Benny Learns Rules, Rather than Sense
What was Benny learning?
Benny was developing flawed math rules in his mind, rather than mathematical sense. This can be seen in the incorrect way he converted fractions to decimals. He seemed confident in his wrong answers, and was unaware of his mistakes.
Benny was also learning the patterns and rules around the tests and the answer key. The program was focused on answers, rather than the mathematical process. Benny began to learn mathematical methods of his own making.
Benny also learned that even if one of his answers is marked wrong, it could still be right. This led him to believe that wrong answers were sometimes correct, they were just not written to match the key. He learned that recognition and promotion did not come unless he could provide an answer that matched the key.
Was Benny an active learner? In what sense? What's your evidence?
He was independently and actively engaged with the material, but he did not have a chance to interact with his teacher or peers about mathematics, or about his methods for achieving an answer. He rarely (possibly never) had a chance to interact with his peers about the process.
Benny moved through the IPI curriculum and passed the tests independently. He was learning, but not learning what we ultimately wanted him to learn: mathematical sense. He seemed convinced that he had figured out the rules, and could communicate them clearly. It wasn’t that he was making random mistakes; Benny was making the same mistakes consistently.
Why was remediation difficult with Benny?
He had practiced and learned the wrong way and developed his own rules and understanding of the procedures. Rules, rather than reasons, have dominated Benny’s understanding of mathematics for a long time. Also there was no one (teacher or peer) there to correct his thinking. Those habits and beliefs about math had already formed. Benny, a sixth grader, had been working with math this way since second grade.
Was Benny's teacher teaching?
I think the teacher was running a program, rather than teaching. He/she did not know his/her student, or identify his serious misconceptions about math. The IPI measurement system was not an adequate assessment of Benny’s understanding of fractions or mathematics. It was not until an adult asked him to clarify his reasoning, and expand on his thinking, that his misconceptions were highlighted. These problems cannot be addressed and corrected until they are diagnosed. It takes an attentive teacher who knows the child, and who can guide the child.
No, there should have been immediate feedback to correct Benny’s misconceptions. Benny’s teacher did not develop a relationship with him that would allow him to ask questions or discuss his thinking.
In the IPI system, the teacher’s role was to use the key to check answers.
“A mastery of content and skill does not imply understanding”
Erlwanger, S. H. (1973). Benny’s conception of rules and answers in IPI mathematics. Journal of Children’s Mathematical Behavior, 1(2), 7–26.
