Examining Students' Quantitative Reasoning in a Virtual Ecosystem Simulation of the Water Cycle

In this study, we designed and tested an instructional module to help students develop their quantitative reasoning while examining the scientific phenomenon of the water cycle. The results from a design experiment in a sixth-grade classroom showed that students exhibited reasoning that integrated science and mathematics and avoided the typical naïve conceptions about the water cycle. Specifically, students engaged in covariational and multivariational reasoning as they coordinated the simultaneous change of the quantities involved in the water cycle. They also exhibited transitive reasoning in terms of those quantities and relational reasoning connecting the water cycle to other science content. These forms of reasoning illustrate the power of mathematical reasoning for helping students construct an advanced understanding of the water cycle phenomenon. The findings also show that the context of science helped students to construct advanced forms of mathematical reasoning about quantities, illustrating the reciprocal relationship between math and science learning. Math and science integration: The case of the water cycle learning Mathematics and science integration efforts illustrate the common practices between math and science and how these commonalities serve as a platform to help students experience meaningful and relevant interdisciplinary learning (e.g., National Research Council [NRC], 2013). A seamless integration of mathematics into science topics can help students develop high levels of understanding about the science phenomena (Roschelle et al., 2007). Although integrated mathematics and science activities have been developed, these activities only encourage students to solve science problems using their existing mathematical knowledge without exhibiting any new forms of mathematical reasoning during the process (Tytler, Williams, Hobbs & Anderson, 2019). In other words, we have limited information about how integrated activities can be designed that would develop students’ reasoning in ways that illustrate a reciprocal relationship (English, 2016) between mathematics and science. A reciprocal relationship would involve developing a deeper understanding of the science phenomenon because of the integration with mathematics while also advancing their mathematical reasoning because of the integration with science. To examine the design of activities that help illustrate such reciprocal relationships between math and science, we designed an instructional module that integrates the mathematics of quantitative reasoning and the science of the water cycle. We chose to focus on the water cycle because it is one of the first, most abstract, and complex scientific phenomena introduced to students (Vinisha & Ramadas, 2013). To understand the complex system of the water cycle, research suggests that its learning must be integrated with other educational contexts in order for students to develop a coherent understanding (Vinisha & Ramadas, 2013). In contrast to this view, the water cycle has been exclusively taught in science classes (Barrutia, Ruíz-González, Villarroel, & Díez, 2019). In the current science curricula, the water cycle is commonly taught using illustrations from science textbooks, where students are asked to describe the water cycle through drawings (Vinisha & Ramadas, 2013). Although the use of textbook illustrations or drawings has been found to help students develop a basic understanding about the water cycle (Cardak, 2009), research shows that students also develop many naïve conceptions about the phenomenon. For example, some students believe that evaporation does not happen in cold temperatures (Benzvi-Assarf & Orion, 2005), or that the amount of water vapor in the air remains unchanged (Cardak, 2009). Moreover, the learning of the water cycle involves students understanding how water goes through a cyclical movement (Ben-zvi-Assarf & Orion, 2005); however, studies show that students do not have this cyclic notion (Barrutia et al., 2019). Instead, they usually over-simplify the cycle, believing that water only travels to the atmosphere from the earth through evaporation and returns to the earth through condensation (Cardak, 2009). To develop the cyclic notion of the water cycle and develop these naïve conceptions to more sophisticated understandings, students must be able to coordinate its different components into a coherent unitary system (Kali, Orion, & Eylon, 2003). The next section describes the theoretical framework that we used to design such experiences for students. ICLS 2020 Proceedings 959 © ISLS Theoretical framework Across the mathematics and science education literature, the development of students’ reasoning is often an isolated process and exclusive to each discipline. To bridge the two disciplines, we first examined their similarities. We found that studying change is predominant in both mathematics and science concepts; thus, we examined its potential for developing students’ integrated reasoning in the two disciplines. In science phenomena, we usually have a single variable changing, or two variables changing simultaneously. Reasoning about these variables typically focuses on cause-and-effect relationships where a student identifies the factors that influence a phenomenon, but this form of reasoning does not explain how these factors interact with each other. In other words, reasoning about the cause-and-effect relationships, such as identifying that humidity is a factor influencing precipitation, is different from reasoning about their quantities and how these quantities covary, such as reasoning that as the humidity increases, the precipitation increases. To investigate students’ reasoning about the quantities embedded in the water cycle, we followed the work of Thompson and colleagues on quantitative reasoning. Quantitative reasoning is a mental process of analyzing a phenomenon, translating it into a system of quantities, and reasoning about the quantities and the relationships between them (Smith III & Thompson, 2008). Hence, by taking a quantitative reasoning perspective, understanding the water cycle includes conceiving its measurable quantities such as air temperature, lake temperature, mountain temperature, land temperature, and humidity, and constructing relationships between those and the processes of the water cycle, such as precipitation, evaporation, infiltration, and runoff. When envisioning two quantities changing as those quantities change simultaneously, an individual is engaging in what research refers to as covariational reasoning (Carlson, Jacobs, Coe, Larsen, & Hsu, 2002; Confrey & Smith, 1994). Reasoning covariationally does not necessarily involve reasoning using numerical values. In fact, research has shown that students can identify and reason about quantities using their magnitudes and not numerical values. For example, when coordinating quantities, students can attend to the direction of their change (Carlson et al., 2002) or to the intensity of change in quantities (Johnson, 2012). In coordinating the direction of change, students may reason about the direct relationships between quantities when quantities change in the same direction or the inverse relationships when quantities change in opposite directions. For example, a student may reason that as air temperature increases, the amount of water that evaporates also increases, illustrating the direct relationship between air temperature and the amount of evaporation as both are increasing. Also, a student may reason that as infiltration decreases, the amount of runoff increases, illustrating an inverse relationship between infiltration and runoff. In attending to the intensity of change in quantities, a student may reason that if the lake temperature is higher or lower, the evaporation is faster or slower, respectively. Although quantitative and covariational reasoning has been significantly studied and found to be an important form of reasoning about quantities, most of these investigations use these forms of reasoning to frame their investigation of a mathematical concept, such as rate-of-change (Johnson, 2012), trigonometric relationships (Moore, 2012), or linear relationships and quadratic relationships (Ellis, 2011). As Thompson and Carlson (2017) pointed out, the great majority of these studies do not contribute directly to defining the covariation construct. In a previous study using quantitative reasoning in the learning of gravity (Basu et al., 2020), we found that sixthgrade students were able to coordinate both the mass and the distance when making judgments about gravity, showing that the science context helped them reason multivariationally about quantities. In mathematics education, multivariational reasoning has been studied as a form of reasoning in advanced mathematics (Kuster & Jones, 2019). Consequently, we started with the conjecture that integrating quantitative reasoning with science will not only help students with the science concept but will also give us an insight into sophisticated mathematical forms of reasoning about quantities that are possible through this integration. Specifically, we aimed to explore: a) How may students reason about quantities in the context of the water cycle? b) How may students’ quantitative reasoning help them develop an understanding of the water cycle?