1. An investigation of college students’ covariational reasoning Kloku (2007) Florida State University Marggie D. GonzalezOctober 17, 2011

2. Author’s Motivation Belief of the importance of the topic Influence instruction Curriculum designer Teacher educator Lack of studies focused on college students’ covariational reasoning.

3. Research Questions What thinking and reasoning processes do college students’ use as they attempt to coordinate simultaneous changes of two variables in continuously changing functional situations? How do college students interpret simultaneous change of two variables in functional situations? How do college students use their interpretations to represent covariation of two variables in graphical context? How do college students interpret covariational of two variables in given graphs of functional events?

4. Literature Review Outline of Literature Review Overview Rate of change Three perspectives to rate of change Images of rate and images of covariation Relationship between function and derivative Static vs dynamic conceptions of functions Reasoning about change Three types of reasoning Covariational reasoning

5. Rate of Change Fundamental to better understand advanced mathematical concepts in calculus. Three perspectives of rate of change (Garnet Hauger) Students’ interpretations of the overall shape of a graph Students’ understanding of average rate of change Students’ understanding of instantaneous rate of change.

6. Image of Rate and Covariation Images of rate and images of covariation (according to Thompson – speed as a rate) Image of change in some quantity A loosely coordinated image of two quantities An image of the covariation of two quantities (car problem) Coordination of two quantities’ values Construction of an image in which both quantities are tracked for some duration

7. Covariational Reasoning Defined as “cognitive activities involved in coordinating two varying quantities while attending to the ways in which they change in relation to each other” (Carlson). “… holding in mind a sustained image of two quantities, values (magnitudes) simultaneously” (Saldanha & Thompson). Necessary for analyzing, interpreting, and representing the patterns of change in continuously changing events (Carlson).

8. Example Imagine this bottle filling with water. Sketch a graph that represents the relationship between amount of water that is in the bottle and the height.

9. Methods Task-based clinical interviews Calculus with Analytic Geometry III Participants Selection Tasks

10. Data Analysis Covariational Framework (Carlson, 1998) Five mental actions of coordinating dynamic situations

11. Findings/Conclusions First, thinking about simultaneous changes between two variables one step at a time and conceiving of functional situations statically leads to difficulties in coordinating the continuously changing rate of change over entire domain. Second, difficulties in graphical representations produce inconsistencies between interpretations and representations of simultaneous changes of two variables. Third, strong procedural tendency hindered reasoning and meaningful interpretations about change in functional situations. Fourth, reasoning based on irrelevant arguments or inappropriate principles leads to erroneous conclusions about simultaneous changes of two variables.

12. Implications Results would provide more information to develop more effective instruction suggest some alternative ways in instruction, and inform educators and curriculum designers on whether or not to consider the inclusion of some aspects of function concept in earlier grades.

13. Critique The author clearly identified the topic of the review Covariational reasoning abilities Significance of the study? Implications? The review was not written as a cohesive essay. Review did not interpreted and/or critiqued the literature, just summarized

14. Critique The author did not explicitly discuss how his study would help to advance our understanding of college students’ covariational reasoning abilities. Areas for future research Students’ covariational reasoning in different representational system (not graphical representation) Role of mental imagery in students’ covariational reasoning.

15. Critique Conclusions and implications were included. Conclusions section was cohesive and easy to read. A table was included Implications section was short.