Because of the complexity of calculus, there are many instructors who
feel that calculus should not be taught at the high school level. But the
NCTM's Curriculum and Evaluation Standards
f School Mathematics (1989) recommends that "in grades
9-12, the mathematics curriculum should include the informal exploration
of calculus concepts from both a graphical and a numerical perspective."
This standard does not advocate the formal study of calculus in high school
but it does advocate the informal investigation of key concepts of calculus,
such as limit, the area under a curve, the rate of change, and the slope
of a tangent line. These concepts can be developed as natural extensions
of topics that students have already encountered (Jockusch and McLoughlin,
1990). The approach should focus on exploring concrete problems in a way
that is sensitive to students' existing understandings and should cause
students to achieve deeper conceptual understanding.
The reason some instructors argue against the teaching of calculus in high school is that at that level, the course is "watered down...stressing manipulations but slighting subtle processes." This practice interferes with the more conceptual emphasis that may be offered in the college course (Ferrini-Mundy and Gaudard, 1992). Another argument against teaching calculus at the secondary school level is that students who are accelerated into calculus in the secondary school lack the time to develop a sound background in algebra, functions, and other traditional pre-calculus topics.
Orton (March 1985, p.ll) argues that the crucial issue is not when calculus should be taught but how teachers should promote the understanding of calculus and pre-calculus according to the level of attainment of the pupil. Fundamental concepts of calculus can be taught as early as middle school. By middle school, students are ready for concrete experiences with the concept of slope. They can take first steps in exploring the concept of rate of change (Jockusch and McLoughlin, 1990, p.532).
Many other researchers and instructors feel that calculus should be taught at the high school level, but they feel the instructors should abandon the manipulative techniques method of teaching calculus. Instead, teachers should develop activities for students that are aimed at providing students with firm conceptual underpinnings of calculus. The conceptions that students bring from their previous mathematical experiences strongly influence how they make sense of the calculus concepts they encounter.
The function concept is a central organizing idea for the study of calculus. The extent of students' understanding of the function concept is a big determining factor of their understanding such calculus ideas as limit, continuity, and the slope of a tangent line (Ferrini-Mundy and Graham, 1991). A number of studies have found that students strongly prefer functions expressed in terms of formulas and are reluctant to admit other representations, such as graphs, tables, and correspondence, as part of their concept image of function (Ferrini-Mundy and Lauten, 1994, p.115). NCTM's curriculum standard emphasizes the importance of helping students make connections between mathematical representations; the most relevant connections in calculus are between the analytic and graphical representations of functions. The standards also encourages not only the exploration of function characteristics but the introduction to students of another mode of mathematical thinking.
Usually in high school calculus classes, students are able to master the procedural components of the course. However, their understanding of the conceptual aspects of calculus are at a minimum. For example, although students can produce correct answers to limit problems, many researchers have found that students are uneasy about the mismatch between their intuitions and the answers they reach through mathematical manipulations. In addressing the problem, Orton (1989, p.15) states that "we must avoid the 'pitfalls', we must avoid producing pupils who have 'learned to apply processes mechanically (and) are mystified about the principles'."
Orton also says that concepts "must be introduced intuitively~ in the first instance and that pupils must be allowed to ~'draw graphs of functions and find rates of change and areas under graphs by drawing." Ferrini-Mundy and Lauten (1994, p.ll7) state that students can be encouraged to deal explicitly with the conflict between their conceptions and the formal concepts by giving them an opportunity to use spreadshee~s to explore sequences, series, convergence, and limits in tabular and graphical representations.
Thinking visually can be extremely useful in many calculus-related contexts, and activities that promote and encourage visual solutions are likely to help students' understanding. Visual thinking in calculus can be promoted through the connections between functions and their graphs. Many researchers believe that if students solve problems visually, they have a deeper understanding than if they solve them only in an analytic mode (Ferrini-Mundy and Lauten, 1994, p. 118).
The NCTM curriculum standard states that computing technology makes fundamental concepts and applications of calculus accessible to all students. Computing technology also permits the foreshadowing of analytic ideas. Graphing calculators and other technological devices eliminate much of the procedural work done in calculus, which leaves the students with the conceptual component of calculus to tackle.
Studying calculus in high school has positive effects for college-bound students. According to Ferrini-Mundy and Gaudard tl992, p.57), high school students who study calculus tend to perform better in first semester calculus courses compared to students who have not studied calculus. "The lack of the year of high school calculus can seriously handicap the first-year university student" (Burton, 1989). The major difference in performance between students who had no secondary school calculus and students who had a brief introduction seemed to be procedural proficiency. The difference in conceptual proficiency is not substantial.
It is very important that students in high school be exposed to fundamental principles of calculus. Whether they are in an algebra course or an AP calculus course, students should be offered the experiences necessary to understand the ideas underlying the concepts of calculus before they meet these concepts in the more abstract setting of a formal calculus course.
Burton, M. (1989). The effect of prior calculus experience in "introductory" college calculus. American Mathematical Monthly, 96, 350-354.
Ferrini-Mundy, J. & Gaudard, M. (1992). Secondary school calculus: preparation or pitfall in the study of college calculus? Journal for Research in Mathematics Education, 23, 57-69.
Ferrini-Mundy, J. & Graham, K. (April 1991). Research in calculus learning: understanding of limits, derivatives, and integrals. Paper submitted to Proceedings of the Special Session on Research in Undergraduate Mathematics Education.
Ferrini-Mundy, J. & Lauten, D. (February 1994). Learning about calculus learning. Mathematics Teacher, 87, 115-120.
Jockusch, E. & McLoughlin, P. (October 1990). Building key concepts for calculus in grades 7-12. Mathematics Teacher, 83, 532-540.
National Council of Teachers of Mathematics, Commission on Standards for School Mathematics. (1989). Curriculum and Evaluation Standards for School Mathematics. Reston, VA: The Council.
Orton, A. (March 1985). When should we teach calculus? Mathematics in School, 14, 11-15.