to teaching and learning 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 used widely since schools began teaching mathematics.

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

- Teacher introduces a rule (students copy the rule).
- Teacher demonstrates the rule 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 approach helps to 'keep the class together'. It allows teachers to ’teach to the middle’, which on the surface, makes the class easier to manage. Only one idea need be presented at a time.
- A rules-based approach is easier to use year after year. It requires little modification over time compared to alternative approaches.
- It’s a teacher-centered approach and therefore a relatively easy one to administer, requiring less skill than a student-centered alternative.
- Most of us were taught using a rules-based approach. Teaching the way we were taught feels natural.
- A rules-based approach can be reasonably effective at enabling students – those gifted with a good math memory – to gain high results in short-term mathematics tests.
- Gaining good results in short term math tests is viewed favourably by many parents, schools and society at large, even if it is at the expense of true understanding by the students.

None of these reasons are educationally sound; rather they are simply reasons why teachers find a rules-based 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 the alternative teaching and learning process is. What's more, professional learning which advocates such change rarely gives participants a tangible roadmap nor sufficient guidance of what the alternative looks like.
- Teachers hold negative misconceptions about the alternative. The main misconception is that any alternative to a rules-based approach will require more time, time that they do not have.

When working with teachers and leading them towards an alternative to their procedural-only approach, they always state the 'but this will take more time' objection. However, by the end of the journey, they all agree that the alternative, in most cases, saves them time and at worst, takes no more time than the procedural version.

I refer to the alternative as a conceptual approach to teaching mathematics, or 'teaching math conceptually'. In a conceptual approach, procedures are still important and indeed taught, but the focus is, first and foremost, on conceptual understanding.

An effective conceptual approach enables the following to occur:

I refer to the alternative as a conceptual approach to teaching mathematics, or 'teaching math conceptually'. In a conceptual approach, procedures are still important and indeed taught, but the focus is, first and foremost, on conceptual understanding.

An effective conceptual approach enables the following to occur:

- Mathematical understanding in students is fostered through the use of activities and strategies specifically designed to engineer ‘aha moments’ to occur for 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-centered approach used.
- Note that a conceptual approach is typically student-centred.

‘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 approach, on the other hand, tends to limit the practitioner in achieving the above outcomes. Yet through adopting an effective conceptual approach the same practitioner will be afforded many more opportunities to bring about collaborative learning, higher-order thinking and true understanding in students.

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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