Mathematics Students’ Characteristics of Basic Mental Models of the Derivative

Abstract

The concept of derivative is characterised with reference to four basic mental models. These are described as theoretical constructs based on theoretical considerations. The four basic mental models—local rate of change, tangent slope, local linearity and amplification factor—are not only quantified empirically but are also validated. To this end, a test instrument for measuring students’ characteristics of basic mental models is presented and analysed regarding quality criteria.Mathematics students (n = 266) were tested with this instrument. The test results show that the four basic mental models of the derivative can be reconstructed among the students with different characteristics. The tangent slope has the highest agreement values across all tasks. The agreement on explanations based on the basic mental model of rate of change is not as strongly established among students as one would expect due to framework settings in the school system by means of curricula and educational standards. The basic mental model of local linearity plays a rather subordinate role. The amplification factor achieves the lowest agreement values. In addition, cluster analysis was conducted to identify different subgroups of the student population. Moreover, the test results can be attributed to characteristics of the task types as well as to the students’ previous experiences from mathematics classes by means of qualitative interpretation. These and other results of students’ basic mental models of the derivative are presented and discussed in detail.

Unpacking mental models, strategies, and schemas pre-service mathematics teacher in solving maximum rectangular areas

“…Research related to mental models has been carried out on several aspects, including analogical problemsolving , deductive and inductive reasoning (Johnson-Laird & Byrne, 1993), probabilistic inference (Kahneman & Tversky, 1982), and mental models of the derivative (Greefrath et al, 2022), However, research on mental models in problem solving, especially finding the maximum rectangular area involving the concept of derivatives, has not been done much. Finding the maximum rectangular area is a mathematical problem because it is challenging for individuals but cannot be solved directly with a specific algorithm (Hoosain, 2004) because constraints limit the problem, namely the size of the given circumference.…”

“…The schema in this study refers to the knowledge structure containing information about the core concepts and the relationships between the concepts. Furthermore, neither Bofferding (2014), Chinnapan (1998), as well as Greefrath et al (2020Greefrath et al ( , 2022 does not disclose and describe the process of accommodation of mental models during problem-solving. Whereas Oleson et al (2010), Piaget (1954), andvan Ments &Treur (2022) state that mental models experience an adaptation process through two forms, namely assimilation and accommodation, but according to Ifenthaler et al (2008) and van Ments and Treur (2022) mental models only experience accommodation processes.…”