# Is Relational Understanding and Instrumental Knowledge of Mathematics a Continuum or a Dichotomy?

**Is Relational Understanding and Instrumental Knowledge of Mathematics a Continuum or a Dichotomy? **

You are in a classroom listening to your mathematics teacher going through a maths problem. Without using any mathematical jargon nor formulae the teacher tackles the problem using real life examples and experiences, and you are convinced that you have understood all the mathematical concepts involved. The month of June quickly approaches and you have to sit your exams. The first question you encounter requires that you show all your workings systematically, using the necessary mathematical rules and procedures, including the provided formulae. Upon seeing the word procedures and formulae, you become confused because as much as you understand the question conceptually, at no point during the lessons were you advised about procedures and rules to follow when tackling that sort of problem. Since it’s an exam, time is not on your side and you are stuck. Wouldn’t you wish you were taught in a certain way, where both the real life experiences and procedures were introduced? That’s why am investigating whether it is a good idea to disregard instrumental teaching altogether, or allow it to complement relational knowledge.

In 1976, in his highly regarded and well received article on mathematical understanding in schools, Richard Skemp concluded that instrumental mathematics should not be regarded as mathematics at all, hinting that the term ‘mathematics’ ought to be used for relational understanding of mathematics only. He described relational understanding as “knowing both what to do and why”, while instrumental understanding entails “rules without reasons” (Skemp, 1978, p. 9), like a formula. He argued that despite the subject matter of relational and instrumental mathematics being the same, “… the two kinds of knowledge are so different that there is a strong case for regarding them as different kinds of mathematics”.

The article touched on a long standing debate on whether the way mathematics is taught in schools is up to scratch, and if not, what is the cause. Hence, the introduction of the two types of understanding, dubbed ‘relational and instrumental’ which, according to Richard Skemp, were borrowed from Stieg Mellin-Olsen, of Bergen University.

On face value, in industries like mechanics and engineering, most of the mathematics that is widely used could be said to be instrumental. But despite this view, a huge chunk of the mathematics being used has to cover computational, reasoning and modelling processes that could only be possible with relational understanding of the mathematical concepts involved. That means both relational and instrumental mathematics are being used in tandem. Practically, in these industries the relational mathematics is being complemented by instrumental knowledge, allowing a diverse application of the mathematics into decision aid, psychology, ranking and spatial reasoning.

Several mathematics education researchers like Hiebert and Carpenter (Hiebert & Carpenter, 1992; Hiebert & Lefevre, 1986), have also raised issues with the dichotomy between relational and instrumental understanding, arguing that both **procedural and conceptual understanding** in mathematics is necessary for mathematical expertise and that there is no need to separate them. Thus these researchers prefer continuum and suggest that absolute separation of the two is impossible.

At the same time, education researchers like Resnick and Ford (1981) and Nesher (1986), suggests that the separation between learning algorithms and understanding it is both misleading and superficial, arguing that research in student performance in mathematics does not elaborate the relationship between mathematical performance and success in mathematical understanding. Here, Nesher disregards and questions the usefulness of relational – instrumental dichotomy proposed by Skemp.

Despite Nesher’s assertion, I would argue that both knowing what to do and the reasons behind doing so (relational knowledge), and the ability to execute mathematical rules and procedures well (instrumental) (Skemp, 1976), are the makings of a good mathematician, and the former could only work if the latter complements it. For example, let’s say a child has been taught a mathematical concept which both the teacher and the child are convinced that the concept has been understood. But the teacher allows the child to find her own way of solving the problems emanating from the concepts she has learnt and understands very well. One of two outcomes may be expected from this. Either the student would excel as she perseveres and work through the concepts to come up with a result, which in turn would strengthen her knowledge of the subject which would stay with her forever. Yes, the mathematical rules and procedures may be muddled up and all over the place but the student has been given a chance to learn, explore and understand the subject. And if she were to come up with an evaluation, a conjecture, and be able to predict, devise and check solutions, then we would all say the art of teaching has been accomplished. In a world where there are no time limits, no curriculum to follow nor exams to sit, this would be a perfect situation and this is what our education system should be aiming for.

But we live in a world where mathematical activity requires the use of three basic dimensions of mathematical knowledge: algorithmic, formal and intuitive, according to Fischbein (1983,1993). Despite Fischbein’s three types of knowledge sounding different from Skemp’s two dimensional dichotomy, the concepts are quite similar. Like with instrumental knowledge, the algorithmic dimension consists of rules, procedures for solving, and their theoretical justifications, while the formal dimension includes axioms, definitions, theorems and proofs, more like in relational understanding. It is hard to determine where the intuitive dimension is leaning to since it is a kind of cognition that comprises the ideas and beliefs about mathematical entities and the mental models that are used for representing mathematical concepts and operations. It is the type of knowledge that we tend to accept confidently as being obvious, with a feeling that it needs no proof. Fischbein argues that intuitive knowledge has an imperative power that tends to eliminate alternative representations, interpretations or solutions. Unlike Skemp’s dichotomy, Fischbein’s three dimensions are continuum. He argues that they are not discreet and that they overlap.

In his book, Intuition in Science and Mathematics: An Educational Approach, H. Fischbein (1983) argues that developing new mathematical and scientific intuition requires exposing students to situations in which the student is asked to evaluate, conjecture, predict, devise and check solutions, so that he can be able to cope in uncertain events. I wonder if the student would be able to achieve all this by mere intuition. I would argue that in order to develop to this level, both relational understanding and instrumental knowledge is necessary.

Teachers have been exposed to quite an elaborate number of understanding from many researchers. All categories have one aim, and that is to give us a deeper insight into the thinking of a student. They include Skemp (1976,1978) with his relational and instrumental understanding, Backhouse (1978) on symbolic understanding, Buxton (1978) on the idea of formal and logical understanding, and Haylock (1982) who suggested that it is important to consider understanding as the creation of connections between new and existing knowledge, while Sierpinska (1994) describes understanding as emerging in response to difficulties encountered when current knowledge meets new, not readily reconcilable experiences. All this categorisation is necessary but we must be careful not to over-categorise but regard and perceive each child as a separate entity with different capabilities of understanding. It could be a mistake to take the route of one size fits all.

The importance of enabling children to learn relationally cannot be overestimated. Teachers and parents alike realise this but schools are under enormous pressure from authorities. Unfortunately, the optimum means of showing progress is through the national exams. No wonder some parents, as recent as two weeks ago, opted to take their children out of classes, demonstrating against government’s SATS tests for ten and eleven year olds. Does this mean that society recognises the importance of learning relationally, as opposed to learning for exams? It is obvious that these frequent exams are pulling education standards down, and the fact that they are being forced on children as young as ten could only result in an increasing amount of rote learning for exam purposes. Unless there is a revamp on how schools’ progression is assessed, then we shouldn’t expect significant improvement any time soon.

From another perspective, we could have been encouraged if governments had the will to retain current maths teachers and attract new good mathematicians into the system. Instead, the talent pool of good mathematics teachers with expertise in relational understanding are being lured by big corporations, whose remuneration system could easily surpass that of a mathematics teacher, tenfold. I have included a link to James Simon’s (a mathematician) interview (https://youtu.be/gjVDqfUhXOY) on this topic.

One solution that may help to resolve this high turnover among experienced maths teachers in the education system is to make significant changes to the way teachers are remunerated and respected in the society, otherwise it will be very hard to reverse the current downward spiral in the education system. Since everyone works for a combination of financial reasons and respect, a significant rise in salaries (not as much as what the private sector does pay) would not only encourage new good mathematicians and scientists to get into teaching, but would also create a community of maths teachers who feel important, respected and love what they do.

Much has been said as to why most of our students are missing out on learning relationally, vis-à-vis, lack of qualified teachers, too much workload, demoralised teachers, government interference, poor curriculum and poor examination methods, among others. But what is the by-product of teaching maths purely instrumentally? Yes, it may overcome some of the above disadvantages of teaching relationally e.g. time saving and meeting examination requirements, but one thing for sure is that the learner will not retain that knowledge in the long term and as soon as the facts are twisted he will become confused (Skemp). That’s when you start hearing students and adults alike proclaim ‘I am not a maths person’. This is what we hear every day, especially here in the west. It is reported that people in the west, especially adults, are more likely to be unmoved and quite proud to wear such a badge, claiming how poor they are in mathematics unlike their counterparts in the east. You wouldn’t hear such proclamations among students or adults in the east (BBC radio 4, 10/05/16). If a parent or any adult whom a child looks up to makes such claims in front of the child, you wouldn’t expect that child to perceive mathematics differently. This kind of mentality trickles down to the young ones who no longer make an effort to improve their grades in the subject. But remember, the cause of this circle is the classroom, where that adult was taught instrumentally as a child, in order to finish the syllabus and pass exams.

To conclude, I agree with Skemp that relational understanding is vital in mathematics, but I would argue that at the same time it needs to be complemented by rules and procedures which, unfortunately may be relayed instrumentally.

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