The Calculus I Student

*David M.
Bressoud, *May 2011 **(corrected December, 2011)**

The data from the MAA survey [1] of mainstream Calculus I classes are in, and we are in the process of preparing it for analysis. The survey was a great success. We had responses from almost 700 instructors and over 14,000 students at all types of post-secondary institutions. While it is too early to report on any associations that appear, I do have some basic summative data that should be of interest.

The following are the characteristics of students who were enrolled in mainstream Calculus I after the second week of classes in Fall term, 2010. (Numbers in parentheses are for all full-time first-year students in four-year undergraduate programs [2].) After showing what we have learned about these students, I will end this column with my own thoughts and reactions.

Demographics and Aspirations

- 55% are men, 45% women (versus 44% men, 56% women for all full-time first-year students in four-year undergraduate programs)
- 97% are full-time students
- 75% are freshmen, 14% sophomores, 6% juniors
- 88% were born in the United States; for 85%, English is the primary language spoken at home; 93% graduated from high school in the US (97% of all full-time first-year students in four-year undergraduate programs are US citizens, and an additional 2% are permanent residents)
- 76% are White, 14% Asian, 5% Black (versus 73% White, 9% Asian, 12% Black)
- 10% are Hispanic (versus 12% Hispanic)
- 79% have a father with at least some college; 61% have a father with at least a Bachelor’s degree; 31% have a father with at least some graduate school (versus 67% for at least some college, 53% at least Bachelor’s degree, and 25% for at least some graduate school)
- 82% have a mother with at least some college; 58% have a mother with at least a Bachelor’s degree; 22% have a mother with at least some graduate school (versus 69% for at least some college, 55% at least Bachelor’s degree, and 22% for at least some graduate school)
- 53% expect at least some difficulty paying for college (versus 65%)
- Career goal

- Medical/Biological Sciences: 30% (versus 11%)
- Engineering: 30% (versus 10%)
- Business: 8% (versus 14%)
- Physical Sciences: 6% (versus 3%)
- Computer Science: 5% (versus 1%)
- Science or Mathematics Teaching: 3%
- Social Sciences: 2% (versus 12%)
- Mathematics: 1% (versus 1%)
- Undecided: 9% (versus 7%)
- 81% are certain of what they want to do after college
The low ratios for the Physical Sciences (6:3) and Mathematics (1:1) probably reflect the large numbers of these students who place directly into Calculus II or higher.

Preparation

- 62.4% took the SAT exams. For Critical Reading, the mean score was 611 with a standard deviation of 86 and an interquartile range of [550,670]. For Mathematics, the mean score was 652 with a standard deviation of 76 and an interquartile range of [610,700].
- 62.5% took the ACT exams. For Mathematics, the mean score was 28.5 with a standard deviation of 4.3 and an interquartile range of [26,31].
- 61% took a calculus class in high school.
Note that this is corrected from the originally reported figure of 68%.Of these students:

- 62% took an Advanced Placement AB course
(corrected from 56%)- 13% took an Advanced Placement BC course
(corrected from 12%)- 34% took the Advanced Placement AB or BC exam and earned a 3 or higher
(corrected from 33%)- 4% studied calculus in an International Baccalaureate program
- 61% earned an A in their high school calculus course
- 13% took AP Statistics
- 17% took precalculus in college
- 11% are taking calculus in college for at least the second time
- 80% are comfortable using graphing calculators
- 13% are comfortable using a Computer Algebra System (Mathematica, Maple, Matlab)
- 32% were always allowed to use graphing calculators on mathematics exams in high school, 58% were sometimes allowed to use them on exams, and 10% were never allowed to use them on exams
- 32% were allowed to use a TI-89 or -92 or comparable calculator with symbolic operation capabilities on mathematics exams in high school

Attitudes

- 95% agree with the statement: “I believe I have the knowledge and abilities to succeed in this course.”
- 90% agree with the statement: “I am confident in my mathematical abilities.”
- 89% agree with the statement: “The process of solving a problem that involves mathematical reasoning is a satisfying experience.”
- 83% agree with the statement: “I enjoy doing mathematics.”
- 65% would want to continue studying mathematics even if it was not required for their major
- Given the choice between “When studying Calculus I in a textbook or in course materials, I tend to memorize it the way it is presented” and “When studying Calculus I in a textbook or in course materials, I tend to make sense of the material, so that I understand it”, 74% chose the latter.[3]
- Given the choice between “The primary role of a mathematics instructor is to work problems so students know how to do them” and “The primary role of a mathematics instructor is to help students learn to reason through problems on their own”, 72% chose the latter [4]
- The combined amount of time that they expect to spend in studying each week for all of their classes is

- 0–10 hours: 19%;
- 11–20 hours: 45%;
- 21–30 hours: 21%;
- > 30 hours: 15%
- 58% expect to earn an A in this course, and 94% expect to earn at least a B
- 69% expect to continue and take Calculus II
The actual distribution of final grades for all of these students was A: 22%, B: 28%, C: 23%, D, F, or withdrew: 27%. [5]

Thoughts and ReactionsWhat strikes me very forcefully in this description of Calculus I students is how accurately it portrays those who have been successful in high school mathematics:

- accelerated into calculus while in high school, earned an A in it, many earned 3 or higher on the AP exam,
- average SAT math score over 650, average ACT math score over 28,
- enjoy mathematics and want to continue studying it,
- understand that learning mathematics is about making sense of it and want instructors who will help them learn how to reason through mathematical problems,
- are confident of their ability to do well in mathematics,
- intend to pursue a career in science or engineering.

I am distressed by how poorly these students do in Calculus I: Over a quarter essentially fail, and only half earn the A or B that is the signal that they are likely to succeed in further mathematics. I know the frustration of high school teachers who see what they consider to be the best and brightest of their students run into mathematical roadblocks in college. I recognize that much of the fault lies on the high school side of the transition. Many students who consider themselves well prepared for college mathematics in fact are not. We need to do a better job of communicating what these students really need and working with their teachers so that they can provide it. I also know that we in the colleges and universities can do a better job of supporting these students after they have arrived on our campuses, moving them forward with challenging and engaging mathematics while bringing them up to the level they need to be at to succeed.

Over the next several months, we will be analyzing our data to identify eight departments across a variety of types of colleges and universities that have particularly effective programs. This does not necessarily mean high pass rates, but rather evidence that they are doing significantly better than their peers in preparing their students for further mathematics. We will then send teams into these institutions to prepare detailed case studies that we hope will promote best practices throughout the country.

[1] Characteristics of Successful Programs in College Calculus, NSF DRL REESE grant no. 0910240.

[2] Cooperative Institutional Research Program at the Higher Education Research Institute at UCLA. 2010. The American Freshman: National Norms 2010. University of California Press. Berkeley, CA.

[3] When instructors were asked at the start of the semester: Given the choice between “When studying Calculus I in a textbook or in course materials, students tend to memorize it the way it is presented” and “When studying Calculus I in a textbook or in course materials, students tend to make sense of the material, so that they understand it”, 64% chose the former. There is a significant disconnect between what students plan on doing and what instructors believe they will do.

[4] When instructors were asked at the start of the semester: Given the choice between “My primary role as a Calculus instructor is to work problems so students know how to do them” and “My primary role as a Calculus instructor is to help students learn to reason through problems on their own”, 90% chose the latter. On the role of the instructor, there is greater, though not perfect, alignment.

[5] At the start of the term, instructors were asked to estimate what percentage of the students would receive D, F or withdraw. The average estimate was 26%.

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David Bressoud is DeWitt Wallace Professor of Mathematics at Macalester College in St. Paul, Minnesota, and Past-President of the MAA. You can reach him at [email protected]. This column does not reflect an official position of the MAA.