Richard Skemp (1919-1995) was the first mathematician to bring together mathematics, psychology and education. After working as a maths teacher for several years in Oundle and Oxford, Skemp became interested in how children learn. It was that question that sparked his interest in studying psychology, which in turn informed his theories.
Schemas and the theory of intelligent learning
‘Intelligent learning,’ Skemp explains in “The Idea of a Schema” (1987), involves building networks of ‘schemas’ or conceptual structures that enable us to achieve our goals. Schemas “act as a tool for future learning”1 so that deep, long-lasting and versatile understanding is possible.
Nearly everything we learn is dependent on something we have learnt previously2, and this is especially true for mathematics. Skemp wrote that “the more…schemas we have available, the better our chance of coping with the unexpected”.3
One of Skemp’s strongest beliefs was that children can learn intelligently from a young age4 and that is why it’s important to help them develop schemas for future learning.
Relational understanding and instrumental understanding
One of Skemp’s keystone theories is the difference between relational and instrumental understanding. He defined relational understanding as “knowing what to do and why.”5 Instrumental understanding, on the other hand, he describes as “rules without reasons.”6
In other words, relational understanding is mastering maths while instrumental understanding is rote learning.
Skemp builds a case for and breaks down the benefits of each type in his article “Relational Understanding and Instrumental Understanding.”
The benefits of instrumental learning are:
- It’s usually easier to understand
- The results are more immediate and obvious
- The right answer is faster to come by
Whereas, the benefits of relational learning are:
- It’s more adaptable for future learning
- It’s easier to remember
- It’s an effective goal in itself
- It acts “as an agent of its own growth”7
Skemp explains that instrumental understanding is so prevalent due to its usefulness for testing and quick rewards of efforts. However, he emphasises the greater benefit of learning for future application. While instrumental or rote learning has more immediate results, longevity is not guaranteed. Relational understanding built from utilising schemas creates better, more dynamic mathematicians who will find learning maths rewarding.
Skemp believed mismatches between these types of learning are the most harmful problems in maths education and challenging to deal with. These mismatches occur when the teacher and pupil don’t want to understand in the same way. For example, a pupil may want to learn the rule instrumentally, whereas the teacher wants the pupil to understand relationally.
Even worse is when a pupil wants to understand relationally and a teacher is teaching instrumentally. This could lead to frustration on both sides. Skemp also identifies “a less obvious mismatch” and that is “between teacher and text.”8
Skemp explains that teachers must be on board for teaching relationally because while a school can supply textbooks that aim to teach this way, if a teacher is focused on instrumental teaching, then the efforts of getting pupils to understand relationally will fail. And the same is true for the opposite scenario, when a teacher aims to teach relationally and the materials do not support that. This shows the importance of unified, whole-class learning and high-quality textbooks.
Maths — No Problem! textbooks are built on Richard Skemp’s theories, as well as other renowned theorists such as Zoltan Dienes, Jean Piaget, Lev Vygotsky and Jerome Bruner, in addition to being aligned with the English National Curriculum. The benefit of high-quality textbooks and an intelligently planned curriculum is that children can develop a relational understanding of maths at a young age. And by teaching children maths relationally, we are supplying them the foundation, support and knowledge to become confident problem-solvers.
1Richard Skemp, “The Idea of a Schema,” in The Psychology of Learning Mathematics: Expanded, 1st ed. (New York: Routledge, 1987), 24.
2 Skemp, “The Idea of a Schema,” 25.
3 Skemp, “The Idea of a Schema,” 24.
4 David Tall, Richard R Skemp.
5 Richard Skemp, “Relational Understanding and Instrumental Understanding,” in Mathematics Teaching in the Middle School, vol. 12, no. 2. (National Council of Teachers of Mathematics, 2006, 89.
6Skemp, “Relational Understanding…,” 89.
7Skemp, “Relational Understanding…,” 92-93.
8Skemp, “Relational Understanding…,” 90.
Skemp, Richard R. “Relational Understanding and Instrumental Understanding.” Mathematics Teaching in the Middle School, vol. 12, no. 2, 2006, pp. 88–95. JSTOR. Accessed 11 Aug. 2022.
Skemp, Richard R. “The Idea of a Schema.” The Psychology of Learning Mathematics: Expanded, 1st ed., Routledge, New York, NY, 1987, pp. 22–34.
Tall, David. Richard R Skemp.