Tel Aviv University
Tel Aviv University
Lester and Sally Entin Faculty of Humanities
Jaime and Joan Constantiner School of Education
Paper for YESS-4
Approaches to the Integral Concept
The case of high school calculus
Submitted by: Anatoli Kouropatov
YESS-4 August 2008 Anatoli Kouropatov
This study is motivated by a very real problem in the high school mathematics classroom:
High ability students have great difficulties with the integral concept. The students do not acquire
comprehension regarding the concept of the integral and are satisfied, in the best case, by formal
techniques to the solution of the problems.
I propose to investigate to what extent this can be attributed to the way integrals are normally
introduced in high school. For this purpose it will be necessary to conceptualize the meaning of
comprehension; to investigate the relation between the inner structure of the integral concept, and
comprehension of the concept; to devise and examine coherent didactical tools for developing
significant concept comprehension in learners; and to investigate which types of problems relate to
the inner structure of the integral concept.
According to my hypothesis, the mathematical idea that enables a deeper comprehension of the
integral and doesn't damage skills is to introduce the integral as an accumulation function
(Thompson, 1994; Thompson & Silverman, 2007).
From here, the main goals of the current study can be formulated in the following way:
- To analyze how the theoretical (didactic and instructional) components of the integral concept
are connected to the deep comprehension of the concept
- According to the results of this analysis, to analyze a common approach to the concept of the
integral and the concept of accumulation function,
- To evaluate the present situation concerning students' knowledge about the integral concept,
- To develop a unit of instruction on the topic of the integral that is based on the initial
presentation of the concept as an accumulation function and to examine, mainly qualitatively, the
progress of students regarding both, the technical skills and deeper comprehension.
This research, based on theoretical considerations and on empirical evidence, may shed light on the
learning of integral calculus in high school and point to ways of improving it.
Learning the concept of the integral is an important part of the high school
mathematics curriculum in almost the entire world, and among others also in Israel. In
my eyes, this importance is absolutely justified: It isn't possible to imagine modern
scientific culture without integrals. This concept (along with its relative, the
derivative) constitutes a mathematical domain that is a language, a device, and a
useful tool that is very important for other fields: physics, engineering, economy, and
statistics. This domain is usually known as differential and integral calculus. Along
with this, the concept of the integral represents a philosophical idea for the
understanding of the world: contemplation of the totality of little parts of a whole
YESS-4 August 2008 Anatoli Kouropatov 1
enables conclusions regarding the whole in its entirety, as well as its internal structure
It is important to point out that the idea of integral was created and grew from
within physics, from within the attempt to invent a mathematical tool that enables
people to describe, to analyze and to explain different physical phenomena, e.g.
volume, mass, work (Newton, 1686).
The most prevalent approach in schools (as well as colleges and universities)
to the instruction of the integral is based on the definition of the indefinite integral as
an inverse operator to the derivative (antiderivative). From here, one arrives at the
definite integral with the help of different formulations of the Fundamental Theorem
of Calculus (Spivak, 1967; Goren, 2005; Aspis, 2006; Meytav, 2006; Yakuel, 2006).
Based on this approach students acquire (or, at least, are supposed to acquire)
technical skills for the treatment of classical problems using integrals: computation of
areas (and less frequently, volumes) of shapes (bodies) whose boundaries are defined
by means of graphs of some elementary functions (whose antiderivatives are also
elementary functions). In this way, do the students also acquire or develop the
comprehension of the concept more deeply? Professional literature (Orton, 1983;
Ashkinuze, 1987; Thomas & Hong, 1996; Sealey, 2006) and my personal experience
show that most of the students do not acquire comprehension regarding the concept of
the integral and are satisfied, in the best case, by formal techniques to the solution of
This begs the question whether another approach to the instruction of the
integral concept exist, which enables, on one hand, not to lose technical skills, and on
the other hand, to develop deeper comprehension of the concept? My hypothesis is:
Yes, such an approach exists. This question is made up of two facets:
- Theoretical aspect (possible potential of the proposed approach): Is it correct that
students can acquire comprehension of the concept by a certain approach?
- Empirical aspect (effectiveness of proposed approach): Do students, who have been
exposed to the proposed approach, indeed acquire the expected comprehension?
Before specifying a hypothesis about the nature of such an approach, one
should ask whether such an approach aiming at deeper comprehension is really
needed. The study proposed here is based on a central proposition: Significant
comprehension of mathematical concepts is at the heart of the instruction of
mathematics. Therefore every effort to develop comprehension is wanted and
required. This proposition becomes almost evident when considering integrals. The
YESS-4 August 2008 Anatoli Kouropatov 2
idea of calculus in general and the idea of integral in particular were born from human
attempts to understand the world, from applications (Newton, 1686). In some way the
integral is the application. So, in my eyes, there is no way to understand such an idea
without understanding the strong connection between the mathematical concept and
its application, in other words, without significant comprehension.
In undergraduate education, usually the same approach is used as in school;
there are, however, a small number of exceptions: Initial presentation of the concept
on the basis of Riemann sums (Taylor, 1992; Courant, 1934; Apostol, 1969), on the
basis of average height (Turegano, 1998), and on the basis of differential equations
( .)5991 ,שטייIndeed, in high schools, any presentation of the integral other than the
common one is extremely rare. However, even superficial contemplation on the
common approach raises doubts:
• If integral "is" antiderivative – it is easy (even at a high school level) to bring
an example for a function, the integral of which cannot be recorded with
elementary functions. So in this case, for high school students, the integral is
unknown and might be considered non-existent…
• If integral "is" area - it seems difficult to deal with situations where integral is
volume, and such situations occur even in high school…
• If integral "is" limit – it is difficult to overcome high school students’ lack of
familiarity with the concept of limit…
• In contrast to the concept of the derivative, for which a language that is rather
rich and varied is developed in schools, regarding the concept of the integral
the language is amazingly poor.
• And apparently, there is simply no place for the use of preliminary intuitions
(Fischbein, 1978; Fischbein, 1987) and it is problematic to identify and to
operate cognitive processes that are unconnected and independent (like, for
example, those that were described in the theory of procept (Gray & Tall,
1994)) that lead to deeper comprehension of the concept.
My hypothesis is that the mathematical idea that enables a deeper comprehension of
integral and doesn't damage skills is to consider the integral as an accumulation
function (Thompson, 1994; Thompson & Silverman, 2007), in the plain meaning: a
sum that has a large number of very small terms.
There are at least three reasons which strengthen this hypothesis:
YESS-4 August 2008 Anatoli Kouropatov 3
- The idea of accumulation allows in a natural way to combine the concepts of
definite and indefinite integral and its connection with the concept of the derivative.
- The idea of accumulation allows in a natural way to represent the connection
between the mathematical idea of integral and its applications.
- The idea of accumulation allows in a natural way to represent the generalization of
the mathematical concept of integral, e. g. to Riemann-Stieltjes integrals, Lebesgue
integrals, Lebesgue-Stieltjes integrals, and Denjoy (gauge) integrals.
I think, and I hope to show that the idea of accumulation can make use of students’
intuitions and enable them to develop and to operate cognitive processes that will lead
to deeper and perhaps wider comprehension of the concept of integral.
From here, the main goals of the proposed study can be formulated in the following
- To analyze how the theoretical components of the concept of the integral are
connected to the deep comprehension of the concept;
- According to the results of this analysis, to analyze a common approach to the
concept of the integral and the concept of accumulation function;
- To design and to offer a unit of instruction for the integral concept based on the idea
of accumulation function;
- To examine, mainly on the qualitative level, the progress of students learning with
this unit regarding technical skills as well as deeper comprehension.
It is important to notice the following:
- Analyzing the comprehension of the integral concepts requires the conceptualization
of the central proposition regarding the concept of the integral;
- Analyzing a common approach to the concept of the integral and accumulation
requires the testing of the present situation: students' understanding regarding the
concept of the integral;
- Analyzing the concept of accumulation function requires the mathematical
consolidation of this concept.
I hope that the results of this study will give mathematics teachers and their
students an additional, perhaps more efficient, possibility to acquaint themselves with,
to learn, and to understand one of the most brilliant mathematical philosophical ideas
- the concept of the integral.
YESS-4 August 2008 Anatoli Kouropatov 4
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YESS-4 August 2008 Anatoli Kouropatov 10
1. (i) Write an integral allowing to calculate the area of the rectangle given in the
(ii) Calculate the area of the rectangle with the help of the integral that you
wrote in (i).
2. If it is possible, calculate the indicated area. If not, explain why it is impossible:
1 x = − y2 +1
3. Compare the results of the following integrals and explain your findings.
∫ x dx and ∫ x dx
∫ x dx and ∫ ( x − 1) dx
c. ∫ x 2 dx and ∫ ( x 2 + 1)dx
d. ∫ π x dx and
∫ π x dx
4. Does the integral ∫ sin x 2 dx exist? Explain your answer.
YESS-4 August 2008 Anatoli Kouropatov 11
5. Consider the sketch of the graph of the function y = cos x 2 given in the figure.
Does the integral ∫ cos x 2 dx exist? Explain your answer.
6. If ∫ f ( x)dx = 10 then ∫ f ( x − 3)dx = ? . Explain your answer.
7. If ∫ f ( x)dx = 1 then ∫ ( f ( x) + 2)dx = ? . Explain your answer.
8. If ∫ f ( x 4 )dx = 11 then ∫ f ( x 4 )dx = ? . Explain your answer.
YESS-4 August 2008 Anatoli Kouropatov 12
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