This year I have commenced training to become a mathematics specialist teacher, specifically called an Extending Mathematical Understanding (EMU) teacher. Through this program there has been a requirement to engage in professional reading. One of the professional readings came from the Richard Skemp, Relational Understanding or Instrumental Understanding (1976). This reading struck me for a number of reasons.
Relational understanding relates to a procedural understanding of mathematics. The example given by Skemp refers to ‘borrowing or trading’ when solving an addition or subtraction algorithm. While students will get to the correct answer they have used a process. If a teacher then proceeds to ask a question which doesn’t fit with their understanding of the rule they have applied they will become stuck. This could be a question around the age difference between two people or given a year of birth asking how many years the person has lived.
In comparison, instrumental understanding or conceptual understanding of mathematics identifies understanding the ‘why’ of what is happening within the mathematics. To explain this reasoning, Skemp conducted an experiment into whether individuals retained information better when learning in a relational way compared to an instrumental understanding. The below pictures were given to two groups. The first group had to memorise them not understanding any words or knowledge about them. Similar to if you tried to learn a new language by rote. The other group was given the words and was able to make connections between to symbols to develop an understanding of why the symbols were chosen. In the rote learnt group, only 8 out of 32 participants could recall what the symbols meant after 4 weeks whereas 58 out of the 69 who experienced the instrumental understanding could recall them after 4 weeks.
Playing “devil’s advocate,” Skemp highlights that there are a three positives to a relational understanding (1976). These included the ‘instant gratification’ received after getting back a page of correct mathematics and often the answer can be achieved quiet quickly. Skemp, however, also outlines four key benefits of instrumental understanding:
- It allows for adaptability across tasks;
- It’s easier to remember – strangely enough when they need to remember all the rules to apply for a relational understanding having a deep understanding of mathematics is easier;
- Relational understanding can be a goal in itself; and
- Students can extend schema into new areas more organically.
While, I am still at the beginning of my understanding and reading on this topic, I am interested to know more. I feel quite strongly that within the classroom I want my students to be achieving an instrumental understanding. Instead of a student using mental strategies to solve 24 x 99 in front of me, I’d love if they could use their understanding of numbers to know they could have multiplied 24 x 100.
I’ll leave you with a thought shared at our EMU professional development day. Think of a London taxi driver in comparison to a London bus driver. The taxi driver has an instrumental (conceptual) understanding of the streets of London; they would be able to tell you all the different ways to get to the one location while taking into consideration the traffic of the time of day. In comparison the bus driver will be able to get you to your destination but only via the one direct route they’ve been trained to follow every single day. If this is the mathematics classroom, would you prefer the taxi driver or the bus driver?
Richard Skemp, Relational and Instrumental Understanding
Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics teaching, 77(1), 20-26.
Richard Skemp, The Need for a Schematic Learning Theory
Skemp, R. R. (1962). The need for a schematic learning theory. British Journal of Educational Psychology, 32(P2), 133-142.