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This paper extends work in the areas of quantitative reasoning and covariational reasoning at the undergraduate level. Task-based interviews were used to examine third-semester calculus students' reasoning about partial derivatives in five tasks, two of which are situated in a mathematics context. The other three tasks are situated in real-world contexts (i.e. weight, revenue, and mortgage). There are three main findings from this study. First, interpreting quantities representing partial derivatives in real-world contexts was problematic for most of the students. Second, students' interpretations of partial derivatives were not consistent across tasks. Third, all the students engaged in several levels of covariational reasoning when interpreting quantities representing partial derivatives in the context of weight, revenue, and mortgage. Implications for calculus instruction and directions for future research are included.

This study used task-based interviews to examine students' reasoning about multivariable optimization problems in a volume maximization context. There are four major findings from this study. First, formulating the objective function (i.e. the function whose maximum or minimum value(s) is to be found) in each task came easily for 15 students who participated in the study. Second, most of the students successfully used the Lagrange Multiplier method or the Substitution method (i.e. eliminating one of the variables from the objective function, finding partial derivatives, and solving the simultaneous equations that result) to find the critical points of the objective function in each task. Third, verifying/justifying extrema (i.e. showing that the dimensions students found in each task are dimensions of largest volume object) was problematic for a majority of the students. Fourth, solving systems of nonlinear equations to find critical points of the objective function in each task was particularly challenging for nearly half of the students. This study has characterized students' difficulties when solving multivariable optimization problems. Further research is needed to explore instructional approaches instructors could adopt to help students develop a robust understanding of multivariable optimization problems and the methods used to solve these problems.

This article reports on a qualitative investigation into students' thinking about a differential equations problem posing task; i.e. an initial value problem. Analysis of written and verbal responses to the task indicate that only four of the 34 students who participated in the study were successful in posing problems. Furthermore, only one of the problems they posed is solvable. In general, findings of this study indicate that posing problems involving initial value problems is particularly challenging for students. In light of the multitude of benefits of problem posing in mathematics education, I recommend the inclusion of problem posing tasks in the teaching and learning of undergraduate mathematics courses, including ordinary differential equations. [ABSTRACT FROM AUTHOR]

While research on the opportunity to learn about mathematics concepts provided by textbooks at the secondary level is well documented, there is still a paucity of similar research at the undergraduate level. Contributing towards addressing this knowledge gap, the present study examined opportunities to engage in quantitative and covariational reasoning, in the context of the derivative concept, provided by two textbooks commonly used in the teaching of calculus in the United States-a regular calculus textbook and an applied calculus textbook. Analysis of expository sections, examples, and exercises related to the derivative provided in these textbooks revealed three main results. First, the applied calculus textbook provides plenty of opportunities to engage in quantitative reasoning. Similar opportunities are limited in the regular calculus textbook. Second, opportunities to engage in covariational reasoning are not only minimal in both textbooks, but also limited to low levels of covariational reasoning, namely coordination, direction, and quantitative coordination. Third, the applied calculus textbook consistently defines ordinary and partial derivatives as rates of change while the regular calculus textbook consistently defines these concepts as limits of a difference quotient. Findings of this research have implications for several stakeholders, including calculus textbook authors and calculus instructors. [ABSTRACT FROM AUTHOR]

Contributing to research on undergraduate students' thinking about problem solving tasks, the present study reports on students' reasoning about two initial-value problems i.e. first-order linear ordinary differential equations with initial conditions. A qualitative analysis of task-based interviews and work written by 34 students revealed that although the students knew the algebraic steps they had to use to solve the two problems, 16 students reported that steps involving integral calculus were particularly challenging. In fact, a majority of these students exhibited lack of facility with the $ u $ u -substution integration technique, something that severed as a stumbling block for them when calculating the integrating factor in their attempt to solve the two problems. Only seven students were successful in mathematizing one of the problems. Five students created and used diagrams to support their thinking when solving one of the problems. These students reported that the creation of the diagrams helped them to solve one of the two problems. Overall, findings of this study suggest that helping students develop greater facility with integration techniques, and more broadly integral calculus, could increase students' success when solving initial-value problems. Implications for instruction and directions for future research are discussed. [ABSTRACT FROM AUTHOR]

Contributing to a growing body of research on undergraduate students' quantitative reasoning, the study reported in this article used task-based interviews to investigate business calculus students' quantitative reasoning when solving two optimization tasks situated in the context of revenue and profit maximization. Analysis of verbal responses and work written by 12 pairs of students during the task-based interviews revealed that nearly all pairs of students created new quantities (e.g., diminishing marginal returns). Students used these new quantities to reason about relationships among computer sales, sales discount, and total revenue in a revenue maximization task. The creation of these quantities helped the students to solve the problem posed in the task. Ten pairs of students interpreted marginal cost as total cost and marginal revenue as total revenue in a profit maximization task. Implications for business calculus instruction and directions for future research are discussed.

Much research has reported on difficulties exhibited by students when working with the concept of the derivative in first-semester university calculus. In an effort to generate and share a resource on effective instructional practices related to the teaching of the concept of the derivative, a 12-item questionnaire was administered to experienced calculus instructors in the United States. Most of the 10 experts who participated in this study considered providing ample examples and practice problems, using graphing utilities such as Desmos and GeoGebra, and employing problem solving strategies in the teaching of the concept of the derivative to be effective in supporting students learn about the concept of the derivative, among other things. These experts further remarked on representations of the concept of the derivative and real-world contexts they typically use in their teaching of the concept, in addition to providing rationales for the choice of real-world contexts they typically use in their teaching of the concept. Recommendations for instruction are included. [ABSTRACT FROM AUTHOR]

Much research has reported on students' difficulties with solving related rates problems in calculus. In an effort to generate a resource that could potentially address some of these difficulties from a teaching standpoint, a questionnaire about effective instructional approaches related to the teaching of related rates problems, among other things, was administered to 14 veteran calculus instructors. Analysis of the responses provided by the instructors revealed that all the instructors considered the use of diagrams to be helpful when solving related rates problems. Furthermore, a majority of these instructors noted that introducing a set of steps (i.e., a guideline), during classroom instruction, that students could follow when solving related rates problems is helpful for students when working with this type of problems. These instructors further identified strengths and weaknesses in the way related rates problems are typically presented in calculus textbooks. Implications for instruction are included. [ABSTRACT FROM AUTHOR]