Skip to content

Two types of teaching mathematics: instrumental and relational#

See also: teaching-mathematics

Skemp (2006) describes two ways of teaching-mathematics: instrumental and relational. Linked to the two types of understanding. A relational understanding is perhaps an example of Boaler's idea of a mathematical-mindset

  • relational - knowing what to do and why
  • instrumental - know the rule, how to use it, but not why

Skemp believes that teaching for relational understanding is better. But aims to provide an analysis of the relative benefits. Helping understand why instrumental approaches are so relevant and perhaps ideas about the difficulty of teaching for relational understanding.

Alfeld cites a letter Newton wrote for

A vulgar Mechanick can practice what he has been taught or seen or seen done, but if he is in an error he knows not how to find it out and correct it, and if you put him out of his road, he is at a stand; Whereas he that is able to reason nimbly and judiciously ... is never at rest till he gets over every rub

Advantages of instrumental mathematics#

  1. Instrumental mathematics is usually easier to understand.

    Some topics are hard to understand relationally (multiplying two negative numbers).

    If what is wanted is wanted is a page of right answers, instrumental mathematics can provide this more quickly and easily (p. 92)

  2. The rewards are more immediate, and more apparent.

  3. The right answer can be achieved more quickly (because less knowledge is involved)

Advantages of relational mathematics#

  1. It is more adaptable to new tasks

  2. It is easier to remember (but harder to learn).

  3. Relational knowledge can be effective as a goal in itself.

    moving away from extrinsic to intrinsic motivation

  4. Relational schemas are organic in quality

    If #3 applies then individuals may start to "act as an agent of their own growth". Looking for relational understanding from the start.

Reasons why instrumental mathematics is taught#

Skemp (2006) offers four reasons why a teacher might decide

  1. That relational understanding would take too long to achieve, and to be able to use a particular technique is all that these pupils are likely to need.
  2. That relational understanding of a particular topic is too difficult, but the pupils still need it for examination reasons.
  3. That a skill is needed for use in another subject (e.g. science) before it can be understood relationally with the schemas presently available to the pupils.
  4. That he is a junior teacher in a school where all the other mathematics teaching is instrumental.

And situational factors

  1. The backwash effect of examinations.
  2. Over-burdened syllabi
  3. Difficulty of assessment - telling the difference from exam paper between instrumental and relational understanding
  4. The difficulty for teachers to restructure their existing and longstanding schemas

References#

Richland, L. E., Stigler, J. W., & Holyoak, K. J. (2012). Teaching the Conceptual Structure of Mathematics. Educational Psychologist, 47(3), 189--203. https://doi.org/10.1080/00461520.2012.667065

Skemp, R. R. (2006). Relational Understanding and Instrumental Understanding. Mathematics Teaching in the Middle School, 12(2), 88--95.