Home - Check out our Videos

Dedicated to maths learning via deep relational understanding where connections are made through investigations, problem solving, practical work and discussion.

Skemp (1976) discussed ‘Relational Understanding’ and ‘Instrumental Understanding’ at length in his paper of that name. Instrumental understanding is a process that necessitates memorising methods and procedures or ‘rules without reasons’ (Skemp 1976). Instrumental rules include, add a zero when you multiply by 10, cancel zeros if you are dividing, minus a minus equals a plus and to divide by a fraction you turn it upside down and multiply. These are easily remembered rules however, if the teacher asks a question that does not quite fit the rule the pupil may get it wrong.

25 x 10 = 250 but 2.5 x 10 ≠ 2.50 (Adding zeros)

350 ÷ 10 = 35 but 305 ÷ 105 ≠ 35 ÷ 15 (Cancelled zeros)

Rather than teaching principles that can have a general application (in this case the deeper understanding of place value and common factors) these rules require more rules to ensure the first rule works. A concept that Skemp (1976) referred to as a multiplicity of rules.

O’Sullivan et al. (2005) considered Skemp’s work to be a ground-breaking paper in the history of mathematics education. However, even Skemp (1976) acknowledged that an instrumental approach can have short term successes and so, many commentators suggest that an instrumental style of teaching and learning is still a process that is adopted by many teachers of mathematics (O’Sullivan et al. 2005, Williams 2008, Boaler 2009 and Mason et al. 2010).

Boaler (2009) expands Skemp’s ideas suggesting that the problem with instrumental understanding is that children often cling to these methods and procedures. They see each problem as having one solution which must be remembered and repeated. For low attaining pupils maths becomes a never ending ladder of rules stretching up to the sky (Boaler 2009). For these students the learning is more difficult; made more so by their rigid approach. They are often asked to practice methods and so become even more rigid. They need to be shown flexibility, that there is seldom only one way to answer a problem and that mathematics is a series of interconnected ideas. In other words pupils need to develop what Skemp (1976) referred to as:

relational understanding;

knowing both what to do and why.