“Benny’s Conception of Rules and Answers in IPI Mathematics”, 1973 (; backlinks):
[commentary: 1, 2; 2014] Presents excerpts from interviews with a bright 12-yr-old 6th-grade pupil “Benny” who was having difficulties in mathematics. He had been in the individually prescribed instruction (IPI) mathematics program since the 2nd grade and earlier had seemed to be making good progress. His imperfect understanding of decimals and fractions revealed weaknesses of IPI, which are discussed in detail.
This study arose from visits made to a 6th grade class using Individually Prescribed Instruction (IPI) Mathematics in order to assist pupils who required remedial instruction and discover the nature of their trouble.
In these terms, a 12 year old boy named Benny did not seem a likely subject for the study. He was making much better than average progress through the IPI program, and his teacher regarded him as one of her best pupils in mathematics. In a structured program like IPI, it was expected by the teacher that Benny could not have progressed so far without an adequate understanding and mastery of previous work.
Benny was willing to talk to me, and I was eager to get started, so we began to discuss his current work.
I soon discovered that Benny understood incorrectly some of the previous work. He could add fractions and multiply decimals correctly in most of the exercises, but he said that 2⁄1 + 1⁄2 was equal to 1, and 2⁄10 as a decimal was 1.2…Benny converted fractions into decimals by finding the sum of the numerator and denominator of the fraction and then deciding on the position of the decimal point from the number obtained.
Subsequent discussions and interviews with Benny led me to an understanding of his concept of decimals and fractions, and his views about rules, relationships, and answers in mathematics.
…Clearly, then, “making good progress” in IPI means something other than what we had thought. Benny was in a small group of pupils who had completed more units (with a score of 80% or better) than any other child in the class. He worked very quickly. When he failed to get 80% marked right by the IPI aide, he tried to grasp the pattern of the correct answers; he then quickly changed his answers in ways which he hoped would better agree with the key, a process which we will examine in more detail later.
Benny’s case indicates that a “mastery of content and skill” does not imply understanding. This suggests than an emphasis on instructional objectives and assessment procedures alone may not guarantee an appropriate learning experience for some pupils. [cf. LLM underperformance on arithmetic unless carefully designed around its tokenization & computational limits]