Teaching mathematics to build a strong conceptual understanding is a monumental challenge when teaching at the elementary or secondary levels. A seminal article in the journal The Arithmetic Teacher titled, Relational Understanding and Instrumental Understanding, by Richard R. Skemp, published in 1978, explored two distinct modes of understanding:
Instrumental understanding: knowing procedures and rules—what to do—without necessarily understanding why they work.
Relational understanding: grasping both what to do and why– making meaningful connections and understanding underlying concepts
The ultimate aim in teaching mathematics is to help the student achieve relational understanding, or mathematical concepts so that they can apply these concepts to the solution of problems. A student who has a relational understanding of a mathematical concept can rediscover a procedure or algorithm they may have forgotten, but a student who has only memorized a procedure will be unable to recover it when it is not remembered.
However, relational understanding an order of magnitude more difficult to achieve than rote memorization, and is much lower on the Bloom taxonomy scale of cognitive difficulty. During a year when I taught mathematics to students of varying abilities, I discovered that those who struggled with mathematics, strongly resisted understanding “why” and would plead, “just tell me how.” The capable students, on the other hand, would delve into the “why” with relish.
In describing the results of his research during the 1960’s and 1970’s, USSR mathematician Vladimir Krutetskii noted innate differences in the cognitive capacities of children, while carefully sidestepping the Soviet prohibition on linking talent to inheritance:
The difference between capable, average, and incapable pupils, as our research permits us to conclude, comes down to the following. In able pupils these associations can be formed “on the spot”; in this sense they are “born,” if one can so express it, already generalized, with a minimal number of exercises. In average pupils these associations are established and reinforced gradually, as a result of a whole series of exercises. They form isolated, concrete associations, related only to a given problem, “on the spot.” Through single-type exercises these associations are gradually transformed into generalized associations. In incapable pupils, even the isolated, concrete associations are formed with difficulty, their generalizations are still more difficult, and sometimes such generalizations do not occur at all.
Though many people would like to believe that we are all equal in our ability to learn mathematics and are distinguished only by the effort we expend, both research and experience teach us that this is not so. Research has shown that some of the concepts in the typical mathematics curriculum are above the cognitive capacity of the students of lower ability and they can apply mathematics only at the rote level. That is why the recently released Common Core State Standards for Mathematics (CCSSM) was rejected by many states in the U.S..
With the remarkable advances in AI, mathematics educators will be re-thinking what mathematics is appropriate for people of different intellectual levels and/or career aspirations. The kind of arithmetic facility that was needed a generation ago, is not required of all people in this technological world. Similarly, there will be fewer people who need to learn calculus, but more who will need to be statistically literate. Processing information intelligently is vital in this era. In all cases, relational understanding is more important than instrumental learning, but we have to make a greater effort to determine what mathematical curriculum is appropriate for students of different academic abilities.
