AND IF SO

HOW SHOULD IT BE TAUGHT?

by

Melanie D. James

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.

WORKS CITED

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.