PreCalc / Calc Survey Results

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The item corresponding to the first question was named “pre-calculus topics”. For this item, you were asked to rate 33 mathematics topics, typically covered in a pre-calculus course, on their importance as prerequisites in a college calculus course. A total of nine composite variables were based on the 33 options in the pre-calculus topics item. Six of these emerged from principal components analysis (PCA): (1) Logs & Exponents, (2) Limiting Process, (3) Factoring / Quadratics, (4) Interpreting Graphs, (5) Excursions on Quadratics, and (6) Composition Decomposition. A seventh scale, “functions”, was constructed using topics that were removed during the PCA. For the final two variables, the 33 topics were categorized as either “procedural” or “conceptual” and corresponding scales were constructed.

The item corresponding to the second question was named “teaching and learning themes”. For this item, participants were asked to rate twelve teaching and learning themes (e.g., group work, study skills). Pre-calculus faculty rated these themes on the likelihood that they would emphasize them in the classroom and calculus faculty on the importance of these themes for success in their calculus course.
Two factors, named ClassroomCentered and StudentCentered, emerged from the PCA on the “teaching and learning themes” item. The first factor contained variables whose values can be directly affected by classroom practices (e.g., discussion of projects and applications, group work) and the second factor included variables that tend to describe affective characteristics of students (e.g., study skills) and are less likely to be directly affected by classroom practices.
Generally, variables in the ClassroomCentered factor tend to be closer to the reform side of the “reform versus traditional divide” and variables in the StudentCentered factor tend to be closer to the traditional side.

The third question pertained to college calculus faculty only. The corresponding item was named “accommodations for underprepared groups”. For the third item, college calculus faculty were asked to rate six accommodations for underprepared groups on the likelihood that they would implement these in their classroom.
Two components, AdjustStandards and Opportunities, also emerged from the PCA on the “accommodations for underprepared groups” item. Overall, options in the first component involve methods through which an instructor can adjust his or her standards and options in the second component involve methods through which the instructor can help students meet the standards he or she has set.

The three groups (HS, PreC, and CALC) were mostly in agreement on the primary purpose of pre-calculus being to prepare students for calculus and on the percentage of material that is often included in pre-calculus courses but is not directly relevant to calculus. Sources, likely to inform instructors’ decisions on what to emphasize in their courses (e.g., literature, personal experience, knowledge of future requirements), received similar ratings by the three groups. Personal experience and literature, for example, were rated consistently as most and least likely, respectively. Thirty-two percent of high school faculty participants had college teaching experience in pre-calculus, or a related course. This characteristic proved to be important. Ratings of items by participants in this subgroup were more closely aligned with ratings by postsecondary faculty than with ratings by high school faculty without college teaching experience. Finally, college calculus faculty indicated modest agreement (mean rating 3.07) with the statement “Students starting my calculus course are adequately prepared to successfully complete the course with a grade of C or better.” When asked to indicate areas in which incoming students were perceived to be lacking, 62% of responses were categorized as belonging either to algebra or to pre-calculus.

Key findings corresponding to the first research question:

1) Significant differences among the three groups were found on most comparisons. Most of these differences were between high school pre-calculus faculty (HS) and college calculus faculty (CALC) with college pre-calculus faculty (PreC) somewhere in the middle. HS and CALC had significant differences on 7 of 9 variables; PreC and CALC on 5; HS and PreC on 4

2) Generally, as the course level increased (HS=1, PreC=2, CALC = 3) the mean rating for the composite variables decreased. (The coding HS=1, PreC=2, and CALC=3, can correspond to the chronological order in which a student might register for the respective courses, or an increasing level of mathematical maturity required for each course).

3) On seven content-based composite variables (excluding procedures and concepts, which overlapped with the other variables), there was some agreement in the relative rankings of the rating means (e.g., Excursions on Quadratics had the least mean rating across the three groups). The variable rated highest by CALC, however, consisted of topics typically covered in a high school Algebra II course and the variable rated highest by HS consisted of topics often discussed at the start of a college calculus course. This confirms the finding in ACT (2006) showing high school faculty having a preference for more content coverage and college faculty having a preference for less content.

4) The variable “concepts” was rated significantly higher than “procedures” by all three groups. For the HS group, however, the difference “concept minus procedure” was significantly higher than the corresponding differences for the other two groups.

5) Knowledge of college expectations was rated highly (mean rating 4.5 of 5) as a source informing high school faculty decisions on what to emphasize in pre-calculus. Yet, ratings by high school faculty who reported having college teaching experience were more closely aligned with ratings of postsecondary faculty. On seven of nine factors there was a significant difference between high school faculty with college teaching experience and those without. The differences between these two subgroups decreased as more time elapsed since the last college teaching experience.

6) Knowledge of subsequent expectations and experience teaching calculus were rated highly (mean ratings 4.5 and 4.3 respectively) by pre-calculus faculty. Despite the closer proximity of PreC to CALC the two groups differed significantly on 5 of 9 variables.

Key findings corresponding to the second research question:

1) Each group rated the StudentCentered factor significantly higher than the ClassroomCentered factor.

2) Probably not surprisingly, the difference “StudentCentered minus ClassroomCentered,” for CALC was significantly higher than the corresponding difference for HS and the effect was large.

3) Mean ratings for the ClassroomCentered factor decreased as the grade level increased.

4) Ratings by high school faculty with college teaching experience were more closely aligned with ratings by postsecondary faculty than with ratings by high school faculty without college teaching experience.

Findings corresponding to the third research question:

Not surprisingly, the key finding based on comparing the mean ratings of the components AdjustStandards and Opportunities was that college calculus faculty are significantly more likely to offer students more opportunities to meet their standards rather than adjust them.

Some conclusions:

This study confirms the existence of an expectations gap, specifically at the level of pre-calculus, with respect preparation for college calculus. Significant differences were found in ratings of pre-calculus content as well as pedagogy. Further, the study found differences that are more granular. The largest number of differences existed between high school faculty who teach pre-calculus and college faculty who teach calculus. Differences, however, existed between high school faculty with college teaching experience and those without, between postsecondary faculty who teach pre-calculus and those who teach calculus, and between faculty teaching the same course (pre-calculus) at the high school or college level. A trend emerged in that ratings of the importance of calculus pre-requisites decreased as the course level increased and as the teaching experience got closer to that of college level calculus. An exception to this trend was, probably not surprisingly, the significantly higher, with large effect, ratings by college calculus faculty on the StudentCentered variable.

These differences seem contrary to the high ratings by high school and college pre-calculus faculty (mean ratings 4.35 and 4.53 out of 5, respectively) of “knowledge of subsequent (college) requirements” as a source informing their decisions on what to cover in their course. With this inconsistency, the question arises as to the nature and source of this knowledge. For high school faculty, this knowledge may be about college access rather than college success (Kirst, 2008) and may come from the administrative branches of the postsecondary community rather than the faculty. Such a finding would be consistent with the misalignment, reported by Achieve (2007), between admissions exams and standards for success, would be supported by the finding from this study showing ratings by high school faculty with college teaching experience being closer aligned with ratings by college faculty, and would confirm claims of “mixed messages” by the college community regarding college readiness (Conley, 2003). Misalignment was less for the two postsecondary groups, likely due to increased contact, yet higher than what might be expected. Eighty-five percent of college pre-calculus faculty, for example, rated their calculus teaching experience as very likely (57% 5 and 28% 4 out of 5) to inform their decisions on what to emphasize in the course. Differences between college pre-calculus and calculus faculty ratings may be due to the multipurpose nature of pre-calculus, or insufficient communication, or agreement, within college mathematics departments. The mathematical profile of students taking pre-calculus in high school and that of students taking pre-calculus or calculus in college may be affecting the ratings by respective faculty. Many of the more talented and motivated students are among those taking advanced mathematics courses in high school (Bressoud, 2009a). As such, a large percentage of students taking calculus for the first time in college may be among those with less talent, motivation, or background preparation than their peers who took pre-calculus or calculus in college. Respective faculty may be rating pre-calculus topics according to the calculus course their students are likely to enroll in and the perceived needs of their students.

The majority of high school faculty who responded to the survey for this study also reported teaching an AP calculus course. An AP recommendation, for high school pre-calculus teachers, is to devote a unit or two introducing limits, towards the end of the course. Traditionally, at the college level, limits are taught at the start of the calculus course. Although no significant difference was found on the mean ratings across levels of AP teaching experience, AP considerations may have affected the ratings by high school participants, given the prominent role of AP in higher-level high school mathematics.

Works cited in this summary.

Achieve (2007). Aligned expectations? A closer look at college admissions and placement tests. Retrieved from http://www.achieve.org/AlignedExpectations

ACT (2006). ACT National curriculum survey 2005 – 2006. Retrieved from http://www.act.org/research/curricsurvey.html

Bressoud, D. M. (2009a). Launchings AP® Calculus: What we know. Retrieved from http://www.maa.org/columns/launchings/launchings_06_09.html Accessed September 2009

Conley, D. T. (2003). Mixed messages: What state high school tests communicate about student readiness for college Eugene, OR: The Center for Educational Policy Research, University of Oregon. Retrieved from http://www.s4s.org/Mixed_Messages.pdf

Kirst, M. W. (2008). Secondary Schools and Colleges Must Work Together. Thought and Action, The Nea Higher Education Journal (Online Journal) Retrieved from http://www.nea.org/

Lutzer, D. J., Rodi, S. B. Rodi, Kirkman, E. E., and Maxwell, J. W. (2007). Statistical abstract of undergraduate programs in the mathematical sciences in the United States, Fall 2005 CBMS Survey. Providence, RI: American Mathematical Society Retrieved from http://www.ams.org/cbms/cbms2005.html

Planty, M., Provasnik, S., and Daniel, B. (2007). High school coursetaking: Findings from The Condition of Education 2007 (NCES 2007-065). U.S. Department of Education. Washington, DC: National Center for Education Statistics. Retrieved from http://nces.ed.gov/programs/coe/2007/analysis/index.asp

Provasnik, S., and Planty, M. (2008). Community Colleges: Special Supplement to the Condition of Education 2008. National Center for Education Statistics, Institute of Education Sciences, U.S. Department of Education. Washington, DC. Retrieved from http://nces.ed.gov/programs/coe/

Seymour, E. and Hewitt, N.M. (1997). Talking about leaving: Why undergraduates leave the sciences. Boulder, CO: Westview Press.

Seymour, E., and Hewitt, N.M. (1994). Talking about leaving: Factors contributing to high attrition rates among science, mathematics, and engineering undergraduate majors. Bolder, CO: Bureau of Sociological Research, University of Colorado.

Pre-Calculus Topics

• Find vertical asymptotes
• Graph a logarithmic function
• Graph an exponential function
• Find the discriminant
• Use the graph to find the domain of a function
• Recognize special products e.g. difference of squares
• Synthetic Division
• Basic operations with complex numbers
• Simplify logarithmic expressions
• Solve a logarithmic equation
• Complete the square
• Use trigonometric Identities
• Graph piecewise defined functions
• Factoring
• Graph basic trigonometric functions
• Find the solutions of a quadratic equation
• Solve an exponential equation
• Draw conclusions about a function from its tabular representation
• Understand composition of functions
• Understand how a transformation affects the graph of a function
• Understand the behavior of polynomials as x --> ± ∞
• Understand the behavior of functions near asymptotes
• Draw conclusions about a function from its graphical representation
• Quadratic models
• Use the rule of a function to find a domain element yielding a specified range element. I.e. given k, find x such that f(x)=k
• Find the domain of radical functions
• Write a linear equation in different forms
• Use the graph to predict the family of a function e.g. polynomial of odd/even degree, exponential, etc
• Understand the roles of domain and range of f as compared to those for f^-1
• Relate the difference quotient to the slope formula
• Understand decomposition of a function
• Understand the concept of infinity
• Understand the concept of “x approaching but never reaching a particular value”