevertheless, I’m confident in suggesting that ‘Procedures-first, Understanding-second’ is still the most widely employed approach by most teachers of mathematics today. I believe there are some compelling reasons for its continued popularity.

- It 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 seemingly difficult simply because it is unfamiliar.
- It can be reasonably effective at enabling students – those gifted with a good memory – to gain high results in short-term mathematics tests. This appears to be the primary outcome sought by most parents and school administrators.

None of these reasons is educationally sound. Rather they are simply reasons why teachers find a procedural approach natural to use and difficult to move away from.

- Teachers typically do not know what a successful alternative looks like. Perhaps they trialled some conceptually-based activities in the past and found them to either fail or not provide any advantages to the procedural approach with which they are familiar. 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 rules-based/procedural approach will require more time, time that they do not have.
- Another is that alternatives almost always involve the extensive use of hands-on materials (which would clearly be problematic 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.

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
- Does not need to involve the extensive use of equipment, and
- Will allow for the explicit teaching of procedures.

- 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 sense of 'directing the learning'.
- Mathematical understanding in students is fostered through the use of activities and strategies which allow students to 'get their hands dirty' early in the unit before procedures are introduced. Allowing students to understand what it is that they are working on before they encounter rules and procedures the all-important ‘aha moments’ are maximised and students gain a greater sense of ownership over their learning.
- Collaboration between students is actively fostered.
- Metacognition and other higher-order thinking processes are encouraged.
- As a result, student engagement is improved (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.’

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 have strong reasons to believe that today the most commonly used approach to teach mathematics is a Procedures-first, Understanding-second approach.

The drives to improve standards across the globe, including the Common Core roll-out in the US, are worthy initiatives. However, in order for real change to occur, teachers require much more than research-backed theories. Teacher require tangible, 'feet-on-the-ground', long-term guidance with easy-to-implement, tried-and-proven strategies in order to make a successful transition.

The drives to improve standards across the globe, including the Common Core roll-out in the US, are worthy initiatives. However, in order for real change to occur, teachers require much more than research-backed theories. Teacher require tangible, 'feet-on-the-ground', long-term guidance with easy-to-implement, tried-and-proven strategies in order to make a successful transition.

Once teachers are presented with sufficient, quality, in-class examples and over an extended period of time, they then trial these with their students and report back on their implementation experiences. It is then that 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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