High School Calculus Research Articles

Effects of Early Acceleration of Students in Mathematics on Taking Advanced Mathematics Coursework in High School

Based on data from the Longitudinal Study of American Youth (LSAY), students were classified into high-, middle-, and low-ability students. The effects of early acceleration in mathematics on the most advanced mathematics coursework (precalculus and calculus) in high school were examined in each category. Results showed that although early acceleration did not promote the majority of accelerated students to participate in the most advanced mathematics coursework in high school, nonaccelerated students were unlikely to pursue the most advanced mathematics coursework in high school at all. Early acceleration helped encourage students from some disadvantaged backgrounds to pursue the most advanced mathematics coursework in high school. Schools mattered to early acceleration, with more incidences of statistically significant variance among schools in taking the most advanced mathematics coursework for accelerated than non-accelerated students.

The Unit of Life How We Live and Why We Die The Secret Lives of Cells by Lewis Wolpert Norton, New York, 2009. 252 pp. $24.95. ISBN 9780393072211. Faber and Faber, London. £14.99. ISBN 9780571239115.

Drawing from 30 years of amiable correspondence with his high school calculus teacher, Strogatz demonstrates the joys to be found in doing mathematics.

Characterizing the gender gap in introductory physics

Previous research [S. J. Pollock et al., Phys. Rev. ST Phys. Educ. Res. 3, 1 (2007)] showed that despite the use of interactive engagement techniques, the gap in performance between males and females on a conceptual learning survey persisted from pretest to post-test at the University of Colorado at Boulder. Such findings were counter to previously published work [M. Lorenzo et al., Am. J. Phys. 74, 118 (2006)]. This study begins by identifying a variety of other gender differences. There is a small but significant difference in the course grades of males and females. Males and females have significantly different prior understandings of physics and mathematics. Females are less likely to take high school physics than males, although they are equally likely to take high school calculus. Males and females also differ in their incoming attitudes and beliefs about physics. This collection of background factors is analyzed to determine the extent to which each factor correlates with performance on a conceptual post-test and with gender. Binned by quintiles, we observe that males and females with similar pretest scores do not have significantly different post-test scores $(p>0.2)$. The post-test data are then modeled using two regression models (multiple regression and logistic regression) to estimate the gender gap in post-test scores after controlling for these important prior factors. These prior factors account for about 70% of the observed gender gap. The results indicate that the gender gap exists in interactive physics classes at our institution but is largely associated with differences in previous physics and math knowledge and incoming attitudes and beliefs.

Developing Reasoning through Proof in High School Calculus

Students develop reasoning skills by constructing mathematical proofs of some nontypical calculus results.

Calculus of functions and their inverses: a unified approach

In this pedagogical article, I explore a unified approach in obtaining the derivatives of functions and their inverses by adopting a guided self-discovery approach. I begin by finding the derivative of the exponential functions and the derivative of their inverses, the logarithmic functions. I extend this approach to generate formulae for the derivatives of inverse trigonometric functions. I strongly believe that this guided self-discovery approach should be adopted before obtaining the derivatives of the inverse functions using the chain rule or the general formula for the derivative of an inverse function. It is important to observe that this article is aimed at making a freshman or high school calculus class more entertaining, and so the algebraic proofs of most of the statements are reserved for future classes.

The Technology Penalty

THIS column has been rattling around in my head for several years now, and I think the time has come to write it. I frequently write about the benefits, or payoffs, of technology. But what about the technology penalty? The technology penalty is what happens when you do something using technology that you could do easier, quicker, or more efficiently without using technology. Allowing students to e-mail you their assignments is a good example of paying the technology penalty. If I collect papers in class, I can simply sit at my desk, read the papers, and scribble notes on them. This task usually takes an hour or so. If I allowed 30 students to e-mail me an assignment, I assure you I could not read and respond to 30 e-mails in an hour. Thus it is a rare occasion when I accept e-mailed assignments. Rather, I usually ask students to print out their assignments and give them to me in class or drop them by my office. Why would I want to do the printing for them? Using calendaring and address book software is another example of incurring the technology penalty. Keeping your calendar on a computer or PDA requires that you start--or wake up--the computer or PDA, go to the application, select the view (day, week, month, or year), and scroll or page to the specific date, time, or event. In contrast, I carry a month-at-a-glance paper calendar that I get free from Hallmark every year. Because the previous month's pages are dog-eared, it takes only about a second to find this month's page. The calendar fits in my shirt pocket, so I don't have to drag around any electronics. I also have a small paper address book that fits into the same shirt pocket. Call me old-fashioned if you like, but the systems work for me, and there is no technology penalty. Instructors who teach students how to use powerful software like Photoshop pay an immense technology penalty. Adobe releases a new version of Photoshop every six to 12 months with all kinds of changes and new features. Instructors must then relearn the program and change their syllabus and handouts. Explanatory handouts usually have screen captures in them, so the screen captures from the old version must be updated. If the instructor uses a textbook on Photoshop, that will usually change with each new release of the software as well. A colleague who teaches computer graphics and Photoshop has to go to daylong professional development seminars on Photoshop a couple of times a year just to stay sharp. Fortunately, my colleague has a technician who can upgrade all the computers in his teaching lab, but many teachers have to do their own upgrades. Another example of software that changes rapidly is Web creation software, such as Macromedia's Studio 8. What started out few years ago as simply Dream weaver morphed into Studio MX and is now Studio 8. The current version of Studio 8 contains Dream weaver, Flash Professional, Fireworks, Contribute, and Flash Paper. Each of these programs has its own learning curve that takes a good amount of time to negotiate. Shallow learning curves are certainly technology penalty. High school teachers are not immune to the technology penalty. Not long ago, I received an e-mail from a high school calculus teacher who had just received a Smart board. In order to take advantage of all the things the Smart board could do, the teacher was spending about two hours a day preparing for the next day's lesson. The payoff for his students was great, but so was the technology penalty he was paying. Still another example of paying the technology penalty is using computer grade book software. Many teachers who use grade book software also keep a paper grade book. Doing any job twice, with and without technology, is a definite technology penalty. An inherent penalty in grade book software is the time it takes to scroll up and down the list of students to enter a grade. Visually scanning a paper grade book page is much faster. …

Why students with an apparent aptitude for computer science don't choose to major in computer science

The statistics show that the number of Computer Science majors is dropping across the United States. Possible reasons include a reduced number of jobs in the field, an incorrect perception of what Computer Scientists do, and the students' disinterest due to a lack of familiarity with the subject. The reasons may be different for males and females. This paper reports on a study in which 836 high school calculus and pre-calculus students were surveyed to try to determine why students with an apparent aptitude for CS did not pursue a major in Computer Science. The results supported some of the hypotheses for declining enrollment, and rejected others. The top reasons for rejecting the major were the same for both genders, but the reasons for choosing the major differed.

Automatic control in first-year engineering study

An introductory engineering course based on the theme of automatic control in the first year of study prepares future engineers in facing multidisciplinary problems. The field of systems and control is by nature multidisciplinary and universal, providing an appealing introduction to systems concepts. Comprehensive courses on automatic control are normally offered at the senior or junior level. Basic engineering concepts related to dynamic models, feedback, stability, and system performance can be introduced and appreciated by students earlier in the program since many students have had calculus and physics classes in high school. Students experience first hand how theory, computer simulation, and experiments are used to develop solutions to engineering problems. In this article, changes in the engineering curriculum that can better prepare future engineers are advocated. A curriculum that is multidisciplinary from the start, has a strong systems viewpoint, and introduces and integrates modern engineering problems, principles, practices, and solutions is also advocated. The results of an experienced five-week automatic control module that can serve as the foundation for an introductory course are detailed.

Student Enrollment in High School AP Sciences and Calculus: How does it Correlate with STEM Careers?

Many high schools offer students the opportunity to take advanced placement (AP) courses in many subjects including science and mathematics. Studies have shown that students who take these classes are more likely to succeed in college and that failure in engineering education is strongly correlated to deficiencies in mathematics and science. This article presents the background of AP classes and their impact on science, technology, engineering, and mathematics (STEM) career choices of college students. The results of this study confirm that students who take AP classes in calculus and the sciences are more likely to select majors in careers such as engineering, science, mathematics, and the medical field. In this study, both minority and nonminority students who were taking AP calculus and/or science courses in high school selected STEM careers at a higher rate than other careers and males selected engineering at a higher rate than females. Females selected science and mathematics and the medical field at a higher rate than males. Furthermore, the size and location of the high school, profession of parents, and socioeconomic level of parents all affect the number of AP classes offered in high schools as well as which ethnic groups primarily take them.

Factors Affecting Mathematically Talented Females' Enrollment in High School Calculus

The purpose of this study was to determine if identifying factors existed that would explain differential mathematics participation among females in high school, specifically the enrollment in high school Calculus. The factors investigated were socioeconomic status, educational aspirations, the education of both parents, and the number of siblings. The database used for this study was taken from the National Educational Longitudinal Study (NELS: 88) and the follow-up conducted in 1992. The sample for the present research was composed of females enrolled in algebra at the onset of this study who scored in the fourth quartile on the mathematics section of the standardized test. The results showed a difference between the two groups in mother's education, SES, and educational aspirations. However, when applying all factors together in a logistical regression, the results of this research determined that the factors did not have predictive value in determining the probability of a mathematically talented female enrolling in high school Calculus. This research suggests that some factors that affect course taking in the general population of students may not be generalized to mathematically talented females.

From the Earth to the Moon: A Freshman Seminar

AbstractThe seminar course for first‐year undergraduate students, From the Earth to the Moon, deals with both technical and non‐technical aspects of space flight, with particular reference to lunar voyages. The goals of the course are to establish a framework for understanding technology and its applications, to present fundamental principles of science and program management, and to motivate students to learn more about the many facets of engineering. As such, the course introduces numerous issues of systems engineering in a broad context, presenting not only science, technology, and mathematics but also the reasons that these subjects are important. Typically, all of the students in the course have studied physics and/or calculus in high school, though half plan to major in the humanities. Thus, the course has dual roles in exposing liberal arts students to details of technology and engineering students to societal impacts of technology.

Maclaurin's Theorem for High School Calculus Students

As an extra enrichment topic in the high school calculus course that I teach, I have occasionally dealt with Maclaurin's theorem. Although Maclaurin's series may be an unusual topic in a high school calculus course, the approximation of such functions as the sine, cosine, and exponential by polynomials is significant and interesting enough for students to learn about and is therefore worthwhile as an extra topic.

Student understanding of basic calculus concepts: Interaction with the graphics calculator

The recent efforts to reform calculus have, in many cases, included increased use of technology. Several studies have explored the interaction between the use of technology and student learning of calculus. Our study seeks to build on this work by exploring the understanding of basic calculus concepts by students who are studying calculus in a graphics calculator-based environment. In particular, the purpose of the study is to developed descriptions of calculus students' understandings of function and limit, to explore the interrelationships among their understandings of these conceptual areas, and to identify instances where there is evidence that the students' understanding seems to have been influenced by the availability of the technology. Data for the study was collected through a series of clinical interviews with a small group of students enrolled in a first semester college calculus course and an advanced placement high school calculus course. The data was used to build descriptions of the students' understanding and the role the graphing technology might play in that development. Implications of these findings for calculus curriculum and instruction are presented.

High School Calculus: Comparing the Content of Assignments and Tests

The purpose of this study was to compare the content of what is taught, as inferred from the content of seatwork and homework questions assigned, with what is tested in a high school calculus course. Data were obtained from a sample of 17 teachers, who provided lists of the homework and seatwork assigned each day and the criteria used to arrive at student grades for the course. Results showed differences in content coverage across teachers and differences in the overlap between content taught and content tested. The discussion focuses on methodological issues in the investigation of teacher grading practices, on the importance of such investigation, and on steps that might be taken to reduce discrepancies.

High School Calculus: Comparing the Content of Assignments and Tests

High School Calculus: Comparing the Content of Assignments and Tests

Around the Sun in a Graphing Calculator

The topics of polar equations and parametric equations are found in almost all precalculus and calculus textbooks. They furnish important a lgebraic representations of real-world problems. For example, polar equations are used to describe the e lliptical orbit of planets around the sun (Kepler's first law), and parametric equations are frequently used to describe the trajectory of moving objects. These topics are seldom given more than superficial treatment in high school and college precalculus and calculus courses because of the difficulty of obtaining appropriate graphical representations.

Solving Extreme—Value Problems without Calculus

Too frequently students who are veterans of high school calculus arrive at the college campus only to fail even to place into first-semester calculus courses or to have difficulties with subsequent courses should they succeed in “placing out” of first-semester calculus.

Report of the CUPM Panel on Calculus Articulation: Problems in Transition from High School Calculus to College Calculus

Report of the CUPM Panel on Calculus Articulation: Problems in Transition from High School Calculus to College Calculus

Report of the CUPM Panel on Calculus Articulation: Problems in the Transition from High School Calculus to College Calculus

Report of the CUPM Panel on Calculus Articulation: Problems in the Transition from High School Calculus to College Calculus

Iteration of an Interesting Function

In the November 1981 issue of Scientific American. Douglas Hofstadter described a problem that could be understood and partially solved by most high school calculus students but was sufficiently open-ended to encourage independent research. The problem could be used to help students understand such mathematical topics as functional notation, composition of functions, the solving of systems of equations, and the derivative.