This post describes foundational reasoning abilities and mathematical knowledge students need to develop before beginning a course in calculus.
1. Covariational reasoning
This type of reasoning involves recognition of two quantities that are changing together. A student who considers how two quantities in a dynamic situation change together is said to be engaging in covariational reasoning. This is a foundational way of thinking that is needed to construct meaningful formulas and graphs to model relationship in applied contexts. There are four mental actions associated with covariational reasoning in the context of making or interpreting a graph of two quantities that change together: i) identifying the quantities in the situation that are to be related; ii) visualizing how the direction of the two quantities change together; iii) visualizing how the amount of change of one quantity changes while considering contiguous fixed amounts of the other quantity on intervals of that quantity; and iv) visualizing how the rate of change of the output variable with respect to the input variable is changing on small contiguous interval of the input variable.
2. Proportional Reasoning
This type of reasoning is related to covariational reasoning as it involves two varying quantities also but this is more specific to those relationships that are related proportionally as well. Studies show that students have difficulty with problems that requires them to recognize that the ratio of two varying quantities is a constant or that two varying quantities are related by a constant multiple. Proportional reasoning is important in the understanding rate of change. If two quantities are changing at a constant rate of change, the changes in the two quantities are proportional. It is this understanding that is needed to determine a new value for one quantity when the constant rate of change and a value of the other quantity is known. Recognizing proportionality of quantities and using proportional reasoning is also key to understanding and using the idea og angle measure in trigonometry.
3. Angle measure and the sine function
Studies show that these two concepts are underdeveloped in precalculus students and even in in-service and preservice teachers. Angle measures are often not conceptualize as an amount of openness between two rays with a common endpoint. Also, the need to use an arc of a circle to measure an angle’s openness is usually not recognized by students. Studies show that when students are able to reason about how an angle’s measure and the vertical coordinate of the arc’s terminus covary, they are better able to understand the sine function and use it meaningfully to model periodic motion. This image of the sine function was also found useful for students in connecting their unit circle conceptions of the sine and cosine function functions to their conceptions of these functions in the triangle trigonometry context.
4. The function concept
Understanding the concept of function is not simply about knwowing when a given relations is a function or not. Understanding function includes understandings of various function types that emerge from examining growth patterns in data, and other understandings that have been identified in research studies to be essential for either constructing or interpreting meaningful function formulas and graphs such as
- interpreting function notation especially on what it means to express one quantity as a function of another;
- the equal as a means of defining a relationship between two quantities that change together rather than a statement of equivalence;
- interpreting the function’s defining formula and its graph as specifying how to process the input values to produce output values, etc.
For additional literature and more detailed explanation: Carlson, M., Madison, B. West, R. (2015). A study of students’ readiness to learn calculus. International Journal of Research in Undergraduate Mathematics Education. DOI 10.1007/s40753-015-0013-y.