In preparing to return to the classroom as a middle school math teacher, I added a few “math education” articles and book chapters to my summer reading pile. One item was the chapter “Learning Mathematics” in the 2011 edition of the Handbook for Research on Learning and Instruction (Edwards, Esmonde, Wagner, & Beattie, 2011). The chapter begins with a reminder of the “Benny article” which appeared in 1973.
IN 1973, S. H. Erlwanger described the case of Benny, a sixth-grade student who was enrolled in a school where Individually Prescribed Instruction (IPI) mathematics was used. Erlwanger described Benny as a student who 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 conversation with the student, Erlwanger found Benny to hold many incorrect concepts of math, and despite being able to answer questions correctly in the prescribed curriculum, Benny did not understand how to perform the operations. It illustrates on the potential that students may be able to provide “correct” answers without truly understanding the material.
Perhaps the most disconcerting aspect of the article is that the arguments made to promote IPI mathematics in the early 1970’s are used to advocate for similar curricula today. IPI mathematics was designed to:
- Identify clear outcomes;
- Assess students’ progress;
- Ensure mastery before proceeding.
An example of the rhetoric that promoted IPI is the commentary by Clyde Yetter (1972) that appeared in Educational Leadership. The similarities between Yetter’s description of the strengths of IPI and modern equivalents are striking (and disturbing given Benny’s case).
As IPI existed before desktop computers arrived in schools, assessments were administered via paper, and (at least in the case of Benny’s classroom) they were scored by a teacher’s aide. With today’s computer-delivered individualized instructional programs, in which correct answers are judged by the program, teachers can “teach” math with these programs without ever actually seeing any answers provided by the student.
The most worrying aspect of Erlwanger’s description of Benny and his math is the lack of understanding he demonstrated. For example, Benny had “divined” rules he used to add decimals, and he applied his rules consistently. Erlwanger supplied examples of the rule Benny followed including those for converting fractions to decimals that are detailed in this screenshot from a digital version of the article:
After describing these rules, Erlwanger concluded, “Benny’s case indicates that a ‘master of content’ does not imply understanding.”
Erlwanger also details that nature of the instruction in IPI classrooms. Among the concerns raised by Erlwanger are the intent that students interact with teachers as needed (interaction Benny indicated was rare and occurred only when he initiated it), but that seemed contradicted by the goal of IPI mathematics curriculum to allow for “independence, self-direction, and self-study. The IPI mathematics “for Benny implies self-study within the prescribed limits…, and there is never any reason for Benny to [discuss] either with hos teacher or peers what he has learned and what his views are about mathematics.”
Benny appears to have had sophisticated understanding of the IPI mathematics program, and even expressed concern over the materials. Benny had concluded the answer keys were incorrect, but the teacher or teacher aide were compelled to use the incorrect answers to score his work. Benny differentiated the scores on IPI tests (scores he could earn by figuring out the pattern of answers and using a trial-and-error approach) and the mathematics he was inventing. Benny mathematics, of course, had little to do with the nature of numbers.
Erlanger concluded the version of individualized learning, which is deeply aligned with behaviorist beliefs, had contributed to the discrepancy between Benny’s IPI performance and his understanding of mathematics. IPI’s prescribed curriculum prevented Benny from “realiz[ing] he has to reason, seek relationships, make generalizations, and verify discoveries by independent means.”
As an educational technologist, I have been in countless classrooms over the last two decades. I can be a “fly on the wall” as teachers proceed as if I am not in the room. I have seen the increasing use of digital versions of IPI, and I have seen much computer-based math instruction that resembles IPI, and I suspect the results are similar to what Erlwanger observed in Benny.
I understand the value of digital instruction in mathematics, I also understand the value of computer-scored tests and quizzes. Any teacher who believes math education ends with those programs is sadly mistaken. Principals, curriculum coordinators, technology coordinators, or other leader who adopts a digital math instruction program without having clear expectations about its minority role in math education and without ensuring a comprehensive mathematics education experience for students is unqualified for the positions they hold.
Edwards, A. R., Esmonde, I., Wagner, J. F., & Beattie, R. L. (2011). Learning mathematics. Handbook of research on learning and instruction, 55-77.
Erlwanger, S. H. (1973). Benny’s conception of rules and answers in IPI mathematics. Journal of Children’s Mathematical Behavior, 1(2), 7-26.
Yetter, C. C. (1972). Do Schools Need IPI? Yes!. Educational Leadership, 29(6), 491-4.