Saturday, November 1, 2014

MAA Calculus Study: Women Are Different


MAA’s study of Calculus I, Characteristics of Successful Programs in College Calculus (CSPCC) revealed some interesting and important differences between the men and the women who study calculus in college. The most dramatic of these is the intended major, but the study also revealed differences in preparation (women calculus students have taken more advanced mathematics courses in high school), standardized test score (women score slightly lower on SAT and ACT Math), persistence (women are less likely to continue in mathematics), and reasons for not continuing (women performing at the same level as men are more likely to consider their grades and understanding of calculus to be inadequate). The overall impression that emerges is that women are much more reluctant than men to pursue a mathematically intensive major, and that any indication that they may not be up to the task is much more influential for them than for men.

Even though women make up the majority of undergraduates, CSPCC found that they account for only 46–47% of the students in Calculus I in four-year undergraduate programs. Even once men and women enter calculus, they do not necessarily have the same goals. Most of the women (53%) in Calculus I intend to pursue the biological sciences or teaching, with only 20% heading into the physical sciences, engineering, or computer science. The situation is reversed for men, where 53% intend to major in the physical sciences, engineering, or computer science, and only 23% are going into the biological sciences or teaching (see Figure 1).

Figure 1: Career goals of all students in Calculus I, by gender
phys sci = physical sciences; eng = engineering; comp = computer and information science; geo = geo sciences; bio = biological & life sciences, including pre-med; social = social sciences
We did find interesting differences in the backgrounds of women and men taking calculus at Ph.D. granting universities (see Table 1). Women calculus students are more racially and ethnically diverse and noticeably less likely to consider themselves to be math people. Women who take Calculus I in college are slightly more likely than men to have been on an accelerated track: Algebra II by 10th grade, Precalculus by 11th, and Calculus by 12th. Their SAT Math scores are slightly lower (about a quarter of a standard deviation) than those of men, echoed almost precisely in their ACT Math scores. It has been documented (Strenta et al, 1994) that even when women and men have comparable high school backgrounds and records of performance, women do tend to score lower on standardized tests in science and mathematics.

Table 1: Characteristics and background of students entering Calculus I
at Ph.D.-granting universities, by gender (N = 3125 women, 3824 men)

women
men
White
72%
81%
Asian
18%
12%
Black
6%
4%
Hispanic
10%
8%
Algebra II by grade 10
87%
83%
Precalculus by grade 11
85%
79%
Calculus by grade 12
74%
70%
SAT Math
mean (SD)
3rd quartile median
1st quartile
654 (70)
700
660
610
670 (69)
720
670
630
See self as math person
42%
55%

Women were almost twice as likely as men to choose not to continue in calculus, even when Calculus II was a requirement of their intended major. Of the men who began the fall term intending to continue on to Calculus II and who successfully completed Calculus I (C or higher), 11% changed their mind by the end of the class. For women, the figure was 20%. We found that the difference between women and men over whether to continue was present irrespective of grade or intended major (see Table 2). In general, bio-science majors are much more likely to switch than are engineering majors, almost certainly because Calculus II is less essential to their intended field. Yet whether the intended major was in the bio sciences or engineering, women were consistently less likely to continue to Calculus II. What was striking was how discouraging a C in Calculus I was to a woman’s intention to pursue engineering, while it barely dented a man’s confidence.

Table 2: Switchers at Ph.D. universities, organized by grade in Calculus I and intended major

women
men
Grade in Calc I
A
10%
6%
B
13%
6%
C
24%
12%
Bio Science Majors
Grade in Calc I
A or B
19%
13%
C
29%
26%
Engineering majors
Grade in Calc I
A or B
4%
2%
C
19%
7%
Note: Among all students who entered Calculus I with a definite intention to continue on to Calculus II (N = 988 women, 1476 men), percentage that had decided not to continue or were undecided whether to continue to Calculus II by the end of that term.

Table 3 looks at the reasons that students gave for switching. The first two reasons, "Too many other courses I need to take," and "Have changed major," are not necessarily indictments of calculus instruction, but they do point to missed opportunities. We see that women and men are equally likely to believe that calculus "takes too much time and effort." A fifth of the A and B students in Calculus I gave a bad experience in the class as one of their reasons for switching. For those earning a C in the class, this was by far the most popular reason for women to switch out, and close to the most commonly cited reason for men to switch out. But the most striking gender differences occur for the last two reasons. Only 4% of the men earning an A or B were dropping calculus because they did not understand calculus well enough to continue its study, but this was true of almost a fifth of the women earning an A or B. Even more notably, not a single man earning an A or B felt that this grade was not good enough to continue the study of calculus, but this was true of 7% of the women who were switching out of the calculus sequence. This is consistent with the findings of Strenta et al (1994) that found strongly significant differences (p < 0.001) between women and men: Women were much more likely to question their ability to handle the course work, and women were much more likely to feel depressed about their academic progress. They also found that women were more likely than men to leave science because they found it too competitive (p < 0.01).

Table 3: Reasons for switching at PhD universities, by grade in course and by gender
(N = 143 women, 109 men)

Reason for switching
Gender
Students earning
A or B
Students earning
C
Too many other courses I need to take
Women
43%
33%
Men
42%
16%
Have changed major
Women
40%
43%
Men
33%
39%
Takes too much time and effort
Women
33%
25%
Men
29%
26%
Bad experience in Calculus I
Women
18%
53%
Men
19%
35%
Don’t understand calculus well enough
Women
18%
38%
Men
4%
26%
Grade was not good enough
Women
7%
15%
Men
0%
13%
Note: Students could select multiple responses.

The picture that emerges is one of women who are as well prepared as men for the calculus sequence but less attracted to the most mathematically intensive fields and much more easily dissuaded from continuing the study of mathematics. The amount of work required to succeed in college-level mathematics is not a factor in the gender differences, but women are bringing a more self-critical attitude toward what they understand.

Reference

Strenta, A. C., Elliott, R., Adair, R., Matier, M., & Scott, J. 1994. Choosing and leaving science in highly selective institutions. Research in Higher Education. 35 (4), 513–547.

With thanks to Cathy Kessel for bringing this reference to my attention.

Wednesday, October 1, 2014

The Pitfalls of Precalculus

Many of the students who aspire to study science or engineering never manage to get through calculus. For many if not most of them, the stumbling block is not calculus as such; it is an inadequate grasp of the mathematics of precalculus. This is why most colleges and universities administer a placement exam and offer a course called Precalculus. This is a high school course, but most colleges and universities offer it for students who want to take calculus but are believed to lack the mathematical foundation needed to succeed. Does precalculus in college actually work? Do the college students who take precalculus go on to succeed in calculus? Does a college course in precalculus even help student performance in calculus? Recent research by Sonnert and Sadler suggests that the benefits of college precalculus are marginal at best. At worst, it can be damaging.

Precalculus, as the name suggests, should be a course that students take to prepare for calculus. In fact, the University of Illinois at Urbana-Champaign has changed the name of this course to Preparation for Calculus, emphasizing the fact that this is not and should not be taken as a course to satisfy a mathematics requirement, but only as preparation for success in calculus. Unfortunately, precalculus often proves to be terminal, even for students who do well. A variety of local studies have shown that large numbers of students who are successful in precalculus choose not to continue on to calculus. A Texas Tech study [Jarrett 2000] found that a third of their students who earned a B or higher in their precalculus course failed to enroll in Calculus I. There are similar data from Arizona State University [Thompson et al 2007]: Among declared Engineering majors who earned a C or higher in Precalculus, 38% failed to enroll in Calculus I. It was worse for other majors: 55% of Physical Science majors, 56% of Mathematical Science majors, and 65% of Life Science majors who earned a C or higher in Precalculus failed to enroll in Calculus I. Herriott and Dunbar [2009] reported comparable data from the University of Nebraska-Lincoln and a collection of colleges in Illinois.
What about the students who do go on to Calculus I? Has a semester of college-level precalculus helped them? Gerhard Sonnert and Philip Sadler have just reported on the first large-scale study to address this question [Sonnert & Sadler 2014]. What they found is that for students whose high school mathematical preparation lies between the mean and one standard deviation below the mean (roughly the 15th to 50th percentile) of all Calculus I students, taking precalculus in college produced a small and not statistically significant improvement in their expected grade: around 1 point on a 100-point scale. More than one standard deviation below the mean, their study showed worse grades in Calculus I if they also took precalculus, but here the numbers are so small that the results are problematic. For students whose high school preparation was 0.3 standard deviations or more above the mean, taking precalculus in college was associated with a reduction in their Calculus I grades by 6 or more points, a result that was statistically significant at p < 0.05 for students at 0.3 standard deviations and numerous higher cut-offs.
The Sonnert-Sadler analysis relies on data collected in their FICS-Math study (Factors Influencing College Success in Mathematics, NSF #0813702) that collected both extensive background information and final grades of approximately 10,500 students enrolled in Calculus I in fall 2009. From their own analysis and that of others, once you control for variables such as gender and socio-economic status, high school preparation is the best predictor of success in college calculus. Sonnert and Sadler used a hierarchical logistic regression to identify six high school variables that were highly correlated with success and from which they could build a score to describe readiness for calculus. The variables they identified are
  1. SAT/ACT math scores
  2. Took precalculus in high school
  3. Grade in high school precalculus
  4. Took non-AP calculus in high school
  5. Took AP Calculus in high school
  6. Grade on AP Calculus exam
Because of the large sample with which they were working, they were able to find substantial numbers of students with comparable readiness scores, many of whom had taken precalculus in college, many of whom had not. At each level of preparation, from 1.8 standard deviations below the mean to 1.0 standard deviations above, in increments of 0.1 standard deviations, they were able to simulate a discontinuity regression, comparable to the one described in my column from January 2012, First, Do No Harm. This demonstrated the results reported above.

The exceedingly modest gains for students below the mean are not very surprising. Across all subjects, effective remediation is tough to pull off. There are some precalculus programs that seem to be helping. Identification and investigation of these will be part of MAA’s next study of the precalculus/calculus sequence, Progress through Calculus (NSF #1420389). Some of the more promising directions include courses that weave precalculus review into the introductory calculus course. The University of Illinois has also had success with its use of ALEKS in their Preparation for Calculus. Unfortunately, the vast majority of college precalculus is still taught as recapitulation of the course students took and failed to master in high school, just coming at them a lot faster.

The harm that appears to be done by putting good students into precalculus is a more intriguing result, perhaps a reflection of the damage done to their self-confidence.

References

Herriott, S.R. & Dunbar, S.R. 2009. Who takes college algebra? Primus. 19:1, 74–87.

Jarrett, E. 2000. Evaluating Persistence and Performance of ‘Successful’ Precalculus Students in Subsequent Mathematics Courses, M.S. Thesis, Texas Tech University.

Sonnert, G. & Sadler, P. 2014. The impact of taking a college pre-calculus course on students’ college calculus performance. International Journal of Mathematical Education in Science and Technology. DOI: 10.1080/0020739X.2014.920532

Thompson, P. et al. 2007. Failing the future: Problems of persistence and retention in Science, Technology, Engineering, and Mathematics (STEM) majors at Arizona State University. Report prepared & submitted by the Provost’s Freshman STEM Improvement Committee.

Monday, September 1, 2014

Beyond the Limit, III

In my last two columns, Beyond the Limit, I and Beyond the Limit, II, I looked at common student difficulties with the concept of limit and explained Michael Oehrtman’s investigations into the metaphors that students use when they try to apply the concept of limit to problems of first-year calculus. The point of this exploration is to identify the most productive and useful ways of thinking about limits so that we can channel calculus instruction toward these understandings. In this month’s column, I will describe Oehrtman’s suggestions for how to accomplish this.

In the MAA Notes volume Making the Connection (Carlson and Rasmussen 2008), Oehrtman focuses on the last of the strong metaphors described in his 2009 paper, that of limit as approximation. The point of building instruction around this approach is that it arises spontaneously from the students themselves, providing what Tall refers to as a cognitive root:

Rather than deal initially with formal definitions that contain elements unfamiliar to the learner, it is preferable to attempt to find an approach that builds on concepts that have the dual role of being familiar to the students and providing the basis for later mathematical development. Such a concept I call a cognitive root. (Tall 1992, p. 497)

While limit as approximation was not always the most commonly employed way of understanding limits—that honor frequently went to limit as collapse—it has several major advantages. First, because it does arise spontaneously from many of the students, we know that it is easily accessible to many of them. Second, it is the metaphor that comes closest to the mathematically correct definition of limit. This is important. Because it comes so close to the formal understanding of limits, it provides a means for students to reason consistently throughout the course, providing coherence and making it easier to transfer this understanding to novel situations. Finally, explorations of how to approximate quantities such as instantaneous velocity or the force on a dam provide direct connections between the concepts of calculus and the modeling situations students will encounter in engineering or the sciences.

In student responses to the eleven problems that Oehrtman posed to the 120 students in his study (reproduced at the end of this column), Oehrtman found that the approximation metaphor was employed by 11% in answering questions #1 and #2, 26% for #6, 35% for #4, 70% for #3, and 74% for #8. This last asked to explain in what sense \( \displaystyle \sin x = x - \frac{1}{3!} x^3 + \frac{1}{5!} x^5 - \frac{1}{7!}x^7 + \cdots . \) Oehrtman quotes at length one of these explanations that I wish to reproduce here because it amply demonstrates how, without any explicit instruction in ideas of approximation, it was the metaphor instinctively seized by a student trying to verbalize what she knew about Taylor series.

When calculating a Taylor polynomial, the accuracy of the approximation becomes greater with each successive term. This can be illustrated by graphing a function such as sin(x) and its various polynomial approximations. If one such polynomial with a finite number of terms is centered around some origin, the difference in y-values between the points along the polynomial and the points along the original curve (sin x) will be greater the further the x-values are from the origin. If more terms are added to the polynomial, it will hug the curves of the sin function more closely, and this error will decrease. As one continues to add more and more terms, the polynomial becomes a very good approximation of the curve. Locally, at the origin, it will be very difficult to tell the difference between sin(x) and its polynomial approximation. If you were to travel out away from the origin however, you would find that the polynomial becomes more and more loosely fitted around the curve, until at some point it goes off in its own direction and you would have to deal once again with a substantial error the further you went in that direction. Adding more terms to the polynomial in his case increases the distance that you have to travel before it veers away from the approximated function, and decreases the error at any one x-value. Eventually, if an infinite number of terms could be calculated, the error would decrease to zero, the distance you would have to travel to see the polynomial veer away would become infinite, and the two functions would become equal. This is a very important and useful characteristic, as it allows for the approximation of complicated functions. By using polynomials with an appropriate number of terms, one can find approximations with reasonable accuracy. (Oehrtman 2008, pp. 72–3)

I am certain that this explanation echoes much of what this student heard and saw in the classroom. I read in it much of what I say when I explain Taylor series. Yet this observation is useful because it demonstrates which images and explanations have resonated with this student.

For Oehrtman, this explanation is classified as approximation not just because that word appears frequently in the student’s explanation, but because the student combines it with a sense of the numerical size of the error. This is very different from the metaphor of limit as proximity. For this student, what is important is not just that the graphs of the polynomials are spatially close but that she can control the size of the error.

Given this observation, Oehrtman has built a series of activities designed to encourage and strengthen student reliance on the metaphor of approximation, several of which are described in his 2008 article, many more of which are coming available on his new website, CLEAR Calculus (contact Mike for access to the posted materials by writing to michael.oehrtman@gmail.com). Thus, instead of introducing the slope of the tangent as the limit of the slopes of secant lines, he chooses a particular point on a particular curve, in this case x = –1 on y = 2x, and introduces the secants as lines whose slopes approximate the slope of the tangent line. He gets students to identify those secant lines that provide an upper bound on the slope of the tangent and those that provide a lower bound and then has them explore secant lines that tighten these bounds until they can approximate the slope of the tangent to within an error of 0.0001. The word limit need never arise.

In a similar vein, integrals are approached via Riemann sums, but not as the limit of these sums. Rather, these sums provide approximations to the desired quantity. One can identify those approximations that overshoot and those that undershoot the true value and adjust the partitioning of the interval to make the error as small as one wishes.

Each of Oehrtman’s activities is built around five questions:
  1. What are you approximating?
  2. What are the approximations?
  3. What are the errors?
  4. What are the bounds on the size of the errors?
  5. How can the error be made smaller than any predetermined bound?
As Oehrtman explains, the last two are intentionally reciprocal: Given a choice of approximation, what are the bounds on the error? Given a bound on the error, what approximation will achieve it?

While this approach provides a route through calculus that does not require the use of the word “limit,” Oehrtman does not avoid it. For those students who will pursue mathematics, it is a term that will come up in other contexts. For the serious student of mathematics, it is absolutely essential. What Oehrtman does recognize is that performing a series of exercises in which one finds limits or explains why they do not exist has little or no bearing on the development of a robust personal understanding of derivatives and integrals.

Not surprisingly, students whose understanding of limits is deeply rooted in the concept of approximations, including the reciprocal processes of determining the bound from the approximation and finding an approximation that will satisfy a particular bound, find it much easier to grasp the formal epsilon-delta definition of limit. In fact, Oehrtman, Swinyard, and Martin (2014) have documented the relative ease with which students schooled in this approach are able to rediscover the mathematically correct definition of limit for themselves.

This is not a new insight. In Emil Artin’s A Freshman Honors Course in Calculus and Analytic Geometry Taught at Princeton University (published by MAA in 1958), he talks about approximations to the slope of the tangent line before introducing limit as “the number approached by the approximations to the slope” (page 23). The Five Colleges Calculus Project, Calculus in Context, also begins with approximations, as do Calculus with Applications by Peter Lax and Maria Terrell and The Sensible Calculus Program by Martin Flashman.



The following are abbreviated statements of the problems posed by Michael Oehrtman (2009) to 120 students in first-year calculus via pre-course and post-course surveys, quizzes, and other writing assignments as well as two hour-long clinical interviews with twenty of the students.
  1. Explain the meaning of \( \displaystyle \lim_{x\to 1} \frac{x^3-1}{x-1} = 3. \) 
  2. Let \( f(x) = x^2 + 1.\) Explain the meaning of \( \displaystyle \lim_{h\to 0} \frac{f(3+h)-f(3)}{h}.\) 
  3. Explain why \( 0.\overline{9} = 1.\) 
  4. Explain why the derivative \( \displaystyle f’(x) = \lim_{h\to 0} \frac{f(x+h)-f(x)}{h}\) gives the instantaneous rate of change of f at x
  5. Explain why L’Hôpital’s rule works. 
  6. Explain how the solid obtained by revolving the graph of y = 1/x around the x-axis can have finite volume but infinite surface area. 
  7. Explain why the limit comparison test works. 
  8. Explain in what sense \( \displaystyle \sin x = x - \frac{1}{3!} x^3 + \frac{1}{5!} x^5 - \frac{1}{7!}x^7 + \cdots . \) 
  9. Explain how the length of each jagged line shown below can be \( \sqrt{2} \) while the limit has length 1. 
  10. Explain what it means for a function of two variables to be continuous.
  11. Explain why the derivative of the formula for the volume of a sphere, \( V = (4/3)\pi r^3 \), is the surface area of the sphere, \( dV/dr = 4\pi r^2 = A. \) 


Artin, E. (1958). A Freshman Honors Course in Calculus and Analytic Geometry Taught at Princeton University. Buffalo, NY: Committee on the Undergraduate Program of the Mathematical Association of America

Callahan, J., Hoffman, K., Cox, D., O’Shea, D., Pollatsek, H., Senechal, L. (2008). Calculus in Context: The Five College Calculus Project. Accessed August 11, 2014. www.math.smith.edu/Local/cicintro/book.pdf

Carlson, M.P. & Rasmussen, C. (Eds.). (2008). Making the Connection: Research and Teaching in Undergraduate Mathematics Education. MAA Notes #73. Washington, DC: Mathematical Association of America.

Flashman, M. (2014). The Sensible Calculus Program. Accessed August 11, 2014. users.humboldt.edu/flashman/senscalc.Core.html

Lax, P. & Terrell, M.S. (2014). Calculus with Applications, Second Edition. New York, NY: Springer.

Oehrtman, M. (2008). Layers of abstraction: theory and design for the instruction of limit concepts. Pages 65–80 in Carlson & Rasmussen (Eds.), Making the Connection: Research and Teaching in Undergraduate Mathematics Education. MAA Notes #73. Washington, DC: Mathematical Association of America.

Oehrtman M. (2009). Collapsing Dimensions, Physical Limitation, and Other Student Metaphors for Limit Concepts. Journal For Research In Mathematics Education, 40(4), 396–426.

Oehrtman, M., Swinyard, C., Martin, J. (2014). Problems and solutions in students’ reinvention of a definition for sequence convergence. The Journal of Mathematical Behavior, 33, 131–148.

Tall, D. (1992). The transition to advanced mathematical thinking: functions, limits, infinity, and proof. Chapter 20 in Grouws D.A. (ed.) Handbook of Research on Mathematics Teaching and Learning, Macmillan, New York, 495-511.