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Describing cognitive orientation of Calculus I tasks across different types of coursework

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Abstract

We discuss the findings of an analysis of cognitive orientation of 4,953 mathematical tasks (representing all bookwork, worksheets, and exams) used by five instructors teaching Calculus I in a two-year college in the United States over a one-semester period. This study uses data from one of 18 cases from the Characteristics of Successful Programs in College Calculus. We found differences in the cognitive orientation by type of course work assigned (graded vs. ungraded) and differences by the instructors who assigned the course work. We discuss implications for practice and propose some areas for further exploration.

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Notes

  1. In our setting, the majority of tasks are not done in class, so looking at tasks as written is our best approximation of cognitive demand.

  2. The cognitive demand of a task can be decreased through regular practice, memorization of specific routines, following examples superficially, or cheating.

  3. Section 3.2.3 describes optimization problems that are not Apply Understanding.

  4. We do not have labs and quizzes from one instructor.

  5. Cohen’s κ allows assessing inter-rater reliability when there are two coders and the variables have several categories. This coefficient calculates agreement taking into account chance agreement. For that reason it is more stringent than calculating the rate of agreements to the total of agreements and disagreements. According to Landis and Koch (1977), κ = 0.40–0.59 reflects moderate inter-rater reliability, 0.60–0.79 substantial reliability, and 0.80 or greater outstanding reliability.

  6. This process included the development of a system that included five other dimensions besides cognitive orientation. For details on the development of this framework see White et al. (2013).

  7. Recall that our categorization includes an extra category not accounted for in the Tallman et al.’s analysis. Our Rich Task category is a subset of their category “required students to demonstrate an understanding of an idea or procedure.” This means that our proportions of Rich Tasks are conservative estimates of what their framework would produce.

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Authors and Affiliations

  1. Department of Mathematics, 530 Church St., Ann Arbor, MI, 48109-1043, USA

    Nina White

  2. School of Education, 4041 SEB, 610 East University, Ann Arbor, MI, 48109-1259, USA

    Vilma Mesa

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  1. Nina White
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  2. Vilma Mesa
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Correspondence to Nina White.

Additional information

This work was partially funded by the National Science Foundation under Grant No. 0910240. The opinions expressed do not necessarily reflect the views of the Foundation. The Undergraduate Research Opportunity Program at the University of Michigan also provided funding for this project. We thank Cameron Blum who assisted with coding.

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White, N., Mesa, V. Describing cognitive orientation of Calculus I tasks across different types of coursework. ZDM Mathematics Education 46, 675–690 (2014). https://doi.org/10.1007/s11858-014-0588-9

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