Differentiating between counterexamples for supporting students’ algorithmic thinking

Abstract

This study examines the use of counterexamples for supporting the development of students’ algorithmic thinking. Working from the premise that some counterexamples are more effective than others for the development of generalized algorithms, the study proposes distinctions between counterexamples in relation to the iterative refinement of student-invented algorithms. Furthermore, the study identifies some factors that may influence differences among counterexamples. Using task-based interviews, data were collected from 23 undergraduate students working in pairs ( n = 8) and individually ( n = 7) on three algorithmatizing tasks. From a thematic analysis of the data, two illustrative cases are presented to show how and why different counterexamples might bring about particular revisions in students’ algorithms. The two illustrative cases highlight two types of counterexamples— set-of-instructions-changing ( SoI-changing) and domain-of-validity-narrowing ( DoV-narrowing)—and their influencing factors. Implications of the findings are discussed with respect to existing literature, further research, and teaching.

What challenges emerge when students engage with algorithmatizing tasks

“…One might say that Liv and Mel, as evidenced by their work in Table 2, made little to no progress because they could not get past their initial algorithm, a 2 + b 2 , which does not yield the correct solution for the general class of problems that they were asked to solve-all pairs of real numbers a and b. However, as alluded to above, this "general class of problems" tends to be pre-determined and fixed by an external source (e.g., teacher, researcher) and does not necessarily align with the problems that students intend to solve with their algorithms (Tupouniua, 2020a(Tupouniua, , 2023. The external source's fixed view of "problems to be solved," especially when it does not align with the students' domain of validity, can be problematic (Tupouniua, 2023).…”

“…The data presented in the three illustrative cases were collected as part of a larger ongoing selfdesigned study (Tupouniua, 2023) exploring various aspects of students' algorithmic thinking across different educational sectors in New Zealand. This larger study is a continuation of work that began during my doctoral study (Tupouniua, 2019) which examined the mechanisms by which students' algorithms emerge.…”

“…In theory, these counterexamples are meant to motivate students to construct a generalized algorithm. However, I have also argued (Tupouniua, 2020(Tupouniua, , 2023 that past studies commonly construe "generalized algorithm" as an algorithm that correctly solves all problems that the teacher/researcher wants the student(s) to solve. Consequently, past studies consider all students' revisions that do not result in a generalized algorithm to be incorrect, and they deem all counterexamples that do not motivate a revision that yields a generalized algorithm as ineffective.…”

“…Liv and Mel had two and five years teaching experience respectively at the primary level. The task (Tupouniua, 2023) at hand invited the students to create an algorithm for expanding the expression (a+b) 2 where a and b can be any two real numbers.…”

“…This paper presents three illustrative cases corresponding to three emergent challenges that some students encountered as they grappled with creating their own algorithm. These three challenges are ones that have been evident throughout my research in mathematics education, which primarily focuses on exploring a diverse range of aspects pertaining to algorithmic thinking (e.g., Moala et al, 2019;Tupouniua, 2019Tupouniua, , 2020Tupouniua, , 2023. The first challenge highlights discrepancies between the method with which students solve a problem versus the algorithm they create, and claim would, when implemented, solve the same problem.…”