## 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

Some reasons why transitioning from a procedural approach to a conceptual approach can be difficult

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 used widely since we began teaching mathematics in schools.

I refer to the procedural process as a ‘rules-based approach’. It looks something like this:

- Teacher introduces a rule/procedure and students write it down.
- Teacher demonstrates the rule/procedure using an example (students copy the example).
- Students work through 5-10 questions of a similar type.
- Teacher introduces the next rule … and …
- Repeat … and repeat …

For many practitioners the shift away from a rules-based (procedural) approach is challenging. This is partly because there are compelling reasons in favour of using a rules-based approach; reasons which, although not educationally sound are, nonetheless. compelling.

- A rules-based/procedural approach helps to 'keep the class together'. It minimises the spread of students allowing teachers to teach to the one level, which on the surface, makes the process easier to manage.
- A rules-based/procedural approach is easier to use year after year. It requires little modification over time compared to alternative approaches.
- The approach is teacher-centred, the approach the majority of teachers are most familiar with and is therefore seemingly easier to administer.
- A rules-based/procedural approach can be reasonably effective at enabling students – those gifted with a good memory – to gain high results in short-term mathematics tests which seems to be the primary outcome sought by most parents and school administrators.

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

- Teachers typically do not know what the alternative looks like. They are familiar with the traditional approach but have no valid insight into what an alternative teaching and learning process looks like. What's more, much of the professional learning (that advocates such change), rarely gives participants a sufficient, easy-to-follow roadmap into the alternatives.
- Teachers hold negative misconceptions about what they perceive as the alternatives.
- 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 rules-based/procedural approach will not be successful at teaching students routines and procedures.

When working with teachers and leading them towards an alternative to a traditional procedural approach, any resistance they have to embracing a (hybrid) conceptual approach always centres around one or more of the above misconceptions. And of course, in each case the opposite is true - a quality conceptual approach can a) save time, does not need to involve the extensive use of equipment and will allow for the explicit teaching of procedures.

In the article 'Teaching Mathematics using an approach that is both conceptual and procedural - 9 keys' I unpack the hybrid procedural-conceptual approach. (The article contains one main page with expansions to each key on separate pages - you choose which expansions you read.. If you are interested in the current article then this 'hybrid approach' article is a must.)

An effective conceptual approach should contain the following elements:

In the article 'Teaching Mathematics using an approach that is both conceptual and procedural - 9 keys' I unpack the hybrid procedural-conceptual approach. (The article contains one main page with expansions to each key on separate pages - you choose which expansions you read.. If you are interested in the current article then this 'hybrid approach' article is a must.)

An effective conceptual approach should contain the following elements:

- The approach will be largely 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 rules and procedures are introduced. Allowing students to 'understand what it is that they are doing' before they encounter rules and procedures the all-important ‘aha moments’ are maximised and students gain a greater sense of ownership over their learning.
- As a result, student engagement is high.
- Collaboration between students is actively fostered.
- Metacognition and other higher-order thinking processes are encouraged.
- Differentiation of instruction occurs easily, a natural by-product of the student-centred approach.

‘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 rules-based/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.’

The drive 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 theory and research – they require tangible, 'feet-on-the-ground', long-term guidance in order to make the 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 this journey.

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