Young Students’ Functional Thinking Modes: The Relation Between Recursive Patterning, Covariational Thinking, and Correspondence Relations

Abstract

A theoretical model describing young students’ (Grades 1–3) functional-thinking modes was formulated and validated empirically (n = 345), hypothesizing that young students’ functional-thinking modes consist of recursive patterning, covariational thinking, correspondence-particular, and correspondence-general factors. Data analysis suggested that functional-thinking tasks can be categorized on the basis of the proposed model. Analysis traced three categories of students that represent different functional-thinking profiles. Category 1 students exhibited a recursive-thinking profile. Category 2 students utilized a combination of recursive and contextual strategies and exhibited an emergent covariational and correspondence-particular thinking. Category 3 students approached functional-thinking situations flexibly, using a combination of covariational and correspondence strategies. A structural model showed two parallel paths from recursive patterning to correspondence-general through correspondence-particular or covariational.

Task Development to Measure Functional Thinking: Focusing on Third Graders’ Understanding

“…An example of repeated numeric patterns is "1, 3, 1, 3, …," in which 1 and 3 are repeated (i.e., ABAB form). The first-and second-grade textbooks in Korea included ABAB, AABB, and ABC forms, whereas prior studies involving recursive patterning tasks used ABBC and AAB patterns as well as the three forms from the textbooks (e.g., Papic et al, 2011;Pittalis et al, 2020).…”

“…Growing geometric patterns cover various forms in which the changing parts of the figure constituting each term increase consistently (see Table 1). These patterns have been popular not only in textbooks but also in related studies (e.g., Pang & Sunwoo, 2016b;Papic et al, 2011;Pittalis et al, 2020). Growing numeric patterns usually have rules of adding or multiplying by a constant number in a sequence of numbers, as in the examples "1, 3, 5, 7, …" and "2, 4, 8, 16, ….…”

“…Another important issue in constructing covariation tasks is to confirm whether students solve the given problem by recognizing the relationship between two covarying quantities. For example, in the task of Pittalis et al (2020) for students in Grades 1 to 3, the problem situation shown in Figure 1a was that the number of blocks increased by 2 each day. In order to assess whether students could employ covariation thinking, the tasks included specific questions, such as "If the snake has 40 blocks in day 20, how many blocks will it have in day 22?"…”

“…For example, in Figure 1b, a student can fill in the output as follows: 2 times 2 plus 1 is 5, and 3 times 2 plus 1 is 7; so, 7 times 2 plus 1 is 15. In such a case, the student's answer is based on the specific cases of the corresponding pairs; this type of answer arises from correspondence-particular thinking (Pittalis et al, 2020). Alternatively, the relationship in Figure 1b can be expressed either by (1) an equation "I × 2 + 1 = O" (I: input value, O: output value) or (2) a generalized statement, like "Multiply the input value by 2 and then add 1; then you get the output value."…”

“…In contrast, tasks measuring functional thinking can be constructed differently depending on the modes of functional thinking, which are recursive thinking, covariation thinking, and correspondence thinking (Blanton et al, 2015;Stephens et al, 2017). In other words, it is possible to identify students' functional thinking by using tasks that can be solved with only a single mode of functional thinking (Pittalis et al, 2020). In addition, students' functional thinking has been found to be different according to the types of functional relationships being studied, such as y = x ± a and y = ax (e.g., Choi & Pang, 2012).…”