Relational Understanding and Instrumental Understanding1 Richard R Skemp Department of Education University of Warwick Faux Amis Faux amis is a term used by the French to describe words which are the same or very alike in two languages but whose meanings are different For example French word histoire libraire chef agr ment docteur m decin parent Meaning in English story not history bookshop not library head of any organisation not only chief cook pleasure or amusement not agreement doctor higher degree not medical practitioner medical practitioner not medicine relations in general including parents One gets faux amis between English as spoken in different parts of the world An Englishman asking in America for a biscuit would be given what we call a scone To get what we call a biscuit he would have to ask for a cookie And between English as used in mathematics and in everyday life there are such words as field group ring ideal A person who is unaware that the word he is using is a faux ami can make inconvenient mistakes We expect history to be true but not a story We take books without paying from a library but not from a bookshop and so on But in the foregoing examples there are cues which might put one on guard difference of language or of country or of context If however the same word is used in the same language country and context with two meanings whose difference is non trivial but as basic as the difference between the meaning of say histoire and story which is a difference between fact and fiction one may expect serious confusion Two such words can be identified in the context of mathematics and it is the alternative meanings attached to these words 1 First published in Mathematics Teaching 77 20 26 1976 1 each by a large following which in my belief are at the root of many of the difficulties in mathematics education today One of these is understanding It was brought to my attention some years ago by Stieg Mellin Olsen of Bergen University that there are in current use two meanings of this word These he distinguishes by calling them relational understanding and instrumental understanding By the former is meant what I have always meant by understanding and probably most readers of this article knowing both what to do and why Instrumental understanding I would until recently not have regarded as understanding at all It is what I have in the past described as rules without reasons without realising that for many pupils and their teachers the possession of such a rule and ability to use it was why they meant by understanding Suppose that a teacher reminds a class that the area of a rectangle is given by A L B A pupil who has been away says he does not understand so the teacher gives him an explanation along these lines The formula tells you that to get the area of a rectangle you multiply the length by the breadth Oh I see says the child and gets on with the exercise If we were now to say to him in effect You may think you understand but you don t really he would not agree Of course I do Look I ve got all these answers right Nor would he be pleased at our devaluing of his achievement And with his meaning of the word he does understand We can all think of examples of this kind borrowing in subtraction turn it upside down and multiply for division by a fraction take it over to the other side and change the sign are obvious ones but once the concept has been formed other examples of instrumental explanations can be identified in abundance in many widely used texts Here are two from a text used by a former direct grant grammar school now independent with a high academic standard Multiplication of fractions To multiply a fraction by a fraction multiply the two numerators together to make the numerator of the product and the two denominators to makes its denominator E g 4 5 2 3 of 3 5 10 13 2 4 3 5 30 65 158 136 The multiplication sign is generally used instead of the word of 2 Circles The circumference of a circle that is its perimeter or the length of its boundary is found by measurement to be a little more than three times the length of its diameter In any circle the circumference is approximately 3 1416 times the diameter which is roughly 3 1 7 times the diameter Neither of these figures is exact as the exact number cannot be expressed either as a fraction or a decimal The number is represented by the Greek letter Circumference d or 2 r Area r2 The reader is urged to try for himself this exercise of looking for and identifying examples of instrumental explanations both in texts and in the classroom This will have three benefits i For persons like the writer and most readers of this article it may be hard to realise how widespread is the instrumental approach ii It will help by repeated examples to consolidate the two contrasting concepts iii It is a good preparation for trying to formulate the difference in general terms Result i is necessary for what follows in the rest of the present section while ii and iii will be useful for the others If it is accepted that these two categories are both well filled by those pupils and teachers whose goals are respectively relational and instrumental understanding by the pupil two questions arise First does this matter And second is one kind better than the other For years I have taken for granted the answers to both these questions briefly Yes relational But the existence of a large body of experienced teachers and of a large number of texts belonging to the opposite camp has forced me to think more about why I hold this view In the process of changing the judgement from an intuitive to a reflective one I think I have learnt something useful The two questions are not entirely separate but in the present section I shall concentrate as far as possible on the first does it matter The problem here is that of a mis match which arises automatically in any faux ami situation and does not depend on whether A or B s meaning is the right one Let us imagine if we can that school A send a team to play school B at a game called football but that neither team knows that there are two kinds called association and rugby School A plays soccer and has never heard of rugger and vice versa for B Each team will rapidly decide that the others are crazy or a lot of foul players Team A in particular will think that B uses a mis shapen ball and commit one foul after another Unless the two sides stop and 3 talk about what game they think they are playing at long