Monday, April 2, 2012

MAA/NCTM Joint Position on Calculus

I am very pleased to report that MAA and NCTM (National Council of Teachers of Mathematics) have just approved the joint position statement on calculus that comes at the end of this column. It was crafted in response to the recognition of three serious problems:
  1. The first problem is the general perception among high school students and their parents that preparation for college should, if at all possible, include studying calculus in high school. The result is that too many students short-change their preparation in algebra, geometry, trigonometry, and other mathematical topics in order to stay on a fast track to calculus. Too many otherwise talented students arrive at university without the mathematical foundation that is needed to succeed in the college-level mathematics required for their intended major.
This problem is documented particularly clearly in two national longitudinal studies conducted by the US Department of Education. From the high school class of 1992, 31% of those who completed a course of calculus in 12th grade or earlier then enrolled in precalculus when they got to college.[1] From the high school class of 2004, 17% (one in six) of those who had completed a calculus course in secondary school reported taking remedial mathematics when they got to college.[2]
The solution is to insist that the first priority should be to establish a solid foundation in mathematics, rather than to get into calculus while still in high school.
  1. The second problem is related to the first. It is that too often the calculus course that is taught in high school is a version that trains students in techniques of differentiation and integration and in procedures for solving certain standard problems without developing their understanding of calculus. While this builds familiarity with the language of calculus, it misrepresents the true nature of college-level mathematics and creates a false sense of confidence. One of the most dramatic findings of the MAA’s national study of Calculus I instruction in college and university is the high confidence level of entering students and how precipitously it drops as a result of encountering the reality of college-level expectations for calculus.[3]
The solution is to insist that when calculus is taught in high school, it is taught as a course whose content really is equivalent to a mainstream college course.
  1. The third problem is an outgrowth of the first two. Today’s reality for most students headed into STEM careers is a double dose of introductory calculus, once on each side of the high school to college divide. At the very least, this wastes student time, especially when the college course is taught as if this material were being experienced for the first time. For those students who, for whatever reason, have had a bad or simply uninspiring experience of calculus in secondary school, the prospect of repeating their experience can dissuade them from continuing their study of mathematics. Even for those who enjoyed their secondary school calculus, repeating this course can lead to boredom and poor study habits, often resulting in poor performance and a move away from mathematics intensive disciplines.
The solution is to insist that college faculty recognize their audience and deal with it, modifying how calculus is taught and offering alternatives to calculus for entering students.
Recommendations that address these problems lie at the heart of this position statement. Most satisfying from my point of view is the recognition that the issue of when and how calculus is taught is important and requires the efforts of both MAA and NCTM.

I want to thank all those who have worked to make this joint position statement possible. First is Mike Shaughnessy, President of NCTM, who agreed to make the adoption of such a statement one of his personal priorities. Next is Gail Burrill, former NCTM President and current Chair of the MAA-NCTM Joint Committee on Mutual Concerns who created the framework in which it was possible to craft this statement and who provided much useful feedback during its creation. And finally I want to thank the three people who worked with me to actually write the statement: Mike Boardman at Pacific University, Tom Kilkelly at Wayzata High School in Minnesota, and Dane Camp at New Trier Township High School in Illinois.
Question: How should secondary schools and colleges envision calculus as the course that sits astride the transition from secondary to postsecondary mathematics for most students heading into mathematically intensive careers?
MAA/NCTM Position Although calculus can play an important role in secondary school, the ultimate goal of the K–12 mathematics curriculum should not be to get students into and through a course in calculus by twelfth grade but to have established the mathematical foundation that will enable students to pursue whatever course of study interests them when they get to college. The college curriculum should offer students an experience that is new and engaging, broadening their understanding of the world of mathematics while strengthening their mastery of tools that they will need if they choose to pursue a mathematically intensive discipline.


In particular, the fact that calculus is a college mathematics course that increasingly is taught in high school requires that—
  1. Students who enroll in a calculus course in secondary school should have demonstrated mastery of algebra, geometry, trigonometry, and coordinate geometry;
  2. The calculus course offered in secondary school should have the substance of a mainstream college-level course;
  3. The college curriculum should acknowledge the ubiquity of calculus in secondary school, shape the college calculus curriculum so that it is appropriate for those who have experienced introductory calculus in high school, and offer alternatives to calculus.
Faculty in our colleges and secondary schools should work together to re-envision the role of calculus in secondary and postsecondary mathematics education.  Faculty on both sides of the transition from secondary to college mathematics should work together to strengthen the mathematics curriculum so that students who intend to pursue a mathematically intensive career can acquire the mathematical knowledge and capabilities needed for such a career. College faculty and secondary teachers should define what it means for a student to be ready for college-level mathematics. After a student has matriculated in college, they should assess the effect of college-level mathematics offered in secondary school.  They also should clarify and broaden what is meant by college-level mathematics for secondary school.  They should also work to achieve a better understanding of the mathematical strengths and weaknesses of matriculating students, assess the effectiveness of placement programs for collegiate mathematics, and clarify and broaden what the first year of college mathematics can and should entail.


MAA and NCTM are committed to taking appropriate action within the structure of their organizations to assist in guiding the implementation of these recommendations.


[1] US Department of Education. 2008. National Education Longitudinal Study of 1988 (NELS:88). nces.ed.gov/surveys/NEL


[2] National Science Board. Science and Engineering Indicators: 2010. National Science Foundation. Arlington, VA. Appendix Table 1-20. www.nsf.gov/statistics/seind10/appendix.htm


[3] The MAA study Characteristics of Successful Programs in College Calculus found that 61% of the students in college Calculus I had studied calculus in high school, and 58% of all the students starting Calculus I expected to earn an A in the course. Confidence was the greatest single casualty of the course, dropping by almost half a standard deviation from the start to the end of the course.

5 comments:

  1. this is a good observation, i appreciate this effort, i m of the victim who took calcus at first time at two years college have grade A, but continue at higher level i was failing. i think according to the statement above they teach us how to integrate and diffreciate without understand the view of calculus. Thanks i will like the council of teacher and instructor in university to restate this issues because is an issues that cause pain in my heart. thanks. My name sulaimon isiaka a student of illinois institute of technology study electrical engineering.

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  2. The natural course to follow trig is analytic geo. It used to be standard. It is where algebra and trig and geo are put to use, reviewed by practice, and it introduces
    some basic ideas used in calculus, such as coordinates, conics, etc. Because it does not involve the "algorithm versus understanding" issue, it is easier to teach and to learn.

    Sherman Stein

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  3. In my teaching experience of 8 years as an online math tutor I find that Calculus is the most difficult in the later stages. When students starts with Calculus I,II... it goes on fine but student really doesn't follow the actual under lying concept. Later when they encounter Riemann and Lebesgue they find it totally abstract compared to what they have been doing before.

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  4. Of course most calculus graduates never understand calculus. How can they understand when most of their educators (professors of mathematics) do not understand calculus? In fact, almost all of their educators don't know what a "number" is.

    Add to this the fact that standard calculus is flawed, you have the classic case of the blind leading the blind.

    Michael: The Riemann integral is ill-defined and the Lebesgue integral is simply nonsense.

    John Gabriel
    http://thenewcalculus.weebly.com

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  5. "The second problem is related to the first. It is that too often the calculus course that is taught in high school is a version that trains students in techniques of differentiation and integration and in procedures for solving certain standard problems without developing their understanding of calculus. While this builds familiarity with the language of calculus, it misrepresents the true nature of college-level mathematics and creates a false sense of confidence. One of the most dramatic findings of the MAA’s national study of Calculus I instruction in college and university is the high confidence level of entering students and how precipitously it drops as a result of encountering the reality of college-level expectations for calculus."



    I took the time to interview people and when they scored highly in AP calculus, usually a 5, there was no indication that the above occurred. I am not saying that these students were done with calculus, but they were in my opinion the best prepared students.

    We know why students have a false sense of confidence. Because schools are giving them a false sense of confidence.

    Why are the cut scores so low in AP calculus?

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