Quantifying exponential growth: Three conceptual shifts in coordinating multiplicative and additive growth

A link between students’ discernment of variation in unidirectional change and their use of quantitative variational reasoning

“…Researchers have found that early secondary students can discern variation in unidirectional change when employing chunky images of change (Ellis et al, 2015;Lobato et al, 2012). Given a situation involving a plant's height growing exponentially with time, Ellis et al (2015) found that early secondary students conceived of variation in fixed amounts of increases in height and time, thereby demonstrating chunky images of change.…”

“…Discerning variation in unidirectional change (e.g., discerning an Bincreasing^or Bdecreasingî ncrease) is challenging even for successful university students (Carlson, Jacobs, Coe, Larsen, & Hsu, 2002), yet important for secondary students' study of key mathematical concepts such as function and rate (e.g., Ellis, Özgür, Kulow, Williams, & Amidon, 2015;Herbert & Pierce, 2012;Johnson, 2012;Lobato, Hohensee, Rhodehamel, & Diamond, 2012). Researchers have identified factors that could mitigate students' discernment, including students' images of change and variation (Castillo-Garsow, Johnson, & Moore, 2013;Thompson & Carlson, 2017) and students' interpretation of graphs (Bell & Janvier, 1981;Clement, 1989;Moore & Thompson, 2015;Slavit, 1997).…”

“…Kaput (1994) argued that dynamic computer environments (DCEs), linking dynamic animations and graphs, could serve as rich settings for students to investigate the mathematics of change and variation. Researchers investigating secondary students' conceptions of function, rate, and variation have made extensive use of DCEs in their studies (e.g., Ellis et al, 2015;Herbert & Pierce, 2012;Johnson, 2012;Lobato et al, 2012;Saldanha & Thompson, 1998). When interacting with DCEs representing varying rates of change, secondary students have shown evidence of discerning variation in unidirectional change (Ellis et al, 2015;Johnson, 2012;Saldanha & Thompson, 1998).…”

“…Researchers investigating secondary students' conceptions of function, rate, and variation have made extensive use of DCEs in their studies (e.g., Ellis et al, 2015;Herbert & Pierce, 2012;Johnson, 2012;Lobato et al, 2012;Saldanha & Thompson, 1998). When interacting with DCEs representing varying rates of change, secondary students have shown evidence of discerning variation in unidirectional change (Ellis et al, 2015;Johnson, 2012;Saldanha & Thompson, 1998). Using Marton's variation theory (2015), we frame our design of filling polygon DCEs and related tasks, which we implemented to provide students opportunities to discern variation in unidirectional change.…”

“…Using Thompson's theory of quantitative reasoning (1994), we incorporated attributes-length, height, and area-which we anticipated students could conceive of as capable of varying and possible to measure. Researchers have shown the utility of variational reasoning for fostering early secondary students' conceptions of function and interpretations of graphs (Ayalon, Watson, & Lerman, 2015;Ellis et al, 2015;Hitt & González-Martín, 2015;Johnson, 2012;Lobato et al, 2012). We use Thompson's theory of quantitative reasoning (1994) to frame our perspective on variational reasoning.…”