Relating students’ units coordinating and calculus readiness

ABSTRACT

Adolescent and children’s concepts of multiplication and fractions have been linked to differences in the number of levels of units they coordinate. In this paper, we discuss relationships between adult students’ conceptual structures for coordinating units and their pre-calculus understandings. We conducted interviews and calculus readiness assessments with 27 introductory calculus students at a public university in the United States. Results indicate that students assimilating multiplicative situations with three levels of units performed significantly better on calculus readiness assessments than students assimilating multiplicative situations with fewer than three levels of units. We discuss implications for instruction.

KEYWORDS:

• Units coordinating
• calculus readiness
• multiplicative reasoning

Notes

1. All names of students are pseudonyms.

2. Thompson and Carlson (2017) created a conceptual framework for variational and covariational reasoning to apply more broadly to situations that do not necessarily involve rate of change. In this paper we focus the M. Carlson et al. (2002) earlier description of levels of the framework because that work informed the PCA.

3. The three items that were excluded were 1, 14, and 21 from Figure 6. The rate of change replacement item was from Carlson’s 1998 study (the ladder task). The other two (measurement and slope) items were developed and validated with secondary mathematics teachers (Byerley & Thompson, 2014, 2017). See Appendix A for verbal descriptions of the 25 tasks.

4. We include the value of the ${\chi }^2$ statistic and p-value for each item in Appendix B.