Grandi's series in education

Studies

Cognitive impact

Around 1987, Anna Sierpińska introduced Grandi's series to a group of 17-year-old precalculus students at a Warsaw lyceum. She focused on humanities students with the expectation that their mathematical experience would be less significant than that of their peers studying mathematics and physics, so the epistemological obstacles they exhibit would be more representative of the obstacles that may still be present in lyceum students.

Sierpińska initially expected the students to balk at assigning a value to Grandi's series, at which point she could shock them by claiming that 1 − 1 + 1 − 1 + · · · = ¹⁄₂ as a result of the geometric series formula. Ideally, by searching for the error in reasoning and by investigating the formula for various common ratios, the students would "notice that there are two kinds of series and an implicit conception of convergence will be born." However, the students showed no shock at being told that 1 − 1 + 1 − 1 + · · · = ¹⁄₂ or even that 1 + 2 + 4 + 8 + · · · = −1. Sierpińska remarks that a priori, the students' reaction shouldn't be too surprising given that Leibniz and Grandi thought ¹⁄₂ to be a plausible result;

"A posteriori, however, the explanation of this lack of shock on the part of the students may be somewhat different. They accepted calmly the absurdity because, after all, 'mathematics is completely abstract and far from reality', and 'with those mathematical transformations you can prove all kinds of nonsense', as one of the boys later said."

The students were ultimately not immune to the question of convergence; Sierpińska succeeded in engaging them in the issue by linking it to decimal expansions the following day. As soon as 0.999... = 1 caught the students by surprise, the rest of her material "went past their ears".^[1]

Preconceptions

In another study conducted in Treviso, Italy around the year 2000, third-year and fourth-year Liceo Scientifico pupils (between 16 and 18 years old) were given cards asking the following:

"In 1703, the mathematician Guido Grandi studied the addition: 1 – 1 + 1 – 1 + ... (addends, infinitely many, are always +1 and –1). What is your opinion about it?"

The students had been introduced to the idea of an infinite set, but they had no prior experience with infinite series. They were given ten minutes without books or calculators. The 88 responses were categorized as follows:

(26) the result is 0

(18) the result can be either 0 or 1

(5) the result does not exist

(4) the result is ¹⁄₂

(3) the result is 1

(2) the result is infinite

(30) no answer

The researcher, Giorgio Bagni, interviewed several of the students to determine their reasoning. Some 16 of them justified an answer of 0 using logic similar to that of Grandi and Riccati. Others justified ¹⁄₂ as being the average of 0 and 1. Bagni notes that their reasoning, while similar to Leibniz's, lacks the probabilistic basis that was so important to 18th-century mathematics. He concludes that the responses are consistent with a link between historical development and individual development, although the cultural context is different.^[2]

Prospects

Joel Lehmann describes the process of distinguishing between different sum concepts as building a bridge over a conceptual crevasse: the confusion over divergence that dogged 18th-century mathematics.

"Since series are generally presented without history and separate from applications, the student must wonder not only "What are these things?" but also "Why are we doing this?" The preoccupation with determining convergence but not the sum makes the whole process seem artificial and pointless to many students—and instructors as well."

As a result, many students develop an attitude similar to Euler's:

"…problems that arise naturally (i.e., from nature) do have solutions, so the assumption that things will work out eventually is justified experimentally without the need for existence sorts of proof. Assume everything is okay, and if the arrived-at solution works, you were probably right, or at least right enough. …so why bother with the details that only show up in homework problems?"

Lehmann recommends meeting this objection with the same example that was advanced against Euler's treatment of Grandi's series by Callet.

Notes

References