Who Died and Made Calculus King?
Ginny Renzi, M.Ed.
Many students experience some difficulty in attaining proficiency with the conceptually difficult and rigorous mathematics adventure that is calculus. As I contemplate the problems they face, both individually and collectively, I am left with many questions, together with some answers. My thoughts reinforce a long-held belief that there are issues that need to be considered when counseling students regarding appropriate courses, and that these issues should also be addressed in future examinations of curricular design and teaching methods for the various calculus options.
There are implied assumptions concerning the teaching and learning of calculus that are not borne out in the experience of all calculus students or in all calculus classrooms. Calculus students are often expected to learn by the "intuitive approach," which demands certain competencies and learning styles, but who is to determine what these competencies are and whether they exist within an individual student? If math students have not experienced this pedagogical methodology in their previous courses, will a magic switch turn on within them, simply because they have walked through the door of the calculus classroom?
My own experiences in learning calculus, along with those of my students, demonstrate the problems inherent in this approach. I have considered the following questions at length:
What are some of the implied assumptions of calculus instruction?
In reality, which students take calculus?
- All calculus students should be expected to learn at the same rate of speed, in this case very quickly.
- All calculus students are active math learners, able to comprehend fully new concepts without the benefit of direct instruction.
- All calculus students can make independent connections of new material with previous learning.
- All calculus students have the ability to handle mathematical insight versus mathematical regurgitation, especially in testing situations.
- All calculus students have experienced the same type of instruction and expectations for independence.
- All calculus students have the time needed to keep up with the amount of work required in order to stay current at all times.
- All calculus students have the same learning style or benefit from the same teaching style.
What should college counselors, students, and parents consider when making the decision for math placement following pre-calculus?
- The path toward calculus is often determined during middle school, when an individual student is deemed proficient enough in math to be placed in an "honors" course. Following the expected progression, this student will usually finish pre-calculus in 11th grade, although I am beginning to see more students who attain this level at the end of 10th grade.
- Mathematically excellent students who demonstrate flexible, insightful problem solving skills and retain mathematical concepts over time clearly belong in calculus. These students enjoy math for its own sake and are likely to major in a math-related subject in college. They are excellent candidates for advanced placement courses.
- Students who wish to apply to highly competitive colleges are likely to opt for one of the calculus choices. College guidance counselors and college admissions officers both imply, if not explicitly state, that calculus is a requirement for acceptance at such schools.
- Very diligent and committed students who have achieved mathematical success through their own hard work and/or excellent teaching in previous math courses will want to attempt calculus. These students, however, may not be intuitive mathematical thinkers and may have experienced some difficulties with math instruction in the past.
- A final group of students appears to end up in a calculus course because a some scheduling constraint that is beyond their control. A few years ago two of my students were "promoted" from a standard Algebra II course directly into Honors Calculus because the school did not have enough students to warrant a pre-calculus course. They succeeded - only because of a wonderful teacher who was committed to filling in all of the gaps in their mathematical instruction.
What are some other options that should be considered by students who have successfully completed pre-calculus?
- Previous PSAT, SAT, and SAT II scores should be examined. Although there are valid concerns about standardized testing, it is important to consider the information that they can provide. PSATs and SATs examine a student's ability to think mathematically on novel, verbally oriented questions. SAT IIs are designed to examine retention and application of previous learning in addition to the mathematical insight required for SATs.
- Any previous math difficulties that the individual student has encountered should be examined and evaluated, no matter how insignificant these appear to be. Consider issues of speed, word and/or application problems, and those recurring errors often classified by teachers as "careless" which are generally related to visual attention to detail.
- The course load and/or other activities the student will carry should be inspected. Individuals who are considering calculus are generally also committed students who are active at school as well as in the community at large. They are often taking multiple honors and/or AP courses, and they participate in athletics and are involved in numerous activities (such as yearbook) that will erode the time and energy they will have to devote to learning calculus.
- The particular math teacher who will be teaching the course should be considered. Math teachers have widely varying approaches to teaching and learning. If a student has enjoyed working with a specific teacher in the past, of if a teacher has a reputation for making math understandable and/or enjoyable, this fact should not be ignored.
What causes students to experience difficulty once they are taking calculus?
- Non-advanced placement calculus courses that place an emphasis on how to learn calculus along with the expected course content. Because such courses are not taught to the AP test, teachers can have the luxury of slowing down when needed.
- Statistics, either the traditional or the new AP course, can be an equally rigorous mathematical expedition. Unlike calculus, which has little apparent connection to everyday life, statistics proficiency is essential to becoming an educated consumer in today's society. Unfortunately, statistics is not always looked upon with the same favor by the aforementioned college counselors and admissions officers.
- Discrete math is the course that is generally recommended to students who need to fulfill their math requirement, who realize that it is not a good idea to take their senior year off from math, or who recognize that calculus is not their best option. The content of this course varies tremendously from school to school but is often not considered by students or counselors to be rigorous enough to warrant consideration for better math students. Whereas calculus builds upon itself continuously, discrete math, as the name implies, includes self-contained units. Consequently, it can be an excellent choice for overly committed seniors.
How can teachers help all students learn calculus?
- Many aspects of algebra and pre-calculus are self-contained. A student's success will not necessarily depend upon his or her proficiency with material in a previous chapter. Calculus, on the other hand, is entirely new material that builds upon itself continuously.
- Calculus students cannot always assimilate the material quickly enough.
- Calculus students can fall behind and find it difficult to catch up.
- Some students have difficulty with the large number of word/application problems.
- Some students do not possess the cognitive flexibility to switch strategies that can be required to solve specific calculus problems.
- Some calculus students have difficulty with active working memory and consequently with manipulating all aspects of a problem without getting lost.
- Some calculus students do not remember all of the necessary material, especially formulas, from algebra and pre-calculus.
- Calculus students are challenged to learn inverse operations in close succession; consequently, they can confuse one type of a problem with its exact opposite.
Finally, I am not arguing that students should not take calculus; indeed, I am convinced that most students can succeed in this rigorous course of study if their learning is properly structured. Even more important, I truly believe that students can enhance their mathematical self-esteem while recognizing the value of their efforts. I wish, however, that I were convinced that this is the case for most high school seniors who take calculus. It would be interesting indeed to see how many of these students continue along the path to higher-level math courses in college.
- Calculus teachers should incorporate explicit strategies for learning calculus into classroom instruction.
- Teachers should identify and examine pre-requisite skills for each unit.
- Teachers should strive to relate new material to material that the students have learned previously.
- Teachers should require students to maintain lists of all important formulas and definitions.
- Teachers should require students to memorize needed formulas, even though these can be programmed into calculators. Attaining automaticity with these formulas will enhance problem solving and processing speed.
- Teachers should give frequent short quizzes to help students determine their own level of understanding.
- Teachers should inspect homework frequently to encourage students to avoid falling behind.
- If there is not enough time to go over homework in class, teachers should have a solution key available to all members of the class.
- Teachers should be careful to assign problems the answers to which are available in the back of the book, and should encourage their students to check their work as they go along.
- Teachers should provide ample numbers of easier problems before proceeding to harder problems, especially those which require applications of the new concepts.
- Teachers should remind their students to refer back to the examples in the book rather than requesting that they reread the text for needed clarification.
- Teachers should consider alternate assessments: journals and projects that include previous AP problems are particularly appropriate.
- Teachers should recognize how learning styles, both theirs and those of their students, relate to the learning of calculus.
Ginny Renzi, M.Ed.
Mathematics and Learning Specialist