Samuel Otten
SME 926
Erlwanger, S.H. (1973). Benny’s conception of rules and answers in IPI mathematics. Journal of Children’s Mathematical Behavior, 1(2), 7-26.
Skemp, R.R. (1977). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20-26.
Benny is a somewhat paradoxical mathematics student. His teachers report that he is above average in the subject and is readily progressing through the Individually Prescribed Instruction (IPI) program, earning the required 80% on the IPI examinations. However, an observant mathematician can readily see that Benny has severe misconceptions and is unable to successfully complete such simple tasks as adding decimals or fractions. How can a student amongst the top of his math class have such shortcomings? The answer lies in the distinction explicated by Skemp between relational understanding and instrumental understanding. It will be argued that Benny is a student attempting to develop instrumental understanding of mathematics within an instrumental system of instruction, and his failures in this domain are manageable. On the other hand, when viewed in the light of relational understanding, more severe and troubling failures are illuminated.
The terms ‘relational’ and ‘instrumental understanding’ have already been used, so a definition is in order. As presented by Skemp, relational understanding can be thought of as knowledge of the content of a particular subject (e.g, rules, procedures) coupled with knowledge of why things are that way. Instrumental understanding is knowledge of the content without knowledge of why it is that way. As an example, a student of the calculus with instrumental understanding will have memorized many methods for finding derivatives and will have experience at recognizing when to apply a particular method. Contrastingly, a student with relational understanding will comprehend the limiting concept behind the derivative and may even be able to justify a few of the differentiating methods. There are several pieces of evidence in Erlwanger’s article that suggest Benny falls in the former of these categories.
First, Benny was often asked to explain why he had done a particular mathematical procedure, and in response he would simply state the rule that he was following. On page 9, Benny responded, “So you have to drop that 10…you have to add these numbers up which will be 5.” There is no mention of why he is doing any of his steps – of what lies behind the rules. Sometimes he was following an erroneous rule, but it was a rule to him nonetheless. Even Benny’s ‘one rule is as good as the next’mentality points to instrumental understanding because there is no schema in place with which to judge the validity of a rule. To employ one of Skemp’s analogies, Benny has directions but no map.
Second, Benny does not have an appropriate sense of whether his work is correct or incorrect. If he has done correct work, he does not know it until he has compared it to the answer key, and sometimes even when he has done incorrect work he still feels he is right as seen in Benny’s following remark from page 15: “if I had this example [i.e.
2 + 8/10], and I put 1.0, I get it wrong. But really they’re the same, no matter what the key says.”A student with relational understanding would be able to independently verify results and, as Skemp points out on page 15, would exhibit “confidence in his own ability” – no matter what the key says.
Third, Benny is extremely prone to overgeneralizations, as in treating the denominators in the addition of fractions the same as in multiplication (), treating the number of decimal places in addition the same as in multiplication (), and flipping fractions according to the commutative law of algebra ( because ). Skemp writes on page 9, “I do not think he was being stupid…He was simply extrapolating from what he already knew. But relational understanding,by knowing not only what method worked but why, wouldhave enabled him to relate the method to the problem, and possibly toadapt the method to new problems.”
There are more examples that could be cited to attest to Benny’s instrumental understanding but need not be. Additionally, the argument could be made that Benny was simply adapting to the IPI system, which is oriented toward instrumental understanding; and that Erlanger’s criticism of the procedure-and-answer driven program is really a cry out in favor of relational understanding, but that will have to wait. What cannot wait is the resolution of the paradox mentioned in the first sentence of this paper. Within the framework of these two types of understanding, we can see that Benny is an above-average achiever when assessed with instrumental measures, while simultaneously is lacking the knowledge of why things work that is so important in relational mathematics.
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