Calculus
MIP faculty collaborations on the Calculus pathway began with an Initiation Workshop in August, 2021. At this workshop, faculty identified the central concepts and skills critical for success in Calculus I. Participants of the workshop identified seven foundations to guide development, testing, and refinement of inquiry-oriented course materials to support the development of the central concepts and skills in the Calculus pathway.
Functions serve as the basic language and notation for students’ experience in Calculus I. A robust understanding of functions is therefore critical for students’ success in the course. Many difficulties students experience while reasoning with functions are based in a static “action view” of evaluating a function for one input at a time, typically based on an algebraic formula. In contrast, a “process view” of function in which a student can conceive of the entire process happening to all input values at once, enables them to conceptually run through a continuum of input values while attending to the resulting impact on output (e.g., see the discussion of action and process views in Oehrtman, Carlson, & Thompson, 2008). This way of thinking about the covariation of input and output values is foundational for constructing meaningful formulas and graphs when modeling relationships in applied contexts, interpreting limits conceptually or formally, and thus reasoning about all concepts defined in terms of limits (Carlson et al., 2002; Moore & Carlson, 2012; Oehrtman, Carlson, & Thompson, 2008). [Explore the Details]
Limits are often the first mathematical operation students encounter that cannot be conceived through finite computation, leaving them to negotiate spontaneous concepts of an actual infinity with infinite processes that do not end. Covariational reasoning about functional dependence is required to first conceptualize, then to coordinate, two such infinite processes quantitatively (e.g., a domain process in which 
%5Cto%09L "f(x)\to L")
Local Linearity, Differentials, Infinity, and Infinitesimals
Researchers (e.g., Thompson et al, 2015) have demonstrated that many calculus students’ understanding of derivatives is not sufficiently grounded in robust meanings of rate of change. When prompted to explain what the derivative at a point represents, students often reply, “The slope of the tangent line.” Limited to such geometric interpretations, students struggle to apply derivatives in novel contexts and to understand more advanced topics in calculus, such as linear approximation, L’Hopital’s rule, implicit differentiation, related rates, Riemann sums, definite integrals, and the fundamental theorem of calculus. To address this problem, several mathematics educators (e.g., Ely, 2021) have documented the affordances of students’ conceptualizing differentials as linear functions (i.e., as infinitesimal changes that vary proportionally). Related recommendations include supporting students’ interpretation of “instantaneous rate of change” as “average rate of change over infinitesimally small intervals where the corresponding changes in the measures of the input and output quantities are proportional.” [Explore the Details]
A critical foundation for reasoning about rates of change is conceiving of changes in quantities as quantities in their own right and distinguishing such changes from the original quantities. From this foundation, students may begin to understand, distinguish, and use the meanings of constant rate of change and average rate of change in various contexts and representations. These concepts, in turn, form a foundation for students’ understanding and reasoning about instantaneous rate of change in calculus. In particular, constant rate of change entails a proportional relationship between corresponding changes in the measures of the two quantities (e.g., see the conceptual analysis in Thompson, 1994). Reasoning about these changes and their proportional relationship across multiple representations can build an important foundation for further development of average and instantaneous rates. Non-quantitative interpretations of constant and average rate of change restrict students to iconic images, such as “steeper is faster.” Although such pseudo-structural reasoning may be sufficient for many procedural applications of derivatives, they prevent students from productively unpacking these ideas when necessary in problem-solving situations. [Explore the Details]
Continuity is a property of functions with several important implications. The conclusions of Rolle’s theorem, the mean value theorem, the intermediate value theorem, the extreme value theorem, and the fundamental theorem of calculus all require a function to be continuous on a closed interval. Continuity is also a necessary condition for integrability and for the algebraic properties of definite integrals. It is essential that students understand the relationship between continuity and differentiability, and leverage their understanding of continuity to make strategic inferences about function behavior. [Explore the Details]
Accumulation, Integrals, and the Fundamental Theorem of Calculus
Several scholars have described consequential ways of understanding the fundamental theorem of calculus and how students might apply definite integrals in modeling problems. Multiple studies (Orton, 1983; Orton, 1984; Serhan, 2015; Rasslan & Tall, 2002) suggest that students might not hold quantitative meanings for the components of an integral despite being proficient with integral calculations. Compounding this potential lack of meaning are various studies documenting the challenges students experience when attempting to apply definite integrals to contexts in physics or engineering (Sealey, 2014; Meredith & Marrongellle, 2008; Jones 2013; Jones 2015; Simmons & Oehrtman, 2017; Chhetri & Oehrtman, 2015; Bajracharya & Thompson, 2014). Other research has documented students’ difficulties with coordinating the product structure f(xi)Δx of an accumulated quantity (e.g., Sealey, 2014). Mathematics educators have responded to these difficulties by demonstrating the effectiveness of engaging students in tasks that require them to consider how to approximate the accumulation of a quantity (or to construct a function that represents the value of an accumulated quantity) by assuming that a varying quantity (a rate, a force, etc.) is constant over some interval of its variation, and then to approximate the change in the accumulated quantity over each successive interval by computing the product of the (assumed) constant quantity and the change in the independent variable. Generally, an important instructional goal is to help students conceptualize the product of the integrand and the change in the function’s independent variable as an approximation of the change of the accumulated quantity. [Explore the Details]
Creating and interpreting mathematical models is a critical path for students to both better understand the underlying mathematics and to be prepared to apply that mathematics in other disciplines. In calculus, students have an opportunity to reason about new types of quantities and quantitative relationships (such as instantaneous rates) and to distinguish them from previous non-calculus quantities (such as constant or average rates). Students may represent quantities and quantitative relationships as ways to mathematize a context or, conversely, to give contextual meaning to mathematical symbols. They may strategically manipulate or interpret these representations to draw inferences about a context or use the context to construct conjectures or arguments about the mathematics. [Explore the Details]
Calculus I CoRDs and ARCs
|
Liz Lane-Harvard, University of Central Oklahoma
|
This ARC provides a pair of activities designed to help students develop an intuitive understanding of the Mean Value Theorem that is grounded in understanding its key foundational ideas (e.g., continuity, average rate of change, instantaneous rate of change). The activities provide opportunities for student engagement with mathematics while also allowing opportunities for collaboration. Active student engagement is promoted through a contextual situation as well as meaningful scaffolding that builds on previous student experience.
|
|
Michael Oehrtman, Oklahoma State University
|
These resources are designed to be used through a first-semester calculus course to help students develop a rigorous and productive understanding of limits, derivatives, and integrals. The activities engage students in problem-solving about approximating unknown quantities using appropriate function values, difference quotients, or Reimann sums. While these ideas reflect the underlying structure of formal ε-δ definitions and proofs, they are framed in more accessible language and ideas of errors, error bounds, and reasoning to achieve any desired degree of accuracy. Representing their reasoning in consistent ways across multiple representations helps students link these situation-specific ideas to the underlying mathematical principals in a coherent way. Then engaging in similar tasks across different contexts supports attention to the general mathematical structure of limits, derivatives, and definite integrals that can support subsequent modeling and interpretation.
|
|
Exploring the Connection between Secant Lines and the Tangent Line at a Given Point (ARC) Michael Fulkerson, University of Central Oklahoma
|
This lesson focuses on the connection between secant lines and the tangent line at a given point. Students will explore the slopes of various lines secant to a given graph by hand, as well as via a Desmos activity. After predicting the slope of the graph at a given point, they compute the actual slope at that point using the limit of a difference quotient. Students then have the opportunity to change the given point or the given function, make new predictions, and check their work in Desmos.
|
|
Building Understanding of the Epsilon-Delta Definition of a Limit via Graphic Representations (ARC) Swarup Ghosh, Southwestern Oklahoma State University
|
This activity supports construction of meaningfulness for the ε–δ definition of the limit of a function through manipulation and interpretation of graphic representations. Students manipulate a dynamic sketch of a function and propose a value for a proposed limit L as x approaches a value a, if they think one exists. The definition of a limit is then applied to determine whether or not for a chosen ε, does a given δ determine a neighborhood about a, such that if x is within δ units of a, then the corresponding f (x) values must be within ε units of L.
|
|
Functions, Limits, and Rate of Change (CoRD) Nathan Drake, Oklahoma Baptist University
|
This CoRD is designed to increase students’ ability to connect function values, limits, derivatives, and second derivatives to graphical representations. Student will create graphs at four different times within Calculus I with information given: function values, limiting behavior, derivatives, and second derivatives. Students will examine the graphs that they and their peers create to identify commonalities and notice differences that still adhere to the information given. A subsequent activity at each of these four junctures will encourage communication among students by having one verbally describe a given graph while a second sketches those details on a given non-standard axis.
|
This work is licensed under CC BY-NC-SA 4.0
